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Theoretical and data-driven analysis of two-sided markets and strategic behavior in modern economic systems

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Theoretical and data-driven analysis of two-sided markets and strategic behavior in modern economic systems

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Content Theoretical and Data-Driven Analysis of Two-Sided Markets and Strategic
Behavior in Modern Economic Systems
by
Aikaterini Giannoutsou
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BUSINESS ADMINISTRATION)
December 2024
Copyright 2024 Aikaterini Giannoutsou

To my family
ii

Acknowledgments
First and foremost, I would like to thank my advisor, Andrew Daw, for his constant support,
guidance, encouragement, and mentorship throughout my Ph.D. journey. I feel incredibly
fortunate to have had the opportunity to work with him and to learn from and be influenced
by his intellect, expertise, patience, and kindness. His mentorship has been invaluable, and
it has been both an honor and a pleasure to have him as my advisor.
I would also like to extend my thanks to my committee members, Kimon Drakopoulos,
Milan Mric, and Afshin Nikzad, for their insightful feedback and for supporting my research
in numerous ways. Their perspectives helped shape this dissertation, and their constructive
critiques were instrumental in improving my work.
Furthermore, I want to thank Greys Soˇsi´c and Hamid Nazerzadeh with whom I collaborated in the early years of my Ph.D.. Working with them taught me valuable lessons, and
their guidance helped lay the foundation for my research.
I am immensely thankful for the friendship, love, and support of my close friends and
loved ones, both in Los Angeles and in Athens. Their presence has brightened my days and
provided me with strength, especially during challenging times.
Finally, I owe my deepest gratitude to my parents, Lena and Giannis, and my brother,
Stavros, for their boundless love and support in every aspect of my life. Having them as my
family is the greatest gift, and their unwavering support is something I cherish deeply.
ii

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter 1: Price-Delay Trade-offs in Services: Customers, Servers, and the
Firm-Platform Distinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Contributions and Organization of the Chapter . . . . . . . . . . . . 2
1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Modeling the Different Sides & Problem Formulation . . . . . . . . . . . . . 8
1.3.1 Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Servers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 The Delay Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Service Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Modeling the Different Revenue Optimization Problems . . . . . . . . . . . . 16
1.4.1 Traditional View of Price-and-Delay Differentiation (Firm’s Optimization Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Gig Economy View of Price-and-Delay Differentiation (Platform’s Optimization Problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Analyzing the Impact of Server Choice on the Service Operation . . . . . . . 25
1.5.1 Revenue and Performance under Negligible Server Opportunity Cost 26
1.5.2 Non-Trivial Server Opportunity Cost and the Value of Server Payoffs 28
1.5.3 High Server Opportunity Cost Leads to Firm-Platform Revenue Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 The Role of the Delay Function . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.1 Contrasting Revenue under Differing Delays: Spatial vs. Non-Spatial
Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6.2 Non-Instantaneous Service Rate . . . . . . . . . . . . . . . . . . . . . 37
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
iii

Chapter 2: Data-Driven Analysis of Driver Availability and Rider Delays:
Insights from Ride-Hailing Data in the New York Metropolitan Area . . 44
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Problem Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1 Problem Description & Relation to Chapter 1 . . . . . . . . . . . . . 47
2.3.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Density Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Open Drivers’ Proxy - Simulation Approach . . . . . . . . . . . . . . . . . . 58
2.5.1 Motivation for Simulation . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.2 Max Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Regression of ETA on Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.6.1 The Regression Model for Each Zone . . . . . . . . . . . . . . . . . . 63
2.6.2 Obtaining Driver Availability from Simulation . . . . . . . . . . . . . 64
2.6.3 The ETA Quantity & Associated Challenges . . . . . . . . . . . . . . 65
2.7 Comparing Taxi & Uber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.7.1 Initial Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.7.2 Accounting for Different Zone Density . . . . . . . . . . . . . . . . . 72
2.7.3 Incorporating Regression Insights to the Comparison . . . . . . . . . 75
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter 3: Experimental Study of Stable Recycling Alliances . . . . . . . . . 82
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Stability concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.1 The LCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.2 The EPCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.3.3 The TSD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 Motivation for Behavioral Experiment and Hypotheses . . . . . . . . . . . . 92
3.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.1 Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5.2 Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5.3 Analysis Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Appendix A:Appendices for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 111
A.1 Proofs of Optimal Solutions when c = 0 . . . . . . . . . . . . . . . . . . . . . 111
A.1.1 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.1.2 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.1.3 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.1.4 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.1.5 Proof of Corollary 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.2 Proofs of Structural Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
iv

A.2.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.2.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2.3 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2.4 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.2.5 Proof of Corollary 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2.6 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2.7 Proof of Corollary 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.8 Proof of Proposition 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2.9 Proof of Corollary 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 Proofs of Optimal Solutions when c > 0 . . . . . . . . . . . . . . . . . . . . . 130
A.3.1 Proof of Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.3.2 Proof of Proposition 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A.4 Supporting Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Appendix B:Appendices for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . 141
B.1 Data Processing & Associated Challenges . . . . . . . . . . . . . . . . . . . . 141
B.2 Density Metrics - Additional Plots . . . . . . . . . . . . . . . . . . . . . . . 144
B.3 Calculating Taxi Zone Adjacencies . . . . . . . . . . . . . . . . . . . . . . . 145
B.4 Max Flow Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 146
B.5 Regression Visualization for Different Locations, Additional Plots . . . . . . 147
B.6 Uber vs Taxi Rankings for Different Density Classifications - Additional Plots 151
B.7 Uber vs Taxi Rankings for Different Density Classifications - Evening Rush . 155
Appendix C:Appendices for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 158
C.1 Mathematical expressions for TSD models . . . . . . . . . . . . . . . . . . . 158
C.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.3 Calculations for Example 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
v

List of Tables
2.1 Field Descriptions for Ride-Hailing Data . . . . . . . . . . . . . . . . . . . . 51
2.2 Top 10 Busiest Subway Stations in 2021 . . . . . . . . . . . . . . . . . . . . 80
C.1 Pareto dominant outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
C.2 Player B prefers No collaboration. . . . . . . . . . . . . . . . . . . . . . . . . 161
C.3 One player gets more without collaboration than in the outcome most preferred by the other player. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
C.4 Both players get more without collaboration than in the outcome most preferred by the other player. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C.5 Two stable outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
C.6 Calculations for Example 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 166
vi

List of Figures
1.1 The Server Groups and the Customer Arrival Rates They Face . . . . . . . . 21
1.2 Optimal server payoffs vs. c for different M1. The rest of the customer
characteristics are α1 = 1.2, α2 = 0.4, M2 = 60, and the delay function has
γ =
1
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3 Optimal revenues vs. c for different M1. The rest of the customer characteristics are α1 = 1.2, α2 = 0.4, M2 = 60, and the delay function has γ =
1
2
. . . 35
1.4 Optimal revenues vs. c for different M1. The rest of the customer characteristics are α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60, and the delay function has
γ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5 Optimal revenues vs. c for various E[S]. The customer characteristics are
α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60, and γ = 1. . . . . . . . . . . . . . . . 39
1.6 Feasible Region of d1, d2 for c = 0 and various E[S]. The customer characteristics are α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60. . . . . . . . . . . . . . . . . 40
2.1 NYC Taxi Zones, excluding Staten Island . . . . . . . . . . . . . . . . . . . . 52
2.2 Visualizations of different urban metrics. Each plot shows a different aspect
of the spatial analysis. (a) Intersection Density, (b) Road Density, (c) Median
Height, (d) Presence of Pedestrian Count Stations, (e) Average Pedestrian
Count (2021), (f) Number of WiFi Hotspots. . . . . . . . . . . . . . . . . . . 56
2.3 Network Graph of Locations & Max Flow Problem Variables. Connecting
edges indicate adjacency between locations. . . . . . . . . . . . . . . . . . . 61
2.4 ETA vs Number of Available Drivers during Morning Rush hours at Financial
District South (Taxi Zone 88) in Manhattan, for different values of c . . . . . 67
2.5 Comparison of α rankings for different c values . . . . . . . . . . . . . . . . . 69
vii

2.6 Comparison of Uber and Taxi Ride Rankings Classified by Borough . . . . . 73
2.7 Comparison of Uber and Taxi Ride Rankings by Location, Classified by Density 75
2.8 Comparison of Uber and Taxi Ride Rankings for c = 0. Density classification
is based on both density metrics and α regression coefficients, with square
markers indicating agreement between the two methods . . . . . . . . . . . . 78
B.1 ETA vs Number of Available Drivers during Morning Rush hours at Jamaica
Estates (Taxi Zone 55) in Queens, for different values of c. . . . . . . . . . . 144
B.2 Neighboring Zones Based on ‘Queen’ Adjacency Matrix . . . . . . . . . . . . 146
B.3 Neighboring Zones Based on ‘Rook’ Adjacency Matrix . . . . . . . . . . . . 146
B.4 ETA vs Number of Available Drivers during Morning Rush hours at Allerton/Pelham Gardens (Taxi Zone 3) in the Bronx, for different values of p.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
B.5 ETA vs Number of Available Drivers during Morning Rush hours at Coney
Island (Taxi Zone 55) in Brooklyn, for different values of p. . . . . . . . . . 149
B.6 ETA vs Number of Available Drivers during Morning Rush hours at Jamaica
Estates (Taxi Zone 131) in Queens, for different values of p. . . . . . . . . . 150
B.7 ETA vs Number of Available Drivers during Morning Rush hours at Financial
District South (Taxi Zone 88) in Manhattan, for different values of p. . . . . 151
B.9 Comparison of Uber and Taxi Ride Rankings for p = 0.5. Density classification is based on both density metrics and α regression coefficients, with square
markers indicating agreement between the two methods . . . . . . . . . . . . 152
B.8 Comparison of Uber and Taxi Ride Rankings for p = 0.2. Density classification is based on both density metrics and α regression coefficients, with square
markers indicating agreement between the two methods . . . . . . . . . . . . 153
B.10 Comparison of Uber and Taxi Ride Rankings for p = 0.8. Density classification is based on both density metrics and α regression coefficients, with square
markers indicating agreement between the two methods . . . . . . . . . . . . 154
B.11 Comparison of Uber and Taxi Ride Rankings when c = 0 for evening rush
hours. Density classification is based on both density metrics and α regression
coefficients, with square markers indicating agreement between the two methods157
viii

Abstract
The first chapter of my dissertation focuses on the rapidly increasing gig economy where
businesses operate more like ‘two-sided’ platforms than traditional ‘one-sided’ firms, taking
into account not just customer preferences but also those of the service providers. This study
examines how these two-sided markets approach pricing and service delays differently from
traditional businesses, because of the need to consider the impact of the service providers’
choices. Using a game theoretic model we represent the participating sides (customers,
servers, managers) and by considering that the service delays experienced by customers
vary with the number of participating service providers we explore various business models
within this context. Interestingly, we find certain conditions where traditional firms and
modern platforms perform similarly, and we identify the factors that make service providers’
preferences either significant or negligible in influencing a platform’s profits. Our research
also provides insight on the competitive edge that transportation-based services hold in the
gig economy. Overall, this study underscores the managerial importance of understanding
these dynamics within the rapidly changing landscape of the gig economy.
The second chapter empirically examines the theoretical assertions of the initial chapter.
Utilizing a comprehensive dataset of ride-hailing activity within the New York metropolitan
area, coupled with geo-spatial data, this analysis aims to validate the relationship between
rider delays (Estimated Time of Arrival, or ETA) and the availability of drivers in specific
locations. This study seeks to ascertain whether the functional relationship observed in
real-world data mirrors the theoretical models commonly referenced in existing ride-hailing
literature (Larson et al. 1981). Additionally, considering the diverse spatial characteristics
of New York’s neighborhoods, this chapter explores variations in the dependency of delays
on driver availability, investigating the extent to which these relationships align with spaix

tial delay theory or diverge towards patterns more indicative of congestion-related delays.
Building on theoretical insights from the first chapter, this inquiry further investigates how
spatial dynamics influence the performance outcomes for ride-hailing platforms. This study
is critical as it not only seeks to bridge theoretical assumptions with empirical findings in the
context of urban mobility, but also contributes to the optimization of ride-hailing operations,
ultimately enhancing service efficiency and consumer satisfaction in metropolitan transport
networks.
The third chapter undertakes an experimental study of farsighted coalitional stability, a
game-theoretical concept that extends beyond traditional equilibrium models by incorporating the agents’ farsightedness. This investigation is inspired by the research of (Tian et al.
2019), who explored the stability of producers’ strategies under environmental legislation
mandating product recycling in markets with multiple, differentiated consumer products.
The aim is to design behavioral experiments that address critical questions: First, whether
theoretically stable outcomes are indeed stable in real-life bargaining scenarios; second, which
behavioral factors significantly influence agents’ decision-making processes. Given the scant
experimental research on Farsighted Stability, this study fills a significant gap by testing
the theory’s applicability in controlled laboratory conditions. We examine the conditions
—such as model parameters and levels of farsightedness— under which outcomes deemed
stable theoretically are realized through actual player interactions. This chapter aims to illuminate the practical implications of farsighted coalitional stability, thereby advancing our
understanding of strategic behavior in complex decision-making environments.
x

Chapter 1
1.1 Introduction
Over the last decade or so, the gig economy has rapidly gained importance as an integral
component of the modern labor market, reshaping the conventional employment landscape
with its flexible, on-demand work structure. Building on this swift growth, the gig economy
has become a more common feature of everyday economic interactions and an increasingly
universal work experience. For example, an August 2021 survey by the Pew Research Center
found that approximately 1 in 6 adults in the U.S. have earned at least some income from
gig work; among younger adults (ages 18 to 29), this fraction rises to nearly 1 in 3 (Anderson
et al. 2021) Furthermore, the flexibility provided by the gig economy is also attractive to
companies, especially in an economic climate characterized by fluctuations, with many incorporating independent or freelance workers into their overall employment strategy (Upwork
2022). Together, this has created an economic juggernaut, with transaction data indicating
a global gig workforce of over 43 million people and over $143B in annual gig wages globally
in 2018 (Mastercard et al. 2020).
Anecdotal evidence and empirical data alike suggest that transportation-based or spatial
services are predominant within the gig economy sector. For example, the data provided by
(Mastercard et al. 2020) shows that spatially-distributed offerings such as ride-hailing and
delivery services constitute the lion’s share of productivity in the gig economy. For example, in that 2018 data set, the global wage disbursements for transportation-based services
totaled $61.3B, whereas the disbursements for the other gig economy categories of professional services, household services and handmade goods, and asset-sharing services were
$6B, $14.1B, and $52.7B, respectively. Naturally, one would expect that delivery services
have only become more heavily patroned in the trends following the Covid-19 pandemic,
1

suggesting that spatial services are perhaps even more important in the gig economy today. Spatially oriented services can also take many forms, ranging from the well-known
examples of ride-hailing and delivery to more unique (but, no doubt, valuable) offerings like
tractor-sharing among smallholder farmers (Adebola et al. 2022).
In general, services are, almost by necessity, customer-centric in their design and execution. These operational frameworks and systems are inherently tuned towards consumer
preferences, offering a level of customization in both the product offerings and pricing structures to closely match market demand and consumer willingness to pay. Within the context
of the gig economy, platform-based services must also navigate the preferences of the workforce. The very essence of gig work offers workers a degree of flexibility that is generally
unattainable in conventional “firm” employment settings. Consequently, platforms that provide gig-based services must factor in this additional layer of decision-making, balancing the
needs and choices of the consumer with those of the gig worker. This careful balance, of
these two sides, is key to how gig economy businesses operate and affects the dynamics of
supply and demand in this growing industry.
In this chapter, we study price-and-delay differentiation in a two-sided platform and examine how its performance stands against that of the traditional firm operational model.
We aim to understand the influence of servers on the platform’s strategic decisions concerning this differentiation, as well as to identify additional levers of control at the platform’s
disposal beyond pricing. The effectiveness of these price and delay differentiation strategies
is measured through their impact on the platform’s revenue.
1.1.1 Contributions and Organization of the Chapter
We consider three modes of operation of a service system. These models vary in the number
of price-and-delay offerings they provide and in their operational focus: one-sided or firm
modes focus solely on customer needs, whereas two-sided or platform modes consider the
interests of both customers and service providers in their revenue maximization strategies.
2

We will analyze the performance of these different service operations in terms of the optimal
expected revenue, and we will characterize this revenue as a function of the underlying
server opportunity cost rate. This leads to the following organization of the chapter and its
associated contributions.
After reviewing related literature in Section 1.2, we devote Section 1.3 to discussion and
motivation of the three different sides — customers, servers, and service operation — that
have agency in our game theoretic model of the service. Tying these three sides together, in
Section 1.3, we also define the customers’ expected delay in the service as a general function
of the number of servers who choose to participate, and we highlight two relevant special cases
of the function that capture waits in spatial and non-spatial services. Then, in Section 1.4,
we employ these components into (non-linear) optimization models (Equations 1.8, 1.10,
and 1.17) for the service operation’s revenue maximization objective subject to constraints
due to the customer and (in the case that the service is a platform) server preferences.
Naturally, accounting for server choice will reduce the service operation’s control over its
offerings, and we propose additional structure for the platform to ameliorate this.
In Section 1.5, we present our main results, which characterize the optimal revenue of
the firm/platform and its relation to the servers’ opportunity cost. In particular, we identify
two critical thresholds of this cost rate: one in which the servers’ payoff hits their breakeven value in each mode of operation (Propositions 5, 6, and 7) and one in which the
firm and platform reduce to an equivalent operation under price-and-delay differentiation
(Proposition 10). These insights are built upon explicit characterization in the tractable base
case when the servers have negligible opportunity cost (Propositions 3 and 4 and Theorem 1).
These findings are all further explained and illustrated through accompanying numerical
experiments. Then, in Section 1.6, we contextualize and extend these results by studying
the functional structure of the customers’ delay for service as dependent on the number of
participating servers. Most notably, as a direct consequence of our results in Section 1.5, we
find the perhaps surprising insight that the firm and platform may be more likely to yield
3

equivalent revenue in spatial services than in non-spatial settings (Corollary 5), which may
offer intuition for the predominance of spatial services in the gig economy.
Finally, we conclude with discussion of possible model generalizations and directions of
future work in Section 1.7. All proofs are contained in Appendix A, as are supporting
technical results.
1.2 Related Work
The phenomena we study are related to the growing literature on the the operations of
two-sided markets, and gig economy platforms in particular, but also to broader work on
price-and-delay differentiation for consumers who are both price and waiting-time sensitive.
Below we discuss some streams of literature to which our work is related and of which we
aim to combine in this chapter.
Price & Delay Differentiation
Our model most closely resembles that of (Maglaras et al. 2018), which is a multiserver queueing model of a revenue-maximizing firm providing a service to a market of
heterogeneous price and delay-sensitive customers with private individual preferences. This
work motivates the equilibrium model we use through a fluid limit of an underlying queueing
theoretical model. It also resembles (Afeche 2013), which studies revenue maximization
problem in a single-server queueing system facing a market with two customer types. This
paper is similar in nature and deals with how should a firm design a price/lead-time menu
and scheduling policy to maximize revenues from heterogeneous time-sensitive customers
with private information about their preferences. (Nazerzadeh et al. 2018) also studies a
service firm that is faced with price and delay sensitive customers who are differentiated on
both their value for the service and the cost of waiting. Using a large system approach, that
work characterizes the firm’s revenue maximizing menu of price and delay quotations and the
value of customer differentiation. Similarly to these papers, we are focused on the analysis
4

of the firm-consumer dynamics in situations where a firm would want to offer differentiated
prices and delays in heterogeneous, time-sensitive customers. Moreover, we consider that
that the delay that the customers experience is directly related to the servers who provide
the services to the customers. In other words, we consider the relation that supply-side
behavior has with the resulting delays in the system. This is something that had not been
addressed in the aforementioned prior works.
Server Supply & Service Delay
The main relationship for servers and delay (or estimated time of arrival, ETA) in spatial
settings, that many of the papers on ride-hailing systems follow, is defined in Chapter 5 of
the textbook by (Larson et al. 1981). This is also what we use to model the relationship
between the density of servers and the resulting delay when we refer to spatial settings. The
celebrated work of (Castillo et al. 2017) offers a steady-state characterization of a ridehailing platform, and shows how dynamic pricing allows a platform to remain efficient after
a severe supply-demand imbalance. (Nikzad 2017) and (Besbes et al. 2022) also study the
relationship between the density of drivers (servers) and the resulting service time. However
these works do not discuss the possibility of multiple prices and multiple ETAs on the same
platform and the impact that such an environment would have on the agents involved. The
recent work by (Freund et al. 2021) touches on both price and delay differentiation as well
as density of drivers in spatial settings but focuses on the dynamic price changes that a
ride-hailing platform uses to efficiently balance supply and demand while our work focuses
on longer-term market outcomes of pricing and delay differentiation. (Benjaafar, Ding, et al.
2022) delves into the consequences of labor pool expansion and the broader implications for
regulators considering the implementation of wage floor policies. This stream of research,
however, does not extend to the strategies for maximizing revenue nor does it explore the
implications of multiple service offerings differntiated by price and delay.
Strategic Agents in Service Systems
Our work focuses on how agents behave in gig economy settings and how their choices
5

impact the other agents. (Afeche et al. 2018) studies the problem of matching demand
(customers) with self-interested capacity (servers) over a spatial network and analyzes optimal operations when there are significant spatial demand imbalances. In general, there is
considerable literature on the behavior of drivers/servers in ride-hailing platforms. (Garg
et al. 2021) considers that servers are strategic, wanting to maximize their long-run earnings
while working for the platform, and studies the effect of surge pricing on server earnings and
their strategies to maximize such earnings. (Castro, Frazier, et al. 2020) considers a setting
where drivers are heterogeneous in their compatibility with different types of jobs in the
platform and shows that their strategic choices, of which types of jobs to choose, can lead to
poor throughput outcomes for the platform if the appropriate matching mechanism is not
used. In a later work, (Castro, Ma, et al. 2021) looks again at the matching of drivers to
different types of trips in a queue (e.g ridesharing drivers in an airport), and shows that
if the appropriate dispatch rule is not used by the platform, the variance in earnings from
different types of trips incentivizes drivers to cherrypick the most profitable ones resulting
in an increase in customers’ waiting times and in a loss of efficiency in the platform. These
are all works that focus on the strategic behavior of servers and the challenges this strategic
behavior may cause to the platform. This stream of literature highlights how important it
is for the platform to understand what the servers incentives are and take the appropriate
action in order to ensure that their strategies are aligned with the platform’s goals. In those
works, however, the demand side is generally fixed and not affected by the decisions of the
server side. In our work, we want to study the interactions of the customers and servers in
such a two-sided market where the behavior of the supply side can impact the decisions of
the demand side.
The Gig Economy and Two-Sided Markets
Being an important and growing field, the gig economy has attracted considerable academic attention, with researchers examining its unique operational dynamics and economic
impact. For instance, (Lian and Van Ryzin 2021) develops a theoretical model to under6

stand optimal growth in two-sided markets, examining how platforms can control supply
and demand through pricing strategies to maximize profit and market sustainability. In
another work, (Lian, Martin, et al. 2022) proposes a theory of gig economies that explores
the balance of pay rates and profit margins, highlighting the diseconomies of scale in labor
costs and the conditions necessary for the formation of a gig economy, including the impact
on worker pool and market pricing strategies. These studies contribute to a deeper understanding of the strategic and economic factors that influence the evolution and functioning
of gig economies.
Many models of platforms draw upon a two-sided market representation. For example,
(Bai et al. 2019) presents an analytical model with endogenous supply (number of participating agents) and endogenous demand (customer request rate) to study an on-demand
service platform. However, this paper does not consider that this platform may offer differentiated prices. (Ahmadinejad et al. 2019) and (Wu et al. 2020) both study two-sided
competition between on demand platforms. (Ahmadinejad et al. 2019) in particular, focuses
on ride-hailing platforms and considers that the driver supply directly impacts the delay the
customers are experiencing. It is very similar in flavor with the way we approach the agents
involved in the problem (platform, servers, customers) and their interactions, as well as in
the way we characterize the resulting equilibria. But, instead of platform competition, which
is their focus, we want to understand how all these sides interact inside the same platform,
when this platform wants to offer differentiated service classes. (Taylor 2018) examines an
on-demand service platform, taking into account customer delay sensitivity and agent independence, and their impact on the resulting prices and wages. However, it does not account
for the price and delay differentiation that our work addresses. (Cachon et al. 2017) also
focuses on two-sided markets with self-scheduling capacity, such as rideshare platforms, but
focuses on how the effect of surge pricing helps regulate the system dynamically, while we
are focused on more long-term implications of price and delay interactions. Finally, (Siddiq
et al. 2022) adds an additional side to the study of two-sided, ride-hailing platforms, by
7

investigating the impact of autonomous vehicles (AVs) on the profitability and welfare outcomes of ride-hailing platforms, driver-workers, and rider-consumers, considering different
AV ownership structures.
In this chapter, we bring together elements from all these aforementioned areas as we
study price and delay differentiation in a service system where delays are directly related
to the available supply of servers and when both customers and servers are strategic and
interact inside the system.
1.3 Modeling the Different Sides & Problem Formulation
The game we study involves three types of agents: the management of the service operation,
the customers, and the servers. Customers arrive over time and, faced with a price and
delay until the start of service, choose whether to request a service or not. Our foremost
attention will be a case in which the service system offers two service classes that each have
a different price-and-delay pair. In this setting, the customers choose to request whichever
they prefer, or they may choose to not request service at all. The servers in the system can
choose whether to participate or not and, in the case of two service classes, they additionally
choose whether to serve one or both of two customer groups. Based on the market conditions
and the characteristics of the customers and servers, the service system management decides
whether to offer a single price at a single delay, or offer two different prices at two different
delays. In this section, we formally present the objectives and strategies for each these three
agents in the game, and we also discuss the functional form of service delay for the customers.
We then formally present price differentiation first in the traditional firm view, and then we
move on to the gig economy platform view of price differentiation, which is the focus of this
chapter.
8

1.3.1 Customers
We will assume that the platform is faced with two types of customers, type 1 and type 2.
For each type i = 1, 2, let us define an overall arrival rate Mi > 0 and a valuation distribution
Fi
, where Fi has a continuous density, and finite mean. We will accordingly let Vi denote
a valuation random variable, which for each customer is an independent and identically
distributed (i.i.d) draw from Fi
. This valuation, Vi
, represents the customers’ willingness-topay for the service. Notice that we suppress any indexing over customers in the Vi notation
for simplicity through the i.i.d. assumption, but each type i customer will (almost surely)
have a different realization from Fi
. Because customers are averse to waiting, we furthermore
consider that they incur an additive linear delay cost, or waiting time sensitivity, αi
.
Therefore, the net utility that customer of type i obtains from the service is equal to the
valuation they have for the service, minus the price they need to pay, minus they delay that
they experience discounted by their waiting time sensitivity. In other words,
Ui(p, d) = Vi − p − αid i = 1, 2 . (1.1)
An incoming customer faced only with one price and one delay will request the service as
long as their utility given by (1.1) is non-negative. If the service offers two price-and-delay
pairs, then a customer will choose the one that gives them the highest net utility; if their net
utility from both pairs is negative then they leave without requesting. Faced with options
j = 1, 2, a customer of type i will prefer (but not necessarily select, depending on the level
of their valuation) option 1 over option 2 if and only if
Vi − p1 − αid1 ≥ Vi − p2 − αid2 ,
9

and this naturally simplifies to
p1 + αid1 ≤ p2 + αid2 . (1.2)
Hence, customers of the same type, who consequently have the same delay-sensitivity αi
, will
choose the same preference between the two options. Incorporating this with the necessary
non-negativity of utility, the resulting arrival rates to each service offering are thus
µ1 = M1F¯
1(p1 + α1d1)1{p1 + α1d1 ≤ p2 + α1d2} + M2F¯
2(p1 + α2d1)1{p1 + α2d1 ≤ p2 + α2d2} ,
and
µ2 = M1F¯
1(p2 + α1d2)1{p2 + α1d2 ≤ p1 + α1d1} + M2F¯
2(p2 + α2d2)1{p2 + α2d2 ≤ p1 + α2d1} .
In the rest of the chapter, we will make the following assumption on the distribution of
the customers’ valuation.
Assumption 1. For the valuations of type 1 and type 2 customers it holds that: V1 ∼ U(1, 2)
and V2 ∼ U(0, 1).
Commonly seen in the related literature (e.g., Afeche 2013; Bimpikis et al. 2019; Wu
et al. 2020), the uniform valuation distributions in Assumption 1 will give rise to a stylized
model that offers tractable yet meaningful analysis. Notice that under this assumption,
type 1 customers (with representative valuation random variable V1) are guaranteed to have
a higher valuation for the service relative to type 2 customers (with V2). As a result of
this stylized representation, we thus consider that there is a separation of the two types of
valuations. We then have the following technical assumption, which relates the delay costs
of the two types of customers.
Assumption 2. For the delay costs of type 1 and type 2 customers it holds that: α1 > 1 >
α2;.
10

Assumption 2 shows that, as one would expect given Assumption 1, type 1 customers are
also more time-sensitive relative to type 2 customers. Hence, the combination of assumptions
tells us that the type 1 customers not only have a higher ability to pay compared to type 2
customers, but also that their sensitivity to waiting is higher than the sensitivity to price.
Notice that the separation point between the sensitivities, 1, is also the the border between
the valuation distributions. This technical condition will enable simplified results, but it
could likely be relaxed. The essential part of Assumption 2 is the intuitive one: type 1
customers are less patient than type 2 customers.
1.3.2 Servers
We assume that servers aim to maximize their long-run payoff. This payoff is given by the
earnings they make from a trip, minus the waiting cost they incur for the time they spend
before being matched with an incoming customer. We will assume that all servers incur the
same cost rate c while waiting for a match with a customer. Therefore a server’s payoff from
serving at price p to a customer population with an effective arrival rate of µ is given by
π =
µ(p − cτ )
n
, (1.3)
where for the expected server waiting time, τ , we have by Little’s Law that τ =
n
µ
. In the
above, n is the number of servers that choose to participate in the platform and are thus
able to serve customers. We want to highlight here the direct dependence of server payoff
on the number of servers participating. This is a variable that will be endogenous in the
model. We assume that this number of participating servers, n, is bounded above by some
number of potentially participating servers, N. That is, N is the size of the “pool” of servers
that are enrolled in the platform and can join the service at any time depending on the
payoff they expect to gain by doing so. In the main results of our work, we assume that
N is infinitely large or, in other words, there is no supply of servers constraint. In Section
11

1.7, we will also discuss an extension of our model to the case where the server pool size N
could be limited, and we will argue that our main insights hold in such a situation as well.
In the following sections, we will also give detailed expressions for the servers’ payoff under
the different platform operation modes we analyze, which we will introduce shortly.
As we consider the dynamics of the servers throughout this chapter, let us emphasize the
cost c as the most important element of the server side. One can think of c as the servers’
opportunity cost rate, as a server will not participate in the service if the expected payoff
from (1.3) is negative. Hence, in our analysis, c will serve as the main characterization of
server behavior and their sensitivity to their own waiting times. In our results, we see how
important this opportunity cost is, and we will identify how it affects the servers’ decisions,
and the resulting outcome for the platform.
1.3.3 The Delay Function
Inherently, the decisions of the customers are impacted by those of the servers, and vice versa
(Benjaafar and Hu 2020). These two sides are perhaps most prominently related through the
customers’ expected delay, which depends on the number of servers able to meet their request.
To capture this, we will suppose that the steady-state delay the customers experience is a
continuous and strictly decreasing function of the number of servers participating in this
service, n. In essence, the more participating servers there are, the shorter the delay for
the customers will be. As a general form of the delay function, we let the delay function be
defined by
d(n) = n
−γ
, (1.4)
for some γ > 0. Because d(n) is strictly decreasing for any γ > 0, there is a bijection between
any delay, d, and the number of participation severs, n, and thus one can be immediately
recovered from the other. Additionally, this form of d(n) offers a natural comparison across
12

