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Substitution and variety, group power in negotiation
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Substitution and variety, group power in negotiation
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Content
SUBSTITUTION AND V ARIETY
GROUP POWER IN NEGOTIATION
by
Feng Chen
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)
May 2009
Copyright 2009 Feng Chen
ii
Table of Contents
List of Tables ...................................................................................................................... iii
List of Figures .................................................................................................................... iv
Abstract .............................................................................................................................. vi
Chapter 1 Substitution and Variety ..................................................................................... 1
1.1 Introduction .............................................................................................................. 1
1.2 Literature review ...................................................................................................... 9
1.3 General inventory problem with substitution ......................................................... 13
1.4 Approximation with a simple inventory policy ...................................................... 17
1.4.1 Demand approximation ................................................................................ 17
1.4.2 Main theorem ............................................................................................... 20
1.4.3 Total number of customers ........................................................................... 25
1.5 Extensions .............................................................................................................. 28
1.5.1 Uncertain demand ........................................................................................ 28
1.5.2 Unequal popularity....................................................................................... 29
1.5.3 Gap between simultaneous and sequential substitution ............................... 30
1.6 The assortment problem ......................................................................................... 32
1.7 About utility model and no-purchase probability ................................................... 42
1.8 Managerial insight and conclusion ......................................................................... 45
Chapter 2 An Experimental Study on Group Negotiation Power ..................................... 48
2.1. Introduction ............................................................................................................ 48
2.2. Literature Review ................................................................................................... 57
2.3. Methods .................................................................................................................. 60
2.3.1 Experiment design ....................................................................................... 60
2.3.2 Hypotheses ................................................................................................... 62
2.3.3 Participants and administration .................................................................... 63
2.4. Result ...................................................................................................................... 64
2.4.1. Descriptive statistics ........................................................................................ 64
2.4.2. Ordinal regression model ................................................................................. 65
2.4.3. Ordinal regression results ................................................................................ 66
2.5 Discussion ................................................................................................................... 69
2.6 Managerial insight ...................................................................................................... 74
2.7 Further research .......................................................................................................... 75
Bibliography ..................................................................................................................... 76
iii
List of Tables
Table 1: Examples of N(K) 28
Table 2: Profit Simulation with Varied Variety Cost 34
Table 3: Profit Simulation with Quadratic Variety Cost Function 35
Table 4: Profit Simulation with Varied No-Purchase Parameter 37
Table 5: Profit Simulation with Varied Purchasing Cost 38
Table 6: Profit Simulation with Varied Selling Price 39
Table 7: Profit at Different K in Figure 3 41
Table 8: Data Summary 64
Table 9: Parameter Estimate 67
Table 10: Confusion Matrix 68
Table 11: Model Fitting 68
Table 12: Goodness-of-fit 68
Table 13: Test of Parallelism 68
iv
List of Figures
Figure 1: Variety Cost at 10 34
Figure 2: Variety Cost at 300 34
Figure 3: Variety Cost at 1000 34
Figure 4: Variety Cost at 5000 34
Figure 5: Result 1 with Quadratic Function 35
Figure 6: Result 2 with Quadratic Function 35
Figure 7: Result 3 with Quadratic Function 35
Figure 8: Result 4 with Quadratic Function 35
Figure 9: Result at q=0.2 37
Figure 10: Result at q=0.4 37
Figure 11: Result at q=0.7 37
Figure 12: Result at q=0.9 37
Figure 13: Result at c=1 39
Figure 14: Result at c=3 39
Figure 15: Result at c=6 39
Figure 16: Result at c=8 39
Figure 17: Result at r=10 40
Figure 18: Result at r=20 40
v
Figure 19: Result at r=30 40
Figure 20: Result at r=50 40
Figure 21: Sellers form coalition 53
vi
Abstract
This dissertation is on two topics. Chapter 1, substitution and variety, focuses two
major issues in retail inventory management: (1) the assortment problem; (2) the
inventory problem: the optimal inventory policy given a set of variants. It is well known
that it is difficult to reach analytically tractable solutions solving these problems because
of the randomness in the total market demand, in consumer choices, and the resulting
consumer substitution behaviors. We start by considering a single product category and a
single profit-maximizing retailer. Demands for each of the variants are uncertain and are
generated from a consumer choice model. For each customer, a realization of the
consumer choice model is a ranking of the variants, from the most desirable variant to the
least desirable one. We assume that each customer purchases at most one unit of the
product, and we are able to determine an inventory policy which results in the optimal
expected cost with a very high probability. We are also able to numerically approximate
the optimal number of variants that a retailer should carry.
The experiment in chapter 2 investigates the behavioral effect on negotiation power.
Negotiation power has been well studied under the two-person bargaining setting.
However, it is much less known how the negotiation power shifts when two or more
players join forces to bargain as one entity. This research explores the perception of
negotiation power of sellers’ alliance and buyers through an experiment. We set up a
vii
situation where both parties have no alternatives if they can not reach agreement. Data
are analyzed through the ordinal regression model. 4The results show that parties with
lower initial negotiation power are more likely to perceive increment of negotiation
power after joining forces than those with higher initial negotiation power. Sellers who
join forces are more likely to perceived increase in negotiation power than buyers who
face the alliance. Additionally, parties without positive impact from the alliance size are
more likely to perceive increment of negotiation power.
1
Chapter 1 Substitution and Variety
1.1 Introduction
In the past few years, the number of products offered by retailers to consumers has
increased dramatically. This suggests that line extension has become a favored strategy of
retailers, which often extend their product lines and offer a large variety aiming to
capture a larger share of the market and extract more consumer surplus. However,
offering more variants to customers tends to complicate the issues regarding the proper
assortment and the stocking level for different products in the category for the retailer.
Indeed, maintaining a large variety is costly, especially when customers’ preferences are
too hard to predict well beforehand.
In this study, we focus on the effects of consumer substitution behavior and variety
on the firm’s market share. Generally speaking, our main objective is to study the
following classical problems:
A. The inventory substitution problem—what is the optimal stocking policy for each of
the variants in a product line when consumer substitution behaviors are taken into
consideration?
B. The assortment problem—what is the optimal subset of variants that the retailer
should carry?
2
In practice, a retailer will first decide how many different products (variants) to carry
in a category (problem B) and then choose a stocking level for each variant (problem A).
In this study, we analyze these problems in a backward sequence. First, we approximate
the optimal stocking policy that leads to close-to-optimal sales given the variety. Second,
based on our heuristic policy, we seek the best assortment the retailers should carry. The
argument for the backward approach is that knowing the solution of problem A helps us
to determine the expected sales for a given set of variants and thus makes it easier to
solve the assortment problem.
An important step in studying the inventory problem with substitution is to capture
consumer substitution behavior. We build up a general model of consumer choices
without prescribing any consumer substitution pattern, but with some limiting
assumptions. We then use the choice model to generate consumer preferences and
demands for different variants. An out-come, or realization, of the consumer choice
model is a ranking of the K variants, from the most desirable (highest utility) to the least
desirable (lowest utility). Each customer has his or her own ranking preferences that may
be different from those of other consumers. We can calculate the demand for each variant
based on the preference rankings of all consumers. In other words, the demand
distributions of different variants are not exogenous, but rather they are a function of the
number of variants and are determined by the choice model. Consumers’ preferences are
assumed to be random, and it is assumed that retailers can only predict customers’
3
preferences and estimate the demand for variants through predicted preferences. Thus,
retailers are only able to infer the demand distribution for each of the K variants. To
make the model more realistic, we include a no-purchase option in the buyer’s choice set.
For a consumer, some variants may have such small utilities that they are inferior to the
option of not purchasing a product at all. In this setting, when a customer tries to buy a
variant and it is not available, he or she will substitute with the next-preferred option
according to the realized preference ranking. If it turns out that the next variant yields
smaller utility than does the no-purchase option, the choice is to not purchase anything
and leave the store.
There are two common ways to model consumer substitution behaviors. One is the
sequential model, in which customers arrive sequentially and the seller satisfies their
demands one by one (sequential substitution). The other is the simultaneous model,
which assumes that all consumers arrive simultaneously and reveal their preference at the
same time (simultaneous substitution). Then the retailer allocates the available inventory
to consumers so that expected profits are maximized and consumers purchase variants
with utilities that are higher than their no-purchase utilities.
Sequential substitution captures reality well but is comparably more difficult to model
and solve. We show that by using the simultaneous substitution model, we are able to
identify the optimal allocation of products to customers subject to their individual ranked
preferences. Clearly, the optimal profit achieved using the simultaneous substitution
4
model is an upper bound of the sequential substitution model, assuming everything else
remains equal. In this study, we use the simultaneous substitution model because it
enables us to obtain structural results. Perhaps more importantly, we prove that when the
market size is large enough, the simultaneous substitution model provides a very tight
upper bound on the optimal profit of the sequential substitution model. That is, the gap
between the resulting optimal profits of the two models is very small—a result that we
are the first to prove analytically. Others have shown the same result using numerical
studies. For example, Gaur and Honhon (2005) show that the gap between the profits of
the two models is small.
In solving these problems, we start by making the following important assumptions.
Variants have identical quality, price, and popularity.
Like many other studies, we assume that different variants have different
characteristics (e.g., flavor, color) but are conceived to be of identical quality, and that all
variants have the same costs and retail prices. We start by assuming that the retailer does
not know the preferences of the consumer but assumes that all consumers have the same
choice model. In other words, the distributions of consumers’ preferences are identical,
but not their realized preferences. This differs from some other substitution studies that
assume variants (products) having different preference rankings or a predefined
substitution order. For example, van Ryzin and Mahajan (1999) analyze the optimal
assortment assuming that the variety choice set consists of variants with different degrees
5
of popularity. Gaur and Honhon (2005) build up a Lancaster choice model that assumes a
predefined substitution order (that is, once a customer’s first choice is realized, the seller
knows the entire preference rankings of the customer). Later, in Section 1.6.2, we relax
this assumption to allow different popularity among variants.
A major limitation of this study is our inability to consider variants with different
costs and retail prices. There is no doubt that this assumption is limiting. Yet, many
products have the same cost and retail price. For example, the cost and retail price of T
shirts is not a function of the color, and similarly the cost and the retail price of yogurt are
identical and are not a function of the flavor.
Inventory level is not known to customers.
Like many other studies, we assume that a customer does not know the inventory
level of each variant. In other words, remaining stocking level of all products will not
interfere with the choice realization of each consumer.
The market has a fixed size.
We start by assuming a fixed market size with N consumers, where N is known with
certainty. The purpose for this assumption is twofold. First, we restrict the source of
uncertainty to the consumer choices but not demand. This will give us better
understanding regarding the role of substitution and its effect on inventory policy. Later,
when relaxing this assumption, we will be able to identify the effects of market
uncertainty on the inventory policy. Second, we make this assumption for exposition and
6
clarity. In Section 1.6.1, we will relax this assumption and allow for a random market
size. We also assume that N is not too small. This assumption is crucial to some of our
results. However, N does not have to be very large, and this restriction is met in most
practical applications.
All customers have at least two variants that they are willing to purchase.
Each customer is assumed to be willing to purchase one of at least two variants. That
is, each customer has a most desirable variant and at least one more variant with utility
higher than the no-purchase utility. Thus, if customers’ first-choice product is not
available, they will try to buy at least one other product from the same category. This
assumption ensures that all customers will make substitutions if their first-choice product
is out of stock. In other words, we exclude those who strictly prefer only one type of
product from the category (one-choice customers) and no-purchase customers. This
assumption helps us to solve the inventory problem with consumer substitution. We will
relax this assumption and consider the no-purchase customer in solving the assortment
problem. In Section 1.7.2, we will see that the number of one-choice customers indeed
becomes very small very quickly as the size of variety increases.
One customer will buy, at most, one unit of product.
This assumption excludes multiple-item purchasing. The purpose is to give us a
simple starting point in analyzing this complex problem.
There is no competition.
7
The firm is assumed to be a monopoly, which helps us to have a better picture of how
variety affects the market share.
Our analysis leads to the following main findings of the study.
1. With a very large probability, it is possible to satisfy all consumers’ demands with
exactly N units of inventory and no safety stock. This result is quite remarkable and of
practical value. It means that while the retail firm faces uncertainty in consumer
preferences and does not carry any safety stock, it is still able to satisfy the demands of
all customers.
2. The fact that no safety stock is needed depends heavily upon the assumption that
the size of the population is known with certainty. When the size of the population is
random, it is necessary to have safety stock. In other words, the safety stock is carried
only because of the randomness in the market size, not because of the randomness in
consumer preferences. We show that when population size is a random variable there is a
simple way to calculate the optimal purchasing quantity of each of the variants. The
optimal solution is obtained by solving a simple news-vendor problem.
3. The above result allows us to model a complicated assortment problem with
consumer substitution. The numerical result shows that it might be optimal for a
monopoly to have a large variety because increasing the variety decreases the probability
of no purchasing, which in turn increases the market size and total profit. This eventually
becomes a trade-off with the concurrently increasing variety cost. In other words, the
8
choice of the extent of the product line (i.e., the number of variants) depends upon
parameters that include no-purchasing probability, price, shortage cost, customer pool
size, and variety cost.
4. When variants are not equally popular, we can still achieve the optimal
simultaneous substitution solution by appropriately stocking different variants and
without keeping any inventory.
5. We are able to show, analytically, that the gap between the simultaneous and
sequential substitutions is small.
In what follows, we briefly review some related literature in Section 1.2, then we
build a general inventory model with substitution in Section 1.3. In Section 1.3, we
approximate the optimal solution with a very simple heuristic inventory policy. We show
that under quite general conditions, the demand of all consumers can be satisfied with
very high probability and with no safety stock by using a simple stocking policy. Based
on the approximated inventory solution, we continue to develop the model with some
relaxed assumptions as extensions in Section 1.5 and analyze the assortment problem in
Section 1.6. In Section 1.7, we discuss some commonly used choice models and show
that the selection of choice model indeed will not affect our main result. We also analyze
the no-purchase probability, which decreases very fast as the variety size go up. Finally,
we wrap up with managerial insights and a summary in Section 1.8.
9
1.2 Literature review
Variety can be costly. Chamberlain (1933), in his seminal paper, assumed that
production costs exhibit economies of scale, and he showed that the higher the economies
of scale the lower, in equilibrium, are the number of variants. This result is intuitive and
very common in the economics literature. Stalk (1988) and Baumol and Ide (1956) claim
that the cost of variety is almost proportional to the square root of the number of variants.
We will use this result as an approximation of variety cost. In an empirical study, Kekre
and Srinivasan (1990) examine over 1,400 business units and show that firms with higher
variety have significantly higher relative product prices, which serves as evidence that
variety is costly. On the other hand, increasing variety gives the customers more options
to choose from when the desired item is not available, increases market share, and
reduces requests for out-of–stock products. The Grocery Manufacturers of America
(2002) demonstrate the importance of substitution in the retailing industry. The authors of
this study argue that on average the percentage of shortages in the retail industry is
5%–10%. Also, on average, 45% of consumers who face shortages will substitute and
purchase another product either from the same or a different retailer.
The operations literature that considers substitution aims to study the effects of
substitution on inventory and procurement policies. In most of this literature, the demand
distribution of each of the variants is exogenous and is independent of the number of
10
variants. For examples, see Bassok et al. (1999), Ernst and Kouvelis (1999), and
Netessine and Rudi (2003).
Pentico (1974) analyzes the deterministic, multi-item inventory-optimization problem
with one-way substitution and shows with dynamic programming that there exists an
optimal stocking policy. Pentico (1976) extends this formulation to concave production
and substitution cost functions and presents algorithms to find optimal policies.
Bassok et al. (1999) build up a model analyzing the one-way substitution model for
the manufacturers and characterize the structure of the optimal policy. They also use an
example of a problem involving only two products to illustrate the large effects of
substitution and the significant gains that can be achieved when substitution is considered.
Smith and Agrawal (2000) study the substitution effect on inventory assuming that a
customer switches to the second-most-desired variant when his or her first choice is not
available and that each customer makes at most one such substitution attempt. They
develop a method of determining the optimal level of inventory of each of the variants.
Netessine and Rudi (2003) analyze the centralized and competitive models for
substitution, and they show the concavity of the objective function in a non-competitive
setting and establish the uniqueness of the equilibrium for the competitive n-product case.
Jordan and Graves (1995) study manufacturing flexibility, which is analogous to the
substitution problem, and they show by simulation that limited flexibility, configured in a
certain way, yields most of the benefits of total flexibility (substitution). Our findings are
11
similar to theirs. Anupindi et al. (1998) focus on only two different products and build up
a model to estimate the demand for each product, assuming substitution and that
consumers arrive sequentially. They find that the model can be practically applied
because parameters can be derived by using the maximum-likelihood estimates (MLEs).
In this paper, we take a different route, starting with a choice model to generate the
demand for each of the products. The consumer choice model is one of the classic topics
in marketing. McFadden (1980) reviewed and discussed several probability models
regarding consumer choice behavior, one of them being the multinomial logit (MNL)
utility model, which is widely applied in many marketing and operations studies. Kim, et
al. (2002) introduces a utility model that accounts for the effect of variety choice and
substitution. In this model, a consumer can choose from a set of variants and decide not
only which variants to buy but also the quantity of each of the variants to purchase. The
model also introduces a budget constraint that limits the number of variants that
consumers can purchase. In what follows, we use this model because of its flexibility and
richness. In this study, we use this model in a limited way in that we assume that
consumers will purchase only one item and that the costs of all items are identical. Thus,
we assume that the budget of each consumer is equal to the cost of the product.
Van Ryzin and Mahajan (1999) set up an assortment model in which individual
consumers make purchase decisions according to the MNL model. In their study, the role
of variety is to minimize the probability of no purchasing. This is achieved by providing
12
consumers with a large number of variants so that they are able to purchase their
first-choice variant. The authors assume that if the most preferred variant is not available
then the consumer does not purchase any variant. In other words, they ignore the effects
of substitution and study only the effects of the length of the product line on market share.
They show the existence of an optimal assortment, which is composed of the most
popular variants. Later, Mahajan and Van Ryzin (2000) present a retailing assortment
model based on the MNL utility-choice framework and sequential substitution, and they
use a sample path algorithm to search for the optimal solution. They show that the
algorithm is only able to find stationary points and that it is difficult to guarantee a global
optimum for inventory stocking levels—although the algorithm appears robust in
numerical testing. In addition, with regard to substitution, a retailer should stock more of
the most popular variants than a traditional newsboy analysis would indicate.
Gaur and Honhon (2005) generalize the Lancaster attribute-space model to analyze
the assortment planning. They show that firms provide greater variety when they consider
the substitution effect and that the optimal assortment need not include the most popular
product. Honhon et al. (2006) modeled consumer preference as consumer types in order
to study the optimal assortment and inventory problems with different prices and costs.
They limit the preferences, or substitution behaviors, to one-sided attempts. That is, either
variant A is preferred over B by all customers or variant B is preferred over A by all
customers. When the number of each type of customer is random, they show that a
13
dynamic programming algorithm yields an upper bound of the optimal solution. Caro and
Gallien (2006) build up a stylized, multi-armed, bandit model with Bayesian learning to
study the multi-period optimal assortment problem in the retailing industry. They assume
a perfect replenishment system, and thus inventory is not an issue in their model. Their
analysis yields a closed-form, dynamic index policy that is proved through numerical
experiments to have solid performance. They later extend the model in order to capture
the substitution effect on variety and show that the index in the policy can be updated
accordingly. A very good review of the substitution and assortment problem can be found
in Kok et al. (2006)
1.3 General inventory problem with substitution
In this section, we will build an inventory model that aims to maximize profit with
general consumer substitution. In other words, we will determine the optimal profit the
retailer can achieve given a certain level of variety. This is problem A discussed in
Section 1.1. Traditionally, retailers first estimate the demand, from which they then
decide upon the optimal inventory levels that result in the maximum expected profit.