γ, as d(1) = 1 for all γ > 0 but n
−γ1 < n−γ2
for any n > 1 and γ1 > γ2.
While most of our results will be given for any γ > 0, we will devote considerable
attention to two specific, interpretable values of γ. First, we take γ = 1 as a congestion
type of delay function. This label is drawn from the leading order of dependence on the
number of servers in classical queueing formulas, such as in the M/M/c steady-state mean
waiting time, in which the proportional dependence on n is no more than 1/n (e.g., Kulkarni
2011, Eq. 6.32). Then, we take γ = 1/2 as a spatial type of delay function. As introduced
and motivated by (Larson et al. 1981), this structure captures the expected waiting time’s
leading order of dependence on n when a two-dimensional space must be traversed in order
to begin service, and it has been commonly used in modeling spatially-based services such
as ride-hailing and transportation (e.g., Yan et al. 2020; Freund et al. 2021; Besbes et al.
2022). Through the general form of these functions, we seek to capture the expected delay’s
first order dependence on the number of servers, with the overarching goal to understand
how the shape of this dependence impacts the performance of the service operation.
Finally, let us note that, modeling the delay as a function of the number of participating
servers, rather than presently idle servers, inherently assumes there is an instantaneous rate
of service. This is similar in spirit to instantaneous relocation or travel time assumptions,
which have been leveraged in the related spatial services literature (e.g., Waserhole et al.
2016; Balseiro et al. 2021; Banerjee et al. 2022). Our analytical findings capitalize on the
tractability of this assumption, and, in Section 1.6.2, we numerically verify that our insights
remain valid even when considering a non-instantaneous service rate.
1.3.4 Service Operation
We assume that the management of the service operation knows the following characteristics
of each customer type: the overall arrival rate Mi
, the waiting sensitivity αi
, and the valuation
distribution Fi
. For the servers, we assume the service operation knows the common server
waiting cost, c. However, the service operation does not know each individual customer’s
13

type and their actual valuation, Vi
, and thus they cannot directly offer them the product
that would be most suitable as individuals. Instead, the service offers the same options to
all incoming customers.
The service operation’s objective is to maximize its revenue. It therefore needs to choose
how many options to offer to its customers (one price-and-delay pair or two distinct pairs)
and to find the optimal price(s) for each case. By the payoffs offered to the servers, the
operation also effectively controls the delay(s) for service. Hence, the service operation’s
objective is to maximize the sum of the price and arrival rate for each type, subject to the
choice of price and delay:
max
p1,p2,d1,d2
p1 · µ1(p1, p2, d1, d2) + p2 · µ2(p1, p2, d1, d2) . (1.5)
Let us note that the indices 1 and 2 above enumerate the two options of price-and-delay
offerings, or service classes, and are not to be confused with the index of the customer types.
We will refer to service class 1 as the high-price, low-wait offering, whereas class 2 is the
low-price, high-wait offering (as will be formalized in the revenue maximization problem).
In the case of a single price-and-delay pair, (1.5) naturally reduces to a problem simply with
two variables, p and d.
We will focus on two ways this service operation can be conducted. The first one is the
“one-sided” version, which focuses primarily on customer choice in the optimization problem.
Hence, the service operation is optimizing as a “firm” that does not have to account for its
servers’ preferences and actions. The second one is the “two-sided” or “platform” version
where both customers’ and servers’ characteristics are considered. In this case, the service
operation has to take into account both sides when it is setting prices and delays to maximize
its revenue. The second case captures the dynamics we would expect to find in a gig economy
setting and is the one we ultimately want to understand. The reason we are also considering
the “firm” version is to essentially provide us with a benchmark on how well the “platform”
14

can do.
Throughout this chapter, we will refer to the different scenarios through notation of the
form a × b. The left hand side (LHS), a, refers to the number of price-and-delay pairs that
are offered. If a = 1, this means that there is a single price-and-delay pair and if a = 2 we
have differentiated price-and-delay pairs. The right hand side (RHS), b, refers to the number
of parties the firm/platform needs to account for. When b = 1, we are in the firm case where
primarily the customer side is taken into account, and, when b = 2, we are in the platform
case when both customer and server side are taken into account. For example, 2 × 1 refers
to price-and-delay differentiation without server choice, and 2 × 2 refers to the same with
server choice between the two offerings.
Let us emphasize here that, in the firm case, the service operation does not account for
the servers’ choices relative to the different offerings, but it does account for the fact that a
server will not participate in the service if they expect a negative payoff. This is based on
the understanding that a traditional firm, despite not offering the same level of flexibility
as a gig economy platform, must still ensure its employees receive sufficient compensation
to incentivize their participation. Therefore, both a firm and a platform would consider
that there is a minimum amount of payment the servers need to receive, but the platform
additionally must consider that the servers can also compare payments between the different
offerings and choose which is best for them.
As we mentioned in Section 1.3.2, the servers’ opportunity cost rate c will be the main
lens through which we will observe the server behavior. We will study how the revenue
maximization problems of each mode of operation are impacted by this servers’ cost rate.
For each problem formulation, we will identify some ¯c thresholds (¯c2×1 > 0, ¯c2×2 > 0, and
c¯1×1 > 0) that correspond to the maximum value of c for which we conduct our analysis. For
c greater than these values, the servers’ opportunity becomes so large that the problem is
restricted by the servers’ payoff constraint in a way that does not enable us to obtain useful
managerial insights. In the proof of each relevant proposition in Appendix A.2, we are
15

able to characterize ¯c2×1 and ¯c1×1 but, given their long expressions, we omit them from the
main body of the chapter. Therefore, throughout the chapter, we simply make the following
assumption on the servers’ opportunity cost rate.
Assumption 3. For the servers’ opportunity cost c it holds that c < min(¯c2×1, c¯2×2, c¯1×1) .
In the remainder of the chapter, we will consider three versions of the above maximization
problem (1.5) under this setup we described. First, we will present the 2 × 1 and 1 × 1
problems where we have a firm optimizing revenue considering only the customer choices
and the servers’ opportunity cost. Then, we will move on to our main problem of interest,
2 × 2, where we have a platform optimizing revenue considering both customer and server
choices, as well as the servers’ opportunity cost.
1.4 Modeling the Different Revenue Optimization Problems
1.4.1 Traditional View of Price-and-Delay Differentiation (Firm’s
Optimization Problem)
We will now review how things evolve when the customers characteristics and the servers’
outside option are included in the analysis, but the servers’ choices inside the platform are
not. This firm, or traditional, or one-sided case corresponds to the customer-centered version
of the revenue maximization problem, which has been studied in the related literature.
Following Section 1.3, each customer’s valuation is their own private information and is
therefore unknown to the platform. By Myerson’s Revelation Principle (Myerson 1981), we
can focus our attention to direct mechanisms that would result in the customers reporting
their valuation type to the platform. By imposing Individual Rationality (IR) and Incentive
16

Compatibility (IC) constraints to hold for the prices and delay times, it would be a Nash
Equilibrium for each customer to truthfully report their type, and choose the “appropriate”
service option. These constraints are
pi + αidi ≤ pj + αidj
, ∀i ̸= j
µi = MiF¯
i(pi + αidi) i = 1, 2
(Incentive Compatibility) ,
(Individual Rationality) ,
(1.6)
where, by Assumption 1, F¯
1(x) = 2 − x for x ∈ (1, 2) and F¯
2(x) = 1 − x for x ∈ (0, 1).
Now, as we mentioned earlier, to ensure sufficient server participation, we would need to
add one additional constraint to the optimization problem to require that the servers earn a
non-negative payoff from participating in the service. Recall as well that the invertibility of
d(n) implies that the number of participating servers would be determined by the delays the
firm chooses. In particular, with n as the total number of servers participating in the service
and with d1 as the smallest delay the platform offers, n would be given by n = d
−1
(d1) = d
− 1
γ
1
.
Therefore the server payoff, which must be non-negative, would become
π2×1 =
p1µ1 + p2µ2 − c · n
n
=
p1µ1 + p2µ2 − c · d
− 1
γ
1
d
− 1
γ
1
. (1.7)
17

Together, we can express the firm’s revenue maximization problem as
max
p1,p2d1,d2
p1 · M1(2 − p1 − α1d1) + p2 · M2(1 − p2 − α2d2) (1.8)
s.t 1 ≤ p1 + α1d1 ≤ 2
0 ≤ p2 + α2d2 ≤ 1
p1 + α1d1 ≤ p2 + α1d2
p2 + α2d2 ≤ p1 + α2d1
p1M1(2 − p1 − α1d1) + p2M2(1 − p2 − α2d2) − c · d
− 1
γ
1 ≥ 0
d1 ≤ d2
p1, p2, d1, d2 ≥ 0 .
We will refer to (1.8) as the 2 × 1 problem. For a fixed servers’ opportunity cost, c, and
delay parameter, γ, we will denote the optimal solution to the 2×1 problem with (p
∗∗∗
i
, d∗∗∗
i
),
i = 1, 2, and the optimal revenue of 2 × 1 with R∗∗∗
2×1
.
Let us now define a classic variation of this problem, in which the firm offers only one
price-and-delay pair. Notice that a single price and delay solution would immediately be
feasible in (1.8) as well, but for the purposes of illustrating our upcoming results we will
consider also the single-price optimization problem under the same conditions. With a
single price and delay, the relevant server payoff would become
π1×1 =
p(µ1 + µ2) − c · n
n
=
p(µ1 + µ2) − c · d
− 1
γ
d
− 1
γ
. (1.9)
18

Therefore, the single price-delay revenue maximization problem would be:
max
p,d
p · M1(2 − p − α1d) + p · M2(1 − p − α2d) . (1.10)
s.t 1 ≤ p + α1d ≤ 2
0 ≤ p + α2d ≤ 1
pM1(2 − p − α1d) + pM2(1 − p − α2d) − c · d
− 1
γ ≥ 0
p, d ≥ 0 .
We will refer to (1.10) as the 1 × 1 problem. For a fixed servers’ waiting cost, c, and delay
parameter, γ, we will denote the optimal solution to the 1 × 1 problem with (p
∗
, d∗
) and the
optimal revenue of 1 × 1 with R∗
1×1
.
These two firm problems, 2 × 1 and 1 × 1, will serve as our benchmark to understand
the platform’s revenue maximization problem, which we will introduce shortly. Throughout
the remaining exposition of this chapter, we will invoke the fact that, under the stated
assumptions on the customer valuations and waiting time sensitivities, the optimal revenue
of the 2 × 1 case, R∗∗∗
2×1
, is no worse than the optimal revenue of the 1 × 1, R∗
1×1
, for any
server waiting cost c.
Proposition 1. Assumption 1 ensures that, for any server cost c, the firm revenue from price
differentiation is no worse than single price-and-delay revenue. Additionally, Assumptions 1
and 2 ensure that, for server cost c = 0, the firm revenue from price differentiation is strictly
better than single price-and-delay revenue.
This proposition allow us to highlight that our focus in this work is on situations where it
would make sense for a firm to offer differentiated prices and delays when ignoring the server
side choices, in order to understand how the addition of the server side would impact whether
or not service differentiation is still better than a single service. Let us also note that the
second statement of Proposition 1 may appear restrictive, given that it exclusively considers
19

the difference of revenues at c = 0. Indeed, understanding the difference in performance
beyond c = 0 will be a central thread in our upcoming analysis. Moreover, in the course of
this analysis, we will recognize that the c = 0 case actually provides a foundation through
which we can characterize c > 0.
1.4.2 Gig Economy View of Price-and-Delay Differentiation (Platform’s Optimization Problem)
We will now continue to the case where the service operation, in addition to the customer
preferences, must account for the servers preferences inside the system. Using the set-up
and intuition discussed in the previous section, we will study this platform setting.
1.4.2.1 Added Challenges & Added Controls
Our primary motivation stems from the fact that the platform scenario introduces additional complexities compared to the firm’s operation. The key challenge arises from the fact
that, in the platform case, where the service operation provides multiple service offerings to
potential customers, the servers have the flexibility to select the class they wish to prefer to
serve. Because the customers’ delay is a function of the number of participating servers, the
servers’ decision of which class to serve directly impacts the delay experienced by customers.
More importantly, this means that, relative to a firm, a platform relinquishes full control
over this aspect by allowing flexibility and choice to the servers. The price-and-delay differentiation that we presented in Section 1.4.1 is therefore no longer applicable for a platform
exactly because the platform does not have the full control of the customer delays. We can
therefore understand that the firm’s optimization problem in (1.8) is not appropriate for a
platform.
To ameliorate this lack of control, we will suppose that the platform introduces additional
structure. First let us define two service groups with the following characteristics: server
20

group 1, G1, serves only class 1, and server group 2, G2, serves class 1 and class 2. In
the language of the structure of the service design, this leads to an “N-model” relationship
connecting the service classes and the server groups (e.g., Tezcan et al. 2010). The structure
of the server groups can be seen in Figure 1.1. Notice that both of the server groups are
always available to serve class 1 service requests. This ensures that there is priority given to
the high-paying, low-wait customers. Moreover, it means that the customers choosing the
high-price, low-wait class are offered the smallest possible delay given the total number of
participating servers, which we will discuss in more detail in the following section.
µ1 µ2
G1 G2
f1µ1
(1 − f1)µ1
µ2
Figure 1.1: The Server Groups and the Customer Arrival Rates They Face
Having this set-up of server groups, the platform can now control the traffic of class 1
requests that will go to each group. We define f1 to be the proportion of class 1 service
requests that is assigned for service to server group 1. Hence, a 1 − f1 fraction of class 1
requests are served by server group 2, which additionally serves the class 2 requests. If we
denote the number of participating servers in each server group by ni
, i = 1, 2, the servers’
payoffs from each server group are
π1 =
f1µ1(p1 − c
n1
f1µ1
)
n1
=
f1µ1p1 − cn1
n1
, (1.11)
21

and
π2 =
((1 − f1)µ1 + µ2)

(1−f1)µ1p1+µ2p2
(1−f1)µ1+µ2
−
cn2
(1−f1)µ1+µ2

n2
=
(1 − f1)µ1p1 + µ2p2 − cn2
n2
.
(1.12)
Essentially, relative to the firm, the proportion f1 is an additional lever introduced by the
platform to account for the additional degree of choice in the operation arising in the servers’
preferences. We analyze how a platform would want to think about setting this control, in
addition to prices and delays, in the upcoming section.
1.4.2.2 Analysis of the Server Indifference Condition
By definition of this gig-based setting, the platform wants the servers to be able to freely
choose which group to join. The servers, faced with the expected payoffs in each group,
would select whichever among G1 and G2 gives them a higher (non-negative) payoff. To
achieve indifference between these groups, the platform must ensure that the server payoffs
in each group are such that no group is strictly more profitable than the other. If this were
to happen, all servers would select the more profitable group, and the platform would no
longer be able to conduct price-and-delay differentiation. Hence, the 2 × 2 formulation must
therefore control the operation so that the servers are indifferent between the two groups,
i.e π1 = π2. This indifference condition is
f1µ1p1 − cn1
n1
=
(1 − f1)µ1p1 + µ2p2 − cn2
n2
,
which by solving for f1 and using that n = n1 + n2 becomes
f1 =
n − n2
n
µ1p1 + µ2p2
µ1p1
. (1.13)
22

If we solve the maximization problem without the servers indifference condition and the
resulting optimal pair (pi
, di) satisfy 0 ≤ f1 ≤ 1, then the platform can offer those “unconstrained” optimal prices and delays and achieve the optimal revenue by setting the f1
accordingly, to account for the server preferences. However, if the resulting f1 is not between
0 and 1 for such unconstrained optimal prices, then the platform must find other prices,
delays, and f1 proportion that feasibly maximizes the revenue. Hence, in the platform setting, we will add the constraints f1 =
n−n2
n
µ1p1+µ2p2
µ1p1
, f1 ≥ 0 and f1 ≤ 1, to the optimization
problem in (1.8), with f1 also added as a decision variable.
However, we can notice that f1 does not appear in the objective of the optimization
problem, and, moreover, we can formulate this problem without including f1 as a variable if
we instead enforce it as a constraint given by (1.13). That is, the conditions that 0 ≤ f1 ≤ 1
become
f1 ≥ 0 ≡ (n − n2)(µ1p1 + µ2p2) ≥ 0 , (1.14)
and
f1 ≤ 1 ≡
n
n2
≤
µ1p1 + µ2p2
µ2p2
. (1.15)
Note that, for any n it holds that n2 ≤ n, and thus (1.14) must be satisfied whenever d1 ≤ d2.
Hence, (1.15) is the important constraint that must be added to the platform’s optimization
problem. We will refer to (1.15) as the traffic conservation constraint.
Hence, the f1 variable can be eliminated and the problem we want to solve is an optimization problem only with the original variables and an additional constraint related to
the server indifference condition reflected through the feasibility of the traffic conservation
constraint in (1.15). If we also use that n = d
−1
(d1) and plug in the arrival rates that result
23

from our model on the customer valuations, then the traffic conservation constraint becomes
d
−1
(d1)p2M2(1 − p2 − α2d2) ≤ d
−1
(d2)

p1M1(2 − p1 − α1d1) + p2M2(1 − p2 − α2d2)

.
(1.16)
By employing the inverse function of the delay, the platform’s revenue maximization problem
thus becomes
max
p1,p2d1,d2
p1 · M1(2 − p1 − α1d1) + p2 · M2(1 − p2 − α2d2) (1.17)
s.t 1 ≤ p1 + α1d1 ≤ 2
0 ≤ p2 + α2d2 ≤ 1
p1 + α1d1 ≤ p2 + α1d2
p2 + α2d2 ≤ p1 + α2d1
d1 ≤ d2
d
1
γ
2 p2M2(1 − p2 − α2d2) ≤ d
1
γ
1

p1M1(2 − p1 − α1d1) + p2M2(1 − p2 − α2d2)

p1M1(2 − p1 − α1d1) + p2M2(1 − p2 − α2d2) − c · d
− 1
γ
1 ≥ 0
p1, p2, d1, d2 ≥ 0
We will refer to (1.17) as the 2 × 2 problem. We will denote the associated optimal solution
as (p
∗∗
i
, d∗∗
i
), i = 1, 2, and the optimal revenue as R∗∗
2×2
.
Now that we have presented the optimization problems, we can see that, the 2 × 2
problem above is the relevant platform problem since only in this one we see the servers’
choice between the service offerings arising. In the 2×1 problem, given by (1.8), this choice is
not taken into account, making this problem the ‘firm’ problem. Since in the 1 × 1 problem,
given by (1.10), there is not a choice between different offerings to be made, we can view
this case as both firm and platform.
In an initial result, let us now show that we can bound the optimal revenue of inter24

est, R∗∗
2×2
, between the other two, as has been suggested by the superscripts in our chosen
notation.
Proposition 2. For customer market characteristics α1, α2, M1, M2, and any servers” opportunity cost c we have that R∗
1×1 ≤ R∗∗
2×2 ≤ R∗∗∗
2×1
.
The importance of Proposition 2 lies in the fact that we can actually bound the performance of the platform in the more complicated and realistic, 2 × 2 regime, by the optimal
performance of it in the cases where the server choices are ignored. Because we can directly
compute R∗
1×1
and R∗∗∗
2×1
for any server opportunity cost rate c (as we will illustrate in the
following section), we obtain a tractable range on which R∗∗
2×2
lies.
To close this section, let us now define notation for an important quantity, the difference
between the firm’s optimal revenue and the platform’s. Let ∆ be such that
∆ = R
∗∗∗
2×1 − R
∗∗
2×2
. (1.18)
While Proposition 2 shows that for any customer and server characteristics the difference
between R∗∗∗
2×1
and R∗∗
2×2
is non-negative, one of the main goals of this work is to analyze ∆
and understand when this difference may be large, and when it may be negligible.
1.5 Analyzing the Impact of Server Choice on the Service Operation
Having defined our optimization problems in Sections 1.4.1 and 1.4.2, let us now look at
the resulting revenues through the lens of server behavior as reflected through c, the server
opportunity cost. To build tractable insight and scaffold it to generality, we will start
our analysis with the idealized assumption that the server opportunity cost is insignificant,
i.e. c = 0. We will then then move on to a positive opportunity cost to see how the strength
25

of this exogenous option impacts the system. Throughout, we will highlight the differences
between the platform and the firm.
1.5.1 Revenue and Performance under Negligible Server Opportunity Cost
In the context of this subsection, let us suppose that c = 0. Here, the server payoff nonnegativity constraint is trivially satisfied in all problems, and, in this form, the firm problems,
(1.8) and (1.10), are tractable and we can express their optimal solutions in closed form. We
now characterize the optimal 2 × 1 and 1 × 1 solutions when c = 0.
Proposition 3. The optimal solution to the 2 × 1 problem at c = 0 is:
p
∗∗∗
1 = 1 , d∗∗∗
1 = 0 , p∗∗∗
2 =
1
2
, and d
∗∗∗
2 =
1
2α1
.
Proposition 4. The optimal solution to the 1 × 1 problem at c = 0 is:
d
∗ =

(α1 − α2)M2 − α1M1
2α1(α1 − α2)M2
+
and p
∗ = 1 − α1d
∗
.
By looking at Propositions 3 and 4, we see that the optimal solution to the 1×1 problem
appears more complicated than that of the 2×1 problem. Although this may initially appear
un-intuitive, we can quickly recognize that it is a reflection of the difference between the two
operations. That is, in the 1 × 1, the firm must balance the preferences of both types of
customers in single price-and-delay pair, whereas, by definition, the optimal 2 × 1 solution
differentiates between the two types.
An important element of the optimal solution to 2 × 1 is that d
∗∗∗
1
, which is the delay
in the high-price, low-wait service offering, is equal to the smallest possible delay, which, in
this case, is 0. This is because there is an infinite pool of servers and, more importantly,
because the server opportunity cost is 0, implying that all servers in the infinite pool decide
26

to participate and are able to serve customers. By comparison, in the 1 × 1 case which can
be viewed as both firm and platform, we could have both an infinite and a finite number
of servers participating in the platform, depending on the model parameters for the market
conditions.
Corollary 1. If c = 0 and M1 <
α1−α2
α1 M2 then, in the 1×1 problem, there is a finite number
of servers opting in, even though the server pool is infinite.
This is an important feature of the 1 × 1 solution, and we will discuss the implications,
contrasts, and connections with the 2 × 2 case in more detail in the following subsection.
In general, the 2×2 is much harder to characterize in closed form compared to 2×1 and
1×1, which is due to the complexity that the traffic conservation constraint (f1 ≤ 1) brings.
In the case that c = 0, however, we are able to simplify the problem in a way that allows us
to characterize the optimal solutions very efficiently and obtain numerical solutions easily.
In addition to our earlier notation, we will let the optimal solution to the 2 × 2 problem
when c = 0 be denoted (ˆpi
,
ˆdi), i = 1, 2. In this negligible server opportunity cost setting,
the full characterization of the optimal solution to the 2×2 problem is given by the following
theorem.
Theorem 1. The optimal solution to the 2 × 2 problem when c = 0 satisfies:
pˆ1 + α1
ˆd1 = 1 and pˆ1 + α1
ˆd1 = ˆp2 + α1
ˆd2 ,
where 0 < ˆd1 < ˆd2 solve the following system of equations:
α1(γ + 1)d
1
γ
1 − d
1−γ
γ
1 − α
2
1
(γ + 1)d
1+γ
γ
2 + α1(γ + 1)d
1
γ
2 + 2α1d
1−γ
γ
1 d2 − 2α
2
1
(γ + 1)d
1
γ
1 d2 = 0 ,
and
α1M1
M2(α1 − α2)
d
1+γ
γ
1 −
M1
M2(α1 − α2)
d
1
γ
1 − α1d
1+2γ
γ
2 + d
1+γ
γ
2 − d
1
γ
1 d2 + α1d
1
γ
1 d
2
2 = 0 .
27

Notice here that, unlike the 2 × 1 optimal solution at c = 0 where we had d
∗∗∗
1 = 0, we
have ˆd1 > 0 in the optimal solution to 2 × 2. This gives rise to an important corollary.
Corollary 2. For c = 0 in the 2 × 2 problem, there is a finite number of servers opting in,
even though the server pool is infinite.
We see here that in the 2×2 platform price and delay differentiation problem, in a similar
manner to what can happen under only certain market conditions when a single price-anddelay pair is offered, only a finite number of servers is needed at the optimal point in the
platform, even though the zero opportunity cost would otherwise imply that all servers would
participate in the firm under price-and-delay differentiation. We will discuss the implications
of this on the servers’ payoffs towards the end of this section, but first we will first investigate
how the problem changes when the servers have a positive opportunity cost.
1.5.2 Non-Trivial Server Opportunity Cost and the Value of Server
Payoffs
As we mentioned above, when we assume that the opportunity cost is c = 0, we are essentially
ignoring any hesitation on the server side, implicitly assuming that all servers available would
participate. This gives us a good starting point on how to analyze these modes of operation,
but we should not expect it to be an entirely realistic case. Let us now look at how a more
selective population of servers impacts the platform, specifically through investigating c > 0.
Naturally, the server opportunity cost affects the servers’ payoff and this informs their
decision of whether to participate in the service or not. The first important set of values
for c that we will discuss will be denoted by cπ1×1
, cπ2×1
, and cπ2×2
, and these will signify the
server opportunity cost value at which the servers’ payoff from the prices and delays offered
by the service becomes 0. For each mode of operation, this value of c constitutes the first
point at which the servers’ opportunity cost becomes consequential for their participation
decisions. That is, for any c > cπ, the server payoff rate will be its lowest feasible value:
28

π = 0.
Starting with the case of price-and-delay differentiation by a firm, we have the following
proposition.
Proposition 5. For any customer characteristics, cπ2×1 = 0.
One can quickly recognize that Proposition 5 naturally relates to the fact that Proposition 3 found that d
∗∗∗
1 = 0 at c = 0. As we discussed, this zero-delay would require
participation from infinitely many servers, and Proposition 5 now shows that any c > 0 will
create friction through the servers’ payoffs hitting a break-even point.
Moving on to the 1 × 1 case, the value of this threshold, much like the solution and
optimal revenue of this problem at c = 0, will depend on the customer characteristics.
Proposition 6. If M1 <
α1−α2
α1 M2, then
cπ1×1 =

(α1 − α2)M2 − α1M1
2α1(α1 − α2)M2
1
γ
·
(α1M1 + (α1 − α2)M2)
2
4α1(α1 − α2)M2
> 0 .
Otherwise, cπ1×1 = 0.
As can be seen above, in the 1 × 1 case, cπ1×1
can be either 0 or non-negative, depending
on the structure of the optimal solution of the optimization problem at c = 0. Hence, we
can extend this insight to the 1 × 1 revenue for a range of c values.
Corollary 3. If M1 <
α1−α2
α1 M2, then for 0 < c ≤ cπ1×1
, the optimal revenue R1×1 is constant
and equal to R1×1 =
(α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
.
Moving now to the case of price-and-delay differentiation by a platform, recall our notation of the optimal solution to the 2 × 2 problem when c = 0, which is (ˆpi
,
ˆdi), i = 1, 2. Let
the optimal revenue of 2 × 2 for c = 0 be denoted by Rˆ
2×2 = R2×2(ˆpi
,
ˆdi). As we now show
in Proposition 7, the value of c at which the servers payoff first becomes restrictive in the
2 × 2 can be characterized in terms of the optimal solution and revenue at c = 0.
29

Proposition 7. For any customer characteristics,
cπ2×2 =

ˆd1
1
γ
Rˆ
2×2 > 0.
An important consequence of the cπ1×1
, cπ2×1
, and cπ2×2
thresholds is that whenever the
servers’ opportunity cost surpasses each threshold, the server payoff constraint from the
respective mode of operation becomes binding. If, for example, we are looking at the 2 × 2
mode of operation and we know that the servers’ opportunity cost is greater than cπ2×2
, then
we know that at the optimal point the platform would choose to operate the net payoff of
the servers is zero. In this sense, each cπ thresholds represents, for the respective mode of
operation, the threshold at which the outside option becomes an immediate consideration
for the servers. This concept is not novel, as prior studies have discussed the idea of a
“reservation wage,” which refers to the minimum payment a server will accept for a job
(e.g., the “Rothchild’s critique,” as outlined in Rothschild 1978).
Given the history of this concept, we believe it is interesting to now compare the cπ
threshold for the different modes of operation. As can be seen from Propositions 5 and 6, cπ2×1
is zero for any market characteristics, and cπ1×1
is zero under certain market characteristics.
When we are looking at a firm therefore, this service operation is able to extract the full,
potential value from the servers and fully utilize the participating labor force. This means
that for any opportunity cost c, even in the case that c is zero, the participating servers
earn a net payoff of zero. In the 2 × 2 platform case, however, we interestingly see from
Proposition 7 that the cπ2×2
is nonzero and, thus, for a range of c values (from 0 until cπ2×2
),
the participating labor force is able to gain a strictly positive payoff. As we mentioned in
Section 1.4.2.2, the 1 × 1 case can be thought of as both firm and platform, and this is also
manifested again here where, depending on the market conditions, the service operation may
or may not extract the full value of servers’ payoffs.
We see here that the distinction between firm and platform that we make in this chapter
30

not only impacts the firm or the platform itself, but also the servers that (potentially)
participate in each environment. Interestingly, the firm, which is able to set prices and
delays and achieve the revenue it wants given some customer demand, does so by offering
the minimum payoff possible to the servers. A platform however, which needs to give its
servers some flexibility and thus must forfeit some control, ends up benefiting the servers in
some cases by allowing them to gain payoff strictly better than the smallest possible.
As we mentioned in the previous section, the solution to the 2 × 2 problem for c = 0 is
an important one. From Proposition 7, for example, we see that characterizing the optimal
solution at c = 0 can also help us directly compute the opportunity cost threshold cπ2×2
that
makes servers’ payoffs, at this version of the problem, equal to zero. The importance of this
for the platform is illustrated in the following corollary.
Corollary 4. For 0 < c ≤ cπ2×2
the optimal revenue in the 2 × 2, R∗∗
2×2
, is equal to Rˆ
2×2,
while there exists a positive gap between R∗∗∗
2×1 and R∗
1×1
.
Recall from the ordering of the optimal revenues in Proposition 2 that R∗∗
2×2 has an
upper bound of R∗∗∗
2×1
. With this last corollary we actually see that for the range of server
opportunity costs in [0, cπ2×2
] the difference between R∗∗∗
2×1
and R∗∗
2×2
is strictly decreasing since
R∗∗
2×2
stays constant. Hence, as the server opportunity cost increases, the performance of the
firm approaches that of the platform. We will explore this further in the next subsection.
Before doing so, let us first gain further understanding of the optimal solutions to these
revenue maximization problems.
While the 2 × 2 becomes more difficult to characterize in closed form when c > 0, we
can characterize the 2 × 1 and 1 × 1 optimal revenues and solutions. Through these, by
identifying server opportunity cost thresholds we can gain understanding for the behavior
of 2 × 2. The following pair of propositions provides the optimal solutions to the 2 × 1 and
1 × 1 when c > 0.
31