However, when consumers are allowed to choose another product in case of out-of-stock,
the demand estimation for a variant becomes complicated when coupled with the flow to
and from other variants. We tackle this problem in two stages backwardly. First, we
model the allocation problem. Given the realized preferences of all consumers and
14
inventory levels, and assuming simultaneous substitution, we determine the maximum
sales the retailer can achieve. That is, the retailer should allocate available inventory to
consumers in such a way that sales are maximized. At the second stage, after solving the
allocation problem, we continue to develop a general expected-profit-maximization
model, solving for optimal stocking levels under the assumptions of randomized
consumer utilities and fixed variety.
To start building the model, we assume that the set of variants the retailer has is K :
} ,..., 1 { K K =
.
Let 0 stands for the no-purchase option. The size of the potential customer pool is N.
As we discussed before, N is assumed to be fixed in order to isolate the source
uncertainty as being only consumer preferences. We count consumers as potential
customers if they are interested in the product (category).
U denotes the utility matrix:
] [
i
j
u U =
} ,..., 1 , 0 { }, ,..., 1 { K j N i ∈ ∈
,
where
i
j
u
is the utility of the
th
i customer consuming variant j. Notice that for a
consumer, his or her preference ranking can be any sequence of these K variants and that
each choice in this ranking is random and generated from the consumer-choice model.
There are several consumer-choice models that can generate a utility matrix for all
customers. In Section 7, we will briefly discuss three commonly used models. Generally
speaking, the selection of a choice model will not affect our analysis below as long as the
15
model is capable of generating preference ranking for each consumer, assuming equal
popularity among variants.
Let the order quantity be ) , (
, 2 , 1 K
Q Q Q Q Κ = , where ) ,..., 1 ( K j Q
j
= is the order for
variant j . With realized utilities (U ) for N customers, the following integer program can
be used to determine the optimal allocation. The maximization solves the best sales a
retailer can get assuming a specific realization of the utility:
∑∑
= =
=
N
i
K
j
i
j
x U Q f
1 1
max ) , (
s.t.
}. 1 , 0 {
) ,..., 1 ( ,
) ,..., 1 ( ,
) ,..., 1 ( , 1
1
0
1
0
∈
= ≤
= >
= =
∑
∑
∑
=
=
=
i
j
j
n
i
i
j
i
k
j
i
j
i
j
k
j
i
j
x
K j Q x
N i u u x
N i x
The first constraint follows the assumption that each customer buys at most one
product. The second constraint prescribes that the product purchased has to have a
realized utility larger than the realized utility of the no-purchase option. The last
constraint is simply the inventory constraint, assuming that the seller orders
) , (
, 2 , 1 K
Q Q Q Q Κ = . Assuming simultaneous substitution in the above integer
programming, a consumer will not necessarily be allocated their first-choice product, or
even their second preferred item. This does not match reality, but such a drawback can be
16
partly overcome. In the next section, we will show that there exists an allocation scheme
such that the customer will have at least their second-choice product.
It is well known that the above integer program is equivalent to a transportation
problem and that it can be solved as a linear program. The above formulation assumes a
specific realization of ) 1 ( + × K N random variables (utilities). Thus, given the utility
preferences, variety, and purchasing quantities, it is easy to allocate the available
inventory to consumers so that sales are maximized and consumers obtain a variant that
has a utility larger than the no-purchase utility.
Given the purchase quantities, theoretically one can solve a stochastic LP and
calculate the expected profit over all possible utility realizations. Let us denote this
expectation as )] , ( [ U Q f E
U
. In the profit-maximization problem, order quantities will
be our decision variable. That is, we need to choose the order quantity given variety such
that the profit can be maximized over uncertain consumer preferences. We can write the
model as follows:
) ( max Q
Q
Π
∑ ∑
= =
− − − = Π
K
i
i
K
i
i U U
Q c U Q f Q h E U Q f rE Q
1 1
))] , ( ( [ )] , ( [ ) ( ,
where
r: Product unit price
c: Purchasing cost/unit
h: Holding cost/unit
17
Such a problem is difficult to solve directly, and it is impossible to get an analytical
solution. One of the main obstacles is that solving the stochastic LP with large size is not
practical. It gets even more intractable if we would like to consider variety as a decision
variable. Instead, in the next section we will approximate the optimal solution using a
simple heuristic to calculate the optimal inventory policy. Once we establish the
approximated solution to the inventory problem, we will continue to analyze the
assortment problem, and we will show that the general model above can be transformed
into a one-dimension maximization problem that is easier to solve.
1.4 Approximation with a simple inventory policy
In this section, we will develop our main theorem by showing that a simple inventory
policy can be used to approximate the optimal stocking policy in the previous section.
In Section 1.4.1, we set up the demand structure and have an estimate for the demand
of each variant. In Section 1.4.2, we prove that demand can be satisfied under some mild
conditions. In Section 1.4.3, we show that required conditions in the main theorem are
rather mild and can be easily satisfied.
1.4.1 Demand approximation
Instead of finding the optimal inventory policy, we propose a stocking heuristic,
which is optimal with a very high probability. Intuitively, because we assume statistically
18
identical variants, the inventory level should be the same across all variants. The next
question will be to decide the total inventory to maintain. Ideally, we wish to purchase
exactly the N units of product—with each variant having N/K units—that will satisfy all
of the N customers. However, due to the randomness in consumer preferences, the
first-level demand for a variant may go above N/K and create a shortage. Even though the
customer is allowed to substitute when he or she encounters out-of-stock, it is not certain
that the customer can have the next preferred item because the second-priority item may
also be in a shortage status. However, we propose that it is sufficient for the retailer to
purchase a total of N units (N/K units of each variant) because by doing so the retailer has
a very high probability of satisfying the demands of all customers. That is, the optimal
sales can be approximated well with this simple stocking policy. Let us formally denote
the heuristic policy as follows:
Simple stocking policy With known total demand of N customers and with K
statistically identical variants, the stocking quantity of each variant is N/K, and the total
number of stocking units is N.
Next we will show that the probability that the retailer can satisfy all demands with
this simple stocking policy is indeed very high. One important assumption for this
heuristic to be optimal with a very high probability is that the number of customers
cannot be too small. An investigation of how large this number should be will be
meaningful before we continue to develop the main result. Lemma 1 in what follows
19
serves this purpose. Let us have some notations that are essential for structuring the
heuristic. The following terms all refer to one realization of customers’ preference
rankings.
}) ,..., 1 { , , ( K j i j i d
ij
∈ ≠ — Pair demand
It represents the group of customers having a first choice for product i followed by
a second choice of product j in the preferences realization.
M A subset of variants.
M
D
This is the sum of all pair demands with at least one choice that belongs to M. For
example,
if } , { j i M = , then
∑ ∑
≠
=
≠ ≠
=
+ + + =
K
i l
l
K
i l j l
l
lj jl li il M
d d d d D
1
,
1
) ( ) ( .
Let m be the number of variants in the set M, and let 2 1 − ≤ ≤ K m be the total number
of pair demands included in calculating
M
D , which is
2
2 m m mK − − . The probability
of choosing any of them is 1/K(K-1). Because the probability of choosing one of the pairs
in
M
D follows the binomial framework,
M
D can be approximated with a normal
distribution with mean
) 1 (
2
2
−
− −
=
K K
m m mK
N
m
μ and variance
)
) 1 (
2
1 (
2
2
−
− −
− =
K K
m m mK
m m
μ σ . Clearly, for any two sets
I
M and
J
M (
J I
M M ≠ ),
I
M
D
and
J
M
D are correlated as N is fixed. Let’s take a closer look at the covariance between
I
M
D and
J
M
D . As N is fixed, the pair demand }) ,..., 1 { , , ( K j i j i d
ij
∈ ≠ follows a
20
multinomial distribution with parameter )) 1 ( / 1 ),..., 1 ( / 1 , ( − − K K K K N . The variance of
a pair demand is
2 2 2 2
) 1 ( / ) 1 ( − − − = K K K K N σ .
The covariance of two different pair demands is
2 2
) 1 ( / cov − − = K K N . By the additive
law of covariance, the covariance of
I
M
D and
J
M
D can be written as
∑
∈
∈
=
J
J
ij
I
I
ij
J I
M d
M d
J
ij
I
ij M M
d d D D
~
~
) , cov( ) , cov( .
Some covariance are negative, while the others are positive. For example, when 3 = K ,
the variance-covariance matrix is
− −
− −
− −
2 1 1
1 2 1
1 1 2
9
N
.
This makes it very difficult to develop theoretical support regarding the sum of the
different random variables
M
D . In what follows, we first present out main finding and
then the supporting evidences.
1.4.2 Main theorem
The following assumptions will be held in the following discussion.
I. We consider K identical variants.
II. For customer ) ,.., 2 , 1 ( N i i = , there are at least two variants, j and l, such that
) ,..., 1 ( ,
0
N i u u
i i
j
= > and ) ,..., 1 ( ,
0
N i u u
i i
l
= > . That is, for each customer,
21
there are at least two variants with realized utility higher than the realized
no-purchase utility.
Notice that these assumptions are fairly mild and practical. In reality, many
consumers will have a third or even a fourth choice. We simply require that each
customer have at least one alternative to his or her first preference. As for the
no-purchase customers and the group of customers who will not substitute, we will not
include them in the following main approximation. One reason is that there is no
substitution behavior among these people, and so they can be analyzed separately. The
other reason is that the group of customers who have only one choice and the group who
are no-purchase customers are very small, and both decrease very fast as variety increases.
This issue will be discussed in detail in Section 7. As for the third condition, we will
show in the next section that when N is large enough, it can be satisfied with high
probability.
At this point, we need to introduce some more terms that are necessary to prove the
theorem. All of these notations refer to one specific preferences revealing of N
customers.
Direct allocation
An allocation of inventory to N customers based on this specific preference
realization involves applying the following steps.
22
Step 1: Variant ) ,..., 1 ( K j j = is allocated only to consumers whose first preference
is variant j .
Step 2: If, at the end of step 1, there are consumers whose first choices cannot be
satisfied and there are some variants having excess inventory, use the excess to satisfy, as
much as possible, the shortage based on the customers’ second preferences. If, at the end
of step 2, there are still shortages and excesses, we say that an incomplete status ( IS ) is
reached.
IS (incomplete status)
An IS is a state in which there is excess inventory, but it cannot be allocated to
consumers because it consists of variants that are not among the first and second choices
of the consumers. In such a case, there are consumers who cannot satisfy their demands
and purchase a variant of their first or second choice.
Communicate
In any IS , we say that variant i and variant j communicate if the demands for
ij
d and
ji
d ( i j ≠ ) are not satisfied by a single variant. That is, at least one customer
gets variant i , and the same is true for variant j . The importance of the communicate
property is as follows. Suppose we have reached the IS and variant i and variant j
communicate. Also suppose that there is an excess of variant i and a shortage of variant
j. Because the two demand sets communicate, it is clear that at least one consumer
purchases variant j but that this demand can also be satisfied by variant i . By satisfying
23
this demand with variant i , it is possible to reduce the shortage for variant j and at the
same time reduce the inventory of variant i . It is easy to see that the communicate
property is transitive.
Theorem 1: Assuming conditions I–II and the simple stocking policy, for a realization of
all consumer preferences, if
) / ,..., / ( ) ,..., (
2 2
1
2 2
1
K N m k N m D D
K
M M
K
K
K − −
≥
− −
,
where ) 2 2 ,..., 1 ( − − = K i m
K
i
is the number of variants in set ) 2 2 ,..., 1 ( − − = K i M
K
i
,
there exists an allocation (assuming simultaneous substitution) such that the demands of
all customers are satisfied.
Proof. We provide an algorithm so that the N units of inventory are allocated to the N
customers. We start with direct allocation, after which two things can happen. One is that
all products are allocated after direct allocation and nothing further needs to be done. The
other applies to an IS, which we will call
0
IS . In
0
IS , let E be the set of variants that have
excess inventory, and let S be the set of variants that are short. Because the retailer stocks
exactly N units of inventory, it is clear that having one of these sets empty while the other
is not is impossible. In addition, the total excess units should be equal to the total
shortage units. We then apply the following algorithm to further allocate the excess units.
1. Select E i∈ . If there does not exist S j∈ such that i communicates with j,
go to step 2. Otherwise, select S j∈ such that i communicates with j, and use
24
one unit of i to fulfill one unit shortage of j. Repeat, using variant i to satisfy
the demand for variant j until it is impossible to continue. Go to step 3.
2. Select ) , ( ,
1 1
j i l S l ≠ ∉ such that
1
l communicate with i and some S j∈ .
Transfer excess of i to j until this process can the transfer no longer continue.
Go to step 3.
3. If there is no more excess, the allocation is finished. If there is excess, we reach a
new IS. Update set E and S, then select E i∈ and return to step 1.
We now show by contradiction that the above process must end with all inventory
allocated and all demand satisfied. Suppose the process cannot continue. In this case, it
must stop in step 2, where we cannot use the excess inventory to satisfy any shortage.
That is, all variants are separated into two groups, E
~
and S
~
, such that E E
~
⊆ and
S S
~
⊆ , and that no variant in E
~
communicates with any variant in S
~
. Therefore, the
total number of customers demanding a variant in set E
~
is less than the total inventory
available for set E
~
. However, this is a violation of the assumption that
) / ,..., / ( ) ,..., (
2 2
1
2 2
1
K N m k N m D D
K
M M
K
K
K − −
≥
− −
, which states that the total number of
customers demanding any subset of variants should be at east as large as the available
inventory of the set. In other words,
E
D ~ < μ
E
m ~ serves as a contradiction to the
assumed condition. □
25
Also, notice that with each step the total excess and shortage decrease. Thus, it is
impossible to have infinite cycles, and so after a finite number of iterations all inventory
is allocated and all demand is satisfied.
A very important condition for this theorem to hold is that
) / ,..., / ( ) ,..., (
2 2
1
2 2
1
K N m k N m D D
K
M M
K
K
K − −
≥
− −
. In the next section we will show that this
indeed can be satisfied easily when N is not too small.
Notice that another way to present the problem is through the matching (assignment)
in graph theory. The N units of product can be considered as N jobs to be assigned to N
workers, which are our customers. A customer having a variant in the first or second
choice is analogues that a worker is able to do the job. There are in total K types of jobs
with N/K units of each type. By the Hall’s theorem, the necessary and sufficient condition
for N jobs to be completely allocated to N customers is that the number of workers able
to the job(s) in any subset of jobs is larger than the available jobs. For our problem, one
can show that this condition can be reduced to the condition
) / ,..., / ( ) ,..., (
2 2
1
2 2
1
K N m k N m D D
K
M M
K
K
K − −
≥
− −
, which states that the available inventory
for any subset of variants is no larger than the total number of customers demanding
variants in that subset.
1.4.3 Total number of customers
Recall that we mentioned earlier that the number of customers (N) cannot be too small.
26
Here, we present a lemma that defines this restriction explicitly.
Lemma 1: For K identical variants, there exists N(K) such that for ) (K N N ≥ ,
99 . 0 ) ,..., Pr(
2 2
1
2 2
1
≥ ≥ ≥ =
− −
− −
K
N
m D
K
N
m D P
K
M M JNT
K
K
K
Proof: Let
i
E be the event that
K
N
m D
i M
i
< ) 2 2 ,..., 1 ( − − = K i
K
. The probability in
interest can thus be written as
Υ
2 2
1
2 2
1
) Pr( 1 ) ,..., Pr(
2 2
1
− −
=
− −
− = ≥ ≥
− −
K
i
i
K
M M
K
K
K
K
E
K
N
m D
K
N
m D .
By the Bonferroni Inequality, we have
∑ ∑
− −
=
− −
=
− −
=
< = ≤
2 2
1
2 2
1
2 2
1
) Pr( ) Pr( ) Pr(
K
i
i M
K
i
K
i
i i
K
i
K K
K
N
m D E E
Υ
.
Thus, we have
∑
− −
=
− −
=
< − ≥ − =
2 2
1
2 2
1
) Pr( 1 ) Pr( 1
K
i
i M
K
i
i JNT
K
i
K
K
N
m D E P
Υ
.
Next we will show that
∑
− −
=
<
2 2
1
) Pr(
K
i
i M
K
i
K
N
m D can be small as we need by choosing
the proper N.
For any set M with m (
2 1 − ≤ ≤ K m
) variants,
M
D
can be approximated by normal
distribution with mean
) 1 (
2
2
−
− −
=
K K
m m mK
N
m
μ and variance
)
) 1 (
2
1 (
2
2
−
− −
− =
K K
m m mK
m m
μ σ . If we standardize the normally distributed demand, we have
the following:
}
/
Pr{ ) Pr(
m
m
m
m M
M
K mN D
K
N
m D
σ
μ
σ
μ −
<
−
= < ,
27
and
m
m
m
K mN
z
σ
μ −
=
/
.
Substituting for
m
μ and
2
m
σ , we get:
) 1 )( 1 2 (
) (
− − − −
−
− =
m K m K
m K Nm
z
m
.
Let
) 1 )( 1 2 (
) (
) (
− − − −
−
=
m K m K
m K m
m g . Taking the derivative with respect to m, we
have the following:
0
) ) 1 ( ) 1 ( 3 2 (
) ) 1 ) ( 2 )( )(( 1 ( ) (
2 2 2
>
+ + + −
+ − − − −
=
∂
∂
m m K K
m m K m K K
m
m g
.
Thus, ) 1 ( )} ( min{ g m g = and
) 2 ( 2
) 1 (
~
max
−
− = ∗ − = =
K
N
g N z z
m
.
Thus we have
)
) 2 ( 2
( ) 2 2 ( ) Pr(
2 2
1
−
− Φ − − ≤ <
∑
− −
=
K
N
K
K
N
m D
K
K
i
i M
K
i
.
Combine all of the above, we have
∑
− −
=
− −
=
− ≥ − =
2 2
1
2 2
1
) Pr( 1 ) Pr( 1
K
i
i
K
i
i JNT
K K
E E P
Υ
)
) 2 ( 2
( ) 2 2 ( 1
−
− Φ − − − ≥
K
N
K
K
,
where ) (• Φ is the cumulative distribution function of standard normal. It is easy to see
that there exists N(K) for fixed K such that 99 . 0 ≥
JNT
P when N>N(K). Specifically, we
can write N(K) as follow
2 1
))
2 2
01 . 0
( )( 2 ( 2 ) (
− −
Φ − =
−
K
K K N
K
.
The table below gives examples of N(K) at different K.
Table 1: Examples of N(K)
28
3 15
4 39
5 68
10 292
15 647
20 1138
25 1764
30 2528
35 3427
40 4465
45 5643
50 6604
55 7292
60 7980
65 8667
70 9358
75 10043
80 10731
1.5 Extensions
1.5.1 Uncertain demand
In the previous discussion, we assumed that N is known with certainty. We will relax
this assumption by the following corollary.
Corollary 1.1: Suppose that the number of consumers, N, is random with CDF
N
F . Then,
the optimal stocking level is ) (
~
1
α
−
=
N
F Q , where
h r
c r
+
−
= α . The stocking level for each
variant is then
K
Q
~
.
Proof: For every realization n of N where n Q ≥
~
, it is clear that all of the demand
is satisfied and that the excess inventory is equal to n Q−
~
. Also, for n Q <
~
it is clear
29
that the demand of Q
~
customers is satisfied and that the shortage is equal to Q n
~
− .
Thus, it is easy to see that the expected profit function is identical to the profit function of
a news-vendor problem. □
Theorem 1, together with Corollary 1.1, illustrates that safety stock is kept because of
the uncertainty in the market size but not because of the uncertainty in consumer choice.