Proposition 8. The optimal solution to 2 × 1 for 0 = cπ2×1 < c satisfies
p
∗∗∗
2 =
1
2
, d∗∗∗
2 =
1
2α1
, and p
∗∗∗
1 = 1 − α1d
∗∗∗
1
,
where d
∗∗∗
1
solves the polynomial
α1M1d
γ+1
γ
1 − (
M2
4
α1 − α2
α1
+ M1)d
1
γ
1 + c = 0 .
Proposition 9. The optimal solution to 1 × 1 for cπ1×1 < c < c¯1×1 satisfies p
∗ = 1 − α1d
∗
and d
∗
solves the polynomial
α1(M1(α1 − 1) − M2(1 − α2))d
2γ+1
γ + (M2(α1 − α2) − α1M1)d
γ+1
γ + M1d
1
γ − c = 0 .
Comparing Propositions 8 and 9 to 3 and 4, we clearly see that the non-linear server
payoff non-negativity constraint, which needs to be taken into account when c is non-zero,
impacts the problems and their optimal solutions significantly. In both the 1×1 and 2×1, the
optimal delay(s) are given by solving a polynomial, which we provide explicitly. Therefore
for each c, we are able to fully characterize the optimal 2 × 1 and 1 × 1 revenues. Since,
however, the 2 × 2 problem has two non-linear constraints (the server payoff and the traffic
conservation constraints), it is not as straightforward to characterize the 2 × 2 revenue for
any given c. By identifying another important threshold for the opportunity cost c, though,
we are able to relate the 2×2 revenue to 2×1 and obtain a good picture of how the optimal
2 × 2 revenue behaves as c increases.
1.5.3 High Server Opportunity Cost Leads to Firm-Platform Revenue Equivalence
As we saw in the previous section, we can characterize the solutions to the 1 × 1 and 2 × 1
problems even when c is not equal to zero. In combination with our bound from Proposition
32

2, this allows us to obtain a range of revenues on which R∗∗
2×2
lies for any c. Moreover, we
are able to identify another important threshold for this server cost: the point at which, and
after which, the optimal platform and firm revenues are equal, meaning ∆ = 0 as defined in
(1.18).
Proposition 10. There exists a unique server opportunity cost c∆ > 0 given by
c∆ =

1
2α1
1
γ M2
4
(α1 − α2)
α1
, (1.19)
such that for c < c∆, we have R∗∗
2×2 < R∗∗∗
2×1 while for c ≥ c∆, we have R∗∗
2×2 = R∗∗∗
2×1
, and the
problems 2 × 1 and 2 × 2 have equivalent optimal solutions.
This c∆ is an important value for a service operation to know, as it shows that after
this level of the server cost, the revenue from the platform and the firm are equivalent.
The fact that for servers with high opportunity cost the platform can match the revenue of
the firm might seem counter-intuitive at first. However, what is happening here is that at
c∆ the traffic conservation constraint is no longer binding in the 2 × 2, whereas the server
outside option constraint remains binding. As a result, the only important constraint in
both problems is the server payoff non-negativity constraint.
Figures 1.2 and 1.3 show the full picture of the range of c. We demonstrate the solutions
to the optimization problems solved using the “Improved Stochastic Ranking Evolution
Strategy” (ISRES) algorithm (Runarsson et al. 2005) for non-linearly-constrained global
optimization, as developed for R Statistical Software by (“The NLopt nonlinear-optimization
package” 2007). The two parameter settings that we are illustrating here are similar in
everything except the arrival rate of type 1 customers, M1, which impacts the 1 × 1 solution
at c = 0 (see Proposition 4). Starting with Figure 1.2 when M1 = 25, we see that the optimal
solution to the 1 × 1 at c = 0 is such that the payoff for the servers is strictly positive and
therefore the cπ1×1 > 0. Contrasting this with the M1 = 50 figure, we see that, in this case,
the optimal solution to 1 × 1 at c = 0 is such that the payoff for the servers is 0 and stays 0
33

for any c > 0. In the same figure, if we look at the 2 × 1 server payoff, we see that it is 0 for
any c, which demonstrates Proposition 5 and exemplifies our insight that a firm can always
extract the full server value when doing price-and-delay differentiation. Finally, we see that
in both cases, the 2 × 2 has a range of c values for which the servers obtain a non-negative
payoff, but, after cπ2×2
is reached, the payoff stays equal to 0.
If we now look at Figure 1.2 and 1.3 in parallel, we can see the connection between the
revenue behavior of 2 × 2 and the cπ2×2
threshold, which is when π2×2 crosses 0 in Figure
1.2. More specifically, as we showed in Corollary 4, we see that the optimal 2 × 2 revenue
is constant for any c ≤ cπ2×2
. Focusing now only on Figure 1.3, the second dotted vertical
line in both plots corresponds to the c∆ threshold, where the traffic conservation constraint
becomes binding. We see that the plot illustrates Proposition 10 where for c < c∆R
, we have
R∗∗
2×2 < R∗∗∗
2×1 while for c ≥ c∆R we have R∗∗
2×2 = R∗∗∗
2×1
, and the problems 2 × 1 and 2 × 2 are
the same at optimality. Hence, the firm and platform collapse into an equivalent operation.
0.0
0.2
0.4
0.6
0 1 2 3 4
Server Cost
Server Payoff
Mode
1x1
2x1
2x2
(a) M1 = 25
0.00
0.05
0.10
0.15
0 1 2 3 4
Server Cost
Server Payoff
(b) M1 = 50
Figure 1.2: Optimal server payoffs vs. c for different M1. The rest of the customer characteristics are α1 = 1.2, α2 = 0.4, M2 = 60, and the delay function has γ =
1
2
.
34

24
28
32
0 1 2 3 4
Server Cost
Revenue
Mode
1x1
2x1
2x2
(a) M1 = 25
40
45
50
55
60
0 1 2 3 4
Server Cost
Revenue
(b) M1 = 50
Figure 1.3: Optimal revenues vs. c for different M1. The rest of the customer characteristics
are α1 = 1.2, α2 = 0.4, M2 = 60, and the delay function has γ =
1
2
.
1.6 The Role of the Delay Function
In this section, we will explore various ways through which the delay function, defined in
Section 1.3.3, can impact our results, and we will investigate how it can be generalized
further.
1.6.1 Contrasting Revenue under Differing Delays: Spatial vs.
Non-Spatial Services
So far, in our analysis, we have kept the delay function general in all the statements, and
the plots we have presented corresponded to case where γ =
1
2
, a spatial-type delay function
that would correspond to a spatially distributed service, such as a ride-hailing app or a
delivery operation. We now want to understand how our insights would be impacted when
contrasting a spatial delay environment with a non-spatial one.
As an immediate consequence of the analysis in the preceding section, we can obtain
insight on the comparison of delay functions through the c∆ threshold for the server opportunity cost.
35

Corollary 5. Fixing the customer characteristics, c∆ in a spatial delay setting (γ =
1
2
) is
strictly lower than c∆ in a congestion delay setting (γ = 1).
26
28
30
32
34
0 1 2 3 4
Server Cost
Revenue
Mode
1x1
2x1
2x2
(a) M1 = 25
48
51
54
57
60
0 1 2 3 4
Server Cost
Revenue
(b) M1 = 50
Figure 1.4: Optimal revenues vs. c for different M1. The rest of the customer characteristics
are α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60, and the delay function has γ = 1.
When looking at the γ = 1 plots in Figure 1.4 and comparing it with the equivalent
plots of γ =
1
2
in Figure 1.3, we see that, as expected, the main insights on the revenue
ordering, and the impact the c thresholds have on the revenue curves remain. What we
notice, however, is that the c∆ in a spatial delay setting is lower than c∆ in a congestion
delay setting. Corollary 5 formalizes this. Indeed, if we denote the c∆ thresholds for each
value of γ as c∆(γ =
1
2
) and c∆(γ = 1), and take their ratio, we can quickly see that
c∆(γ= 1
2
)
c∆(γ=1) =
1
2α1
. Hence, for highly impatient or delay-sensitive type 1 customers (meaning, α1
is large), the range of server costs in which a firm outperforms a platform in a spatial service
is vanishingly small relative to that in a non-spatial service.
This result, which is an unexpected takeaway of our analysis, suggests that in the spatial
delay setting there is a bigger range of server characteristics in which the platform can
perform equally well to the firm. This is an insight that could help explain the great success
of ride-hailing platforms and general transportation-based dominance in the gig economy
world (e.g. Mastercard et al. 2020). Furthermore, in this model, we do not capture any
overhead costs that a firm may incur relative to a platform. Hence, our results additionally
36

suggest that a platform could be actually be more profitable than the firm for an even larger
range of c, and Corollary 5 implies that these advantages of the platform over the firm would
be even more pronounced in spatial services, relative to non-spatial operations.
1.6.2 Non-Instantaneous Service Rate
For all our results and analysis in the chapter we have followed the assumption, as we
discussed in Section 1.3.3, that there is an instantaneous rate of service. This is, of course,
not realistic, although it has lent valuable analytical tractability. In this subsection, we
relax this assumption and explain how non-instantaneous service would affect our model
and insights.
We now suppose that the mean service time is not zero, which implies that the server
has non-zero busy-ness. Hence, we must update the service capacity, which in turn impacts
the customers’ delay. This significantly complicates the analysis, but we can nonetheless
investigate numerically. We will denote this expected service time by E[S]. In the 2 × 2
problem, the delay for class 1 service requests will be given by
d1 = d(n − µ1E[S]) , (1.20)
which maintains the construction of the two server groups where there is complete priority
to class 1 services. Here, the available capacity to serve class 1 does not depend on class 2,
but it does depend on the prior class 1 services. The delay for class 2 service requests will
be given by
d
2×2
2 = d(n2 − (1 − f1)µ1E[S] − µ2E[S]) , (1.21)
where f1, as explained in Section 1.4.2.2, needs to make the payoffs from the two groups
equal and is given by (1.13). Here, the capacity now depends on both the prior class 2
services and the fraction of class 1 served by group 2.
37

In the 2 × 1 problem, d1 is given by (1.20) and d2 is given by
d
2×1
2 = d(n2 − µ1E[S] − µ2E[S]) . (1.22)
Notice that if E[S] = 0, then the definitions are equivalent to those of our model so far. When
the delays are affected by the expected service rate as above, then the server payoff nonnegativity constraint in all problems (1.8, 1.10, 1.17) and the traffic conservation constraint
in problem (1.17) are also affected.
First, the server payoff non-negativity constraint π ≥ 0, or p1µ1 +p2µ2 −c ·n ≥ 0, now no
longer can use the instantaneous service relationship that n = d
−1
(d1). In order to express
the optimization problem only in terms of our variables, and not n, we want to express n
in terms of d1. To do this, we need to apply the inverse on both sides of (1.20) and obtain
d
−1
(d1) = n − µ1E[S] which, by solving for n, becomes n = d
−1
(d1) + µ1E[S]. This can then
be substituted into the payoff constraint.
The next constraint that is impacted is (1.15), the traffic conservation constraint, f1 ≤ 1
or nµ2p2 ≤ n2(µ1p1 + µ2p2). Similarly to the payoff non-negativity constraint, this must
change because it no longer is the case that n = d
−1
(d1). Furthermore, in this constraint
in the non-instantaneous service case, it is no longer true that n2 = d
−1
(d2). To get n2 in
terms of d2, we take the delay as given by (1.21) and we apply the inverse on both sides of
(1.21) to obtain
d
−1
(d2) = n2 − (1 − f1)µ1E[S] − µ2E[S] ,
which by using the expression for f1 becomes
d
−1
(d2) = n2 −
n2(p1µ1 + p2µ2) − np2µ2
np1µ1
µ1E[S] − µ2E[S] .
38

By solving for n2, we obtain that
n2 =
n(p1(µ2E[S] + d
−1
(d2)) − p2µ2E[S])
np1 − E[S](p1µ1 + p2µ2)
.
Now, plugging n and n2 back in, we find the new f1 ≤ 1 constraint:
p2µ2(d
−1
(d1) + µ1E[S]) ≤ (p1µ1 + p2µ2)(d
−1
(d2) + µ2E[S]) . (1.23)
Let us compare the shape of the optimal revenues when we incorporate the service rate,
E[S], in the constraints as described above. By looking at various possible values for E[S],
we can see how relaxing the instantaneous service assumption impacts our results for a range
of server waiting cost, c. Figure 1.5 shows the optimal revenue plotted versus the servers’
cost for all modes of operation and multiple values of the mean service duration.
45
50
55
60
0 1 2 3
Server Cost
Revenue
E[S]
0
0.05
0.1
0.2
Mode
1x1
2x1
2x2
Figure 1.5: Optimal revenues vs. c for various E[S]. The customer characteristics are
α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60, and γ = 1.
In this plot, we can observe that the cπ2×2
threshold depends on the value of E[S], which
should not be surprising. What is particularly interesting and surprising in this plot is that
we observe that, for c < cπ2×2
, the 2 × 2 revenues increase as the expected service time E[S]
39

increases. This is somewhat counter-intuitive and is not what happens in the revenues when
c > cπ2×2 where, as we would expect, all revenues decrease as the expected service time E[S]
increases. For c < cπ2×2
the reason for this result seems to be tied to the f1 ≤ 1 constraint
that is active in that specific region of c. We observe that by incorporating E[S] in this
constraint, whose final version is given by (1.23), the feasible region actually increases. We
visualize this in Figure 1.6.
(a) γ = 1 (b) γ =
1
2
Figure 1.6: Feasible Region of d1, d2 for c = 0 and various E[S]. The customer characteristics
are α1 = 1.2, α2 = 0.4, M1 = 50, M2 = 60.
Recall that the 2 × 2 problem has a single, common optimal solution for all c < cπ2×2
,
the one that we obtain when c = 0. In order to see that the feasible region gets larger as
E[S] increases, for c < cπ2×2 we plot the feasible d2 and d1 when c = 0 in Figure 1.6. As
can be seen in Figure 1.6, the feasible region of d1 and d2 when c = 0 becomes larger as
E[S] increases, and this is true for both values of the γ delay function parameter that we
investigate. In the problem context, we can reason that this is because the decreased idleness
of the servers actually creates more opportunity for other servers, and this is manifested in
the region in which the platform does not extract the full value of the server payoffs, which
is precisely c < cπ2×2
.
40

1.7 Conclusion
This work examines how a platform’s revenue and overall performance under price-anddelay differentiation are subject to the choices and preferences of both the customers and
servers that participate in the system. We show that, in addition to pricing, the platform can
capitalize on the dispatching of its service offerings to its server groups as a manner of control
in the service operation. Our results provide upper and lower bounds on the platform’s
optimal revenue through contrast with, respectively, price-and-delay differentiation by a firm,
which does not need to account for server preferences, and a service operation that offers only
one price-and-delay pair. Our study provides valuable insights into the revenue potential of
two-sided platforms under different market conditions, based on the characteristics of both
customers and servers. In particular, we characterize the optimal platform revenue across
a range of server opportunity cost rates. We find that this range is partitioned by two
important threshold values: one where the platform first extracts the full potential value of
the servers, and one where the platform’s revenue coincides with the revenue of the price-anddelay differentiated firm, rendering the two modes of operation equivalent. These findings
were then underscored by analysis of the manner in which the delay depends on the number
of servers. Perhaps unexpectedly, we found that this model offers an explanation for the
dominance of spatial services in the gig economy, as spatial-type delays condense the range
of costs on which the firm outperforms the platform.
Clearly, there is potential for further development and future research in this area. Our
model offers valuable insights, yet it is stylized and does not encompass all the subtle details
of the actual environment. While the simplified nature of our model allows us to appreciate
the big picture, it might be worthwhile to incorporate more precise assumptions for more
particular settings, such as non-uniform valuations and different payoff structures. Additionally, it should be noted that in our study, we’ve assumed that the price paid by customers
directly becomes the earnings of servers, without any division of revenue between the service
41

operation and the servers. That is, the service’s revenue is exactly equal to the total payment
to the servers. Although this might appear to be a basic assumption, our findings actually
can be immediately generalized beyond this. As motivated by practice, suppose that the
revenue is split between the service operation and the servers, with the management taking
a θ ∈ (0, 1) fraction of the earnings and the servers taking the remaining 1−θ. In the model,
this simply would mean that the objective function is multiplied by θ, and the earnings term
in the server payoff would be multiplied by 1 − θ, i.e. π = ((1 − θ)pµ − cn)/n in the 1 × 1.
Hence, we can notice that the θ coefficient will not change the direction of the objective,
and the 1 − θ coefficient in the constraint will simply make c more impactful. Hence, this
revenue-splitting arrangement will shift the range of c thresholds we have identified, but it
should not change the nature of our insights.
Another assumption that we have kept for the entirety of the chapter is that the server
pool is infinitely large. It would also make sense, therefore, to examine what is the impact
on our results when the number of potentially participating servers is equal to N < ∞. Our
findings are consistent even in this context. Interestingly, the initial version of our analysis
was the exploration of the revenues of the different modes of operation as N increases, fixed
at a constant server opportunity cost rate c. For the purposes of preserving space, however,
and because of the similarity of this case with the already discussed findings, we decided not
to include this as an additional section in our work. Our analysis reveals that the presence
of a finite server pool does not significantly alter the behavior of these problems. For fixed
customer characteristics, there exists a server pool size Nˆ(c) such that for N > Nˆ(c), the
platform’s optimization problem does not depend on N. Essentially, after a certain number
of servers, it does not matter if the server pool size increases, because the payoffs do not make
it worthwhile for any additional servers to participate. As a result, for any given c, after a
corresponding Nˆ(c), the revenue maximization problem has a constant optimal solution and
constant revenue, and there exists no benefit of adding additional servers to the pool.
A key area for future research is the investigation of how the system behaves with a
42

non-instantaneous service rate. Despite our numerical evidence supporting the robustness
of our insights, a more in-depth analysis on this would be an important direction of future
work. Furthermore, it would be interesting to delve deeper into the strategic behaviors
of the agents involved. Investigating the implications of our findings when a portion of
the population does not behave strategically, or when there is heterogeneity in the servers’
opportunity costs, would enhance our understanding. Finally, throughout this work, we
have broadly assumed an equilibrium model based on steady-state quantities. The logical
progression from here, albeit potentially complex, would be to build a queueing theoretic
model to capture these interactions in a more dynamic fashion. We look forward to exploring
such directions in future work.
43

Chapter 2
2.1 Introduction
In recent years, ride-hailing platforms have emerged as a central component of the gig economy, reshaping traditional transportation services with their flexible, on-demand model.
These platforms, such as Uber and Lyft, dominate the transportation sector within the gig
economy, driving a substantial share of its productivity. According to data from Mastercard, global wage disbursements for transportation-based services, which include ride-hailing,
reached 61.3 billion in 2018, significantly surpassing other gig sectors like professional and
household services (Mastercard et al. 2020). This trend has likely intensified in the wake of
the Covid-19 pandemic, which further accelerated the adoption of ride-hailing and delivery
services. Given their widespread usage and economic impact, understanding the operations
and performance of ride-hailing platforms is crucial, particularly in terms of how they manage
driver availability, pricing, and delays across different urban environments.
A key aspect of understanding the effectiveness of ride-hailing platforms lies in examining
how they manage driver availability and customer delays across different urban environments.
Unlike traditional taxi services, which operate under a more fixed structure, ride-hailing
platforms utilize dynamic matching algorithms and flexible routing to respond to real-time
demand, often allowing them to efficiently allocate drivers even in areas with fluctuating
demand. However, this flexibility raises important questions about how these platforms
perform relative to traditional taxi services, particularly in diverse urban settings where
factors such as density, traffic congestion, and geographical spread can vary significantly. By
comparing the behavior of ridehailing platforms to traditional taxi operations, we aim to
uncover how differences in driver availability and delays manifest across densely populated,
congestion-prone areas versus more spatially dispersed, less dense zones. This comparison
44

not only sheds light on the strengths and limitations of ride-hailing platforms but also helps
us understand where and when they outperform traditional taxis, offering insights on the
conditions under which these platforms can operate most effectively.
In this chapter, we aim to empirically examine how ride-hailing platforms perform relative to traditional taxi services across different urban environments. By analyzing driver
availability and customer delays, we seek to understand how the dynamics of these two operational structures vary between densely populated, congestion-prone areas and more spatially
dispersed zones, highlighting the conditions under which ride-hailing platforms may hold a
competitive advantage.
2.2 Related Work
This chapter examines dynamics that are closely linked to an expanding body of research on
the functioning of ride-hailing platforms. For a more comprehensive review of the literature
related to two sided markets and ride-hailing dynamics you can refer to Section 1.2 in Chapter
1. As an empirical, data-driven, study, our approach is particularly aligned with research
that leverages real-world data to analyze and model the behavior of these platforms, (Garg
et al. 2021), (Cohen, Fiszer, et al. 2023), (Castillo et al. 2017). In this section, we discuss
some important past works that have addressed similar problems to those studied in our
research. We will highlight how our approach is influenced by some of these studies, as well
as how it builds upon and contributes to existing literature, offering new insights into some
specific aspects of the dynamics of ride-hailing platforms.
The study that contains elements of analysis, most similar to what we aim to undertake
in this work, is the paper by (Yan et al. 2020). This paper reviews matching and dynamic
pricing techniques in ride-hailing, highlighting their role in minimizing customer waiting
times. It also examines a pool-matching mechanism, showing that jointly optimizing dynamic
pricing and waiting can improve capacity utilization and overall efficiency. In this paper the
45

authors motivate the use of their delay function for ETA, d(O) = Oα
, with α = −
1
2
, by
fitting a regression model on the logarithmic scale of ETA as a function of available drivers,
using data from downtown San Francisco, during morning rush hours. Their regression
closely matches the theoretical value for a spatial delay, with only minor deviations. They
attribute these deviations to potential violations of assumptions inherent in the spatial delay
function, such as that, in reality, drivers move on a road network, travel speeds vary across
road segments, and available drivers are not uniformly distributed across the area. Thanks
to high-quality data directly from Uber, they were able to fully separate matching time from
actual time. Our approach is strongly influenced by this method of fitting the data. We
will, similarly, attempt to fit a regression model, but adapted to out dataset, and using
the actual time delays experienced by customers rather than the estimated times employed
in this study by (Yan et al. 2020). This adjustment reflects the real-world experience of
ride-hailing users, providing a more direct measure of the delays that customers face.
Our work is also closely related to a body of research that examines the relationship
between the density of servers and delay (or estimated time of arrival, ETA) in spatial
settings, as defined in Chapter 5 of the textbook by (Larson et al. 1981). This is the
theoretical relationship we aim to test empirically, to determine if it holds in real-world
settings. The seminal work of (Castillo et al. 2017) provides a steady-state characterization
of a ride-hailing platform, demonstrating how dynamic pricing helps maintain efficiency
after severe supply-demand imbalances. Similarly, studies by (Nikzad 2017) and (Besbes
et al. 2022) explore the connection between driver density and service time. Additionally,
(Benjaafar, Ding, et al. 2022) investigate the impact of labor pool expansion and the broader
implications for regulatory measures, such as wage floor policies. While all these studies rely
on the theoretical delay function, they do not use data to empirically verify whether this
relationship is actually observed in practice. Our approach fills this gap by using real-world
data to test if the spatial delay function holds in actual ride-hailing scenarios.
Our study is also related to research comparing traditional taxi services with ride-hailing
46

platforms like Uber. For instance, (Wallsten 2015) examines the impact of Uber’s rise on
the traditional taxi industry by analyzing a large dataset of NYC taxi rides, complaints,
and trends in Uber’s popularity. The authors find that the growth of Uber led to a decrease
in consumer complaints about taxis, suggesting that the competition may have incentivized
taxis to improve service quality. Unlike this study, which focuses on consumer behavior and
service improvements, our work examines the operational aspects of ride-hailing and taxi
services, particularly how driver availability and spatial characteristics affect performance.
(Angrist et al. 2021) takes a different approach by comparing the compensation models
for Uber and traditional taxis from a driver’s perspective. Through an experiment offering
virtual taxi medallions to Boston Uber drivers, the authors explore labor supply responses
and drivers’ preferences for commission-based versus lease-based payment models. While
this study provides insights into economic incentives for drivers, our research focuses on
the spatial dynamics of ride-hailing operations and how these platforms perform relative to
traditional taxis in various urban areas. Our approach provides a complementary view by
analyzing how these differences in service models translate to performance across different
city zones.
2.3 Problem Overview
2.3.1 Problem Description & Relation to Chapter 1
The first chapter of this dissertation explores the rapidly growing gig economy, where businesses function more as ‘two-sided’ platforms rather than traditional ‘one-sided’ firms, addressing not only customer preferences but also those of service providers. It investigates
how these two-sided markets differ from traditional businesses in their approaches to pricing and service delays, due to the necessity of considering the choices made by service
providers. Through a game-theoretic model, we represent the various participants (customers, providers, managers), and by analyzing how service delays experienced by customers
47

change with the number of available service providers, we examine different business models
within this framework. Specifically, part of the chapter compares gig-economy companies to
traditional ones, focusing on how customer delays differ depending on the functional form
of the delay.
As is common in some of the literature concerning industries where customers experience
delays in obtaining services, these delays are modeled as being influenced by the number of
service providers available on the platform. In particular, this important relationship has
been extensively studied in the ride-hailing industry ((Castillo et al. 2017), (Nikzad 2017),
(Besbes et al. 2022)). In essence, the more service providers there are, the shorter the delay
for customers. For a general delay function, we define it as:
d(O) = O
α
, (2.1)
for some α < 0. In chapter 1, while all results were given for any α < 0, we devoted
considerable attention to two specific, interpretable values of α. The first one was α = −1, a
congestion type of delay function. This label is drawn from the leading order of dependence
on the number of servers in classical queueing formulas, such as in the M/M/c steady-state
mean waiting time, in which the proportional dependence on O is no more than 1/O (e.g.,
Kulkarni 2011, Eq. 6.32). The second one was α = −1/2, a spatial type of delay function.
As introduced and motivated by (Larson et al. 1981), this structure captures the expected
waiting time’s leading order of dependence on O when a two-dimensional space must be
traversed in order to begin service, and it has been commonly used in modeling spatiallybased services such as ride-hailing and transportation (e.g., Yan et al. 2020; Freund et al.
2021; Besbes et al. 2022).
One key takeaway from the analysis in Chapter 1 is that, in the spatial delay setting,
there is a broader range of server characteristics in which the platform can perform just
as well as a traditional firm. Building on this insight, we pursue two main objectives in
48

this chapter using a data-driven approach based on data from the NYC Taxi & Limousine
Commission, which we will detail in Section 2.3.2. Broadly, the dataset includes ride data
for both ride-hailing companies (representing gig platforms) and traditional taxi services
(representing conventional firms). In this context, delay is measured by the estimated time
of arrival (ETA) for ride-hailing drivers, which represents the time from when a ride is
requested to when the driver arrives to pick up the passenger. The first objective is to
identify areas where delays are primarily congestion-driven (corresponding to |α| = 1) and
areas where delays are more spatially-driven (corresponding to |α| =
1
2
). Once these areas
are classified, the second objective is to investigate whether gig companies indeed have a
stronger advantage over traditional firms in spatially-driven areas, while traditional firms
may dominate in congestion-driven areas.
We will assess whether a specific area is more congestion-driven or spatially-driven using
two primary methods. First, we will use the area’s ‘density’ as a proxy for its characterization,
relying on several density metrics, which will be explained in detail in Section 2.4. Based
on these metrics, we want to explore whether denser areas are more likely to experience
congestion-driven delays, while less dense areas may be more likely to exhibit spatiallydriven delays. Second, we will implement a data-driven simulation and regression model
to estimate the α value of the delay function in different areas. By combining these two
approaches, we aim to gain a comprehensive understanding of the underlying drivers of
delay in both congestion-driven and spatially-driven areas.
2.3.2 The Data
As we mentioned in Section 2.3.1, the way will represent gig and traditional firms in this
work is through ride-hailing and taxi respectively. The data we will be using for these comes
from the New York City Taxi and Limousine Commission (TLC), (“NYC Taxi & Limousine
Commision (TLC) Trip Record Data” 2021), which has been releasing extensive datasets
on ride activities within the metropolitan area annually since 2009. The code and analysis
49

for this project are publicly available in a GitHub repository (“Ridehailing Spatial Analysis
Codebase” 2024).
The TLC datasets provide detailed insights into Yellow Taxi services, encompassing data
such as pick-up and drop-off times and locations, trip distances, detailed fare breakdowns,
types of rates, methods of payment, and driver-reported passenger numbers. Starting in
mid-2013, the TLC introduced Green Taxi data, which include vehicles licensed to pick up
passengers from street hails in the outer boroughs and parts of Manhattan, thus providing
similar datasets to the Yellow Taxi. In our analysis, ‘Taxi’ will refer to both Yellow and
Green taxi rides combined. In 2015, data on For-Hire Vehicles (FHV) were also released,
covering services like livery cabs and black cars, which require pre-booking and do not pick
up street hails. The dataset expanded further in 2019 with the introduction of High-Volume
For-Hire Vehicle data, pertaining to ride-hailing services that handle a significant volume of
passengers daily. Initially, the data for these services was somewhat limited, providing basic
information such as pricing and the associated ride-hailing company. However, by 2021, the
data became much more detailed, including trip distances, times of requests, pickup and
dropoff zones, and extensive pricing details (including base fare, tips, tolls, and surcharges).
Given the richness of this dataset, we will use the High-Volume For-Hire Services (HVFHS)
data to represent the gig economy in our analysis. The ridehailing companies included
in the 2021 dataset are Uber (126,129,064 observations), Lyft (47,575,769 observations),
and Via (891,819 observations). However, we immediately excluded Via due to its smaller
data volume and more complex business model, which would make comparisons challenging.
During our initial exploration, we found that the data from Lyft did not capture relevant
dynamics, likely due to its limited size, compared to Uber. Therefore, in this study, Uber
will serve as the representation of the gig economy.
The dataset we are using is structured at the ride level, where each row corresponds to
a specific Uber ride that took place during the year. Key details related to each ride are
summarized in Table 2.1. It is important to note that the dataset does not include detailed
50