It also demonstrates that when the market size is random it is possible to solve the
problem by solving a simple news-vendor problem.
1.5.2 Unequal popularity
In the above discussion, we assumed that all variants are “equally popular,” by which we
mean that all variants have the same probability of being at a certain position in the utility
ranking of each of the consumers. We will now show that this assumption can be relaxed.
We say that variant i is more popular than variant j if the probability of i being on the
top of the utility list is higher that of j. That is,
j i
p p ˆ ˆ > . Without loss of generality,
assume that
K
p p p ˆ ˆ ˆ
2 1
> > and let p ˆ be a base probability such that p e p
i i
ˆ ˆ = and for
j i, ,
j i
e e > . Notice that there are many choices of base probability, p ˆ , depending upon
the users’ preferences. For example, one can set
K
p p ˆ ˆ = , but the resulting stocking level
for each variant will not be altered by having a different p ˆ . From now on, we will say
that there are
∑
=
K
i
i
e
1
“variants” (and not K). Clearly, some of these “variants” are
identical to each other, but we will consider them as being different. Notice that now we
30
have a problem with
∑
=
K
i
i
e
1
variants and that these variants are identical in the sense that
they have the same probability to have the highest utility in the utility-ranking list.
Stocking
∑
=
K
i
i
e N
1
/ units of each of the variants and applying Theorem 1 ensures that we
can satisfy all of the demand without keeping any safety stock.
1.5.3 Gap between simultaneous and sequential substitution
Theorem 2: Assume a sequential substitution problem in which N is known with
certainty. Also, assume that N/K units of each variant are stocked. Let Λ be the number
of customers whose demand is not satisfied (the number of units in inventory), and let Λ
~
be the percentage of customers whose demand is not satisfied. Then N
2
3
< Λ and
N 2
3 ~
≤ Λ .
Proof: Notice that the direct allocation provides a feasible allocation to the
sequential substitution problem. First, customers satisfy their demand with their
first-priority variant, and when the first-priority variant is not available they choose their
second-priority. If the second-priority variant is not available, they do not purchase any
product and leave the system.
Therefore, the maximum unsatisfied demand, after direct allocation, indeed serves
as a gap between the sequential substitution and simultaneous substitution. Let us denote
this gap by Λ .
31
We will now estimate the magnitude ofΛ . Suppose that after direct allocation we
have ) 1 1 ( − ≤ ≤ K m m variants that are in shortage. By the demand approximation in
Section 4.2 (and assuming that P
0
= 0), we know that with high probability (over 99%)
the maximum shortage for these m variants is
m
σ 3 . Therefore, we have
m
K m
m
σ 3 max
1 1 − ≤ ≤
= Λ ,
where
2
2
) (
K
m K m
N
m
−
= σ .
Obviously, the variance is maximized when
2
K
m= and N
2
3
= Λ .
□
Assuming our simple ordering policy and sequential substitution, we then know that
the maximum shortage or excess, as a fraction of the total customers, is
N 2
3
. For
example, when 000 , 10 = N , then in the worst case only 1.5% of the customers are facing
shortages, and at the same time only 1.5% of the goods are in excess.
Theorem 2 tells us that when the size of the market is large and known with certainty
the expected shortage and excess inventory is relatively small. The above numbers
present the worst possible case, and we conjecture that, on average, expected inventory
shortages and excesses will be lower.
Another extension is to study the problem under a specialized structure, which tends
to make the problem easier. Jordan and Graves (1995) studied manufacturing flexibility
analogous to the substitution problem. They configured limited flexibility in what we
32
have called the chain substitution structure. By simulation, they show that the chain
substitution structure yields most of the benefits of total flexibility (substitution). This
can also be proved by the main theorem in this paper with slightly modified lemma 1. For
the length purpose, we omit the details.
1.6 The assortment problem
In this section, we will solve the optimal assortment (problem B in the Introduction)
by applying the approximated scheme of the inventory problem in the previous section.
As we mentioned earlier, variety is costly. Major sources of variety cost include
management overhead, supplier production cost, and consumer searching cost. On the
other hand, increasing variety has advantages mainly in decreasing the no-purchase
probability and thus increasing the market share. In what follows, we take a simplified
approach and assume that the mean market size is known to be ) 1 (
0
P N − , where
0
P is
the probability of no-purchase and a function of variety. Whether a customer is a
no-purchase or not follows a binomial framework, and the number of customers who are
willing to purchase, denoted as D, can be approximated by a normal distribution with
mean )) ( 1 (
0
K P N
D
− = μ and variance )) ( 1 )( (
0 0
2
K P K NP
D
− = σ .
By Corollary 1.1, for a specific K, we should set ) (
1
α
−
=
D
F Q , where
h r
c r
+
−
= α and
stocks K Q q K i Q
i
/ ) ,..., 1 ( = = = for each variant, which will satisfy these customers’
33
demands with very high probability. With these, we can write the variety problem as
follows:
0
) (
~
max rNP K K c
K
− Π + −
β
cQ D Q hE Q D E r K
D
− − − − − = Π
+ +
] [ ) ] [ ( ) ( μ ,
where
D D D
z F Q σ μ α
α
+ = =
−
) (
1
,
h r
c r
+
−
= α and K Q q K i Q
i
/ ) ,..., 1 ( = = = .
Here, c
~
is the constant factor of variety cost, which can be thought of as consumer
shopping cost. The more variants that a consumer has to examine and consider, the higher
is the cost. Also, we make variety, K, the decision variable, and we introduce no-purchase
customer in the profit function in order to capture the tradeoff between market size and
variety to be offered. If the retailer offers too little variety, it may lose a fairly large
market share as ) (
0
K P gets larger. We can see that there are two types of shortages in
the model. One is the no-purchase customers, who are not willing to purchase because
they do not have any preferred product in the category. The other is the shortage incurred
by insufficient inventory. Both will be considered as lost sales.
Clearly, it is very difficult to have closed-form solution to this problem because the
change of K will change the demand distribution. In what follows, we will investigate
numerically how the profit shifts with different parameters, and we will also assume that
K
q K P = ) (
0
, while q can be determined by a consumer-choice model or simply
estimated by the retailer. In general, it gives the odds of one variant having utility lower
than the no-purchase option.
34
5 10 15 20 25 30
40000 60000 80000
Variety
ExpectedProfit
5 10 15 20 25 30
40000 60000 80000
Variety
ExpectedProfit
5 10 15 20 25 30
40000 60000
Variety
ExpectedProfit
5 10 15 20 25 30
30000 50000
Variety
ExpectedProfit
1.6.1 Variety cost
In this test, we hold everything fixed except the cost of variety. In modeling the
variety cost, we first assume that 5 . 0 = β , which was introduced in 1956 by Baumol and
Ide. Table 8 summarizes the results, and Figures 1 to 4 plot the profit curves in each
setting.
Table 2: Profit Simulation with Varied Variety Cost
5 . 0 = β
q n r c h c
~
Max Profit Best
K
Figure 1 0.5 10,000 10 2 1 10 79,956 17
Figure 2 0.5 10,000 10 2 1 300 78,911 12
Figure 3 0.5 10,000 10 2 1 1,000 76,651 10
Figure 4 0.5 10,000 10 2 1 5,000 65,333 7
Holding everything else unchanged, when variety cost becomes larger, the best
variety to offer and the maximum profit obtained at the best K becomes smaller. However,
the choice of K looks somewhat insensitive to c
~
. That is, even when c
~
increases 100
Figure 3: Variety Cost at 1000 Figure 4: Variety Cost at 5000
Figure 1: Variety Cost at 10 Figure 2: Variety Cost at 300
35
0 5 10 15 20 25 30
0 20000 60000
Variety
ExpectedProfit
0 5 10 15 20 25 30
-200000 -100000 0
Variety
ExpectedProfit
0 5 10 15 20 25 30
-8e+05 -4e+05 0e+00
Variety
ExpectedProfit
0 5 10 15 20 25 30
-4e+06 -2e+06 0e+00
Variety
ExpectedProfit
times from 10 to 1,000, the best K only drops from 17 to 10. One important reason is the
way we model the variety cost, which varies across different industries. For example,
offering a t-shirt with a new style may cost much less than a car dealer offering a car with
a new style. If we let 2 = β and hold all other parameters the same as the ones in Table
9, we will have the following result. Figures 5 to 8 plot the profit curves in each setting.
Table 3: Profit Simulation with Quadratic Variety Cost Function
2 = β
q n r c h c
~
Max Profit Best
K
Figure 5 0.5 10,000 10 2 1 10 78,822 9
Figure 6 0.5 10,000 10 2 1 300 66,811 5
Figure 7 0.5 10,000 10 2 1 1,000 52,662 4
Figure 8 0.5 10,000 10 2 1 5,000 14,842 2
Figure 5: Result 1 with Quadratic Function Figure 6: Result 2 with Quadratic Function
Figure 7: Result 3 with Quadratic Function Figure 8: Result 4 with Quadratic Function
36
Compared with the results in Table 2, the maximum profit obtained at the best K
drops much faster as the cost of variety increases. The best choices of variety are much
smaller than previous ones. Interestingly, even when the c
~
is large—at 5,000 in this
case—the seller is still better off by offering the minimum variety, 2, than by
standardizing the product (K=1). The reason is that by offering 2 variants, the number of
no-purchase customers drops substantially, which leads to larger market share and thus
outweighs the additional cost incurred by this new variant. In the next section, we will
see how the no-purchase probability affects the choice variety and profit.
1.6.2 No-purchase probability
The no-purchase probability is calculated by
K
q K P = ) (
0
. In general, q gives the odds of
one variant having utility lower than the no-purchase option. That is, with probability
(1-q), a variant will be selected in a customer’s choice set (i.e., the set of variants that the
customer is willing to buy). Clearly, the larger the q, the more the customers are not
willing to buy anything in the category. The value of q may indicate the quality,
appearance, and attractiveness of the product, and this can be estimated from a consumer
choice model or simply by consulting with the marketing manager. Table 10 summarizes
the results with different values of q, holding everything else unchanged. For illustration
purposes, we fix c
~
at 5,000, which makes it easier to observe the curve of the function.
Note that even if we decrease c
~
, the profit function will still have the maxima in the
numerical test, but will result in a lager K.
37
Table 4: Profit Simulation with Varied No-Purchase Parameter
5 . 0 = β
q n r c h c
~
Max Profit Best K
Figure 9 0.2 10,000 10 2 1 5,000 69,867 3
Figure 10 0.4 10,000 10 2 1 5,000 66,992 6
Figure 11 0.7 10,000 10 2 1 5,000 60,192 13
Figure 12 0.9 10,000 10 2 1 5,000 45,891 36
2 4 6 8 10
64000 66000 68000 70000
Variety
ExpectedProfit
2 4 6 8 10 12 14
45000 55000 65000
Variety
ExpectedProfit
5 10 15 20 25 30
0 20000 60000
Variety
ExpectedProfit
10 20 30 40 50
-60000 -20000 20000
Variety
ExpectedProfit
The results in Table 10 show that the best choice of K increases with q, while the
maximum profit obtained at the best K decreases. This is because when q increases the
product becomes less desirable for the customers and the retailer has to offer more
variants in order to reduce the loss of market share incurred by the larger q.
Figure 9: Result at q=0.2 Figure 10: Result at q=0.4
Figure 11: Result at q=0.7 Figure 12: Result at q=0.9
38
1.6.3 Purchasing cost and selling price
In this test, we will study how the best choice of variety and the resulting profit
changes with the purchasing cost and the selling price. Clearly, the change from either of
these parameters will change the profit margin and the shortage cost. We first vary
purchasing cost holding everything else unchanged and seek the best profit and variety.
Table 5 summarizes this result. We then select different selling prices while holding
everything else fixed in order to observe the result on maximum profit at best variety.
Table 6 summarizes the second result. Figures 13 to 20 are plots of the profit functions at
different parameters.
Table 5: Profit Simulation with Varied Purchasing Cost
5 . 0 = β
q n r c h c
~
Max Profit Best
K
Figure 13 0.5 10,000 10 1 1 5,000 75,261 7
Figure 14 0.5 10,000 10 3 1 5,000 55,406 7
Figure 15 0.5 10,000 10 6 1 5,000 25,641 7
Figure 16 0.5 10,000 10 8 1 5,000 5,842 6
39
2 4 6 8 10
40000 60000
Variety
ExpectedProfit
2 4 6 8 10
20000 35000 50000
Variety
ExpectedProfit
2 4 6 8 10
0 10000 20000
Variety
ExpectedProfit
2 4 6 8 10
-15000 -5000 0 5000
Variety
ExpectedProfit
Table 6: Profit Simulation with Varied Selling Price
5 . 0 = β
q n r c h c
~
Max Profit Best K
Figure 17 0.5 10,000 10 5 2 5,000 35,558 7
Figure 18 0.5 10,000 20 5 2 5,000 134,442 8
Figure 19 0.5 10,000 30 5 2 5,000 233,884 9
Figure 20 0.5 10,000 50 5 2 5,000 433,223 10
Figure 15: Result at c=6 Figure 16: Result at c=8
Figure 13: Result at c=1 Figure 14: Result at c=3
40
2 4 6 8 10
5000 15000 25000 35000
Variety
ExpectedProfit
2 4 6 8 10 12 14
60000 100000
Variety
ExpectedProfit
2 4 6 8 10 12 14
100000 160000 220000
Variety
ExpectedProfit
2 4 6 8 10 12 14
200000 300000 400000
Variety
ExpectedProfit
It is easy to see that the choice of variety is not sensitive to the change in profit
margin or the shortage cost. One reason is that, with fixed variety K, the variance of the
distribution is relatively small as compared with the mean. In turn, the expected shortage
and excess units are also small. Thus, the results are insensitive to the fluctuation of these
parameters.
To summarize, the total profit function is concave in most of the numerical study,
although in some extreme cases it still becomes a monotone decrease. For example, when
the cost of variety is extremely high, offering just one more variant will cost much more
than the gain in market share. Thus, it is better to standardize the product. However, in
Figure 17: Result at r=10 Figure 18: Result at r=20
Figure 19: Result at r=30 Figure 20: Result at r=50
41
Section 6.1, we see that even when the variety cost parameter is 500 times the selling
price, the seller is better off not to standardize the product. Thus, the extreme case in
which the cost of variety is so high that it is better to standardize the product rarely exists.
Though profit function exhibits a concave shape, the top of the function is fairly flat
in many cases. This tells us that offering somewhere around the best K will give us a
profit roughly around the maximum one. For the example shown in Figure 3, offering 12
variants results in the max profit of 78,911. The following is the list of profit obtained at
different K in this example.
Table 7: Profit at Different K in Figure 2
K 8 9 10 11 12 13 14 15
Profit 78,426 78,732 78,864 78,909 78,911 78,892 78,864 78,831
Clearly, even if the manager does not choose the best one, 12, but selects a number
around 12, the resulting expected profit will not become substantially lower. However,
the profit function does not always have a flat top. It is hard to say when the shape looks
smooth at the top.
At times, when the variants are not equally popular, we can apply the result in Section
5.2 to find the total expected profit. As the more popular items capture more market share,
it is easy to see that it is always better to include the most popular items in the category.
42
1.7 About utility model and no-purchase probability
1.7.1 Consumer choice models
Two commonly used choice models are MNL and Lancaster’s location model. Here, we
will use a third one as an example. Notice that our main result in Section 4 does not rely
on the choice of any model but simply requires using a model that is able to generate a
preference ranking for each customer independently. The choice model below was first
presented by Kim et al. (2002). Because it is an analogue to the MNL model, the nominal
and random terms in this model are assumed to be log linear. Also, the final choice can be
constrained to the customer’s budget limit. The
th
i consumer’s total utility is expressed
in the following way:
∑ ∑
× + = =
j
j j j
j
j
i
j
i
i
j j
e x x u x u
ε γ
α ψ ) ( ) ( ) (
,
where ) 1 , 0 ( ~ N ε .
In this model,
j j
γ ψ , , and
j
α are parameters describing the
th
j variant, and
i
ε is
a random variable representing the uncertainty that the retailer is facing. The realization
of the random variable becomes known to the retailer only when the
th
i customer comes
to shop and reveals his or her preference. That is, the utility of consumer i is the sum of
utilities that results from consuming
j
x units of variants j.
Every consumer solves the following problem:
max ) (x u
i
43
subject to E RX ≤ ,
where R = (
K
r r ,...,
1
) is the price of variant 1,…,j,…K, and E is the total budget available
for purchasing the variants.
We chose this model as an example because of its richness—it has the capability to
solve for multiple selections with a budget constraint for each customer. However, as a
starting step, we assume for the sake of simplicity that each consumer buys only one
product. This is equivalent to assuming that the consumer’s budget is equal to the price of
the product. We believe that by limiting the model by ignoring the issue of consumers,
the purchasing of multiple units is justified by the simplicity of our model and the strong
result we obtain.
1.7.2 No-purchase probability
In this section, we will use the consumer choice model in the previous section as an
example to develop the relation between no-purchase probability and variety. Let
j
p
( } ,..., 2 , 1 { K j∈ ∀ ) denote the probability of having variant j in the choice set—that is,
} Pr{
0
u u
j
> . We denote
j
u as the utility of consuming one unit of variant j . A
customer is called a “no-purchase customer” if the choice set of the customer is empty.
That is, the probability that a customer is a no-purchase customer is:
∏
=
− =
K
j
j
p P
1
0
) 1 ( .
44
We can interpret
0
P as the loss of market share, and it is clear that this probability is
decreasing with the number of variants, K.
For identical customers, we assume that the consumer uncertainty of choosing variant
j is captured by
j
ε , which is independent and identically distributed following a standard
normal distribution. In line with all of the previous assumptions, the utility for the
th
i
customer to choose item j is:
j j
e u
j j j
ε γ
α ψ × + = ) 1 (
.
The no-purchase utility for this customer is given by:
0 0
) 1 (
0 0 0
ε γ
α ψ e u × + =
.
If we let
0
) 1 (
) 1 (
0 0
γ
γ
α ψ
α ψ
δ
+
+
=
j
j j
j
, we have
) 2 / (log ) log ( ) (
0 0 j j j j j
P u u P p δ δ ε ε Φ = < − = > = ,
where Φ is a standard, normal cumulative-distribution function (CDF). Because we
assume identical products, ) ( j i
j i
≠ = = δ δ δ .
K
K
i
i
p P )) 2 / (log 1 ( ) 1 (
1
0
δ Φ − = − =
∏
=
.
It is easy to see that
0
P decreases as K increases. For example, 1 = δ means that
before customers’ realization, the retailer predicts that each variant yields the same utility
as the no-purchase option. In other words, when considering one specific variant and the
no-purchase option, a customer is equally likely to purchase the single variant or decide
not to purchase at all. In this case, if the seller increases variety from 9 to 10,
0
P will
45
decrease by 0.1% (from 0.2% to 0.1%). In most cases, the utility of purchasing a variant
will be larger than the no-purchase utility ( 1 > δ ), and we can see that for a relatively
small number of variants the no-purchase probability is quite small.
In a similar fashion, we can calculate the probability that a customer is not willing to
substitute. That is, if his or her first choice is not available, the customer will leave and
not buy at all. Let us denote this probability as
1
P .
1
1
)) 2 / (log 1 ( * ) 2 / (log *
−
Φ − Φ =
K
K P δ δ .
Notice that both
0
P and
1
P drop quickly as variety increases. In other words, the
probability (
1 0
1 P P − − ) of a customer being willing to substitute at least one time before
deciding not to purchase and to leave the system increases very quickly as variety
increases. The fact that
1 0
1 P P − − is large is important because we start by assuming
that all consumers are willing to substitute at least one time. This is a valid assumption
when the number of variants is relatively large.