Field Name Description
hvfhs license num The TLC license number of the HVFHS base or business.
pickup datetime The date and time of the trip pick-up.
dropoff datetime The date and time of the trip drop-off.
PULocationID TLC Taxi Zone in which the trip began.
DOLocationID TLC Taxi Zone in which the trip ended.
request datetime Date/time when the passenger requested to be picked
up.
on scene datetime Date/time when the driver arrived at the pick-up location (Accessible Vehicles-only).
trip miles Total miles for the passenger trip.
trip time Total time in seconds for the passenger trip.
base passenger fare Base passenger fare before tolls, tips, taxes, and fees.
Table 2.1: Field Descriptions for Ride-Hailing Data
location data, such as geographical coordinates for pick-ups and drop-offs. Instead, the only
available information regarding pick-up and drop-off locations is the ID number of the Taxi
Zone where each ride began and ended.
Since this is the only available location information, and because these zones naturally
segment NYC, we will use them to define the areas we aim to classify as more or less dense.
The zones we focus on include all taxi zones in Manhattan, the Bronx, Brooklyn, and Queens.
Although Staten Island was included in our preliminary analysis, it will be excluded from
the final discussion due to its much smaller data volume and its unique geographic isolation
as an island. Since it does not share the same connectivity dynamics or cross-borough travel
patterns as the other boroughs do, it becomes less relevant for our analysis. After excluding
Staten Island we are left with 237 taxi zones that are depicted in Figure 2.1.
In addition, we excluded from our analysis rides marked as ‘shared’ in the dataset, since
these rides are not matched using a first-dispatch protocol. This is important because our
ETA model assumes a first-dispatch matching system, where the closest available driver
is dispatched to fulfill the ride request. In contrast, shared rides often involve multiple
passengers being picked up and dropped off along a predetermined route, which does not
align with our functional form of the ETA function. By focusing on non-shared rides, we
51

ensure that the analysis remains consistent with our underlying assumptions and that the
results are meaningful for the intended matching system.
Figure 2.1: NYC Taxi Zones, excluding Staten Island
2.4 Density Exploration
As we discussed in Section 2.3.1, our hypothesis is that denser areas are more likely to
experience congestion-driven delays, while less dense areas are more likely to exhibit spatiallydriven delays. To classify the taxi zones as either ‘more dense’ or ‘less dense’, our first
approach will involve using several density metrics related to key factors that contribute
to higher density in urban areas. To account for density, we utilized several metrics from
diverse data sources that capture various aspects of urban density. The choice of metrics
was influenced by several studies that deal with urban density and congestion ((Erdeli´c et al.
2021), (Kumakoshi et al. 2021), (Momeni et al. 2022)) and enabled by publicly available
52

datasets.
Density from Land Use Variety
First, we used the Shannon and Simpson diversity indices (Fedor et al. 2019), that are
common indices to measure biodiversity. In our case, the indices were derived from ‘land
use’ (Primary Land Use Tax Lot Output) data that we obtained from the Department of
City Planning, DCP, of NYC, (“Primary Land Use Tax Lot Output (PLUTO)” 2021).
These indices measure the diversity of land use types within a zone, offering insight into the
complexity and intensity of activities in the area. Zones with higher diversity in land use
are likely to experience more varied and frequent human activity, contributing to increased
density.
Density from Built Environment
Next, we considered the median height of buildings, using data from NYC Open Data’s
Building Footprints dataset, (“NYC Building Footprints” 2021). Taller buildings often
indicate higher population or activity concentration in a given zone, as they can house more
residents, businesses, and services, making this metric a strong proxy for vertical density.
Density from Pedestrian Traffic
We also incorporated data from New York’s bi-annual pedestrian count survey, which
tracks foot traffic at specific locations across the city, (“NYC Bi-Annual Pedestrian Counts”
2021). The variables we used include whether or not a zone contains a pedestrian count
station, the number of count stations within the zone, and the average pedestrian count.
These metrics help capture human activity density, as zones with more pedestrian traffic or
count stations are likely hubs of higher movement and interaction.
Density from Digital Infrastructure
Two additional metrics were drawn from NYC Open Data’s ’Wi-Fi Hotspot Locations’
dataset, (“NYC Wi-Fi Hotspot Locations” 2021), which includes the presence and number
of public Wi-Fi hotspots in each zone. The availability of public WiFi often correlates
with areas of high foot traffic and activity, making this a relevant indicator of technological
53

infrastructure density and the need to serve a densely populated area.
Density from Vehicular Traffic
Finally, we utilized two spatial metrics from OpenStreetMap, (“OpenStreetMap: New
York City” 2021); road density, calculated as the total road length per square kilometer, and
intersection density, calculated as the number of intersections per square kilometer. These
metrics reflect the structural connectivity and accessibility of each zone. A higher density of roads and intersections generally suggests greater urban development and movement
capacity, which directly ties to higher physical and functional density in the area.
To ensure comparability across different metrics, we performed a normalization process,
scaling all metric values to a range between 0 and 1. For each continuous metric, we applied
the empirical cumulative distribution function (CDF), transforming the value at each zone
into its corresponding percentile rank. This allowed us to reduce the influence of extreme
values, ensuring that outliers did not disproportionately affect the analysis. For discrete
metrics, we normalized each value by dividing it by the maximum observed value across all
zones for that specific metric, bringing these values into the same range. This normalization
step is crucial as it ensures all metrics are treated equally, regardless of their original scales
or units. Without this, metrics with larger ranges or more extreme values (e.g building
height) could dominate the results, leading to skewed interpretations of density in different
zones. By standardizing the metrics within the same range, we allow each to contribute
meaningfully to the overall analysis.
To further refine our analysis and identify the most significant contributors to density,
we performed a Principal Component Analysis (PCA), (Abdi et al. 2010). PCA is a dimensionality reduction technique that transforms the original set of metrics into a smaller set of
uncorrelated components, which capture the maximum variance in the data. This method
allows us to reduce the complexity of the dataset while retaining the most important information. By examining the contribution of each metric to the principal components, we were
able to determine which metrics had the greatest influence on the variability across zones.
54

The results of the PCA revealed that the most important metrics for characterizing density
were
• Median Height,
• Road Density,
• Intersection Density,
• Presence of Pedestrian Count Stations,
• Average Pedestrian Count,
• Presence of WiFi Hotspots,
• Number of WiFi Hotspots.
Notably, the Shannon and Simpson indices that we calculated from the Land Use data and
measure diversity, were not selected as key metrics. While these indices can be useful for
capturing diversity in land use, we observed that they often produced high values in more
remote, suburban-like areas of Queens and the Bronx—zones that would not typically be
characterized as dense urban environments. This suggests that these indices may be picking
up aspects of land use variety that are more relevant in suburban contexts, where there is a
mix of residential, commercial, and green spaces, rather than urban density. Consequently,
they did not align well with our objective of identifying dense, highly trafficked areas within
the city. This outcome underscores the importance of carefully selecting metrics that are
not only statistically significant but also conceptually appropriate for the urban context we
aim to analyze.
These metrics were selected because they consistently accounted for a significant proportion of the variance in the data. By focusing on these key metrics, we can streamline
our analysis and better understand how different factors contribute to urban density, while
minimizing the noise from less impactful or redundant variables.
Figure 2.2 shows visualizations of various urban density metrics across NYC, normalized
to allow for consistent comparisons. Each map highlights a different aspect of spatial characteristics within the city’s taxi zones. Intersection density captures how interconnected a
55

(a) Intersection Density
(number per km2
)
(b) Road Density (km per
km2
)
(c) Median Height (m)
(d) Presence of Pedestrian
Count Stations
(f) Average Pedestrian
Count (2021)
(e) Number of WiFi
Hotspots
Figure 2.2: Visualizations of different urban metrics. Each plot shows a different aspect
of the spatial analysis. (a) Intersection Density, (b) Road Density, (c) Median Height, (d)
Presence of Pedestrian Count Stations, (e) Average Pedestrian Count (2021), (f) Number of
WiFi Hotspots.
56

zone is based on the number of intersections per square kilometer. Road density indicates
the density of road infrastructure, also measured per square kilometer. The median height
of buildings, reflects the vertical density of each zone. The presence of pedestrian count stations, highlights zones with higher levels of pedestrian activity monitoring and the Average
Pedestrian Count offers a measure of actual foot traffic within each zone, reflecting how busy
or frequented different areas are. Finally, the Number of WiFi Hotspots serves as a proxy
for digital infrastructure, indicating zones with better connectivity and potentially higher
urban activity,
The six metrics visualized in 2.2 offer a comprehensive overview of urban density and
highlight both the commonalities and distinctions across different aspects of spatial analysis.
While many of these metrics appear to align in their identification of Manhattan as one
of the most dense and developed regions—particularly seen in Road Density, Intersection
Density, and Median Height—subtle differences also emerge when examining other areas.
For instance, both Road Density and Intersection Density show a clear correlation, as they
inherently measure similar aspects of infrastructure. However, Median Height adds another
dimension by capturing vertical development, which is especially pronounced in downtown
Manhattan and key areas of Brooklyn but less so in other boroughs with lower building
heights.
The presence of Pedestrian Count Stations and the Average Pedestrian Count provide an
important behavioral aspect to density, indicating where foot traffic is concentrated. This is
particularly noticeable in pedestrian-heavy zones of Manhattan, where transportation hubs
and commercial districts are prevalent. Lastly, the Number of WiFi Hotspots serves as a
proxy for digital infrastructure, which is heavily concentrated in Manhattan but also extends
to areas like Brooklyn and Queens. While WiFi Hotspots correlate well with pedestrian and
road density in certain regions, they also reveal digital connectivity in less dense zones, such
as tech-driven developments in Brooklyn. Together, these maps demonstrate that while the
metrics often overlap in highly developed areas, each provides a distinct lens through which
57

to view urban density, emphasizing different spatial and infrastructural aspects of the city.
2.5 Open Drivers’ Proxy - Simulation Approach
2.5.1 Motivation for Simulation
The exploration of density metrics for the classification of zones was our first approach.
Moving on to the second approach, the focus shifts to a regression analysis examining the
relationship between the estimated time of arrival (ETA) and the number of available drivers.
This analysis allows us to estimate the α coefficient of the delay function, which in turn helps
classify zones as either congestion-driven or spatially-driven. In the upcoming Section 2.6, we
will present the detailed formulation of the regression model within our framework. However,
in this section, we will briefly present the high-level idea of the regression below in order
to motivate the simulation we had to employ, a critical preparatory step that enabled us to
perform this regression.
This regression approach is motivated by a similar analysis from (Yan et al. 2020),
who also use that, traditionally, we use a model for the en route time function of the form
η(O) = τ ·Oα
, where O is the number of open drivers. The α and τ are then estimated using
the following linear regression on the logarithmic scale:
log(η) = log(τ ) + αlog(O) + ϵ , (2.2)
where ϵ is a normally distributed random variable with mean zero and with the assumption
that the observations are independent from each other.
An important distinction to note about the model from (Yan et al. 2020) is that their
dataset provides the number of open drivers, O, at any given time. In our case, however,
this is not true, as our data is structured at the ride level, and there is no identifier for the
drivers completing the rides. As a result, we are missing, O, which is a crucial variable in
58

(2.2). To address this, we need to develop a proxy for the number of open drivers, O, based
on the data we do have.
Our initial approach was to use the number of rides completed (dropped off) in a zone
within a specified time interval (e.g., 5 or 10 minutes) as a proxy for the number of open
drivers in that zone. However, this method overlooks the role of demand, which strongly
influences driver availability in such settings. Demand can be more easily estimated by
the number of ride requests made within the same time interval. It is important to note,
though, that we lack data on rides that were requested but not completed, or instances
where customers opened the app but did not request a ride. While this presents its own
limitations, we can still use the number of ride requests as a reasonable proxy for demand
and the number of completed rides as a proxy for supply.
The challenge then becomes how to combine these two proxies to estimate the number
of open drivers. At this stage, we attempted to impose a specific functional relationship
between supply, demand, and the number of open drivers. This led us to modify (2.2) into
a multivariate regression, incorporating both the number of completed rides and requested
rides. However, preliminary results were not very responsive to the data, likely due to the
strong assumptions required by these functional forms.
The challenges we encountered—namely, the difficulty of accurately estimating the number of open drivers using only completed rides and ride requests, and the strong assumptions
required for the functional forms—highlighted the limitations of these initial approaches.
These obstacles led us to pursue a more general and flexible solution, which brings us to
the simulation approach we will now present. This simulation allows us to better capture
the dynamic nature of driver availability and demand without making overly restrictive
assumptions.
59

2.5.2 Max Flow Simulation
To address the challenge of estimating driver availability, we developed a Monte Carlo simulation based on a max-flow optimization model. At a high level, this simulation dynamically
allocates drivers to ride requests in various locations over time, while also accounting for the
availability and movement of drivers between zones. The model captures key dynamics of
the ridehailing market by incorporating both demand (ride requests) and supply (completed
rides) in each time window, which allows us to simulate the fluctuating number of open
drivers in each zone.
Given that our data only provides taxi zone IDs rather than precise geographical coordinates for ride pickups, requests, and drop-offs, it is crucial to account for the adjacency
of locations when estimating driver availability. Initially, as we mentioned in the previous
sections, we considered models where drivers available in one zone would only be matched
to rides requested within that same zone. However, this approach does not fully capture the
real-world dynamics of ride-hailing platforms, where drivers frequently move across zones to
respond to demand. In practice, drivers stationed near the border of a zone can easily be
matched to rides requested in adjacent zones, especially during periods of high demand or
when availability within a specific zone is limited. Therefore, in our max-flow optimization
model, we incorporate the adjacency of locations, allowing drivers to be allocated not only to
requests in their current zone but also to those in neighboring zones. This approach better
reflects the flexibility and fluid movement of drivers in urban environments, leading to more
realistic estimates of driver availability across the network. The detailed way in which we
create the adjacencies is described in section B.3 of Appendix B.
At the core of the max-flow step, we represent the city’s zones as nodes in a network, with
edges connecting adjacent zones. These edges allow for the movement of drivers between
neighboring areas, reflecting the real-world dynamics of how drivers relocate to fulfill ride
requests. The flow of drivers along these edges depends on demand in one zone and the
60

availability of drivers in neighboring zones. By modeling the problem as a flow network, this
approach ensures that driver allocation reflects both the spatial layout of the city and the
dynamic nature of driver mobility. The works of (Aveklouris et al. 2024) and (Braverman et
al. 2019) served as inspiration for this simulation approach. This network-based formulation
makes the max-flow model particularly well-suited for capturing the movement of drivers
between zones and their real-time availability across different locations.
s
1
2
3
262
.
.
.
.
.
.
1
2
3
262
.
.
.
.
.
.
t
s
start
j,t dj,t
Figure 2.3: Network Graph of Locations & Max Flow Problem Variables. Connecting edges
indicate adjacency between locations.
Given the following notation, the max-flow simulation algorithm is given by Algorithm
(1). The Max Flow Optimization problem, implemented at each time window, is formulated
in Section B.4 of Appendix B.
• s
start
j,t : number of drivers available at location j, at start of time window t,
• s
end
j,t : number of drivers available at location j, at end of time window t,
• f
j,t
s
: number of drivers originating from location j matched to a ride at time window
t,
• f
j,t
d
: number of requests at location j that are fullfilled at time window t
61

• dj,t: number of requests at time t − 1,
• aj,t: added drivers at location j, at time window t, after max-flow solution,
• uj,t: unmatched drivers at location j, at time window t, after max-flow solution,
• bj,t: abandoning drivers at location j, at time window t, after max-flow solution,
• dropoffsj,t: number of dropoffs at time window t,
Algorithm 1: Max Flow Driver Availability Simulation
Notation: Time window t in [0, T], location j in zones
Data: Load Data
Data: Set s
start
j,0 = 0
for t = 1 to t = T do
s
start
j,t = s
end
j,t−1
Solve Max Flow with s
start
j,t , dj,t
aj,t = dj,t − f
j,t
d
, ∀j
uj,t = sj,t − f
j,t
s
, ∀j
bj,t ∼ Bin(uj,t, p)∀j
s
end
j,t = s
start
j,t − dj,t + aj,t − bj,t
s
end
j,t + = dropoffsj,t
end
For each time window, the simulation starts by setting the number of available drivers
in a location based on the previous time window. It then solves a max-flow problem that
matches available drivers to incoming ride requests, simulating how drivers are assigned to
rides across the different zones. Any remaining unmatched drivers and those completing
drop-offs are then factored into the availability for the next time period. Additionally, some
unmatched drivers may leave the system, with some abandonment probability p, reflecting
real-world conditions where idle drivers abandon the platform if they do not receive a ride
for an extended period. The value of the abandonment probability p is a value we vary in
the model to test the robustness of our results for different driver patience levels.
This iterative process continues for each time window, updating the number of available
drivers at the start and end of each interval based on demand, completed rides, and driver
abandonment. By accounting for these dynamic factors, the max-flow simulation provides
a realistic proxy for driver availability, a key variable that is otherwise missing from the
62

dataset. This simulation enables us to proceed with the regression analysis by providing an
estimate of open drivers in each location at any given time, addressing a significant challenge
in the data.
2.6 Regression of ETA on Drivers
2.6.1 The Regression Model for Each Zone
As we mentioned in Section 2.5, the regression model we want to run, closely following (Yan
et al. 2020), comes from (2.2). In their setting they are fitting that equation to rides from
downtown San Francisco during morning rush hours over a period of several weeks. In our
year-long dataset we similarly focus on rush hours, both morning and evening rush. Any
results we will be presenting on the main body of this chapter are related to morning rush
hours and on section B.7 of Appendix B we also present the final insights of performing the
same analysis on evening rush hours. Morning rush was defined as the hours 6 to 10am and
evening rush as the hours 4-8pm. Since we want to compare different zones, our goal is to
perform this regression for each of the taxi zones we have available. For each location j, we
run the regression
log(¯ηj ) = log(τj ) + αj
log(Oj ) + ϵj
. (2.3)
In this regression model, ¯ηj represents the mean estimated time of arrival (ETA) for location
j, conditional on the average number of available drivers. Specifically, for each location j,
we calculate ¯ηj (Oj ) by taking the mean ETA across all ride requests grouped by the average
number of available drivers, Oj
. This means that ¯ηj (Oj ) is computed as:
η¯j (Oj ) = 1
|Rj,Oj
|
X
i∈Rj,Oj
ηi,j , (2.4)
63

where Rj,Oj denotes the set of rides in zone j with an average number of available drivers
equal to Oj
, and ηi,j represents the ETA for each ride i in that set. This grouping allows
us to analyze how the mean ETA changes with different levels of driver availability, forming
the basis for the regression analysis.
2.6.2 Obtaining Driver Availability from Simulation
The values used for Oj are derived from the simulation described in Section 2.5.2. Specifically,
in that simulation, for each zone j and at each time window t, we obtain the number of
available drivers at the start (s
start
j,t ), and a number of available drivers at the end (s
end
j,t ) of
the time window. The value we can use from Oj can be any of these values or it can be the
average of these, defined as s
avge
j,t =
s
start
j,t +s
start
j,t
2
. For any of the results we will present in this
or the following section we will refer to the regression where Oj = s
avge
j,t . In the Appendix we
provide additional plots where s
start
j,t and s
end
j,t where used as Oj to show that our results are
robust and do not vary significantly based on the choice of which values to use for Oj
.
To obtain a reliable estimate of the mean ETA conditional on driver availability, we
aggregated results across multiple simulation runs. For each simulation, the corresponding
dataset was loaded, and data from the first 25 days was filtered out to allow the system to
stabilize. We then focused on a specific zone and time segment, filtering the data accordingly.
Depending on the selected variable, driver availability was set as either the drivers available
at the start, or at the end of a time window, or as an average of the two.
To ensure the reliability of the mean ETA calculations, we applied an additional filtering
that eliminates data points with insufficient representation across time windows. We begin
by calculating the total number of unique ‘Request Time Window’ values in the dataset. We
set a threshold equal to 0.5% of the total unique time windows to determine the minimum
required coverage. Next, we assess how many unique time windows values are associated
with each level of driver availability and identifies which levels of driver availability meet
or exceed the threshold, effectively ensuring that there are enough data points to provide a
64

meaningful estimate of the mean ETA. Only those driver availability values that pass this
criterion are retained in the dataset, while the others are filtered out. By applying this filter,
we improve the robustness of the regression analysis by ensuring that each level of driver
availability is backed by sufficient observations, reducing the risk of skewed results due to
sparse or unreliable data.
For each simulation, we grouped the data by the chosen driver availability metric and
calculated the mean ETA for each level of driver availability. Before proceeding, checks were
performed to identify and exclude any zero or negative values, ensuring that the logarithmic
transformations required for regression were valid. After processing the data from each
simulation, we combined the results to obtain an overall average of the mean ETA for each
level of driver availability. By averaging across multiple simulation runs, we ensured that
the estimates were more robust and less susceptible to variability from any single run. This
approach provided a clearer, more reliable understanding of how mean ETA changes with
driver availability across different locations and time segments.
2.6.3 The ETA Quantity & Associated Challenges
Another challenge arising from the available data is the choice of how to measure ETA. In
their study, (Yan et al. 2020) obtain the estimated en route time for each rider request
directly from Uber app data, which provides a clear fit for the spatial relationship between
ETA and driver availability. However, our data presents a different challenge, as we are
unable to distinguish between the time it takes to match a driver to a ride request and the
actual time it takes for the driver to arrive at the pickup location. This distinction is crucial
because it affects whether the observed ETA can be attributed to spatial factors or not.
The timestamps we have access to include request time, pickup time, and drop-off time.
For some vehicles, particularly accessible ones that offer this feature, we also have a timestamp indicating when the car arrives at the pickup location, called ‘on scene time’. Therefore, for our ETA calculation, we use the difference between ‘on scene’ time and request
65

time, which was available for all Uber rides.
The method described above allows us to better isolate the actual travel time it takes
for a driver to reach the pickup location from the time spent on matching the driver to the
ride request, which is the aspect we aim to capture. However, it is not entirely perfect,
as there may still be a portion of the time interval that includes the matching process.
Specifically, the difference between the request time and the time the driver arrives at the
location includes the time to match the rider with the driver. At the time of this data uber
was using a batching window approach to matching, (“Batched Matching” 2023). This may
introduce some additional delay purely for informational technology purposes.
In theory, as the number of available drivers approaches infinity, delays should decrease
to zero, reflecting an ideal scenario where there is always a driver immediately available for
any request. However, in our data, we observe that even with a high number of available
drivers, the delays do not drop to zero. Of course, we also do not reach an infinite number
of drivers in practice, which is a limitation of our real-world data. This suggests that there
are additional factors, possibly related to unmodeled constraints, and related matching time
considerations mentioned above, that are not fully captured in our current setup. To address
this, we introduce a variation of(2.3) that incorporates an additional parameter, c. This
parameter c, serves to model the matching time, enabling the regression to more accurately
distinguish between the time spent on matching and the actual travel time. The revised
regression equation becomes:
log(¯ηj − c) = log(τj ) + αj
log(Oj ) + ϵj
. (2.5)
We vary the parameter c to take values of 0, 1, and 2. The values 1 and 2 were chosen based
on observed patterns in how Uber’s batch matching system operates, as well as related
estimates of typical matching times within batching windows, (“Batched Matching” 2023).
Essentially, c represents an adjustment factor that accounts for the average time spent on
66

matching a driver to a ride, which may vary depending on operational conditions.
The choice of c influences how we distinguish between matching time and travel time,
adjusting what portion of the total time is attributed to each. As we vary c, the resulting
regression coefficients, especially α, change accordingly. Generally, as c increases, α also
tends to increase, which aligns with expectations from the model. This suggests that in
some cases, using a higher c brings the α values closer to what we would anticipate based on
theory and the characteristics of the location. This suggests that adjusting c can help refine
the accuracy of our model, at least for certain locations.
Figure 2.4: ETA vs Number of Available Drivers during Morning Rush hours at Financial
District South (Taxi Zone 88) in Manhattan, for different values of c
Figure 2.4 displays the regression results for a specific location (Financial District South,
Taxi Zone 88) during morning rush hours, showing the relationship between the mean ETA
and the average number of available drivers for different values of the parameter c. The
number of available drivers, in this set of regressions was the average of drivers available at
the start and drivers available at the end of each time window in the simulation, and the
67

abandonment probability of the drivers was p = 0.5. Each color corresponds to a different c
value: c = 0, c = 1, and c = 2. The parameter c represents a portion of the ETA that we
attribute to the matching time, which we subtract to better isolate the actual travel time
component.
By introducing the parameter c to represent the matching time, we adjust the model to
better align with theoretical expectations. As shown in the plots, increasing the value of
c appears to bring the regression closer to what we would expect from theory. The fitted
regression line adjusts downward as c increases, reflecting a reduction in the estimated delay
attributable to travel time. This indicates that including c helps to account for part of the
delay that persists even when driver availability is high, making the model more realistic
and consistent with the theoretical understanding of how driver supply should affect ETA.
In Section B.5 of the Appendix, we provide similar comparisons for a location on each of the
other boroughs to show that the same insights remain.
To further analyze how changes in c affect our classification of zones, we include Figure
2.5, which compares the rankings of the absolute values of α coefficients in descending order
for c = 0 and c = 2. Higher absolute values of α indicate denser, more congestion-driven
zones, while lower absolute values suggest more spatial, less dense areas. The plot shows
that while the overall shape of the rankings is preserved across different c values, there are
some noticeable reorderings. This indicates that adjusting c not only shifts the regression
line but also changes the relative classification of zones, affecting how we interpret density
in our analysis. The idea behind this comparison is to show that although the distributions
are similar, the change in c introduces variations that can alter our understanding of specific
zones’ characteristics. These re-orderings highlight the sensitivity of the α coefficients to the
parameter c and emphasize the importance of considering matching time in the regression
model. This further underscores how different assumptions about the underlying components
of ETA can lead to different insights about zone density, reinforcing the need to carefully
interpret the results of our regression analysis
68

Figure 2.5: Comparison of α rankings for different c values
2.7 Comparing Taxi & Uber
Before diving into the interpretation of the regression results and the comparisons across
zones that were central to our analysis, it is important to highlight the strengths and weaknesses of this method. The max-flow simulation and regression model were developed to
address key challenges posed by information not captured in the available data, providing a
data-driven framework to estimate driver availability and analyze the relationship between
driver supply and ETA. The overall approach has several key strengths. The max-flow
simulation process, which dynamically allocates drivers to ride requests while accounting
for movement between adjacent zones, is crucial for overcoming the absence of direct data
on open drivers. By incorporating geographical adjacencies, this approach closely mirrors
69

real-world conditions, where drivers frequently move between nearby areas to fulfill demand.
Additionally, this method does not rely on imposing strong functional assumptions about
how drivers should be distributed, allowing it to remain flexible and responsive to the data,
in the regression model.
However, there are also notable weaknesses. The max-flow simulation, while effective,
does not fully replicate the actual matching and routing mechanisms used by platforms like
Uber. It likely misses certain nuances of driver behavior, such as strategic positioning, reallocation, and various decision-making processes that drivers may employ beyond simple
patience (or abandonment). Additionally, the data-driven approach is still subject to significant noise, as our estimates of driver availability and the effects of varying demand can
fluctuate based on numerous external factors not captured by the model.
Because of this inherent noise, it is essential to interpret the results with caution. While
the findings can provide useful insights into general patterns and relationships between driver
supply and ETA, they should not be taken as precise reflections of real-world behavior.
Instead, they can serve as valuable guides for understanding trends and potential areas for
further investigation, even if some of the finer details are missed.
2.7.1 Initial Insights
We now turn to interpreting the results from the simulation and regression analysis described
in Sections 2.5 and 2.6, focusing on how these results manifest across zones in different boroughs. Our focus will be to base this zone density comparison around the central objective
of this study and the motivation behind our approach from the outset: comparing the performance of gig economy platforms, such as Uber, to that of more traditional firms, such
as Taxi services, across different types of zones. In Chapter 1, we noted that in more ‘spatial’ settings—where delays are less influenced by congestion and more by the geographical
dispersion of demand—gig economy platforms can perform as well as, if not better than,
traditional firms.
70

Our analysis does not account for overhead costs or fixed expenses that traditional taxi
firms might incur, suggesting that in real-world scenarios, gig economy platforms could
potentially outperform traditional services in spatial settings. This insight prompted us to
investigate whether we could observe a similar pattern using real-world data. And this is
the reason we aim to look at Uber and Taxi performance under some classification of zone
density. By establishing this classification, we can make meaningful comparisons between
Uber and traditional taxi services, examining how each performs across different types of
zones.
Earlier, in Section 2.4, we presented basic density maps of NYC based on various metrics,
which provided an initial understanding of which zones are relatively denser. As expected,
these maps indicate that Manhattan is notably more dense compared to the other boroughs,
reflecting higher levels of building density, pedestrian activity, and infrastructural concentration. Figure 2.6 displays a comparison of relative rankings for Uber and traditional taxi
services across various locations (taxi zones), based on the volume of rides. For each plot,
locations are ranked separately for Uber and taxi services, with the rankings determined by
the number of rides requested in each location, sorted in descending order. For example, the
top-ranked location for Uber is the one with the highest number of Uber rides, and similarly,
the top-ranked location for taxis is the one with the most taxi rides. By focusing on relative
rankings, rather than absolute ride counts, for example, we can better understand the comparative strengths of Uber and taxi services across different areas. This approach mitigates
the potential skew that could arise from the inherently larger scale of Uber’s operations,
which might otherwise overshadow valuable insights. Instead, it allows us to see where each
service is performing at its best relative to other locations, highlighting patterns of strong
performance that are more meaningful for understanding their operational dynamics. In
essence, comparing relative rankings gives a clearer picture of how Uber and taxis position
themselves across the city, revealing strategic hotspots and areas of competitive advantage
for each service.
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The color coding in this plot signifies the borough each zone belongs to: yellow for Bronx
(BNX), green for Manhattan (M), purple for Queens (Q), and blue for Brooklyn (BKL).
The 45-degree line serves as a reference, where points along this line indicate equal rankings
for Uber and taxi services in a given location. From the plot, we observe a clear pattern
where Manhattan locations (green) tend to have higher relative rankings for taxis compared
to their relative rankings for Uber. This aligns with our earlier insights and expectations
that dense, highly populated zones—such as those in Manhattan—tend to favor traditional
taxi services. In contrast, the other boroughs—Bronx, Queens, and Brooklyn—show a more
mixed distribution of rankings, with several points below the 45-degree line, indicating zones
where Uber, relatively, outperforms taxis. This suggests that in less dense, more spatially
dispersed areas, the flexibility of Uber’s dynamic matching and routing system may give it
an operational edge.
These observations already hint at the qualitative insights discussed in the previous
chapter of this thesis, where we explored how gig economy platforms like Uber can match or
outperform traditional services in more spatially diverse settings. However, it is important
to recognize that there may be other idiosyncratic factors contributing to the prominence
of taxis in Manhattan—such as regulatory frameworks, customer habits, and established
routes—that are not fully captured in this analysis. To better understand these patterns,
we will now delve deeper into the density characterization of each zone, examining both the
density metrics we presented earlier and the results from the regressions we performed.
2.7.2 Accounting for Different Zone Density
When examining the α values from the regressions for each borough alongside the density
metrics, we must proceed with caution. As highlighted earlier, our results are influenced by
specific characteristics of the available data, unknown platform operations that we attempt
to estimate, and consequently, a significant amount of noise. Therefore, we do not anticipate
a perfect match between the characterization of zones as ‘more’ or ‘less’ dense based on α
72