1.8 Managerial insight and conclusion
This study should be extremely helpful for managers in making assortment and inventory
decisions for the following reasons. Typically, managers must make the inventory and
assortment decisions under the assumption of demand uncertainty. We show that there
are two sources of demand uncertainty: (1) uncertainty that is related to tastes and choices
of consumers (consumer-choice uncertainty), and (2) uncertainty that is related to the size
46
of the market (market-size uncertainty). Different sources of uncertainty must be dealt
with differently. Consumer-choice uncertainty is dealt with by choosing the right
assortment of variants, while market-size uncertainty is addressed by choosing the right
inventory levels and safety stocks. For example, if the market-size uncertainty is
negligible, then despite the choice uncertainty it is possible to satisfy the demand of all
consumers, with high probability, with no safety stock. In this case, the right assortment
is crucial to satisfying consumer demands and maximizing profits. In addition, to the best
of our knowledge we are the first to offer a simple, closed-form solution to the inventory
problem that ensures that the optimal solution is achieved with a very high probability.
There are several possible venues for extending this work: (1) take into account the
“different product cost” issue and develop an appropriate inventory policy addressing the
substitution effect; (2) investigate the role of variety in competition, in which profit
margin may change with the product-line length; (3) collect real demand and inventory
data from some stores to verify the heuristic.
In conclusion, the major contribution of this paper has been to show that uncertainty
in consumer preference can be eliminated by the substitution effect with a simple
heuristic. Such an approximation approach is fruitful in proving that when the market size
is large, not keeping safety stock will result in a small percentage of consumers who are
unable to obtain the variants of their desire. Inventory thus becomes much less of an issue
than the right choice of product-line length. We are also able to extend this result to
47
situations in which the market size is random. In this case, we show that it is sufficient to
solve a simple news-vendor problem to obtain the optimal stocking level. Finally, we are
able to develop a very simple model to calculate the optimal number of variants. This
simple approach seems to provide a promising tool for the analysis of variety decisions,
and it can be extended to address future research issues in this area.
48
Chapter 2 An Experimental Study on Group Negotiation Power
2.1.Introduction
Negotiation power can be viewed as the ability of the negotiator to influence the
behavior of another negotiator, or the ability of the negotiator to increase his or her share
of the profits. From the perspective of social psychology, there are two sources driving
the power—individualistic and relational influence. Individualistic theories explain
power from the negotiator’s personal traits, such as knowledge, risk-averseness, and
communication skills. Relational theories of power analyze a negotiator’s influence over
or impact upon the other party. A typical example of such influence is the best alternative
a negotiator has if the agreement cannot be reached with the other party. This is
equivalent to the disagreement point in the Nash (1950, 1953) bargaining model. Clearly,
the higher the value of the best alternative for party A over party B, the more powerful
party A will be over party B. Another case showing the relative influence of power is
formation of a coalition, which is common in supply chain contract negotiation. To date,
most research work on negotiation has focused on analysis of the one-to-one bargaining
scenario. In contrast, in this study we are interested in seeing how the action of joining
forces and forming coalitions affects the perceived negotiation power of both sides. The
situation we consider is when several sellers coalesce to negotiate on the wholesale price
49
of a single product with a single buyer. Both parties have no alternative but to reach
agreement—that is, if they do not strike a deal, both parties will have zero profits.
The very important starting step is to model negotiation power because it is
intangible and does not have a direct scale. There are fruitful works in economics on
two-person bargaining, and one of the best-known models is the Nash bargaining solution.
Nash (1950, 1953) models two persons bargaining over the split of profit and can deal
with unequal negotiation powers. Negotiation power in the Nash model is defined as risk
averseness—the more risk averse a negotiator is, the less negotiation power he or she has.
Thus, if the two negotiators are risk neutral, according to Nash, they have the same
negotiation power. For two risk-neutral players, the Nash solution is extended to allow,
even in this case, different negotiation powers. Specifically, the asymmetric (generalized)
Nash bargaining solution is a unique pair of payoffs that solves the following
maximization problem.
π = + = + > >
− − =
B S
q
B B
p
S S x x
x x q p q p
d x d x S
B S
, 1 , 0 , 0
) ( ) ( max arg
) , (
*
.
Here,
S
x and
B
x are the payoffs for the seller and the buyer, respectively;
S
d and
B
d are the disagreement values for the seller and the buyer, respectively; p is the
negotiation power of the seller; and q is the negotiation power of the buyer. Clearly,
negotiation power directly affects the final outcome for the buyer and seller.
Another famous bargaining framework is Rubinstein’s (1982) alternating-offer
bargaining model. In this setting, two players take turns making an offer at time
50
,... ,..., 3 , 2 , , 0 Δ Δ Δ Δ t The discount rate for the seller and buyer are
S
δ and
B
δ ,
respectively. When the two players reach agreement, their agreed payoffs will be
discounted by the time passed at their discount rate. The model has subgame perfect
equilibria as follows:
B S
S
B
B S
B
S
B B
S S
x
x
δ δ
δ
μ
δ δ
δ
μ
π μ
π μ
−
−
=
−
−
=
=
=
1
1
1
1
*
*
.
The discount rates for seller and buyer can be viewed as their negotiation power. A
higher discount rate infers a higher cost of delay and thus lower negotiation power. One
can show that there is a one to one connection between the discount rate in this model
and the power parameters in the Nash model such that the equilibrium solutions of these
two models are equivalent. For simplicity, we refer to the Nash model when indirectly
measuring the negotiation power from the final payoff in the agreement. With a fixed
profit pie, we assume that the mapping from a payoff pair to a negotiation power pair is
unique. Thus, knowing the change in final payoffs will give us information regarding the
change in negotiation power of both parties.
These two models assume no coalition on either side of the negotiation. In order to
study the bargaining scenario involving alliance, researchers usually assume that the
coalition can be viewed as one entity. However, to apply any of the above bargaining
51
frameworks, one has to decide the appropriate negotiation power for this entity, which
turns out to be a very difficult question. For example, when each member of a coalition is
risk averse, we may ask: What is the negotiation power of the coalition? And: What is the
risk-averseness of the coalition? Similar questions can be asked about a coalition of
risk-neutral players. The existing literature on the negotiation power of coalitions
generally assumes one of the following as the negotiation power of the group: (1) the
average negotiation power of its members, (2) the maximum negotiation power of its
members, or (3) the minimum negotiation power of its members. Under one of these
assumptions, researchers can continue to analyze the problem of interest based on either
the Nash or the Rubinstein model. Unfortunately, there is no work addressing the validity
of any of these assumptions. Our study should serve as a first step in understanding the
behavioral effect on group negotiation power.
Ideally, we can run the experiment by asking each pair of subjects to have an
unstructured bargain, in which subjects will “role play” and negotiate with the other party
face to face. However, after a pilot study, we found that it is fairly difficult to have the
subjects be in their roles and control the negotiation process. As a first step, we decided
to design an experiment to investigate the perception of the negotiation power of the
coalition under different circumstances. Specifically, we set up a scenario in which a
monopoly buyer needs to source material from local suppliers in several different
locations. There is one and only one supplier in each location. Suppliers can be
52
considered identical in terms of material quality and cost. Originally each of them
accepted a certain wholesale price by entering into supply contracts that will soon
become obsolete. This price conveys the negotiation power of buyer and the sellers under
the Nash bargaining model assuming one-to-one bargaining. At some point in time, the
suppliers decide to join forces as one entity to negotiate with the monopoly buyer for a
new supply contract. Both parties must reach an agreement, if they don’t their profit is
assumed to be zero. Thus, if the players are rational they should agree to accept any
contract that gives them a positive profit. Figure 1 gives a graphical illustration of the
story. The scenario on the left is the buyer negotiating separately with each seller in each
city. As we assume identical sellers, all contracts between the buyer and all sellers reach
the same wholesale price by the Nash bargaining solution. This price will be given in the
story. Then all sellers decide to form a coalition, which is depicted as the graph on the
right in Fig. 21. A delegate for the sellers will bargain with the buyer for the new
wholesale price, and as before, it is a negotiation between two monopolies, the buyer and
the sellers’ alliance. Subjects are told to be in role play (buyer or sellers’ delegate) and to
perceive the new wholesale price, which in turn translates to the perceived negotiation
power by the Nash bargaining model.
The following assumptions are made in each individual negotiation.
• The buyer is a monopoly and each seller is a monopoly in his or her own city.
• It is legal to join forces.
53
• Once the sellers form a coalition, they cannot break the alliance unless they go
out of business.
• All market conditions and cost parameters remain unchanged, including the
interest rate. In other words, the total profit pie is fixed at the given parameters.
The only new condition is the sellers joining forces. With this new condition,
both parties need to decide what the new split for the profit pie should be.
• Sellers are identical. Each seller negotiated separately with the same buyer to
reach the original wholesale price. We further assume that the market conditions
are the same across different locations. Under these conditions, we set the same
original wholesale price for all sellers.
• All information is disclosed to both parties.
An interesting research question is how the formation of the coalition affects the
negotiation power of both parties. In particular, we may ask which factors are related to
Buyer
Seller 1 in
City 1
Seller 2 in
City 2
Seller N in
City N
Buyer
Seller 1 in
City 1
Seller 2 in
City 2
Seller N in
City N
Join
forces
Figure 21: Sellers form coalition.
54
the change in bargaining power. These also are practical and important issues helping
managers to analyze and define proper strategies in supply chain management. However,
little work has been done on trying to understand the group negotiation power versus the
negotiation power of individuals. To begin, we will investigate the main effect of the
following three factors.
First, we would like to see the perception from perspective of different “roles.” As we
set up a scenario in which all parties have no alternatives, it will be interesting to observe
the behavior of sellers, who join forces, and the buyer, who faces the alliance. Intuitively,
sellers form a coalition with the objective of increasing their profit share, and thus they
tend to become more powerful as a group. However, it is not clear that this is always the
case because the buyer in this scenario is a monopoly. As a starting step, we would like to
observe the behavioral thinking of both parties given the new situation.
Second, we consider the initial profit margins defined by the original wholesale price.