Figure 2.6: Comparison of Uber and Taxi Ride Rankings Classified by Borough
values and the classifications derived from our density metrics. This discrepancy is partly
because the density metrics themselves do not represent a definitive ‘ground truth.’ They
capture certain observable features of density, but there are undoubtedly many underlying
factors and nuances that they do not account for.
Moreover, defining a consistent method for characterizing zones as more or less dense,
whether using the density metrics or α values, presents its own set of challenges. Comparing
these two approaches adds an additional layer of complexity, as each method captures different aspects of density and congestion. Our goal is to explore these relationships, acknowledging that the classifications may not align perfectly but can still offer valuable insights
into the interplay between spatial characteristics and service delays across the city.
Figure 2.7 displays illustrates a comparison of ride rankings for Uber and traditional
taxi services across various locations, following the same ranking methodology as in Figure
2.7. Locations are ranked based on the volume of rides, with the top-ranked location for
each service being the one with the highest number of rides. Unlike the previous figure, the
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coding in each plot highlights zones that are classified as ‘more dense’ or ‘less dense’, using
a straightforward thresholding approach. To determine density, we considered the density
metrics identified by PCA, as described in Section 2.4. Locations with a density metric
above a chosen threshold are classified as ‘more dense’ (solid circles), while those below the
threshold are classified as ‘less dense’ (open circles). The four plots correspond to different
density thresholds—0.5, 0.6, 0.7, and 0.8—showing how the classification changes as the
criteria for density become stricter.
The plots reveal several interesting patterns. Overall, we see a general trend where denser
areas tend to have higher rankings for taxi services compared to Uber. This suggests that
traditional taxis may perform better or be more popular in zones that are densely populated,
which could be due to several factors. In densely populated areas, the demand for rides may
be higher, and the structure of traditional taxi operations (e.g., taxis stationed at fixed
locations or cruising popular streets) could be better suited to serve these concentrated
demands. Additionally, taxis do not rely on app-based matching, which might give them an
advantage in areas with more foot traffic or where passengers frequently hail rides directly
from the street. In contrast, Uber rankings are more dispersed, indicating that the platform
may have a stronger presence or relatively higher usage in less dense areas, where its dynamic
matching and flexible routing system can better adapt to more spatially spread-out demand.
This could reflect how ridehailing services effectively fill gaps where traditional taxi networks
might not be as concentrated or efficient.
One particularly notable observation is that, if we look more closely at a density threshold
of 0.7, for instance, nearly all the zones classified as ’more dense’ are located in Manhattan.
This aligns with our earlier insights, reinforcing the idea that Manhattan remains the core of
dense, urban activity in NYC, where traditional taxi services continue to thrive. Interestingly,
these zones include key commercial and business districts such as zones in Downtown NY,
Midtown, and in the southern corners of Central Park where the demand for immediate and
street-hailable transportation remains high. This is consistent with our earlier discussion
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that taxis, due to their established presence and ability to serve spontaneous street hails,
may have an edge in these high-demand areas.
(a) Classification threshold 0.5 (b) Classification threshold 0.6
(c) Classification threshold 0.7 (d) Classification threshold 0.8
Figure 2.7: Comparison of Uber and Taxi Ride Rankings by Location, Classified by Density
2.7.3 Incorporating Regression Insights to the Comparison
We now want to incorporate the α we obtain from the regression analysis to this comparison
of zones. Figure 2.8 provides a more nuanced comparison of Uber and taxi ride rankings
across locations by incorporating both the density metrics classification and the α coefficient
from our regression analysis. The c parameter is equal to c = 0 in this set of regressions
and in Section B.6 of Appendix B we provide the same plot for different c and abandonment
probability, p, values. As before, locations are ranked based on the volume of rides, with
denser zones (according to the density metrics) represented by solid circles and less dense
zones by open circles. Additionally, we use squares to highlight points where the density
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metrics classification aligns with the classification based on the α coefficient. Specifically, a
zone is classified as dense using the α coefficient if its α value exceeds a certain threshold,
reflecting more congestion-driven characteristics. Because the range of α values can vary
depending on the choice of the c parameter in the regression, this classification is based on
the percentile of zones that fall above that threshold, ensuring consistency across different c
values.
As observed earlier, denser areas tend to have higher rankings for taxi services, suggesting an operational advantage for traditional taxis in these zones. In contrast, Uber shows
stronger performance in less dense, more spatially dispersed areas, where its dynamic matching system can adapt more effectively. What is particularly interesting in this new plot is
the agreement between the two classification methods, especially in non-dense areas but also
in some dense ones, as indicated by the square markers.
Notably, depending on the threshold, the majority of these dense zones, where both
classifications agree, are concentrated in Manhattan, with areas such as Murray Hill and
Garment District (Midtown), World Trade Center and Greenwich Village (Downtown), Lincoln Square and Upper East Side South (Southern Corners of Central Park) emerging as high
alpha and high density across different values of p. This consistency reinforces the robustness of our classification approach, suggesting that these Manhattan zones exhibit strong
congestion-driven characteristics regardless of how we adjust the matching dynamics in our
simulations. It underscores the city’s core as a central hub of dense activity, where both
traditional taxis and ride-hailing services contend with high demand and congestion. The
recurring agreement between classifications highlights stable congestion patterns that align
well with real-world urban behavior, illustrating how both services adapt to the challenges
of dense urban centers.
This agreement suggests that the α coefficient from the regression does capture some of
the congestion dynamics we aim to measure, even though there is still a considerable amount
of noise in the α values across different zones. This noise may stem from the complexity
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of ride-hailing operations, platform-specific matching rules, or unobserved external factors
influencing delays. Additionally, since our data is derived from ride-hailing services—which
are inherently spatial and flexible in how they respond to demand—it makes sense that we
observe a stronger spatial signal in our regression results. Despite these sources of variability,
the observed alignment between the two classification methods in many zones reinforces
the validity of our approach and offers valuable insights into how density influences the
performance of ride-hailing and traditional taxi services.
The classification method used in this analysis is based on a simple threshold approach,
which, while straightforward, may not be the most sophisticated way to differentiate between dense and less dense zones. We explored more advanced techniques, such as k-means
clustering and Gaussian Mixture Models (GMM), to classify the zones based on the α coefficients. However, these approaches did not provide a clear advantage in this context, where
the clustering is performed on a single feature. As a result, we opted to continue with the
threshold method due to its simplicity and interpretability. Despite its straightforward nature, this approach effectively highlights the key patterns we seek to uncover, allowing us to
draw meaningful conclusions about the relationship between density and the performance of
ride-hailing and traditional taxi services.
What is particularly compelling about the zones identified as high alpha and high density,
across the all the different p values, is the notable presence of major transit hubs within
these areas, indicating strong alignment between the density characterizations and realworld demand dynamics. Table 2.2 provides a list of the 10 busiest subway stations in 2021,
according to the Metropolitan Transportation Authority (MTA) of New York, (“Subway
and Bus Ridership” 2021), highlighting key points of high rider volume. For instance, the
Garment District zone (as defined by the TLC Taxi Zones) contains four of the busiest
stations: Times Sq-42 St and 34 St-Herald Sq, which rank first and second respectively,
as well as two separate lines serving the 34 St-Penn Station area. Additionally, this zone
includes Port Authority, a major transit hub that not only serves subway lines but also a vast
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(a) p = 0.2
(b) p = 0.5
(c) p = 0.8
Figure 2.8: Comparison of Uber and Taxi Ride Rankings for c = 0. Density classification is
based on both density metrics and α regression coefficients, with square markers indicating
agreement between the two methods
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network of buses. In Murray Hill, Grand Central-42 St—another crucial transit station—is
situated, supporting the notion of a high-density, high-traffic area. Greenwich Village North
borders Union Square, connecting it to the 14 St-Union Sq station, while Lincoln Square East
contains 59 St-Columbus Circle, another major intersection of subway lines. Furthermore,
even though it does not appear in the top 10 stations table, the World Trade Center, a
significant transit center connecting multiple subway lines and the PATH line to New Jersey,
is located in the World Trade Center zone, which is also adjacent to the Financial District
North zone that contains the Fulton Street station, listed among the busiest.
A particularly interesting case is Elmhurst in Queens, which emerges as both high alpha
and high density when the abandonment probability is set to p = 0.2. This is the only zone
outside Manhattan where both methods—the density metrics and regression-based alpha coefficients—indicate high alpha and high density. The TLC Taxi Zone Elmhurst contains the
74-Broadway station, which is among the busiest transit locations outside Manhattan, further reinforcing the robustness of our approach in identifying high-demand areas accurately,
even in different boroughs.
These observations are encouraging as they suggest that our model captures essential
demand dynamics. The alignment between our density classifications and the locations of
major transit hubs implies that the regression model is effectively detecting zones with concentrated demand, validating the use of alpha as a proxy for congestion-driven delay. By
successfully identifying these transit-dense zones, our approach demonstrates its potential to
reflect real-world patterns. Interestingly, our methods captured 9 out of the 10 busiest transit stations of 2021 within the taxi zones where both the density metrics and the alpha-based
regression classifications give the same insights. This strong alignment between our model’s
outputs and actual high-demand locations underscores the robustness of our approach, despite its simplicity, and highlights its relevance in identifying zones of concentrated demand.
Such agreement suggests that our combined methodology can effectively pinpoint critical
urban areas that are likely to experience high congestion.
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Table 2.2: Top 10 Busiest Subway Stations in 2021
Rank Station/Complex Lines Ridership
1 Times Sq-42 St N, Q, R, W, S, 1, 2, 3, 7, A, C, E 29,507,558
2 34 St-Herald Sq B, D, F, M, N, Q, R, W 15,037,793
3 Grand Central-42 St S, 4, 5, 6, 7 14,002,142
4 14 St-Union Sq L, N, Q, R, W, 4, 5, 6 13,165,975
5 34 St-Penn Station A, C, E 9,855,288
6 Fulton St A, C, J, Z, 2, 3, 4, 5 9,728,874
7 74-Broadway 7, E, F, M, R 9,437,073
8 59 St-Columbus Circle A, B, C, D, 1 9,310,678
9 Flushing-Main St 7 9,206,396
10 34 St-Penn Station 1, 2, 3 8,935,671
2.8 Conclusion
This study provides an empirical exploration of how ridehailing platforms, such as Uber,
compare to traditional taxi services across different urban environments. By analyzing driver
availability, customer delays, and ride volumes, we sought to understand the conditions under
which each service type excels. Our findings indicate that urban density seems to be related
to these dynamics. In densely populated zones, such as those in Manhattan, traditional taxi
services tend to perform better, on a relative basis, perhaps due to their ability to remain
readily available for street hails and quickly navigate high-demand areas. In contrast, Uber
appears to have a relative advantage in less dense, more spatially dispersed zones, where its
flexible matching and routing system can better adapt to fluctuating demand across larger
areas.
The use of a max-flow simulation to estimate driver availability allowed us to address
the challenge of missing data on open drivers, providing a more accurate picture of how
supply fluctuates in different locations. By incorporating adjacency between zones, we captured more realistic movement patterns of drivers, reflecting real-world ride-hailing behavior.
However, our analysis also highlighted limitations, such as the presence of noise in the data
and the need to make adjustments to the model, like the introduction of the parameter c to
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account for matching time. This adjustment brought our results closer to theoretical expectations, suggesting that the model could be further refined to better align with real-world
observations.
While our analysis provides valuable insights into the dynamics between ride-hailing platforms and traditional taxi services, it is important to acknowledge several limitations. Firstly,
the relationships we have identified are purely observational and should not be interpreted
as causal. The patterns we observe may be influenced by various external factors, including
policies, infrastructure, and local economic conditions, which are not explicitly captured in
our model. Furthermore, traditional taxis have historically been more prominent in Manhattan, a trend that continues today. This prominence could be driven by factors beyond what
our density metrics capture, such as established customer habits, regulatory frameworks, or
operational practices specific to taxi services. Nevertheless, our findings highlight intriguing
associations that contribute to the narrative we seek to explore, providing a foundation for
future research to delve deeper into these complexities.
Overall, our study underscores the importance of understanding spatial characteristics
when analyzing the performance of ridehailing platforms versus traditional taxi services.
While taxis continue to dominate in dense urban cores, the flexibility of ride-hailing services
offers clear advantages in more spread-out areas. These insights not only contribute to
the broader literature on two-sided markets and gig economy platforms but also provide a
foundation for future research to further investigate how factors like pricing, driver behavior,
and infrastructure may affect the efficiency and effectiveness of these services in varying urban
contexts.
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Chapter 3
3.1 Introduction
In their recent paper, (Tian et al. 2019) (hereafter referred to as TSD) study the stability of
producers’ recycling strategies in communities that have passed legislation under which the
producers are responsible for the proper disposal (i.e., recycling) of the products they bring
to the market. The authors assume that producers compete with multiple, differentiated
products in consumer markets, but may consider cooperating when recycling those products
in order to benefit from economies of scale. Products made by different producers or sold in
different markets might still be recycled jointly. The authors study when firm-based recycling
strategies (i.e., separately recycling products falling under same brand) or market-based
recycling strategies (i.e., separately recycling products falling in the same product category)
emerge as stable outcomes by analyzing two simple producer-market configurations. They
first look at an asymmetric market model with two producers making a total of three products
in two markets, and then at a symmetric market model with two producers competing
with a total of four products in two markets. The authors show that with intense market
competition and differentiated market sizes, producers may recycle their products on their
own without cooperating with others. In some instances, they can add a product from their
competitor to their recycling mix. However, with less intense competition or more equitable
market shares, all-inclusive (resp., market-based) recycling is most common stable outcome
with high (resp., low) scale economies.
In their analysis, TSD assume that any firm can choose to have their products recycled
together with the other firm’s products. However, any firm is free to change its decision,
and the other firm may react by changing its decision in return. In order to identify stable
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recycling structures in which no firm has the power or incentive to further defect, the authors
use a dynamic stability concept, the largest consistent set (LCS; see (Chwe 1994)) in order
to better capture the possible actions and reactions of every firm.
The LCS assumes that the players are infinitely farsighted; that is, it assumes that players
will consider possibly very long sequences of potential defections when making their decisions.
It is not clear that in practice players actually exhibit such a high level of farsightedness.
While myopic stability concepts such as the core assume that players only consider immediate
consequences of their moves and disregard possible reactions by others, which is not likely
to happen in real life, one of the question that we wanted to analyze in this chapter is how
farsighted are the players actually in practice? In other words, if we consider theoretical
results from TSD, is it likely that they will be replicated in an experimental setting? In
addition, the LCS can be rather inclusive, in that it includes all potentially stable outcomes.
In our experiments we also want to study the settings wherein the LCS contains more
than one potentially stable outcome to see if we can identify outcomes that are more likely
than others to emerge as stable in practice. In this work, we also introduce an alternative
stability concept, the equilibrium process of coalition formation (EPCF), which evaluates
the level of farsightedness required for outcomes to be stable, and we investigate if there is
a relationship between the levels of farsightedness required for stability of specific outcomes
and the frequency at which they are observed as stable in practice.
The rest of the chapter is organized as follows. In Section 3.2 we review some relevant
literature. In Section 3.3 we introduce two farsighted stability concepts, the LCS and the
equilibrium process of coalition formation (EPCF), while in Section 3.3.3 we briefly introduce the one of the two models from TSD that we will be considering as the setup for our
experiment and provide some examples of players’ payoffs and corresponding stable outcomes. In Section 3.4 we introduce our hypotheses and the questions we are interested in
investigating through our experiment. The the following Section 3.5 contains a description of
our experiment which is composed of a main task and three control tasks, a brief mention of
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the procedures, and finally our plan for the analysis of the experiment results. We conclude
in Section C. C.1 contains mathematical expressions for models from TSD, C.2 contains
several examples of possible payoffs and corresponding stable outcomes, while C.3 provides
a detailed example of profit calculations.
3.2 Literature Review
The experimental study of coalition formation has a long history, going back to (Kalisch
et al. 1954), who conducted experiments with n-person coalition games and investigated
relationships between observed outcomes of the games and various theoretical concepts.
Since then, there have been many experimental studies aimed to explore the relationship
between theoretical results and people’s behavior. Most of these papers were concerned with
myopic concepts, such as the core. For instance, (Vinacke et al. 1957) studied the degree
to which interaction among three players corresponds to expectations stemming from the
perceptions and motives of the participants, compared to those which would correspond to
the rational outcomes expected by game theorists. The authors found that the coalitions that
actually formed were different than those predicted by game theorists. Some other examples
include (Murnighan et al. 1977), who studied the effects of communication and information
availability in three-person games, (Maschler 1965), who studied players’ behavior and its
relationship to the bargaining set in games with three and four players, (Leopold-Wildburger
1992), who studied payoff division in the grand coalition, (Nash et al. 2012), who studied a
bargaining coalition formation game wherein players form coalitions and transfer bargaining
rights to another player, who distributes payoffs to the coalition, (Dechenaux et al. 2015),
who provided a survey of experimental studies that analyze how structures, heterogeneity,
and information affect group formation in contests, and many others.
The abovementioned papers are concerned with myopic stability concepts. While there
have been some experimental studies looking at farsighted stability, their number is signifi84

cantly smaller and seems to focus on network formation games. One example is (Mantovani
et al. 2013), who adopt the von Neumann-Morgenstern pairwise farsightedly stable set (VNMFS) of networks, which predicts which networks one might expect to emerge in the long
run when individuals are farsighted. This work is the first experimental test of farsightedness
versus myopia in network formation, and the authors found that the agents exhibit a limited level of farsightedness (around two steps). The second example is (Teteryatnikova and
Tremewan 2019), who use two myopic stability concepts (pairwise stability, (Jackson et al.
1996), and the pairwise myopically stable set (Herings et al. 2009), two farsighted stability
concepts (the pairwise farsightedly stable set and VNMFS), and three cautious farsighted
concepts (the largest pairwise consistent set, (Herings et al. 2009) , the largest farsightedly
consistent set, (Page Jr et al. 2005), and the cautious path stable set, (Teteryatnikova et al.
2015)). They showed that there exist environments wherein farsighted stability concepts
identify empirically stable networks that are not identified by myopic concepts, and hence
myopic stability concepts may not be enough to predict all stable outcomes. To the best of
our knowledge, our work is the first experimental test of a farsighted stability concept, in
our case the LCS and the EPCF, outside the network formation games.
3.3 Stability concepts
TSD used the largest consistent set (LCS), introduced by (Chwe 1994), as their stability
concept. In this section, we first introduce the LCS in their setting, and then we introduce the
equilibrium process of coalition formation (EPCF), introduced by (Konishi et al. 2003), as an
alternative farsighted stability concept that enables us to measure the level of farsightedness
required to obtain stability of specific outcomes.
Let N denote the set of all firms and X denote the set of all partitions of N, also referred
to as structures, which are, essentially the alternative options that the firms are faced with.
For every firm i ∈ N, let ΠX
i denote i’s payoff under structure X ∈ X. Denote by ⇀S
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defection by S ⊆ N: X1 ⇀S X2 if structure X2 is obtained when S deviates from structure
X1. In the context of coalition formation, S denotes the coalition that if and when is formed
leads the firms from structure X1 to X2.
3.3.1 The LCS
Let us denote by ≺i the players’ strong preference relations, described as follows: for two
structures, X1 and X2 ∈ X, X1 ≺i X2 ⇔ Π
X1
i < Π
X2
i
.
For S ⊆ N, if X1 ≺i X2 for all i ∈ S, we write X1 ≺S X2. Note that the preference
relation refers only to the firms that are directly involved in the coalition/defection S that
leads from X1 to X2.
We say that X1 is directly dominated by X2, denoted by X1 < X2, if there exists an
S ⊆ N such that X1 ⇀S X2 and X1 ≺S X2. So X1 is directly dominated by X2, if, when at
X1, there is a coalition that can be formed and lead to X2, and all the firms in this coalition
prefer to move to the new structure X2 than staying at X1.
We say that X1 is indirectly dominated by Xm, denoted by X1 ≪ Xm, if there exist
X1, X2, X3, . . . , Xm and S1, S2, S3, . . . , Sm−1 ⊆ N such that Xi ⇀Si Xi+1 and Xi ≺Si Xm for
i = 1, 2, 3, . . . , m − 1.
The definition of indirect dominance is essentially the concept that captures the “farsightedness” of the firms in the sense that they are considering the final outcomes that their
actions lead to. An alternative structure Xm will indirectly dominate another alternative
X1 if Xm can replace X1 in a sequence of moves, such that at each move the active coalition
prefers (the final alternative) Xm to the alternative it faces at that stage. Firms will move
from a structure only if this move leads to a better alternative, either now (direct dominance)
or in the future (indirect). In between, there may be structures that are visited for which the
possible payoffs may be less than in the previous. However, they are farsighted in the sense
that they have considered that even though they may lose something at an intermediate
step, that intermediate step will not in itself be stable, because other coalitions will make it
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less desirable and eventually lead to a better outcome.
A set M ⊆ X is called consistent, if X ∈ M if and only if for all Y ∈ X and S ⊆ N such
that X ⇀S Y , there is a Z ∈ M, where Y = Z or Y ≪ Z, such that X ⊀S Z. The LCS is
the largest consistent set. As Chwe writes, and explains the meaning of the consistent set:
“If M is consistent and X ∈ M then the interpretation is not that a will be stable but that
is is possible for X to be stable. If an outcome E is not contained in any consistent M the
interpretation is that X cannot possibly be stable: there is no consistent story in which E
is stable.”
The LCS assumes that the actual payoff is received only when firms reach a stable
recycling structure, and the defections that they consider when making their decisions might
be seen as a mental exercise in which firms contemplate possible impacts of their moves.
The main idea of the LCS is that a move to another structure, in which defecting firms can
see an increase in their payoffs, is deterred if it triggers a sequence of subsequent moves that
eventually ends in a stable structure in which some of the originally deviating firms are worse
off than in the starting structure. Likewise, a move to another structure in which defecting
firms observe a decrease in their payoffs can happen if it triggers a sequence of subsequent
defections that eventually ends in a stable outcome in which all of the originally deviating
firms are better off than in the starting structure.
3.3.2 The EPCF
While the LCS assumes that the payoffs are generated only once the stable outcome is
reached, an alternative stability concept, the equilibrium process of coalition formation
(EPCF), introduced by (Konishi et al. 2003) , assumes that players receive payoffs after every move, and considers the infinite horizon discounted value of their payoffs. One of
the benefit of this approach is that we can investigate the level of farsightedness required to
achieve each of the stable outcomes.
Let δi ∈ [0, 1] denote the discount factor for i’s future payoffs. Then, i’s payoff from a
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sequence of structures {Xt} can be written as P∞
t=0 δ
t
iΠ
Xt
i
. When δi = 0, only immediate
payoffs are considered, which corresponds to myopic stability concepts such as the core.
Since we are interested in farsighted results, in this work, we will be interested in evaluating
lowest levels of the the discount factors under which some outcomes are stable.
A process of coalition formation (PCF) is a transition probability ψ : X × X → [0, 1]
such that P
Y ∈X ψ(X, Y ) = 1 for ∀X ∈ X. A PCF ψ induces a value function Vi
for every
firm i, which represents i’s infinite horizon payoff starting from the structure X under ψ and
is the unique solution to the equation Vi(X, ψ) = ΠX
i + δi
P
Y ∈X ψ(X, Y )Vi(Y, ψ).
We say that a defection X ⇀S Y is profitable under ψ if Vi(Y, ψ) ≥ Vi(X, ψ) for ∀i ∈ S;
we further say that the defection is strictly profitable if the above inequality is strict. We say
that the defection is efficient if there is no other move X ⇀S Z such that Vi(Z, ψ) > Vi(Y, ψ)
for ∀i ∈ S. (Konishi et al. 2003) consider that a defection from one structure to another
occurs only if all members of the deviating set agree to move and they cannot find a strictly
better alternative structure. In addition, a defection from a structure must occur if there
is a strictly profitable move. Then, the Equilibrium PCF (EPCF) is defined as follows. A
PCF ψ is an Equilibrium PCF if the following holds:
• whenever ψ(X, Y ) > 0 for some Y ̸= X, there exists S ⊆ N such that X ⇀S Y is
profitable and efficient;
• if there is a strictly profitable defection from X, then ψ(X, X) = 0 and there exists a
strictly profitable and efficient defection X ⇀S Y such that ψ(X, Y ) > 0.
We say that a PCF ψ is deterministic if ψ(X, Y ) ∈ {0, 1} for ∀X, Y ∈ X. A structure
X is absorbing if ψ(X, X) = 1, while a PCF ψ is absorbing if, for every structure Y , there
is some absorbing structure X such that ψ
(k)
(X, Y ) > 0, where ψ
(k) denotes the k-step
transition probability. (Konishi et al. 2003) show that the set of all absorbing states, under
all deterministic absorbing EPCFs, is a subset of the LCS. Thus, the absorbing states of the
EPCF provide a refinement of the LCS.
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3.3.3 The TSD Model
As we mentioned earlier, TSD study two models: an asymmetric model with two producers
making a total of three products in two markets, and a symmetric model with two producers
competing with a total of four products in two markets. In this study, we focus on the result
of the asymmetric model and the experiment that will be later described on Section 3.5 was
built around this structure of the market. We believe it is a model rich enough to capture
all the factors that firms may take into account when faced with such decisions, and at the
same simple enough to construct, and move to a laboratory environment.
In the asymmetric model, two firms make a total of three products in two markets. In
one of the markets, the two firms compete as duopolists, each with one product. In the
other market, one of the firms is the monopolist with the remaining product. We assume
that firm A makes product 1 and firm B makes products 2 and 3. Each firm has to decide
what quantity of each product to produce and bring to market, with the goal of maximizing
their individual profits. To comply with EPR-type legislation, firms have to appropriately
recycle all products they bring to the market and are responsible for the recycling costs. The
firms can decide whether they want to cooperate with the other firm when recycling their
products. While the two firms compete in the primary market for products 1 and 2, in order
to take advantage of the economies of scale, products need to be recycled together and firms
may need to cooperate in the recycling market.
Another important factor that influences the producers’ recycling cost when recyclers
contract with multiple producers or recycle multiple products, is the diseconomies of scope
that may arise by recycling differentiated products together. The disassembly of valuable
components and raw materials is more labor intensive when there are more variations in the
way these components and raw materials are connected with each other, and this increase in
task difficulty is one of the main determinants of the producers’ unit recycling costs. TSD
considered the case where firm B produces two different products in order to capture the
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effect of this task heterogeneity that will influence its recycling costs and consequently its
decisions.
In general, firms contract with third-party recyclers to collect and process their products.
Each firm can contract with one or more recyclers. The resulting recycling structure can
belong to one of the following five cases:
• All products are recycled by one recycler. We refer to this case as all-inclusive recycling,
denoted by (123).
• Competing products 1 and 2 are recycled by one recycler; standalone product 3 is
recycled by another recycler. As products from the same market are recycled together,
we refer to this case as market-based recycling, denoted by (12, 3).
• Firm A’s product 1 is recycled by one recycler; Firm B’s products 2 and 3 are recycled
by another recycler. As products made by the same firm are recycled together, we
refer to this case as firm-based recycling, denoted by (1, 23).
• Products 1 and 3 are recycled by one recycler; product 2 is recycled by another recycler.
As the two products that are recycled together are from different markets and made by
different firms, we refer to this case as cross-market/firm recycling, denoted by (13, 2).
• Each product is recycled by an individual recycler. We refer to this case as productbased recycling, denoted by (1, 2, 3).
We can note here that, given this market model considered here (the multi-product fim
B controls two products, 2 and 3, while the specialized firm A controls only product 1,) gives
more power to firm B. More precisely, if the current structure is product-based or firm-based
recycling, specialized firm A cannot change it unilaterally, while multi-product firm B can.
Consequently, whenever product-based or firm-based generates the highest payoff for firm
B (compared to other structures), this structure is uniquely stable—B does not want to
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defect from it, A cannot change the structure on its own, and B can defect to either of these
structures from any of the remaining possible structures.
Depending on their payoffs under different recycling structures1
, each firm has its most
preferred structure. TSD use the LCS to determine stable outcomes. For the model with
economies of scale, their main results are given in the following two propositions.
Note that by using the LCS as a solution concept, multiple structures may arise as
stable and (TSD) found that, in this setup, up to two structures could be stable given
the parameters considered. Proposition 1 captures the cases where a unique structure may
arise as stable, and Proposition 2, the cases of multiple (i.e. two) possible stable recycling
structures.
Proposition 11. (TSD) In the asymmetric model with economies of scale, all structures
may emerge as stable: When economies of scale are moderate to high or cost increase is
low, the most common stable structure is all-inclusive recycling, (123); when economies of
scale are low and cost increase is moderate to high, the most common stable structure is
market-based recycling, (12, 3);
firm-based recycling, (1, 23), can be uniquely stable when products are highly substitutable,
market size of the standalone product 3 is significant compared with market sizes of products
1 and 2, market size of product 2 dominates that of product 1, economies of scale are low,
and cost increase is low;
cross-market/firm recycling, (13, 2), can be stable when products are highly substitutable,
market size of product 1 dominates that of product 2, and either economies of scale are
moderate to high, or economies of scale are low and market size of the standalone product 3
is low compared with products 1 and 2;
product-based recycling, (1, 2, 3), can be uniquely stable when products are highly substitutable, market sizes of products are significantly different, and economies of scale are low.
Proposition 12. (TSD) In the asymmetric model with economies of scale, all-inclusive,
1See Appendix A for more details.
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(123), and market-based recycling, (12, 3), can both emerge as stable when economies of scale
are low, or when economies of scale are moderate and cost increase is high;
all-inclusive, (123), and cross-market/firm recycling, (13, 2), can both emerge as stable
when market size of product 1 dominates that of product 2, the standalone product 3 has small
market size compared to product 1, products are moderately substitutable, and economies of
scale are moderate.
3.4 Motivation for Behavioral Experiment and Hypotheses
Behavioral experiments have been widely used in the Operations Management literature
in order to see whether people would behave in real-life as the analytical model would
describe. Given how important decision-making is in Operations Management but also how
closely it is related to a specific context and environment, behavioral experiments aim, by
carefully controlling the environment, so that it matches the theoretical assumptions as close
as possible, to find the connection between theory and reality, as well as the possible gaps
between the two.
Consider, for example, one of the most important models in inventory theory, the newsvendor model. (Schweitzer et al. 2000) show that people do not actually solve the model correctly and make somewhat systematic and predictable deviations from the optimal decisions.
In the, not so rare, case that there is high evidence of a gap between theory predictions and
actual behavior, behavioral experiments also strive to find what could possibly be the other
factors that affect individuals’ decisions under a particular scenario.
Our main goal for this study is to test whether the theoretical results that (TSD) derived
by using farsighted stability theory, on the specific context of recycling alliances between
competing firms, are actually likely to emerge as stable in a laboratory environment. Many
behavioral studies that have been done in order to experimentally test other theories. One of
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the most striking examples would be that of Standard Utility Theory for example, according
to which decisions are made based on the final payoff. Extensive behavioral experiments have
shown that several behavioral factors affect decision-making such as reference dependency
and loss aversion ((Kahneman 1979), (Thaler 1985), (K˝oszegi et al. 2006)). Motivated by
the extensive evidence that, many times, analytical models may not entirely capture all the
factors that may affect individuals’ decisions and by the fact that, as argued in Section 3.2,
not much experimental work has been done in this particular branch of farsighted stability
theory, we hope to test whether this theory can predict outcomes that individuals would
actually agree upon. At the same time to analyze what could be some of the possible
behavioral traits that lead to the results we get, and, especially in the case where we find
that there is indeed a gap between the theoretical model and reality, to try and provide some
intuition as to why that may be the case.
In order to study the relationship between the theoretical results and players’ behavior
in practice, we have designed an experiment, that we describe in detail in section 3.5, in
which we have chosen some payoff combinations that mimic the relationship between firms’
profits under different scenarios described above. Theses payoffs are presented to players,
and they have to decide which of the options to chose. Notice that there are three possible
outcomes in which players collaborate—the all-inclusive recycling, market-based recycling,
and cross-market/firm recycling. In the remaining two cases, the firms recycle their products
individually, and Firm B can recycle its products jointly or separately, depending on which
option is more profitable. As Firm A does not make any decision in this case, in order to
simplify players’ decision making process, we have decided to only focus on the option that
maximizes Firm B’s profit in a specific case and remove the other option from consideration.
Thus, the players are presented with a total of four possible options, three collaborative and
one non-collaborative, which is the default outcome if a collaborative outcome cannot be
agreed upon.
In the settings with Pareto-dominant choices, we predict that collaboration will be easily
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achieved and that these examples can be used to study participants’ understanding of the
task on hands. The situations that are of more interest are those instances in which we
observe an inconsistency among structures that are most preferred by different firms. Thus,
a player may not end up in his most preferred structure because it requires the participation
of the other player, which may have different preferences. There are several possible types of
relationships between payoffs that can occur, and we illustrate them in a number of examples
in Appendix B. Through the design of our experiment that gives players the context of the
recycling framework and allows them to interact and bargain over which recycling option
to choose we want to investigate: (i) presented with the payoff matrices that correspond
to Proposition 11, whether the recycling structure that is predicted as the stable one, will
actually be the one chosen after the interaction of the players, (ii) presented with the payoff
matrices that correspond to Proposition 12, which of the two stable outcomes will be the
most frequent, and also, (iii) what is the effect of the level of farsightedness, that is required
to, possibly, reach a stable outcome. For this reason, our hypotheses are as follows:
Hypothesis 1. Given the payoff matrices that correspond to each of the scenarios in Proposition 11, the most commonly agreed-upon recycling structure across participants will be the
recycling structure predicted as stable from TSD.
Hypothesis 2. Given the payoff matrices that correspond to each of the scenarios in Proposition 12, the most commonly picked recycling structures across participants will be the two
recycling structures predicted as stable.
In the case of Hypothesis 2, we are also very interested in seeing if there is one of the
two outcomes that is more likely to occur in reality, and if so, if there is a reason why one of
the two will be preferred to the other. This is of primary interest when comparing between
stable outcomes, but also in the case of one stable outcome, if that outcome may not be
reached but preferred over another. Through our laboratory experiments are very interested
in examining is the players’ notion of fairness and inequity and how these behavioral traits
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affect their decisions, their in-between communication, and ultimately, the selected outcome.
Our main motivation is the inequality aversion model of distributive fairness proposed by
(Fehr and Schmidt 1999). According to the model, people in a given situation do not only
care about their own payoff, but also about the difference between their own payoff and the
payoff of the others involved. When the others’ payoff is different from their own, the players
suffer a disutility. The loss of utility caused by this difference in payoffs is larger when the
inequality in payoffs is disadvantageous to the player than in the case where the inequality
is advantageous.
As can be seen in the payoff matrices in Appendix B that we have created in order
to mimic the firms’ profits under the various scenarios described in Section 3.3.3 there is
a substantial discrepancy in the range of payoff difference from table to table. We notice
that while in some payoff matrices the payoffs for both firms are in about the same range
(see example 1.1 in C.1 while in others, the difference in payoffs between the two firms is
substantial (see example 2.2 in C.2. In both these tables, The stable outcomes are bolded.
It is natural to suspect, therefore, that the players’ interaction and decisions in these
situations will be quite different. In the second case for example, the predicted stable outcome
from our theory is 4 which is, however, the outcome with the greatest difference in payoffs
between the players. From our theory it corresponds to a scenario where firm B receives the
highest payoff without cooperation and firm A cannot make them cooperate, but it seems
very likely that, in reality, the player who is assigned the role of Firm B will be willing to
give up some monetary payoff so that the other player receives something better.
Subsequently, we want to investigate what is the level of farsightedness needed to reach
a stable recycling structure, and for that reason we incorporate an infinitely repeated game
structure (which will be described in section 3.5), and use the EPCF (section 3.3.3), to
get insights on the required level of farsightedness for the various scenarios. In general,
we suspect that a higher level of farsightedness has greater potential of leading to a stable
outcome, and this motivates our third hypothesis:
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Hypothesis 3. Convergence to a stable outcome is more likely to occur at high levels of
farsightedness.
Furthermore, besides seeing the effect of the level of farsightedness on whether or not a
stable outcome will be chosen, motivated by (Dal B´o et al. 2011) who study the evolution of
cooperation in infinitely repeated prisoners’ dilemma games, we are also interested in seeing
the effect on cooperation at different levels of farsightedness. That is, do we see individuals
trying to reach a cooperative recycling structure, or do they more often choose to pick the
independent recycling structure, without the collaboration of the other firm?
3.5 Experiment
3.5.1 Experiment Design
The experiments are designed to be implemented as a sequence of “tasks”. In order to be able
to rigorously test our hypotheses and analyze any results through an experimental approach
it is essential to incorporate three key factors (Donohue et al. 2018). First is Theoretical
Guidance since we need to be able to compare results with a theoretical benchmark. That is
why created A “Main Task”, which is the game in which we tried to keep the key elements of
the (TSD) recycling model clear enough so that insights from the experiment can be related
to the theoretical results. In addition, since we are interested in viewing the social preferences
that are relevant with the players’ decision-making in this context, we incorporated three
additional “Control Tasks”, that have been used in several experimental studies ((Charness
et al. 2002), (Kraft et al. 2018), (Kosfeld et al. 2009)). Another important factor is
the Control of Institutional Structure, meaning that options and information available in
the experiment should match what is assumed by model. In order to try to capture the
bargaining process in an experiment for example, a typical approach used to be the exchange
of alternating offers between the bargainers ((Rubinstein 1982)). In our design, we employed
96

a live chat interface to match bargaining procedures as closely as possible which we believe
allows for more freedom and communication during the bargaining process instead of just
having players exchange counter-offers, and at the same time, it will prove as a useful tool in
providing some insight on the decision process. Finally, the factor of induced valuation is also
essential in such studies because rewarding performance in the context of the experiment can
give a cleaner view on how individuals pursue their goals. That is why we also employed a
monetary reward to the participants so that their earnings based on the decisions/outcomes
on each of the parts of the experiment will worth a certain amount in U.S. dollars.
3.5.1.1 The Main Task
The main task, which is to be played first, is the Stable Recycling task which is designed
to test our theory In the main task, the players are initially presented with information on
the context of the story, and the “scenario” of the task: They are told that there are two
possible roles, Firm A that makes only Product 1, and Firm B that makes both Product 1
and Product 2, and both of whom are required by regulation to recycle all products they
make. Both firms contract with third-party recyclers to collect and process their products
and each can contract with one or more recyclers. The two firms can also choose to contract
with a common recycler to process some or all of their products. However, the use of a
common recycler must be jointly agreed upon by both firms. The players are then randomly
matched to a partner and randomly assigned the role of one of the two firms, A or B. They
are given a payoff table that describes their corresponding earnings under different recycling
options (as described in Section 4). The two participants are given 5 minutes to discuss
through the computer to jointly decide which one of the 3 joint recycling options in the
table to use. During the discussion, each participant can both send free texts and propose
an option to the other participant. Once an option proposed by one participant is accepted
by the other participant the discussion ends and the accepted option will be implemented.
Each participant will earn the payoff corresponding to the implemented option given his or
97

her role and in the case the two participants cannot agree on the same option by the end
of 5 minutes, then they will individually arrange their own recycling and earn the payoffs
shown in the last row of the table. At the end of a round, there is a probability equal
to δ (TBD) that they will play the same task, with the same partner again, given the
same payoff table but if the players agree on the same recycling structure twice in a row,
then they will not keep playing with the same partner and same payoff table. After they
have reached this point, or after a specified number of such rounds (TBD), the players are
randomly rematched to a new partner, and randomly assigned a role again and this process
is repeated for a total of five times. We have created different treatments that will allow
us to examine the key elements we want under the possible recycling scenarios , namely
the distinction between one and two stable outcomes, and the distinctions between levels of
farsightedness, which is incorporated in the experiment through the probability of playing the
game with the same partner and same payoff table. These treatments are: high continuation
probability with one stable outcome, high continuation probability with two stable outcomes,
high continuation probability with one stable outcome, and low continuation probability with
two stable outcomes. The assignment, within each treatment, of tables to pairs of players is
also done randomly.
3.5.1.2 The Control Tasks
In order incorporate in our analysis, possible factors that may affect individuals’ considerations and decisions, we tried to find what are the most relevant social preferences in
bargaining contexts. We decided on trying to measure our participants’ levels of fairness,
altruism, and reciprocity, as these are relevant in many bargaining contexts ((Leider et al.
2016), (Breitmoser et al. 2013), (Fehr, G¨achter, et al. 1997)).
A typical approach to quantifying the players’ relevant social preferences with the goal
of using them to shed further light on decision-making, is by incorporating “control tasks”
in the experiment and for this reason we included an ultimatum game to measure fairness, a
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dictator game to measure altruism, and an investment (trust) game to measure reciprocity.
After completing the rounds of the main task, the players then play a sequence of some
variations of the control tasks mentioned above.At each treatment session, the order in which
the control tasks will be displayed is random. The players do two rounds of each task, in
order to obtain observations on their behavior in both roles of each task. At each task, and
at each round within a task they are randomly and anonymously re-matched with a new
partner.
To measure the players’ fairness we employ a variation of the ultimatum game ((Forsythe
et al. 1994)) that contains the following steps: The player that is assigned the role of Player
1 makes an offer (any whole number between 0 and 100) to Player 2 that he/she would
transfer T out of the 100 tokens to them. Player 2, without having observed Player 1’s offer,
indicates what is the minimum amount of T such that he or she would accept Player 1’s
offer. There are two possible outcomes: If the amount T offered by Player 1 is greater than
or equal to the minimum amount that Player 2 would accept, then the offer is accepted,
Player 1 earns (100 - T) Tokens, and Player 2 earns T Tokens. If the amount T offered by
Player 1 is less than the minimum amount that Player 2 would accept, then the offer is not
accepted, and both players earn 0 Tokens.
To measure the players’ altruism, we employ a variation of the dictator game ((Forsythe
et al. 1994)) as follows: at a given round, the player who is assigned the role of the dictator
(displayed as ”Player 1” to the players, in order to avoid -bias-) is allocated 100 tokens and is
asked to choose any whole number between 0 and 100 tokens that she would like to transfer
to Player 2. If Player 1 decides to transfer T tokens to Player 2, then Player 1 receives (100
– T) tokens, and Player 2 receives T tokens. Player 2 does not make any decisions in this
task.
Finally, we include a version of the investment game ((Berg et al. 1995)), in order to
measure our the participants’ reciprocity. This game proceeds as follows: Player 1 is asked
to transfer any whole number T out of 100 tokens to Player 2. The amount T transferred
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is tripled to (3T) tokens and Player 2 receives this tripled amount. Player 2 then decides
how many out of the tripled amount, (3T) tokens, to transfer back to Player 1. We use X
to represent Player 2’s decision. Player 2 can choose to transfer any whole number between
0 and (3T) tokens back to Player 1. The two players earn the following payoffs in this task:
Player 1 earns (100 - T + X) tokens Player 2 earns (3T - X) tokens.
At the end of all rounds of all games, the players are then asked to complete a postexperimental survey which has four parts: (i) some demographic questions (ii) questions
about participants’ decisions in the main game (iii) the Ten-Item Personality Measure (TIPI
- a 10-item measure of the Big Five (or Five-Factor Model) dimensions and (iv) two sections
of the NBE scale ((Cohen, Wolf, et al. 2011)).
3.5.2 Procedures
We plan on running experimental sessions in the computer laboratories of the Marshall Behavioral Research Lab at the University of Southern California. The sessions will encompass
a 2 (Continuation Probability: High versus Low) x 2 (Stable Outcomes: One versus Two)
factorial design. In all four treatments the Main Task is to be played first, followed by a sequence of the control tasks whose order will be randomly assigned per session. We inform all
participants from the beginning that there will be 4 tasks and a small survey to be completed
during the session however we provide participants with the details of each task only after
they complete the previous one. Furthermore, participants do not observe the outcomes of
the tasks or their cumulative earnings until they complete all tasks in order to eliminate the
possibility that outcomes from previous tasks will affect their current and future decisions.
3.5.3 Analysis Plan
After conducting the experiments described in the previous section, and by having gathered
information for our four treatments of interest there are several aspects we want to include
100

in our analysis. The first one, and the main reason for creating this study in the first place,
is to compare our findings with the outcomes that our predicted as stable from TSD, and
therefore testing Hypotheses 1 and 2.
Related to this, there are several questions we would like to answer through the data
we get from the experiment, as for example: In the one stable outcome case, do we see the
participants converging to the outcomes that are predicted as stable by (TSD)? For the cases
where (TSD) predicts that two stable outcomes may emerge as stable, we would like to see
if there is a recycling structure that occurs more often in the experiment, and then more
closely investigate what this structure really means in this specific context, and under the
respective combination of parameters of the model.
Subsequently, we interested in studying the effect of the continuation probability, which
corresponds to participants’ level of farsightedness, on the emergence of stable outcomes,
as well as on the cooperation of players. Note that we have only four possible recycling
structures in the setting we are considering, and therefore, the number of possible actions
and counter-actions to consider is not very big so we suspect that the players do not need
a very high level of farsightedness in order to chose the farsighted stable outcome. Still,
understanding if there is a significant difference between the different continuation probability
treatments can help us understand better the theoretical model.
Next, we believe that the fact that our experiment design allows participants to interact
via a chat interface, will provide us with a very rich content to analyze. We are primarily
interested in studying how does the quality and quantity of communication affect: the occurrence of stable outcomes, but also cooperation (in the sense that ”joint” recycling structures
will be picked instead of independent ones). Furthermore, the process of finding the most
preferred outcome in the LCS, is assumed to function as a “mental exercise” for the involved
agents. It would be interesting to see therefore if farsightedness in this sense is incorporated
within each given round, and if it is reflected in the chat data. We would like to investigate,
for example, what are the other recycling structures the players are considering at a given
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round before making a decision, and how much the other player’s preferences and attitude
affect the final decision. In order to be able to analyze the communication data between the
players we aim on following the example of many similarly structured studies, which follow
the methodology of content analysis (see (Krippendorff 2018) for a detailed presentation of
the methodology), and essentially use two independent researchers to code and interpret the
communication data. We can then check for consistency among the coders’ decisions on the
tasks they will have (for example inferring which outcome was selected based on the conversation) and use only the identified codes for which agreement is strong. This is one possible
way but especially for the analysis of the bargaining data, we really do think that we first
need to conduct some experiments first in order to observe some samples, and understand
which approach would be the most appropriate for the analysis in our case.
Lastly, as mentioned earlier in the chapter, we are interested in understanding the role
of various social preferences in the context of farsighted coalition formation and how it is
related to cases where players freely interact via messages and need to decide on whether to
jointly choose a collaborative option or not. We want to study therefore how the convergence
to an outcome is related to behavioral factors as these are captured from the Control Tasks.
A typical approach would be to group participants according to their behavior as this is
reflected from their actions in the Control Tasks. For example, as (Kraft et al. 2018) do,
who after incorporating a variation of the dictator game in their experiment, classify the
individuals who transferred up to 20 tokens as low prosocial and those who transferred 40 or
more tokens are as high prosocial and then proceed on analyzing the effects of interest under
each classification. Since we plan on employing three such control tasks, we believe that such
an analysis, based on the classifications from each of the tasks, will provide a good summary
of the factors that either aid or impair communication and ultimately, the occurrence of a
stable outcome.
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3.6 Concluding Remarks
Given that Farsighted Stability has little experimental study we believe that conducting a
behavioral experiment to test this theory will allow us to: see under which conditions (model
parameters, and levels of farsightedness) outcomes predicted as stable can occur from players’
interaction in the laboratory. Furthermore, by recognizing that farsightedness is not the only
factor that drives individuals’ and firms’ actions, we think that the investigation of other
behavioral traits under this setting, will help us understand the decision process better and
maybe raise questions on whether the original model may be reformulated to account for
these factors.
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110

Chapter A
Appendices for Chapter 1
This appendix is organized as follows. First, in Section A.1, we present the proofs of the
optimal solutions at c = 0. Then, leveraging these results, in Section A.2, we prove structural
results, and, in Section A.3, we prove the results for the optimal solutions at c > 0. Finally,
in Section A.4, we state and prove two supporting, technical lemmas.
A.1 Proofs of Optimal Solutions when c = 0
A.1.1 Proof of Proposition 3
Proof. Proof. In this proposition, we look at the solutions to the 2 × 1 problem when the
servers’ opportunity cost c = 0. This means that the server payoff non-negativity constraints
is not binding. Therefore when c = 0 we want to find the optimal solution to the following
111

equivalent problem:
max
p1,p2d1,d2
p1 · M1(2 − p1 − α1d1) + p2 · M2(1 − p2 − α2d2) (A.1)
s.t p1 + α1d1 ≤ 2
−p1 − α1d1 ≤ −1
p2 + α2d2 ≤ 1
−p2 − α2d2 ≤ 0
p1 − p2 + α1d1 − α1d2 ≤ 0
−p1 + p2 − α2d1 + α2d2 ≤ 0
d1 − d2 ≤ 0
p1, p2, d1, d2 ≥ 0 .
The Lagrangian for the standard form equivalent of this problem is
L2×1 = − p1 · M1(2 − p1 − α1d1) − p2 · M2(1 − p2 − α2d2)
+ λ1(p1 − p2 + α1d1 − α1d2) + λ2(−p1 + p2 − α2d1 + α2d2)
+ λ3(p1 + α1d1 − 2) + λ4(−p1 − α1d1 + 1) + λ5(p2 + α2d2 − 1) + λ6(−p2 − α2d2)
+ λ7(d1 − d2) − η1p1 − η2d1 − η3p2 − η4d2 ,
where η1, η2, η3, η4 are non-negative real numbers. Taking the Karush–Kuhn–Tucker (KKT)
conditions, e.g., Boyd et al., 2004, Ch. 5.5.3, we have:
∂L2×1
∂p1
= 2M1p1 − 2M1 + α1M1d1 + λ1 − λ2 + λ3 − λ4 − η1 = 0 , (A.2)
∂L2×1
∂d1
= α1M1p1 + α1λ1 − α2λ2 + α1λ3 − α1λ4 + λ7 − η2 = 0 , (A.3)
112

∂L2×1
∂p2
= 2M2p2 − M2 + α2M2d2 − λ1 + λ3 + λ5 − λ6 − η3 = 0 , (A.4)
∂L2×1
∂d2
= α2M2p2 − α1λ1 + α2λ2 + α2λ5 − α2λ6 − λ7 − η4 = 0 . (A.5)
By Lemma 1, we will use that the constraints p1 = 1 − α1d1 and p1 + α1d1 = p2 + α1d2 are
tight at the optimal solution. Furthermore, we will consider that 0 ≤ d1 < d2 to find the
optimal pair of differentiated prices and delays. From the above, and from complementary
slackness, we have that
p2 + α2d1 < p1 + α2d1 ⇒ λ2 = 0 ,
p1 + α1d1 < 2 ⇒ λ3 = 0 ,
p2 + α2d1 < 1 ⇒ λ5 = 0 ,
p2 + α2d1 > 0 ⇒ λ6 = 0 ,
d1 − d2 < 0 ⇒ λ7 = 0 ,
We will consider p1, p2, d2 > 0 meaning that η1 = η3 = η4 = 0. We will first show this
solution, and then show why any of these variables being zero leads to a solution that
cannot be optimal.
We will look at the KKT conditions. Equation (A.2) becomes:
α1M1d1 = λ1 − λ4 , (A.6)
(A.3) becomes:
α1M1 − α
2
1M1d1 + α1λ1 − α1λ4 − η2 = 0 , (A.7)
113

(A.4) becomes:
2M2p2 − M2 + α2M2d2 − λ1 = 0 , (A.8)
and (A.5) becomes:
α2M2p2 − α1λ1 = 0 . (A.9)
If we take the case that d1 = 0, then p1 = 1, and, to solve for p2 and d2 we will use that
p2 + α1d2 = 1, which comes from p2 + α1d2 = p1 + α1d1 being active. By substituting into
(A.9) that p2 = 1 − α1d2 and that λ1 = 2M2p2 − M2 + α2M2d2 (which comes from A.8) we
obtain that d2 =
1
2α1
and therefore p2 =
1
2
. If we consider the case that d1 > 0, then we will
look at the KKT conditions when p1 + α1d1 = 1 and p2 + α1d2 = 1. Equation (A.2) becomes
(A.6). Equation (A.3) becomes:
α1M1 − α
2
1M1d1 + α1λ1 − α1λ4 = 0 , (A.10)
(A.4) becomes:
(2α1 − α2)M2d2 = M2 − λ1 , (A.11)
and (A.5) becomes:
λ1 =
α2
α1
M2 − α2M2d2 . (A.12)
By substituting (A.12) into (A.11) we get that d2 =
1
2α1
and therefore p2 =
1
2
. However,
if we now substitute (A.12) into (A.6) we obtain that λ4 =
α2
4α1M2 − α1M1d1, which if we
finally substitute into (A.10) to obtain d1, we get α1M1 = 0 which is impossible, meaning
that the KKT are not satisfied in the case where d1 > 0 and so such a solution is not optimal.
114

Therefore the optimal solution of the 2 × 1 when c = 0 is:
p
∗∗
1 = 1 , d∗∗
1 = 0
p
∗∗
2 =
1
2
, d∗∗
2 =
1
2α1
We will now examine the cases where each of the variables that we consider to be non-zero,
p1, p2 and d2, becomes zero to show that the solution above is the optimal solution to the
problem.
We first consider that p1 = 0. Notice that by this we have that η1 > 0 and by Lemma
1, we immediately have that d1 =
1
α1
. Then, if we look at the KKT conditions above, (A.2)
becomes λ1 = M1 + λ4 + η1, and (A.3) becomes α1λ1 −α1λ4 −η2 = 0. Then, equation (A.4),
using also that p2 = 1 − α1d2 by Lemma 1, becomes (A.11) and (A.5) becomes (A.12).
Substituting (A.11) into (A.12) we can solve for d2 and obtain that d2 =
1
2α1
which means
that p2 =
1
2
. We see that the p2 and d2 we obtain in this case are the same as in the case
that we claim is optimal above. This means that the M2p2(1 − p2 − α2d2) portion of the
objective is the same in the two solutions. However, the M1p1(1 − p1 − α1d1) portion of the
objective here is equal to zero while in the solution we showed above it is clearly non-zero.
Therefore this solution is sub-optimal.
We now consider that p2 = 0. Notice that by this we have that η3 > 0 and by Lemma 1,
we immediately have that d2 =
1
α1
. Then, if we look at the KKT conditions above, equation
(A.2) becomes (A.6). Equation (A.3), using also that p1 = 1 − α1d1 by Lemma 1, becomes
equation (A.7). Equation (A.4) becomes
M2 −
α2
α1
M2 + λ1 + η3 = 0 , (A.13)
and (A.5) becomes
α1λ1 = 0 . (A.14)
115

The last condition (A.14) combined with (A.6) would give that α1M1d1 = −λ4 < 0 which is
impossible, since we assume that M1 > 0. Therefore such a solution cannot be optimal.
We finally consider that d2 = 0. Notice that by this we have that η4 > 0 and by Lemma
1, we immediately have that p2 = 1. Then, if we look at the KKT conditions above, equation
(A.2) becomes (A.6). Equation (A.3), using also that p1 = 1 − α1d1 by Lemma 1, becomes
equation (A.7). (A.4) becomes
M2 = λ1 , (A.15)
and (A.5) becomes
α2M2 − α1λ1 − η4 = 0 . (A.16)
If we substitute (A.15) into (A.16) we obtain that η4 = (α2 − α1)M2 < 0, since we assume
that α1 > α2, and this is impossible. Therefore such a solution cannot be optimal.
A.1.2 Proof of Proposition 4
Proof. Proof. In this proposition, we look at the solutions to the 1 × 1 problem when the
servers’ opportunity cost c = 0. This means that the server payoff non-negativity constraint
is not binding. Therefore when c = 0 we want to find the optimal solution to the following
116

equivalent problem:
max
p,d
p · M1(2 − p − α1d) + p · M2(1 − p − α2d) .
s.t p + α1d ≤ 2
−p − α1d ≤ −1
p + α2d ≤ 1
−p − α2d ≤ 0
p, d ≥ 0 .
Suppose that p + α1d = 2. Then, d needs to satisfy d ≥
1
α1−α2
in order to be feasible to the
p + α2d ≤ 1 constraint. In this case, the resulting revenue is equal to 0.
Then consider that p + α2d = 0. This is only possible when p = d = 0 in which case the
resulting revenue is also 0.
Instead consider that p + α2d = 1. Then this means that p + α2d > 0 and p + α1d > 1. If
d = 0 then p = 1 and the resulting revenue is M1. If d > 0 then d needs to satisfy d < 1
α1−α2
in order to be feasible to the p + α1d < 2 constraint. In this case, the resulting revenue
would be R1×1 = M1(α1 − α2)d.
Finally suppose that p + α1d = 1. If d = 0 then p = 1 and the resulting revenue is M1.
If d > 0 then we can use p = 1 − α1d to re-write the objective as
max
d
(1 − α1d)(M1 + M2(α1 − α2d)) ,
which by taking the First Order Condition (FOC) is maximized at d =
(α1−α2)M2−α1M1
2α1(α1−α2)M2
and
the resulting revenue is (α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
.
We argue that only p
∗ = 1, d
∗ = 0 and d
∗ =
(α1−α2)M2−α1M1
2α1(α1−α2)M2
, p
∗ = 1 − α1d
∗
can be
optimal. Comparing to the solutions that lead to a revenue equal to zero (when constraint
p + α1d = 2 or constraint p + α2d = 0 is binding) is trivial. All is left is to compare to the
117

solution that would arise when p + α2d = 1 and d > 0. Call this solution (p
†
, d†
). We take
cases. If M1 >
(α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
then R1×1(1, 0) > R1×1(
1
2 +
α1
2(α1−α2)
M1
M2
,
(α1−α2)M2−α1M1
2α1(α1−α2)M2
) and
immediately we have
R1×1(p
†
, d†
) = M1(α1 − α2)d < M1 = R1×1(1, 0)
because for feasibility < d†
(α1 − α2) < 1.
If M1 <
(α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
then R1×1(1, 0) < R1×1(
1
2 +
α1
2(α1−α2)
M1
M2
,
(α1−α2)M2−α1M1
2α1(α1−α2)M2
) but
then we have
R1×1(p
†
, d†
) = M1(α1 − α2)d < M1 < R1×1(
1
2
+
α1
2(α1 − α2)
M1
M2
,
(α1 − α2)M2 − α1M1
2α1(α1 − α2)M2
)
so again this solution is better compared to (p
†
, d†
).
Furthermore, if we want to check if there is an optimal solution with no constraint being
binding we can formulate the Lagrangian for the standard form equivalent of this problem
which is
L1×1 = − pM1(2 − p − α1d) − pM2(1 − p − α2d)
+ λ1(p + α1d − 2) + λ2(−p − α1d + 1)
+ λ3(p + α2d − 1) + λ4(−p − α2d) − η1p − η2d ,
where η1, η2, η3, η4 are non-negative real numbers. Taking the KKT conditions we have
∂L1×1
∂p = 2(M1 + M2)p + (α1M1 + α2M2)d − 2M1 − M2 + λ1 − λ2 + λ3 − λ4 − η1 = 0 ,
(A.17)
118

and
∂L1×1
∂d = (α1M1 + α2M2)p − α1(λ2 − λ1) − α2(λ4 − λ3) − η2 = 0 . (A.18)
If we solve the above with no constraint being binding we obtain the solution d = 0 and
p =
2M1+M2
2(M1+M2) which is infeasible so the only solutions are the one provided above.
Lastly, with algebraic manipulation we could simplify the condition M1 <
(α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
to have that:
If M1 <
α1−α2
α1 M2 then, the optimal solution is
d
∗ =
(α1 − α2)M2 − α1M1
2α1(α1 − α2)M2
and p
∗ =
1
2
+
α1
2(α1 − α2)
M1
M2
,
and if M1 >
α1−α2
α1 M2 then, the optimal solution is d
∗ = 0 and p
∗ = 1.
A.1.3 Proof of Corollary 1
Proof. Proof. Under the customer market conditions of the corollary, we have by Proposition
4 that the optimal solution (p
∗
, d∗
) when c = 0 satisfies d
∗ > 0. Recall our definition of the
delay function in relation to the number of servers available which is d(n) = d. This means
that d
−1
(d
∗
) = n < ∞, hence the number of servers needed to participate in the platform to
offer delay d
∗
is finite and equal to n.
A.1.4 Proof of Theorem 1
Proof. Proof. To prove this theorem about the optimal solution to the 2 × 2 problem, we
will transform the problem to an optimization problem over just the delay variables d1 and
119

d2 by consequence of Assumption 2. The 2 × 2 problem with c = 0 can be written as
max
p1,p2d1,d2
p1 · M1(2 − p1 − α1d1) + p2 · M2(1 − p2 − α2d2)
s.t 1 ≤ p1 + α1d1 ≤ 2
0 ≤ p2 + α2d2 ≤ 1
p1 + α1d1 ≤ p2 + α1d2
p2 + α2d2 ≤ p1 + α2d1
d1 ≤ d2
d
−1
(d1)p2M2(1 − p2 − α2d2) ≤ d
−1
(d2)

p1M1(2 − p1 − α1d1) + p2M2(1 − p2 − α2d2)

p1, p2, d1, d2 ≥ 0 .
By Lemma 1, we consider that p1 +α1d1 = 1 and p2 +α1d2 = 1. Equivalently, we will set
p1 = 1 − α1d1 and p2 = 1 − α1d2. Note that, if feasible, this is immediately a lower bound
on the true problem and in Lemma 1 we also show that it is optimal in the original.
This simplifies the constraints as follows. The constraint 1 ≤ p1+α1d1 ≤ 2 is immediately
true by definition. The constraint 0 ≤ p2 + α2d2 ≤ 1 becomes 0 ≤ 1 − (α1 − α2)d2 ≤ 1.
The upper bound follows from the α1 ≥ α2 assumption, and the lower bound becomes a
bound on d2. This is now: 0 ≤ d2 ≤
1
α1−α2
. Since there is also a non-negativity constraint
on p2, however, and since p2 + α1d2 = 1, this latter constraint becomes 0 ≤ d2 ≤
1
α1
. The
constraint p1 + α1d1 ≤ p2 + α1d2 is also immediately true. The constraint p2 + α2d2 ≤
p1 + α2d1 becomes 1 − (α2 − α2)d2 ≤ 1 − (α1 − α2)d1, or, more simply, (α1 − α2)d1 ≤
(α1−α2)d2, making this constraint equivalent to d1 ≤ d2. The traffic conservation constraint
d
−1
(d1)p2M2(1−p2 −α2d2) ≤ d
−1
(d2)

p1M1(2−p1 −α1d1) +p2M2(1−p2 −α2d2)

simplifies
to d
−1
(d1)(1 − α1d2)M2(α1 − α2)d2 ≤ d
−1
(d2)

(1 − α1d1)M1 + (1 − α1d2)M2(α1 − α2)d2

.
Similarly, the objective simplifies to p1 · M1(2 − p1 − α1d1) + p2 · M2(1 − p2 − α2d2) =
M1(1 − α1d1) + (1 − α1d2)M2(α1 − α2)d2.
120

We are left with a problem that is entirely determined by the delays, meaning that the
decision variables are now only d1 and d2. For simplicity, let us also define ρ = M1/M2 and
divide the objective and traffic-conservation constraint so that the problem can be expressed
as
max
d1,d2
ρ
α1 − α2
(1 − α1d1) + (1 − α1d2)d2
s.t. 0 ≤ d1 ≤ d2 ≤
1
α1
1
d
−1
(d2)
(1 − α1d2)d2 ≤
1
d
−1
(d1)

ρ
α1 − α2
(1 − α1d1) + (1 − α1d2)d2

.
To simplify notation, let us create notation for the two sub-revenues. Let R1(d1) =
(1 − α1d1)ρ/(α1 − α2) and let R2(d2) = (1 − α1d2)d2. Note that R′
1
(d1) = −α1ρ/(α1 − α2)
and R′
2
(d2) = 1 − 2α1d2. Furthermore, since we have that d(n) = n
−γ
for some γ > 0 we
have d
−1
(d) = d
−1/γ. In addition, we can omit the 0 ≤ d1 ≤ d2 ≤
1
α1
constraint because in
Lemma 2 we show that at an optimal solution this constraint will be satisfied. This helps
us notice that the optimization problem has a pretty special form:
max
d1,d2
R1(d1) + R2(d2)
s.t. d
1/γ
2 R2(d2) ≤ d
1/γ
1
(R1(d1) + R2(d2)) .
We can quickly see that the 2 × 1 solution won’t be feasible here: d1 = 0 and d2 =
1
2α1
will not satisfy the constraint. So, the constraint must be tight, leading us to the following
equivalent problem:
max
d1,d2
R1(d1) + R2(d2)
s.t. d
1/γ
1
(R1(d1) + R2(d2)) − d
1/γ
2 R2(d2) = 0 .
121

The Lagrangian for the standard form equivalent of the above problem is
L2×2 = −R1(d1) − R2(d2) + λ

d
1/γ
1
(R1(d1) + R2(d2)) − d
1/γ
2 R2(d2)

.
By taking the KKT conditions we have
∂L2×2
∂d1
= −R
′
1
(d1) + λ

1
γ
d
1−γ
γ
1 R1(d1) + d
1
γ
1 R
′
1
(d1)

= 0 ,
which leads to
λ =
R′
1
(d1)
1
γ
d
1−γ
γ
1 R1(d1) + d
1
γ
1 R′
1
(d1)
. (A.19)
For d2, this is
∂L2×2
∂d2
= −R
′
2
(d2) − λ

1
γ
d
1−γ
γ
2 R2(d2) + d
1
γ
2 R
′
2
(d2)

= 0 ,
which leads to
λ =
−R′
2
(d2)
1
γ
d
1−γ
γ
2 R2(d2) + d
1
γ
2 R′
2
(d2)
. (A.20)
By now setting (A.19) and (A.20) equal to each other and with some algebraic manipulation,
we obtain the following equation in terms of d1 and d2:
α1(γ + 1)d
1
γ
1 − d
1−γ
γ
1 − α
2
1
(γ + 1)d
1+γ
γ
2 + α1(γ + 1)d
1
γ
2 + 2α1d
1−γ
γ
1 d2 − 2α
2
1
(γ + 1)d
1
γ
1 d2 = 0 .
(A.21)
Then, the other equation we know d1 and d2 need to satisfy is the only constraint of the
122

transformed problem
d
1/γ
1
(R1(d1) + R2(d2)) − d
1/γ
2 R2(d2) = 0 ,
which, by substituting the R1(d1) and R2(d2) expressions we find
α1M1
M2(α1 − α2)
d
1+γ
γ
1 −
M1
M2(α1 − α2)
d
1
γ
1 − α1d
1+2γ
γ
2 + d
1+γ
γ
2 − d
1
γ
1 d2 + α1d
1
γ
1 d
2
2 = 0 . (A.22)
Therefore the optimal ˆd1 and ˆd2 of the 2 × 2 at c = 0 can be found from the system of
equations that comes from (A.21) and (A.22), and the corresponding ˆp1 and ˆp2 will be given,
as shown above, by ˆp1 = 1 − α1
ˆd1 and ˆp2 = 1 − α1
ˆd2.
A.1.5 Proof of Corollary 2
Proof. Proof. We have by Theorem 1 that the optimal solution (ˆpi
,
ˆdi) when c = 0 satisfies
ˆd1 > 0. By analogous arguments to those in the proof of Corollary 1 we have that d
−1
(
ˆd1) =
n < ∞, hence the number of servers needed to participate in the platform to offer delay ˆd1
is finite and equal to n.
A.2 Proofs of Structural Results
A.2.1 Proof of Proposition 1
Proof. Proof. For the first part of this statement we consider Assumption 1. In order to show
this we compare the two firm optimization problems, 1 × 1 given by (1.10), and 2 × 1 given
by (1.8). Fix some c ≥ 0 and consider the corresponding optimal solution (p
∗
, d∗
) to 1×1 for
this c which yields an optimal revenue R∗
1×1
. This solution is always feasible in 2×1 because
it satisfies all the constraints when p1 = p2 = p
∗
, d1 = d2 = d
∗
. The Individual Rationality,
non-negativity constraints, and server payoff non-negativity constraints hold trivially and it
123

is easy to verify that the Incentive Compatibility constraints hold as well. Therefore the
optimal solution to 1 × 1 is feasible but not necessarily optimal in 2 × 1 which makes R∗
1×1
a lower bound for R∗∗∗
2×1
, the optimal revenue in 2 × 1.
For the second part of this statement we consider Assumptions 1 and 2, and suppose that
c = 0. We will show that the firm revenue from price differentiation is strictly better than
the firm revenue from a single price and delay.
From Proposition 3 we have that the optimal revenue R∗∗∗
2×1
, at c = 0, would be equal to
R
∗∗∗
2×1 = M1 +
M2
4
α1 − α2
α1
.
Then, from Proposition 4 we have that the optimal revenue R∗∗∗
2×1
, at c = 0, would be equal
to
R
∗
1×1 = max( M1 ,
(α1M1 + (α1 − α2)M2)
2
4α1(α1 − α2)M2
) .
Suppose that M1 >
α1−α2
α1 M2. In this case the optimal 1 × 1 revenue is R∗
1×1 = M1, so it is
immediately clear that R∗∗∗
2×1 > R∗
1×1
.
Suppose that M1 <
α1−α2
α1 M2. In this case, the optimal 1 × 1 revenue is R∗
1×1 =
(α1M1+(α1−α2)M2)
2
4α1(α1−α2)M2
. Therefore we need to show that
R
∗∗∗
2×1 = M1 +
M2
4
α1 − α2
α1
>
(α1M1 + (α1 − α2)M2)
2
4α1(α1 − α2)M2
= R
∗
1×1
.
With algebraic manipulation, the inequality above reduces to
α1M1(2(α1 − α2)M2 − α1M1) > 0
which is true by consequence of this case’s assumption.
124

A.2.2 Proof of Proposition 2
Proof. Proof. Fix customer characteristics α1, α2, M1, and M2 and servers’ opportunity cost
c, and consider the 1 × 1, 2 × 1, and 2 × 2 optimization problems. Any solution to the 2 × 2
problem is feasible in the 2 × 1 problem, and these problems have the same objective, with
2 × 2 having additional constraints. Therefore the optimal solution to the 2 × 2 problem is
feasible in the 2 × 1 problem but not necessarily optimal. Hence, the optimal value to the
2×2 problem is a lower bound for the optimal value to the 2×1 and we obtain R∗∗
2×2 ≤ R∗∗∗
2×1
.
Now, any solution to the 1 × 1 problem is feasible in the 2 × 2 problem. This can be
seen by setting p1 = p2 = p, d1 = d2 = d and verifying that such a solution satisfies all the
constraints of the 2 × 2 problem as long as it is feasible in the 1 × 1.
Similarly to the reasoning in the proof of Proposition 1, all the constraints of 2 × 2 that
are common with 2×1 are satisfied by a feasible solution to 1×1. The ony added constraint,
which is the traffic conservation constraint d
1
γ
2 p2M2(1−p2−α2d2) ≤ d
1
γ
1

p1M1(2−p1−α1d1)+
p2M2(1−p2−α2d2)

, becomes d
1
γ pM2(1−p−α2d) ≤ d
1
γ

pM1(2−p−α1d)+pM2(1−p−α2d)

which, since d, p ≥ 0, reduces to M2(1−p−α2d) ≤ M1(2−p−α1d) +M2(1−p−α2d) which
reduces to M1(2 − p − α1d) ≥ 0 which holds since p − α1d ≤ 2 from being feasible in 1 × 1.
Therefore the optimal solution to 1×1 is feasible in the 2×2 problem but not necessarily
optimal. Hence, the optimal value to the 1 × 1 is a lower bound for the optimal value to the
2 × 1 and we obtain R∗
1×1 ≤ R∗∗
2×2
.
A.2.3 Proof of Proposition 5
Proof. Proof. We have by Proposition 3 that the optimal solution to 2 × 1 satisfies d
∗∗∗
1 = 0.
Since by (1.7), π2×1 is given by
π2×1 =
p1µ1 + p2µ2 − c · d
− 1
γ
1
d
− 1
γ
1
,
125

we immediately see that at c = 0, π2×1 = 0. It is immediate that for any c, the server payoff
non-negativity constraint is binding.
A.2.4 Proof of Proposition 6
Proof. Proof. First consider the case when M1 <
α1
α1−α2M2. We have by Proposition 3 that
in this case the optimal solution to 1 × 1 satisfies d
∗
1 > 0. In this case, by (1.9), π1×1 is given
by
π1×1 =
p(µ1 + µ2) − c · d
− 1
γ
d
− 1
γ
,
we clearly see that π1×1 will be positive when evaluated at that solution. Consider the server
payoff as a function of c, π1×1(c) , and fixed at the optimal solution of 1 × 1 problem when
c = 0, (p
∗
, d∗
) . Since d
∗ > 0 we have that the denominator d
∗− 1
γ > 0 and so we will ignore
the denominator as it won’t affect the function’s sign. By taking the derivative with respect
to c, we find
∂π2×2(c)
∂c = −d
∗− 1
γ < 0 ,
and so π2×2(c) is strictly decreasing in c for c > 0.
We earlier argued that for c = 0 the function is positive. By taking the limit of π1×1(c)
as c → ∞ we see that it is −∞ therefore by the Intermediate Value Theorem there exists a
cπ1×1 where the function crosses 0, which we call ¯π1×1. The uniqueness follows from π1×1(c)
being a strictly decreasing function in c. Moreover, since the function is strictly decreasing,
then for any c > cπ1×1
, the function would be negative. This means that the optimal solution
needs to be such that it makes the constraint feasible, hence for c > cπ1×1
the server payoff
non-negativity constraint is binding.
If we now consider the case when M1 >
α1
α1−α2M2, we have that under these conditions the
126

optimal solution to 1 × 1 satisfies d
∗
1 = 0. Since, by (1.9), π1×1 is given by
π1×1 =
p(µ1 + µ2) − c · d
− 1
γ
d
− 1
γ
,
and we clearly see that at c = 0 we have π1×1 = 0. It is immediate then that for any c, the
server payoff non-negativity constraint is binding.
A.2.5 Proof of Corollary 3
Proof. Proof. By Proposition 6, we have that for any c < cπ1×1
the server payoff is positive.
Therefore the constraint is not binding for the optimization problem and for any c < cπ1×1
the problem is equivalent to the 1 × 1 problem when c = 0, and therefore we would have the
same optimal solution (p
∗
, d∗
) and thus a constant revenue R∗
1×1 = R1×1(p
∗
, d∗
).
A.2.6 Proof of Proposition 7
Proof. Proof. We have by Theorem 1 that, in this case, the optimal solution to the 2 × 2
satisfies ˆd1 > 0. Hence, by definition of the platform’s optimization problem, π2×2 is given
by
π2×2 =
p1µ1 + p2µ2 − c · d
− 1
γ
1
d
− 1
γ
1
,
we clearly see that π2×2 will be positive when evaluated at that solution. Consider the server
payoff as a function of c, π2×2(c) , and fixed at the optimal solution of 2 × 2 problem when
c = 0, (ˆpi
,
ˆdi) . Since ˆd1 > 0 we have that the denominator ˆd
− 1
γ
1 > 0 and so we will ignore
the denominator, as it won’t affect the function’s sign. By taking the derivative with respect
to c:
∂π2×2(c)
∂c = −

ˆd1
− 1
γ
< 0 ,
127

and so π2×2(c) is strictly decreasing in c for c > 0.
We earlier argued that for c = 0 the function is positive. By taking the limit of π2×2(c) as
c → ∞ we see that it is −∞ therefore by the Intermediate Value Theorem there exists a cπ2×2
where the function crosses 0. The uniqueness follows from π2×2(c) being a strictly decreasing
function in c. Moreover, since the function is strictly decreasing then for any c > cπ2×2
the
function would be negative. This means that the optimal solution needs to be such that it
makes the constraint feasible, hence for c > cπ2×2
the server payoff non-negativity constraint
is binding.
A.2.7 Proof of Corollary 4
Proof. Proof. By Proposition 7 we have that for any c < cπ2×2
, the server payoff is positive
when evaluated at (ˆpi
,
ˆdi). Therefore the constraint is not binding for the optimization
problem and for any c < cπ2×2
, we would have the same optimal solution (ˆpi
,
ˆdi) and thus a
constant revenue R∗∗
2×2 = R2×2(ˆpi
,
ˆdi). The positive gap in R∗∗∗
2×1
and R∗
1×1
comes immediately
from the second statement of Proposition 1.
A.2.8 Proof of Proposition 10
Proof. Proof. For this proof, we will compare the 2 × 1 and 2 × 2 problems. Recall that the
only difference between these two problems is the f1 ≤ 1 constraint in the 2 × 2. We will
look at the f1 ≤ 1 constraint evaluated at the optimal solution of 2 × 1, and as a function
of c. We will show that there exists a unique c∆ such that the constraint switches from
not holding to holding, resulting in a unique c∆ so that the constraint switches from being
binding and not being binding in 2 × 2.
Consider some integer γ and some c > 0, and let (p
∗∗∗
i
, d∗∗∗
i
) denote the optimal solution
to 2×1. At this solution, the server payoff non-negativity constraint, π2×2(c) ≥ 0, is binding,
128

therefore
R
∗∗∗
2×1
(c) = c · d
∗∗∗
1
− 1
γ (c) . (A.23)
Now let us look at the f1 ≤ 1 constraint evaluated at this optimal solution to 2 × 1. Here,
d
∗∗∗
2
1
γ (c) · M2p
∗∗∗
2
(c)(1 − p
∗∗∗
2
(c) − α2d
∗∗∗
2
(c)) − d
∗∗∗
1
1
γ (c)R
∗∗∗
2×1
(c) ≤ 0 ,
where for the R∗∗∗
2×1
(c), we will use the expression from (A.23) in order to express this in
terms of c:
d
∗∗∗
2
1
γ (c) · M2p
∗∗∗
2
(c)(1 − p
∗∗∗
2
(c) − α2d
∗∗∗
2
(c)) − c ≤ 0 .
Now, note that since we are looking at the 2×1 solution, we have that for any c, the optimal
solution satisfies d
∗∗∗
2
(c) = 1
2α1
, and p
∗∗∗
2
(c) = 1
2
by Remark 1. Therefore the above inequality
becomes

1
2α1
1
γ M2
4
(α1 − α2)
α1
− c ≤ 0 .
The LHS of the constraint is a decreasing linear function in c that crosses 0 when c = c∆ =

1
2α1
1
γ M2
4
(α1−α2)
α1
. Therefore, if c < c∆, the f1 ≤ 1 constraint does not hold for the 2 × 1
solution meaning that the constraint cannot be ignored in the 2 × 2 problem. If, however,
c ≥ c∆, then f1 ≤ 1 is satisfied for the optimal solution to 2 × 1 and therefore the constraint
is redundant for the 2 × 2 problem, making it equivalent to 2 × 1 and thus with the same
optimal revenue.
A.2.9 Proof of Corollary 5
Proof. Proof. This follows immediately from the expression of c∆ which we obtain in Propo129

sition 10.
A.3 Proofs of Optimal Solutions when c > 0
A.3.1 Proof of Proposition 8
Proof. Proof. In this proposition we look at the solutions to 2×1 problem when the servers’
opportunity cost c > 0. This means that the server payoff non-negativity constraints needs
to be taken into account so we want to find the optimal solution to the problem given by
(1.8) where now the server payoff non-negativity constraint pM1(2 − p − α1d) + pM2(1 − p −
α2d) − c · d
− 1
γ ≥ 0 cannot be ignored. The Lagrangian for the standard form equivalent of
this problem is
L2×1 = − p1 · M1(2 − p1 − α1d1) − p2 · M2(1 − p2 − α2d2)
+ λ1(p1 − p2 + α1d1 − α1d2) + λ2(−p1 + p2 − α2d1 + α2d2)
+ λ3(p1 + α1d1 − 2) + λ4(−p1 − α1d1 + 1) + λ5(p2 + α2d2 − 1) + λ6(−p2 − α2d2)
+ λ7(d1 − d2) + λ8(−p1 · M1(2 − p1 − α1d1) − p2 · M2(1 − p2 − α2d2) + c · d
−1
γ
1
)
− η1p1 − η2d1 − η3p2 − η4d2 ,
where η1, η2, η3, η4 are non-negative real numbers. Taking now the KKT conditions we have:
∂L2×1
∂p1
= (2M1p1 − 2M1 + α1M1d1)(1 + λ8) + λ1 − λ2 + λ3 − λ4 − η1 = 0 , (A.24)
∂L2×1
∂d1
= α1M1p1 + α1λ1 − α2λ2 + α1λ3 − α1λ4 + λ7 − η2 + λ8(α1M1p1 −
c
γ
d
−(γ+1)
γ
1
) = 0 ,
(A.25)
130

∂L2×1
∂p2
= (2M2p2 − M2 + α2M2d2)(1 + λ8) − λ1 + λ3 + λ5 − λ6 − η3 = 0 , (A.26)
∂L2×1
∂d2
= α2M2p2(1 + λ8) − α1λ1 + α2λ2 + α2λ5 − α2λ6 − λ7 − η4 = 0 . (A.27)
By Lemma 1, we will use that the constraints p1 = 1 − α1d1 and p1 + α1d1 = p2 + α1d2
are tight at the optimal solution. Furthermore, we will consider that d1 < d2 to find the
optimal pair of differentiated prices and delays. From this we have that
p2 + α2d1 < p1 + α2d1 ⇒ λ2 = 0 ,
p1 + α1d1 < 2 ⇒ λ3 = 0 ,
p2 + α2d1 < 1 ⇒ λ5 = 0 ,
p2 + α2d1 > 0 ⇒ λ6 = 0 ,
d1 − d2 < 0 ⇒ λ7 = 0 .
Because of the server payoff non-negativity constraint, in this case, unlike the 2 × 1 at
c = 0 case, we need to have d1 > 0. Therefore we consider p1, p2, d1, d2 > 0, meaning that
η1 = η2 = η3 = η4 = 0. We will first show this solution, and then show why any of these
variables being zero leads to a solution that cannot be optimal.
Using that p1 + α1d1 = p2 + α1d2 = 1 we now look at the KKT: (A.24) becomes
(1 + λ8)α1M1d1 = λ1 − λ4 , (A.28)
(A.25) becomes
α1M1(1 − α1d1) + α1λ1 − α1λ4 + λ8

α1M1(1 − α1d1) −
c
γ
d
−(γ+1)
γ
1

= 0 , (A.29)
131

(A.26) becomes
(1 + λ8)M2(1 − (2α1 − α2)d2) = λ1 , (A.30)
and (A.27) becomes
(1 + λ8)M2
α2
α1
(1 − α1d2) = λ1 . (A.31)
Now, by setting (A.30) and (A.31) equal and solve for d2 we obtain d2 =
1
2α1
and so p2 =
1
2
.
Having the p2 and d2 we found and that p1 = 1 − α1d1 we can now use the binding payoff
constraint to solve for d1. If we substitute these for p1, p2, d2 the constraint becomes
(1 − α1d1)M1 +
1
4
M2
(α1 − α2)
α1
− cd
− 1
γ
1 = 0 ,
which by rearranging becomes
α1M1d
γ+1
γ
1 −

M2
4
α1 − α2
α1
+ M1

d
1
γ
1 + c = 0 . (A.32)
which by solving will give us the optimal d1 and then p1 = 1 − α1d1.
We can get the maximum c such that the analysis above holds by taking (A.32) and
optimizing it w.r.t d1. Set d1 = x, (A.32) is written as
α1M1x
γ+1
γ −

M2
4
α1 − α2
α1
+ M1

x
1
γ + c = 0 .
Taking the FOC and solving for x, we obtain that
x
∗ =

(α1 − α2)M2 + 4α1M1
4α
2
1M1(γ + 1) 1
γ
,
132

and then by substituting into (A.32) we get that
c¯2×1 =

(α1 − α2)M2 + 4α1M1
4α
2
1M1(γ + 1) 1
γ2

(α1 − α2)M2 + 4α1M1
4α1
− α1M1

(α1 − α2)M2 + 4α1M1
4α
2
1M1(γ + 1) γ+1!
.
We will now examine the cases where each of the variables that we consider to be nonzero, p1, p2, d2, becomes zero to show that the solution above is the optimal solution to the
problem.
We first consider that p1 = 0. Notice that by this we have that η1 > 0 and by Lemma 1,
we immediately have that d1 =
1
α1
. Then, if we look at the KKT conditions above, (A.24)
becomes
M1(1 + λ8) − λ1 + λ4 + η1 = 0 , (A.33)
(A.25) becomes
α1λ1 − α1λ4 − λ8
c
γ
d
−(γ+1)
γ
1 = 0 , (A.34)
(A.26) becomes
(1 + λ8)(1 − (2α1 − α2)d2)M2 = λ1 , (A.35)
and (A.27) becomes
(1 + λ8)
α2
α1
(1 − α1d2)M2 = λ1 . (A.36)
By setting (A.35) and (A.36) equal we get that d2 =
1
2α1
and p2 =
1
2
. We see that the p2, d2
we obtain in this case are the same as in the case that we claim is optimal above. This
means that the M2p2(1−p2 −α2d2) portion of the objective is the same in the two solutions.
However, the M1p1(1 − p1 − α1d1) portion of the objective here is equal to zero while in the
133

solution we showed above it is clearly non-zero. Therefore this solution is sub-optimal.
We now consider that p2 = 0. Notice that by this we have that η3 > 0 and by Lemma 1,
we immediately have that d2 =
1
α1
. Then, if we look at the KKT conditions above, (A.24)
becomes (A.28), (A.25) becomes (A.29), (A.26) becomes
(1 + λ8)(α2 − α1)M2 = α1(λ1 + η3) , (A.37)
and (A.27) becomes
α1λ1 = 0 . (A.38)
From (A.38) we would get that λ1 = 0, and then from (A.37), we have that that α1η3 =
(1 + λ8)(α2 − α1)M2 < 0, since α1 > α2, which is impossible. Therefore such a solution
cannot be optimal.
We finally consider that d2 = 0. Notice that by this we have that η3 > 0, and, by
Lemma 1, we immediately have that p2 = 1. Then, if we look at the KKT conditions above,
(A.24) becomes (A.28), (A.25) becomes (A.29), (A.26) becomes λ1 = (1 + λ8)M2 and (A.27)
becomes equal to η4 = (α2 − α1)(1 + λ8)M2. From the last equation we get that η4 < 0 since
α1 > α2, which is impossible. Therefore such a solution cannot be optimal.
Remark 1. The optimal solution to 2 × 1 satisfies p2 =
1
2
and d2 =
1
2α1
for any c ≥ 0.
A.3.2 Proof of Proposition 9
Proof. Proof. In this proposition, we look at the solutions to the 1 × 1 problem when
the servers’ opportunity cost is cπ1×1 < c < c¯1×1. This means that the server payoff nonnegativity constraints needs to be taken into account so we want to find the optimal solution
134

to the following problem:
max
p,d
p · M1(2 − p − α1d) + p · M2(1 − p − α2d) .
s.t 1 ≤ p + α1d ≤ 2
0 ≤ p + α2d ≤ 1
pM1(2 − p − α1d) + pM2(1 − p − α2d) − c · d
− 1
γ ≥ 0
p, d ≥ 0 .
The Lagrangian for the standard form equivalent of this problem is
L1×1 = − pM1(2 − p − α1d) − pM2(1 − p − α2d)
+ λ1(p + α1d − 2) + λ2(−p − α1d + 1)
+ λ3(p + α2d − 1) + λ4(−p − α2d)
+ λ5(−pM1(2 − p − α1d) − pM2(1 − p − α2d) + cd− 1
γ
− η1p − η2d ,
where η1, η2 are non-negative real numbers. Taking the KKT conditions we have
∂L1×1
∂p = (2(M1 + M2)p + (α1M1 + α2M2)d − 2M1 − M2)(1 + λ5) + λ1 − λ2 + λ3 − λ4 − η1 = 0 ,
(A.39)
and
∂L1×1
∂d = (α1M1 + α2M2)(1 + λ5)p − α1(λ2 − λ1) − α2(λ4 − λ3) − η2 − λ5
c
γ
d
−
1+γ
γ = 0 .
(A.40)
Since, this is the case where c > cπ1×1
, by Proposition 6, the server payoff constraint is
binding. Therefore, to solve for the optimal solution we take p = 1−α1d and solve the server
135

payoff non-negativity constraint for d in the resulting polynomial which is
−α1(α1 − α2)M2d
2γ+1
γ + (M2(α1 − α2) − α1M1)d
γ+1
γ + M1d
1
γ − c = 0 . (A.41)
We can get the maximum c such that the analysis above holds by taking (A.41) and optimizing it with respect to d. If we set d = x, then (A.41) is written as
−α1(α1 − α2)M2x
2γ+1
γ + (M2(α1 − α2) − α1M1)x
γ+1
γ + M1x
1
γ − c = 0 .
Taking the FOC and solving for x, we obtain the following polynomial
−(2γ + 1) − α1(α1 − α2)M2x
2γ + (γ + 1)(M2(α1 − α2) − α1M1)x
γ + M1 = 0 .
By by setting x
γ = z, and calling A = −α1(α1 − α2)M2, B = M2(α1 − α2) − α1M1, and
C = M1, the resulting quadratic we want to optimize is
(2γ + 1)Az2 + (γ + 1)Bz + C = 0 . (A.42)
Optimizing (A.42) gives us two possible solutions
z =
−(γ + 1)B ±
p
(γ + 1)2B2 − 4(2γ + 1)AC
2(2γ + 1)A
, (A.43)
where we keep the one that gives the highest value of the quadratic and is non-negative, call
it z
∗
. The resulting x is then given by x
∗ = (z
∗
)
1
γ . By then substituting the resulting x
∗
into
(A.41), we find the corresponding ¯c1×1:
c¯1×1 = −α1(α1 − α2)M2x
∗
2γ+1
γ + (M2(α1 − α2) − α1M1)x
∗
γ+1
γ + M1x
∗
1
γ .
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A.4 Supporting Technical Results
Lemma 1. The optimal solution to the 2 × 2 and 2 × 1 problems satisfies p1 + α1d1 = 1 and
p1 + α1d1 = p2 + α1d2.
Proof. Proof. We will consider the 2 × 2 problem. The proof of the 2 × 1 case is analogous
since the two problems have the same objective function. We will compare two feasible solutions of the problem. We call the first one (ˆpi
,
ˆdi) which satisfies the relevant constraints with
equality and we call the second one (¯pi
,
¯di), i = 1, 2 which does not satisfy the constraints
with equality. We will do this separately for each of the constraints above.
First, we look at the constraint p1 + α1d1 = 1. Take some feasible solution to the problem (ˆp1,
ˆd1, pˆ2,
ˆd2), that satisfies ˆp1 + α1
ˆd1 = 1. Now consider a solution (¯p1,
¯d1, p¯2,
¯d2) =
(ˆp1,
¯d1, pˆ2,
ˆd2) with ¯d1 > ˆd1 so that ˆp1 + α1
¯d1 > 1 is still feasible. We will show that the
(ˆp1,
ˆd1, pˆ2,
ˆd2) yields a larger objective. We look at the revenues from each solution that
under the above assumptions become
R2×2(ˆpi
,
ˆdi) = ˆp1M1(2 − pˆ1 − α1
ˆd1) + ˆp2M2(1 − pˆ2 − α2
ˆd2) = ˆp1M1 + ˆp2M2(1 − pˆ2 − α2
ˆd2) ,
and
R2×2(¯pi
,
¯di) = ¯p1M1(2 − p¯1 − α1
¯d1) + ¯p2M2(1 − p¯2 − α2
¯d2)
= ˆp1M1(2 − pˆ1 − α1
¯d1) + ˆp2M2(1 − pˆ2 − α2
ˆd2) .
We have that R2×2(¯pi
,
¯di) < R2×2(ˆpi
,
ˆdi), because
pˆ1M1(2 − pˆ1 − α1
¯d1) < p¯1M1 ,
since the ˆp2M2(1 − pˆ2 − α2
ˆd2) parts are equal in both revenues. Now, this simplifies to
2 − pˆ1 − α1
¯d1 < 1 or ˆp1 + α1
¯d1 > 1 which is true by construction of the new solution,
137

therefore the claim is true.
Since we proved the above, for the second part of the proof we will use that, at an optimal
solution, p1 + α1d1 = 1 needs to be satisfied. This means that, by using p1 = 1 − α1d1 the
original problem is equivalent to:
max
p2,d1,d2
M1(1 − α1d1) + p2 · M2(1 − p2 − α2d2)
s.t 0 ≤ p2 + α2d2 ≤ 1
1 ≤ p2 + α1d2
p2 + α2d2 ≤ 1 − (α1 − α2)d1
d1 ≤ d2
d2p2M2(1 − p2 − α2d2) ≤ d1

M1(1 − α1d1) + p2M2(1 − p2 − α2d2)

p2, d1, d2 ≥ 0 .
We want to show that, in the original problem, p1 + α1d1 = p2 + α1d2 is satisfied at an
optimal solution which means that we need to show that at the equivalent problem above
an optimal solution satisfies p2 + α1d2 = 1. We will use the same strategy as the first part of
this proof, and start with some feasible solution to the problem (ˆp1,
ˆd1, pˆ2,
ˆd2) that satisfies
pˆ2 + α1
ˆd2 = 1. Now consider a solution (¯p1,
¯d1, p¯2,
¯d2) = (ˆp1,
ˆd1, pˆ2,
¯d2) for some ¯d2 > ˆd2
so that ˆp2 + α1
¯d2 > 1 is still feasible. We will show that the (ˆp1,
ˆd1, pˆ2,
ˆd2) yields a larger
objective. We look at the revenues from each solution that under the above assumptions
become
R2×2(ˆpi
,
ˆdi) = M1(1 − α1
ˆd1) + M2pˆ2(1 − pˆ2 − α2
ˆd2) ,
and
R2×2(¯pi
,
¯di) = M1(1 − α1
¯d1) + M2p¯2(1 − p¯2 − α2
¯d2) = M1(1 − α1
ˆd1) + M2pˆ2(1 − pˆ2 − α2
¯d2) .
138

We have that R2×2(¯pi
,
¯di) < R2×2(ˆpi
,
ˆdi), because
M2pˆ2(1 − pˆ2 − α2
¯d2) < M2pˆ2(1 − pˆ2 − α2
ˆd2) ,
since the M1(1 − α1
ˆd1) parts are equal in both revenues because ˆd1 = ¯d1. Now, the above
simplifies to 1 − pˆ2 − α2
¯d2 < 1 − pˆ2 − α2
ˆd2 or, ˆd2 < ¯d2, which is true by construction of the
new solution, therefore the claim is true.
Lemma 2. The optimal solution to the 2 × 2 problem when c = 0 satisfies d2 ≤
1
α1
.
Proof. Proof. We will fix a solution to the 2 × 2 problem that satisfies d2 =
1
α1
and show
that it cannot be optimal. When d2 is fixed at d2 =
1
α1
, then R2(
1
α1
) = 0. Now, if we want to
maximize the revenue R2×2(d1,
1
α1
) w.r.t d1, we get from standard calculus approaches that
d
∗
1 = 0 hence R2×2(0,
1
α1
) = ρ
α1−α2
, which means that the f1 ≤ 1 constraint holds with strict
inequality here, because the LHS is zero while the RHS is non-negative.
Suppose that a solution with d2 =
1
α1
is optimal. We have that
R2×2(d1,
1
α1
) ≤ R2×2(0,
1
α1
) = ρ
α1 − α2
R2×1(0,
1
α1
) = R
∗
1×1
,
where R∗
1×1
is the optimal objective if we were to maximize the transformed objective with
respect to only one delay which is maximized at d
∗ = 0:
max
d
(1 − α1d)
ρ
α1 − α2
+ (a − α1d)d ,
a problem which is equivalent to the original 1 × 1 problem, at c = 0, when dividing the
objective by M2(α1 − α2) since the constraint p = 1 − α1d holds for the optimal solution to
1 × 1 by Proposition 4.
This would mean that R2×2 = R∗
1×1
, which would contradict what we show in proposition
1, that under Assumptions 1 and 2, when c = 0, R2×1 > R∗
1×1
. Therefore, d2 =
1
α1
is not
optimal.
139

Furthermore for d2 >
1
α1
the conclusion above would be the same since R2 which has a
derivative of R′
2
(d2) = 1 − 2α1d1 is decreasing on d2 ∈ (
1
2α1
,∞) and we have that 1
α1
>
1
2α1
since α1 + α2 > 0. Thus for d
′
2 >
1
α1
we would get that R2×2(0, d′
2
) < 0, hence
R2×2(d1, d′
2
) ≤ R2×2(0, d′
2
) = ρ
α1 − α2
+ R2×2(0, d′
2
) <
ρ
α1 − α2
= R
∗
1×1
,
which would, again, contradict the result we mentioned above. Thus, we conclude that an
optimal solution would satisfy d2 <
1
α1
, and therefore the constraint 1
α1
is redundant in the
final version of the problem.
140

Chapter B
Appendices for Chapter 2
B.1 Data Processing & Associated Challenges
The primary difficulties encountered in processing and analyzing the data stem from its sheer
volume. The Taxi and Limousine Commission (TLC) releases annual datasets for ride-hailing
services, with each year comprising approximately 180 million rides. While this extensive
dataset is beneficial as it provides a robust foundation for analysis, enhancing the reliability
of our findings, it introduces substantial challenges at every stage of the analytical process.
These challenges span the initial data cleaning phase, the creation of basic descriptive plots,
various data operations and manipulations, and extend to the regression analysis we aim
to perform. The large dataset requires sophisticated data handling to ensure accurate and
efficient processing. In the following section, we will delve into the specific challenges encountered at each stage of our analysis and discuss the strategies we employed to address
them effectively.
In this report, we detail the analysis process based on one year of data, starting with the
year 2021. This methodology has been systematized to facilitate identical analysis for data
from subsequent years. The Taxi and Limousine Commission (TLC) supplies the annual
data in parquet format, distributed across 12 files—each representing one month’s data.
Our analysis was conducted using Python, with preliminary testing on data subsets carried
out on personal computers. However, due to the extensive size of the dataset, all major
data processing tasks were executed on full datasets and required the robust capabilities of
Google Cloud Console Virtual Machines (VMs). For data storage, we utilized Google Cloud
Storage Buckets. This setup was essential to handle the large-scale data efficiently and to
141

perform the comprehensive analysis required for this study.
In our data cleaning process, and to facilitate the computation of basic plots, we encountered significant memory constraints while handling the full year’s dataset on Google
Cloud VMs. Specifically, we needed to merge our main ride data dataset, which includes
pickup and drop-off location IDs, with the taxi zones dataset. The latter provides crucial
geographic information, assigning each location ID to a specific borough, and includes the
borough names directly associated with each location ID, which is information not included
in the main dataset. The complexity of executing a left join, which combines records from
both datasets based on matching location IDs to append borough information for each ride,
was unmanageable with the entire dataset due to memory limitations.
To address such challenges, we, firstly, utilized the Dask (“Dask: Library for dynamic
task scheduling” ) library, a flexible parallel computing library for analytic computing. Dask
is particularly well-suited for tasks that exceed memory capacity because it processes large
datasets by breaking them into manageable chunks and executing operations in parallel,
contrasting with Pandas (McKinney 2010) which operates in-memory and can become
inefficient with large data volumes. This approach was essential for our project as it allowed
us to handle larger datasets than Pandas could manage by distributing computations and
memory across multiple machines. Implementing Dask required significant adjustments to
our coding practices. Unlike Pandas, which performs operations immediately, Dask operates
lazily, computing results only at the moment they are needed (typically at the end of a
series of operations). This necessitates a different approach to coding, ensuring all intended
computations are correctly lined up before triggering the execution phase. The switch to
Dask involved meticulous re-coding and optimization of our data manipulation scripts to
ensure compatibility and efficiency in a distributed environment. This strategic choice not
only resolved our memory issues but also enhanced the overall scalability and speed of our
data processing tasks.
Even when utilizing Dask, some operations remained too memory-intensive to execute
142

on the full dataset, making it challenging to produce even basic descriptive plots. Therefore,
to clean and prepare the data for plotting, we did not process the entire dataframe at
once. Instead, we performed all data operations, required for the cleaning of the dataset,
on the monthly datasets, cleaning each and exporting them as ‘cleaned-data-month-*’, for
∗ = 1, .., 12. This facilitated the process and we were able to then access the monthly data
to proceed with the analysis we want.
Moreover, we faced challenges when even having to calculate our initial driver proxy (the
number of rides completed within a specified amount of time of a ride request), which is an
essential variable as it later served as the supply proxy we use in the max flow simulation.
Initially, we attempted to compute this proxy by directly counting, for each ride, the number of rides that had finished at the same location within five minutes before the request.
However, this method proved inefficient due to the large size of the dataset, as it required
significant memory and computational resources, making the calculation impractical for the
entire dataset. To improve efficiency, we needed a more optimized approach.
Given the inefficiencies of our initial approach, we adopted an alternate method to estimate the number of open drivers at the time of each request. We discretized each request and
drop-off time into 5-minute intervals, assigning each event to the start of the corresponding time window. This allowed us to create a new dataframe, ‘dropoff-counts-loc’, which
aggregated the number of drop-offs for each location within each time window. With this
setup, we could efficiently calculate the number of rides that finished in each time window,
facilitating different methods for estimating the number of open drivers. For example, if we
wanted to consider as open drivers those who finished a ride in the same time window as
the request, we merged the two dataframes on the matching time windows and locations.
If, however, we wanted to consider as open drivers those who finished a ride in the previous
time window, we shifted the ‘dropoff-counts-loc’ dataframe by 5 minutes and then performed
the merge. Alternatively, if we wanted to consider as open drivers those who finished a ride
in either the same or the previous time window, we utilized a combined approach where we
143

merged the dataframes for both time windows.
B.2 Density Metrics - Additional Plots
(a) Number of Public Wi-Fi Hotspots (b) Presence of Public Wi-Fi Hotspots
(c) Average Pedestrian Count
Figure B.1: ETA vs Number of Available Drivers during Morning Rush hours at Jamaica
Estates (Taxi Zone 55) in Queens, for different values of c.
144

B.3 Calculating Taxi Zone Adjacencies
Incorporating this additional layer of geographical detail is crucial, as it provides a more
nuanced understanding of driver availability and ETA relationships across different neighborhoods. By considering the proximity of taxi zones, we can account for potential overlaps
in driver coverage areas, which is important for accurate modeling. This enhanced spatial
analysis will help us identify patterns and insights that might be missed with broader, less
detailed geographical categorizations. Moreover, this approach allows us to test our model’s
sensitivity to different spatial groupings, thereby strengthening the overall robustness and
validity of our findings.
One of the key components we have developed is an adjacency matrix for the New York
City taxi zones. By leveraging the taxi-zones shapefile provided by the NYC Taxi and
Limousine Commission (TLC) and utilizing Python’s Libpysal library “libpysal: Library for
Spatial Analysis in the PySAL project” n.d., we were able to create both ‘queen’ and ‘rook’
matrices for these zones.
The queen and rook matrices are common types of spatial weight matrices used in geographical analysis. The ‘queen’ matrix considers zones to be adjacent if they share either a
boundary or a corner, similar to the movements of a queen in chess. This means that a zone
is considered adjacent to any other zone that it touches either along an edge or at a vertex.
On the other hand, the ‘rook’ matrix considers zones to be adjacent only if they share a
common boundary, similar to the movements of a rook in chess. This means that a zone is
adjacent only to those zones that it touches directly along one of its edges, excluding any
that only touch at a corner.
These matrices help us understand the spatial relationships between taxi zones, which
can be crucial for our analysis. For example, they allow us to consider how the availability of
drivers in adjacent zones might affect the estimated time of arrival (ETA) for a ride request
in a particular zone.
145

In Figures B.2 and B.3, we provide an example of the adjacencies for a specific taxi
zone, as calculated using both methods. By illustrating these spatial relationships, we can
better understand how interconnected the taxi zones are and how this inter-connectedness
might influence ride-hailing dynamics in New York City. The adjacency matrices form an
important basis for our spatial analysis, allowing us to incorporate geographic proximity into
our regression models and thereby enhancing the robustness and precision of our findings.
Figure B.2: Neighboring Zones Based on
‘Queen’ Adjacency Matrix
Figure B.3: Neighboring Zones Based on
‘Rook’ Adjacency Matrix
B.4 Max Flow Optimization Problem
Maximize X
j
f
j,t
d
subject to f
j,t
s ≤ s
start
j,t , ∀j (Capacity from source to X nodes)
f
j,t
d ≤ dj,t, ∀j (Capacity from Y nodes to sink)
X
i
f
i,t
s = f
j,t
d
, ∀j (Flow conservation)
f
j,t
s ≥ 0, fj,t
d ≥ 0, ∀j (Non-negativity constraints)
146

• Objective: The goal is to maximize the total flow from X nodes (drivers at each
location j) to Y nodes (fulfilled requests) during the time window t. This represents
utilizing as many available drivers as possible to meet demand.
• Capacity Constraints: The constraints ensure that the flow does not exceed the
available number of drivers at each j (from source to X) or the number of requests at
each j (from Y to sink).
• Flow Conservation: The total flow into each Y node (fulfilled requests) must match
the flow from the X nodes, reflecting driver matching to requests.
• Non-Negativity Constraints: The flows cannot be negative, reflecting real-world
scenarios.
B.5 Regression Visualization for Different Locations,
Additional Plots
In this section we present variations of Figure 2.4 for different p values, along with alternative
regression plots using a different proxy for open drivers. Recall that in the main body, all
results are based on the ‘average number of drivers at each time window’ as the driver proxy
in the regression. Here, we also provide results using ’drivers available at the start of the
time window’. The time window refers to the discretization intervals used in our simulation,
and this metric is obtained by averaging the number of drivers available at the start of
each window across the 20 simulation replications. We also examined regression plots using
’drivers available at the end of the time window’, but found them to be almost identical to
those based on ’drivers available at the start of the time window’ Therefore, to streamline
the presentation, we have chosen to omit these plots. In each of the plots of this section,
the left column represents ‘average number of drivers at each time window’, while the right
column shows ’drivers available at the start of the time window’.
147

(a) p = 0.2, average available (b) p = 0.2, start available
(c) p = 0.5, average available (d) p = 0.5, start available
(e) p = 0.8, average available (f) p = 0.8, start available
Figure B.4: ETA vs Number of Available Drivers during Morning Rush hours at Allerton/Pelham Gardens (Taxi Zone 3) in the Bronx, for different values of p.
148

(a) p = 0.2, average available (b) p = 0.2, start available
(c) p = 0.5, average available (d) p = 0.5, start available
(e) p = 0.8, average available (f) p = 0.8, start available
Figure B.5: ETA vs Number of Available Drivers during Morning Rush hours at Coney
Island (Taxi Zone 55) in Brooklyn, for different values of p.
149

(a) p = 0.2, average available (b) p = 0.2, start available
(c) p = 0.5, average available (d) p = 0.5, start available
(e) p = 0.8, average available (f) p = 0.8, start available
Figure B.6: ETA vs Number of Available Drivers during Morning Rush hours at Jamaica
Estates (Taxi Zone 131) in Queens, for different values of p.
150

(a) p = 0.2, average available (b) p = 0.2, start available
(c) p = 0.5, average available (d) p = 0.5, start available
(e) p = 0.8, average available (f) p = 0.8, start available
Figure B.7: ETA vs Number of Available Drivers during Morning Rush hours at Financial
District South (Taxi Zone 88) in Manhattan, for different values of p.
B.6 Uber vs Taxi Rankings for Different Density Classifications - Additional Plots
151

(a) c = 0
(b) c = 1
(c) c = 2
Figure B.9: Comparison of Uber and Taxi Ride Rankings for p = 0.5. Density classification
is based on both density metrics and α regression coefficients, with square markers indicating
agreement between the two methods
152

(a) c = 0
(b) c = 1
(c) c = 2
Figure B.8: Comparison of Uber and Taxi Ride Rankings for p = 0.2. Density classification
is based on both density metrics and α regression coefficients, with square markers indicating
agreement between the two methods
153

(a) c = 0
(b) c = 1
(c) c = 2
Figure B.10: Comparison of Uber and Taxi Ride Rankings for p = 0.8. Density classification
is based on both density metrics and α regression coefficients, with square markers indicating
agreement between the two methods
154

B.7 Uber vs Taxi Rankings for Different Density Classifications - Evening Rush
As mentioned earlier in this chapter, all results and insights presented in the main body
correspond to the morning rush hours. This focus aligns our analysis with (Yan et al. 2020)
who fitted their regression model specifically for morning rush hours. However, we have also
conducted our regression analysis on data from evening rush hours, which yielded results
largely consistent with the morning rush. In the evening, the regression of ETA on the
number of open drivers produced similar patterns to those observed in Figures B.4 through
B.7 where, once again, increasing the parameter c brought the α values closer to theoretical
expectations.
An interesting observation in the evening rush analysis is that the zones identified as
high alpha and high density are both similar to and slightly different from those identified
during the morning rush hours. These are pictured, for c = 0 in Figure B.11. As before,
the majority of high alpha and high density zones are located in Manhattan, and there is
notable robustness across different abandonment probability values, p. This consistency
across p values reinforces the reliability of our approach, indicating that zones classified as
high α in the regression remain largely stable regardless of the specific p value used in the
simulation.
While several key Manhattan zones appear in both morning and evening analyses—such
as Financial District South and North, World Trade Center, Greenwich Village North, and
Lincoln Square East—we also see some variation in high α zones in the evening. Additional
Manhattan areas like Upper West Side South and North, Seaport, and East Chelsea emerge
as high-density zones in the evening, alongside prominent Brooklyn areas like Brooklyn
Heights and Williamsburg. These locations are intuitively expected to be dense, as they
represent vibrant residential and commercial zones with significant foot traffic, transit access,
and nightlife. This consistency in identifying such high-traffic areas across different times of
155

day underscores the robustness of our framework, reinforcing its utility in capturing realistic
urban dynamics and identifying zones of concentrated demand.
156

(a) p = 0.2
(b) p = 0.5
(c) p = 0.8
Figure B.11: Comparison of Uber and Taxi Ride Rankings when c = 0 for evening rush
hours. Density classification is based on both density metrics and α regression coefficients,
with square markers indicating agreement between the two methods
157

Chapter C
Appendices for Chapter 3
C.1 Mathematical expressions for TSD models
We start with the asymmetric model. γ ∈ [0, 1] is the competition intensity between products
1 and 2: γ → 1 implies that products 1 and 2 are perfect substitutes and the market
competition is intense; γ → 0 implies that products 1 and 2 are not substitutable and there
is no market competition. We use qi ⩾ 0 to denote the output of product i, i = 1, 2, 3. The
market surplus brought by the three products is:
U(q1, q2, q3) = X
3
i=1
αiqi −
1
2
X
3
i=1
q
2
i
!
− γq1q2, (C.1)
where αi
is the market size of product i reduced by the unit production cost of product i,
i = 1, 2, 3. By taking the partial derivatives, the prices of the three products are
pi =
∂U
∂qi
= αi − qi − γqj
, i, j = 1, 2, i ̸= j; and p3 =
∂U
∂q3
= α3 − q3, i = 1, 2, 3.
If products are recycled separately, or if products from the same market are recycled
together, we assume that their unit recycling costs are c1, c2 and c3. Due to the increasing
requirements on hardware (machines) and software (technology) when processing products
from different markets together, we assume that the unit recycling cost increases. Thus, if
products 1 and 3 are recycled together, we assume that their unit recycling costs become
λc1 and λc3, with λ > 1.
A recycler’s operations can generate (dis)economies of scale, based on the quantity of
158

products it recycles. TSD assume that such adjustments to the recycling-related costs are
changed quadratically with the product quantity. We use κ to denote the factor of the
quadratic form; positive (negative) κ means a decrease (increase) to the overall recycling
cost, indicating (dis)economies of scale. For instance, if product 1 is recycled alone, its
overall unit cost is adjusted by −κq2
1
; if products 1 and 2 are recycled together, their overall
unit costs are adjusted by −κ(q1+q2)
2
. If different products are recycled together, the change
stemming from scale economies is apportioned to products by their quantities: if products
1 and 2 are recycled together, product i’s cost is adjusted by −
qi
q1+q2
κ(q1 + q2)
2
, i = 1, 2. In
this paper, we focus on the model with economies of scale, so we use κ > 0.
Under a given recycling structure X ∈ X, we let Z
X
i be the set of products recycled
by product i’s recycler: Z
X
i
is a set of products that are recycled together by the same
recycler and i ∈ Z
X
i
. If we combine the two cost drivers described above, under the recycling
structure X, the cost for recycling product i is
C
X
i
(q1, q2, q3)
.= λ
X
i
ciqi − P
qi
j∈ZX
i
qj
κ


X
j∈ZX
i
qj


2
, (C.2)
where λ
X
i =



1, if Z
X
i = 1, 2, 3 or 12,
λ, otherwise.
Firms determine their equilibrium quantities and payoffs under different recycling structures. If we denote the equilibrium quantities by q
X
i
, i = 1, 2, 3
1
, for a given structure,
X ∈ X, the equilibrium payoffs are
Π
X
A = (1 − κ)(q
X
1
)
2
; (C.3)
Π
X
B =



(1 − κ)(q
X
2
)
2 + (1 − κ)(q
X
3
)
2
, if X = (12, 3),(13, 2), or (1, 2, 3)
(1 − κ)(q
X
2
)
2 + (1 − κ)(q
X
3
)
2 − 2κqX
2
q
X
3
, otherwise.
1The exact expressions for q
X
i under all recycling structures can be found in TSD.
159

C.2 Examples
Asymmetric model
For the asymmetric model we consider five different types of relationships between players’
payoffs in different structures. A unique Pareto dominant outcome, which should
emerge as stable. This can be illustrated by payoffs as the ones described in Table C.12
. In
Example 1.1, the Pareto dominant outcome is (12, 3); in Example 1.2, the Pareto dominant
outcome is (123). Both examples correspond to the first item in Proposition 11.
Structure Example 1.1 Payoffs Example 1.2 Payoffs
Player A Player B Player A Player B
(123) 570 700 1180 529
(12, 3) 620 900 390 342
(13, 2) 350 360 670 400
No collaboration 380 680 230 350
Table C.1: Pareto dominant outcomes.
Player B gets the highest profit without cooperation, and Player A cannot
make him cooperate, hence No collaboration should emerge as stable. This can be illustrated
by payoffs as the ones described in Table C.2. In Example 2.1, Player A prefers (123); in
Example 2.2, Player A prefers (13, 2). Neither of these structures may be realized without
participation of Player B, who prefers No collaboration. These examples correspond to the
second and fourth item in Proposition 11.
One player gets more without collaboration than in the outcome most
preferred by the other player, but both players prefer collaboration to the noncollaborative outcome; this can be illustrated by payoffs as the ones described in Table C.3.
In Example 3.1, the outcome most preferred by Player A is (13, 2), and Player B can get
more without collaboration. The LCS predicts that the stable outcome in this case will be
2See Appendix C for a more detailed example of payoff calculations.
160

Structure Example 2.1 Payoffs Example 2.2 Payoffs
Player A Player B Player A Player B
(123) 220 600 400 810
(12, 3) 200 350 350 710
(13, 2) 215 30 900 705
No collaboration 140 715 5 830
Table C.2: Player B prefers No collaboration.
Structure Example 3.1 Payoffs Example 3.2 Payoffs
Player A Player B Player A Player B
(123) 585 315 200 960
(12, 3) 580 815 70 430
(13, 2) 600 120 1400 410
No collaboration 555 790 420 70
Table C.3: One player gets more without collaboration than in the outcome most preferred
by the other player.
(12, 3), the outcome most preferred by Player B. To see this, note that the non-collaborative
outcome is the least favorite for Player A, but the second-highest preferred by Player B, and
Player B does not need collaboration of Player A in order to achieve this outcome. As (12, 3)
is better for both players than non-collaboration, and it is also the outcome most preferred
by Player B, (12, 3) is uniquely stable. This case corresponds to the first item in Proposition
11.
In Example 3.2, the outcome most preferred by Player B is (123), and Player A can get
more without collaboration. The LCS predicts that the stable outcome in this case will be
(13, 2), the outcome most preferred by Player A. To see this, note that the non-collaborative
outcome is the least preferred by Player B, but the second-highest preferred by Player A,
and Player A does not need collaboration of Player B in order to achieve this outcome. As
(13, 2) is better for both players than non-collaboration, and it is also the outcome most
preferred by Player A, (13, 2) is uniquely stable. This case corresponds to the third item in
Proposition 11.
If we look at Example 3.1 from the EPCF perspective, consider the PCF under which
161

Player B always defects from (123) or (13, 2) to a non-collaborative outcome, and both
players jointly move from the non-collaborative outcome to (12, 3). All of these moves are
profitable, hence (12, 3) is stable for any level of farsightedness. Similar conclusion holds for
Example 3.2 with the PCF under which Player A always defects from (123) and (12, 3), and
both players jointly move from the non-collaborative outcome to (13, 2).
Both players get more without collaboration than in the outcome most
preferred by the other player, but No collaboration is not the most preferred outcome
for either player. This can be illustrated by payoffs as the ones described in Table C.4. In
both examples the outcome most preferred by Player A is (13, 2), and the outcome most
preferred by Player B is (12, 3). The LCS predicts that the stable outcome will be No
collaboration. To see this, note that Player B can achieve more without collaboration than
in any other outcome except (12, 3). As (12, 3) is dominated by No collaboration for Player
A, A would not want to join B in (12, 3). Player A would need participation of Player B for
any outcome other than the non-collaborative one, so the non-collaborative outcome emerges
as stable.This case corresponds to the fourth item in Proposition 11. In a similar way as in
the Examples 3.1 and 3.2, we can show stability of non-collaboration under the EPCF for
any level of farsightedness.
Structure Example 4.1 Payoffs Example 4.2 Payoffs
Player A Player B Player A Player B
(123) 300 700 170 350
(12, 3) 250 970 180 765
(13, 2) 600 560 265 300
No collaboration 270 900 190 760
Table C.4: Both players get more without collaboration than in the outcome most preferred
by the other player.
Two possible stable outcomes—the top choice of Player A is the second choice for
Player B, the top choice for Player B is the second choice for Player A, and either of these
two outcomes can be stable. This can be illustrated by payoffs as the ones described in Table
C.5.
162

In Example 5.1, the outcome most preferred by Player A is (12, 3), and the outcome most
preferred by Player B is (123); in Example 5.3, their preferences are reversed. Note that both
players prefer the other player’s favorite outcome to the non-collaborative outcome, hence
both (12, 3) and (123) can be stable. This case corresponds to the first item in Proposition
12.
In Examples 5.2 and 5.4, the outcome most preferred by Player A is (13, 2), and the
outcome most preferred by Player B is (123). Note that both players prefer the other
player’s favorite outcome to the non-collaborative outcome, hence both (13, 2) and (123) can
be stable. This case corresponds to the second item in Proposition 12.
Structure Example 5.1 Payoffs Example 5.2 Payoffs Example 5.3 Payoffs Example 5.4 Payoffs
Player A Player B Player A Player B Player A Player B Player A Player B
(123) 770 200 575 920 990 845 3500 5100
(12, 3) 780 175 450 535 910 940 2500 1400
(13, 2) 745 70 960 700 495 300 5000 3400
No coll. 765 60 325 490 510 500 1300 3200
Table C.5: Two stable outcomes.
In examples with two potentially stable outcomes, we want to investigate if one outcome
is more likely to emerge as stable than the other. Consider again Example 5.1, but now from
the perspective of the EPCF. In order to obtain (12, 3) as a stable outcome, we could use
the PCF that requires both players to jointly move to (12, 3) if the current outcome is either
(13, 2) or the non-collaborative outcome; this move is profitable for both players. However,
if the current outcome is (123), the PCF requires Player A to defect to the non-collaborative
outcome (from which both players jointly move to (12, 3) in the next step). This move
reduces Player A’s payoff in one period, but he receives a higher payoff over infinitely many
subsequent periods. This move is profitable for Player A only if
770
1 + δ + δ
2 + . . .
≤ 770 + δ · 765 + 780
δ
2 + δ
3 + . . .
⇔
770
1 − δ
≤ 770 + δ · 765 + 780
δ
2
1 − δ
⇒ δ ≥
1
3
.
1

Similarly, we can analyze when is (123) likely to emerge as stable, and use the PCF that
requires both players to jointly move to (123) if the current outcome is either (13, 2) or
the non-collaborative outcome; this move is profitable for both players. However, if the
current outcome is (12, 3), we require Player B to defect, which leads to the non-collaborative
outcome (from which both players jointly move to (123) in the next step). This move reduces
playerB’s payoff in one period, but he receives higher payoff over infinitely many subsequent
periods. This move is profitable for Player B only if
175
1 − δ
≤ 175 + δ · 60 + 200
δ
2
1 − δ
⇒ δ ≥ 0.82.
If we conduct a similar analysis for Example 5.3, then for stability of (123) Player A needs
910
1 − δ
≤ 910 + δ · 510 + 990
δ
2
1 − δ
⇒ δ ≥ 0.83,
while for stability of (12, 3) Player B needs
845
1 − δ
≤ 845 + δ · 500 + 940
δ
2
1 − δ
⇒ δ ≥ 0.784.
In both examples, stability of (12, 3) would require a lower level of farsightedness compared
to stability of (123), hence (12, 3) may be more likely to emerge as stable (although the
difference is much smaller in Example 5.3).
Next, consider Example 5.2. In order to obtain (13, 2) as a stable outcome, we could
use the PCF that requires both players to jointly move to (13, 2) if the current outcome is
either (12, 3) or the non-collaborative outcome; this move is profitable for both players. If
the current outcome is (123), the PCF requires Player A to defect to the non-collaborative
outcome (from which both players then jointly move to (13, 2)). This move reduces Player
A’s payoff in one period, but he receives a higher payoff over infinitely many subsequent
164

periods. This move is profitable for Player A only if
575
1 − δ
≤ 575 + δ · 325 + 960
δ
2
1 − δ
⇒ δ ≥ 0.394.
Similarly, to obtain (123) as stable, we can use the PCF that requires both players to jointly
move to (123) if the current outcome is either (12, 3) or the non-collaborative outcome; this
move is profitable for both players. If the current outcome is (13, 2), the PCF requires Player
B to defect to the non-collaborative outcome (from which both players then jointly move to
(123) ). This move reduces Player B’s payoff in one period, but he receives higher payoff
over infinitely many subsequent periods. This move is profitable for Player B only if
700
1 − δ
≤ 700 + δ · 490 + 920
δ
2
1 − δ
⇒ δ ≥ 0.49.
If we conduct a similar analysis for Example 5.4, then for stability of (123) Player B needs
3400
1 − δ
≤ 3400 + δ · 3200 + 5100
δ
2
1 − δ
⇒ δ ≥ 0.105,
while for stability of (13, 2) Player A needs
3500
1 − δ
≤ 3500 + δ · 1300 + 5000
δ
2
1 − δ
⇒ δ ≥ 0.595.
From this analysis, stability of (13, 2) requires a lower level of farsightedness compared
to stability of (123) in Example 5.2, while the opposite is true for Example 5.4, hence (13, 2)
may be more likely to emerge as stable in the first case, and (123) in the second case.
165

C.3 Calculations for Example 1.1
Assume α1 = α2 = 300, α3 = 100, c1 = c2 = 2, c3 = 5, γ = 0.5, λ = 1.59, κ = 0.2. Then,
we can calculate equilibrium quantities and corresponding prices and profits for all players
under all possible structures, as follows:
Recycling structure (123) (12, 3) (13, 2) (23, 1) (1, 2, 3)
q1 121.93 122.13 121.02 121.03 121.14
q2 122.61 122.13 121.17 121.57 121.14
q3 50.71 48.47 46.55 49.45 48.47
p1 116.76 116.80 118.40 118.19 118.29
p2 116.42 116.80 118.32 117.91 118.29
p3 49.29 51.53 53.45 50.55 51.53
ΠA 14,569 14,617 14,353 14,355 14,380
ΠB 17,004 17,218 16,665 16,640 16,977
Table C.6: Calculations for Example 1.1.
Note that Player B generates a higher payoff in (1, 2, 3) than in (1, 23), hence the noncollaborative outcome in this case is (1, 2, 3). To make it easier for players to compare the
payoffs in different structures and to have payoffs in different examples in a similar order of
magnitude, we modified the numbers while preserving the order between payoffs in different
structures. For Player A, we subtracted 14,000 from each payoff and rounded numbers to
tens, which lead to 570, 620, 350, 360, and 380 (note that we did not use 360 in Example
1.1, as the non-collaborative choice for Player B in this case would be to recycle his products
independently). For Player B we did a similar thing, but deducted 16,300, which led to
payoffs of 700, 900, 360, 340, and 680.
166

Abstract (if available)

Abstract The first chapter of my dissertation focuses on the rapidly increasing gig economy where businesses operate more like ‘two-sided’ platforms than traditional ‘one-sided’ firms, taking into account not just customer preferences but also those of the service providers. This study examines how these two-sided markets approach pricing and service delays differently from traditional businesses, because of the need to consider the impact of the service providers’ choices. Using a game theoretic model we represent the participating sides (customers, servers, managers) and by considering that the service delays experienced by customers vary with the number of participating service providers we explore various business models within this context. Interestingly, we find certain conditions where traditional firms and modern platforms perform similarly, and we identify the factors that make service providers’ preferences either significant or negligible in influencing a platform’s profits. Our research also provides insight on the competitive edge that transportation-based services hold in the gig economy. Overall, this study underscores the managerial importance of understanding these dynamics within the rapidly changing landscape of the gig economy.
The second chapter empirically examines the theoretical assertions of the initial chapter. Utilizing a comprehensive dataset of ride-hailing activity within the New York metropolitan area, coupled with geo-spatial data, this analysis aims to validate the relationship between rider delays (Estimated Time of Arrival, or ETA) and the availability of drivers in specific locations. This study seeks to ascertain whether the functional relationship observed in real-world data mirrors the theoretical models commonly referenced in existing ride-hailing literature (Larson et al. 1981). Additionally, considering the diverse spatial characteristics of New York’s neighborhoods, this chapter explores variations in the dependency of delays on driver availability, investigating the extent to which these relationships align with spatial delay theory or diverge towards patterns more indicative of congestion-related delays. Building on theoretical insights from the first chapter, this inquiry further investigates how spatial dynamics influence the performance outcomes for ride-hailing platforms. This study is critical as it not only seeks to bridge theoretical assumptions with empirical findings in the context of urban mobility, but also contributes to the optimization of ride-hailing operations, ultimately enhancing service efficiency and consumer satisfaction in metropolitan transport networks.
The third chapter undertakes an experimental study of farsighted coalitional stability, a game-theoretical concept that extends beyond traditional equilibrium models by incorporat- ing the agents’ farsightedness. This investigation is inspired by the research of (Tian et al. 2019), who explored the stability of producers’ strategies under environmental legislation mandating product recycling in markets with multiple, differentiated consumer products. The aim is to design behavioral experiments that address critical questions: First, whether theoretically stable outcomes are indeed stable in real-life bargaining scenarios; second, which behavioral factors significantly influence agents’ decision-making processes. Given the scant experimental research on Farsighted Stability, this study fills a significant gap by testing the theory’s applicability in controlled laboratory conditions. We examine the conditions —such as model parameters and levels of farsightedness— under which outcomes deemed stable theoretically are realized through actual player interactions. This chapter aims to il- luminate the practical implications of farsighted coalitional stability, thereby advancing our understanding of strategic behavior in complex decision-making environments.

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Creator Giannoutsou, Aikaterini (author)

Core Title Theoretical and data-driven analysis of two-sided markets and strategic behavior in modern economic systems

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Degree Conferral Date 2024-12

Publication Date 12/02/2024

Defense Date 10/29/2024

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

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