As the total profit pie is assumed to be fixed, the original wholesale price decides the
profit shares for both parties, as well the profit margin. Based on the Nash bargaining
model, these original profit shares reflect the negotiation power for both parties—that is,
the given initial profit margins can be translated into the bargaining power of both parties.
A typical example is the labor union in which workers join to form one entity to bargain
with the employer. Each employee’s negotiation power before the alliance is negligible,
while the gain in power by forming a coalition is substantial. However, it is not clear how
55
this gain compares with the situation in which each member has relatively strong
negotiation power before joining forces. In particular, is there a relation between the
initial negotiation power and the group negotiation power? In a supply chain contract
negotiation, answering such a question is crucial in helping to decide whether or not to
join forces and form coalitions.
Third, the size of the coalition may affect the perception of the coalition by the other
party. In economics, the benefit from a large buyers’ coalitions is called countervailing
power, an effect by which large buyers have an advantage in extracting price concessions.
However, the issue tends to be complicated in practice because there are usually other
factors, such as alternatives. The study by Ellison and Snyder (2001) shows that hospitals
and HMOs obtain substantially lower wholesale prices for antibiotics than do chain
drugstores as they can restrict formularies to increase their substitution opportunities. In
other words, forming a coalition simply to have a size impact has limited benefit
compared with small competitors who have good alternatives in the negotiation. By
designing an experiment, we can isolate the size impact and observe its effect on either
side of the negotiation table. Specifically, we choose two levels of size, small and large.
For the seller’s alliance, it is natural to assume that the impact increases with the coalition
size, and thus we call it a positive impact with the larger coalition size. However, we
propose reverse coding for the buyer who faces the coalition instead of forming the
56
alliance. That is, for the buyer the positive impact of size happens when the sellers’
alliance is small.
The outcome is the new wholesale price perceived to be reached in the negotiation.
We separate this price into three categories for both the sellers’ alliance and the buyer
based on the change of profit margin. For either the seller or the buyer, if the new
wholesale price results in an increase in his or her profit margin, this outcome is in the
“Up” category. That is, under the given conditions, the player perceives that the seller’s
coalition will increase his or her negotiation power. Similarly, if the profit margin
remains unchanged under the new price, the outcome is in the “Unchanged” category,
and it will be in the “Down” category if the profit margin will decrease.
Clearly, the three categories of the dependent variable, which is the change in profit
margin, is ordered from “Down” to “Up.” Later, we will fit the data by ordinal regression
to capture the ranking of these three categories. With our regression results, we have the
following main findings, which should serve as a first step toward understanding the
factors affecting group negotiation power:
• Initial profit margin (profit share of a fixed pie) is negatively related to the
perceived change in profit margins, which in turn translates into the perceived
negotiation power. Sellers, who form the coalition, or the buyer, who faces the
coalition, that start with a substantially lower profit margin are more likely to
57
perceive a profit margin increase than those with a higher initial share of the fixed
profit pie.
• Sellers who join forces are more likely to perceive a profit margin increase than
the buyer who faces the coalition.
• Finally, a sellers’ alliance with fewer members is more likely to perceive a profit
margin increase than one with more members, while a buyer facing a smaller
coalition is more likely to perceive a profit margin decrease than a buyer facing a
larger coalition.
In the next section, we will briefly review some related literature. Section 2.3 will lay
out the experiment design and procedure, and data analysis will be presented in Section
2.4. Section 2.5 will summarize the study and outline some possible future research on
this topic.
2.2.Literature Review
In cooperative game theory, there are fruitful studies on pure bargaining problems
considering a group of two or more participants facing a set of feasible outcomes and a
given disagreement outcome. All participants need to reach unanimous agreement on one
feasible outcome or else the disagreement outcome will be the result. All of these studies
assume that each individual acts based on his or her own interest and does not form
coalition. Nash (1950, 1953) presented a two-person bargaining model in which players
bargain over the partition of a cake of fixed size. Each player has his or her own
58
disagreement utility, which defines what the players will get if they fail to reach
agreement, and the solution has been shown as a unique pair of utilities. Rubinstein (1982)
set up a procedure for bargaining that allows players to take turns making offers
regarding the partitioning of a fixed-size cake. Unlike the disagreement utility in the Nash
model, the Rubinstein model prescribes for the players discount factors, or cost of delay
in reaching agreement. A unique subgame perfect equilibrium is characterized and
proved as a solution to this model. Horn and Wolinsky (1988) develop a game theoretic
bargaining model and study wage bargaining under trade unionism. They assume that if
no parties join forces the employer takes turns negotiating with the workers, and the total
payoff pie of all parties increases with the number of successful negotiations. Through
the subgame perfect equilibrium, they show that joining forces may not be always
beneficial.
Ellison and Snyder (2001) conduct an empirical study in the healthcare industry and
show that forming a large buyer alliance may not result in substantially lower prices
when facing limited substitution choices of suppliers. In this case, buyers forming a
coalition simply does not change the monopoly status of the supplier or the alternative
choices of any buyer. In other words, joining forces solely for the purpose of size may not
gain additional power for its members. Chae and Heidhues (2003) use a variant of the
two-person, alternating-offer bargaining model to study the effect of buyers joining
59
forces across markets. They show that forming a buyers’ alliance among risk-averse
buyers can be advantageous in their model.
In marketing, Zwick and Weg (1996) conduct a controlled bargaining experiment to
explore the degree to which motives affect bargaining behavior. They found that
bargainers are motivated first and foremost by individual incentives and moderarted by
other-regarding motives, such as fairness. In psychology, Beest et al. (2005) study the
formation of coalitions through lab experiments, and they find that participants are more
reluctant to exclude in order to minimize their losses than to maximize their gains.
Additional analyses have revealed that fairness is more cognitively accessible when
payoffs are negative rather than positive.
There is very little work in operations management studying the value of joining
forces. Mahesh and Bassok (2008) study the assembly bargaining problem (A-B-P)
assuming suppliers have the choice to coalesce. A-B-P describes a supply chain in which
several (N) suppliers sell components to an assembler who faces random demand for the
assembled (final) product. The final product requires a single unit from each of the
suppliers (the components are complementary). The assembler must purchase the
required components, assemble them in expectation of the demand, and sell the final
product to meet demand at the end of the period. The authors develop a negotiation
framework to determine the exact allocation of the total profit (pie) that each player gets.
They show that, in general, when the assembler is weak (i.e., has little negotiation power)
60
it is in the best interest of the suppliers to join forces in one big alliance. In this case, the
assembler prefers to negotiate with as many suppliers as possible. But when the
assembler is powerful it is the interest of the suppliers to stay independent, while the
assembler would like to negotiate with a small number of suppliers. They also show that
even if the negotiation power of the coalition is a function of its members, the structure of
the results does not change.
These results hold when assuming that the negotiation power of the coalition remains
the same or increasing slowly with the size of the coalition.
2.3.Methods
2.3.1 Experiment design
The main aim of this experiment is to test which factor(s) is/are related to the change
of perceived negotiation power, leading to expected change in the realized negotiation
power. However, negotiation power is an intangible term and cannot be quantified
directly. In this study, we settle the measurement of negotiation power by the Nash
bargaining solution, in which it is not difficult to see that different solutions of the
maximization correspond to different pairs of power parameters. In other words, if the
solution changes, so do the power parameters of the two parties, holding everything else
equal. In this experiment, subjects have been randomly assigned scenarios with different
settings indicating given individual bargaining power. We then ask subjects to be in role
61
play and perceive the new bargaining result after the sellers join forces. If they perceive
changes in the new price, we believe that they indeed expect change in their negotiation
power because all other conditions are held constant. Instead of measuring the scale of
change, we separate the changes into three categories that can be ordered from low, or
decrease, to middle, or unchanged, to high, or increase. Accordingly, an individual will
perceive his or her profit margin as going up or down or remaining unchanged. The
factors we focus on in analyzing the perception of negotiation power are role, alliance
size, and initial profit margin before coalition.
With two levels in each of the factors, this is a 2 (role) X 2 (initial profit margin) X 2
(alliance size) between-subject experiment. We ask subjects to assume the role as told in
the story and to write down their answers. For initial profit margin, we select prices that
give either high profit margin or low profit margin to infer the high and low initial
negotiation power. For alliance size, we choose either few sellers (2) or many sellers (10)
in the story. Roles are either a buyer who will face the coalition or a sellers’ delegate who
will represent the alliance. The payments to subjects are the same and fixed and are not
related to the outcome the parties negotiate.
Specifically, we set the initial wholesale price to be either 17 or 7, with product retail
price being 25 per unit, cost per unit for the buyer being 3, and cost per unit for each
seller being 2. This defines a profit pie of 25-3-2 = 20 per unit of product to share
between the buyer and each seller. When the wholesale price is at 17, the buyer receives a
62
low profit margin (25-3-17 = 5) and each seller reaches a high profit margin (17-2 = 15).
It is just the opposite when the wholesale price is at 7. All cost and price parameters are
explained to the participants.
As for size effect, we need to separate the discussions for the buyer and the sellers’
alliance. For the seller, we consider the coalition with larger size a positive impact as it
tends to put greater pressure on a buyer than does a small coalition size. Comparatively,
for the buyer, a positive impact occurs when he faces a smaller-size alliance. However, it
is not clear if the positive size impacts always benefit the negotiation power of the party.
One concern is risk-averseness. For sellers, the larger-size coalition may become more
risk-averse as more members’ interests are at stake. Similarly, the buyer facing a
large-size coalition may gain more power simply because the other side becomes more
risk-averse.
2.3.2 Hypotheses
Based on the above analysis, we propose the following hypotheses.
H1 The parties who have a lower initial profit margin are more likely to perceive a
profit margin increase than are those with a higher initial share of the fixed profit
pie.
H2 A sellers’ alliance is more likely to perceive a profit margin increase than the buyer
facing the coalition.
63
H3 Parties without a positive size impact are more likely to perceive a profit margin
increase than parties with a positive size impact.
2.3.3 Participants and administration
Participants were undergraduate students taking the core course “Operations
Management,” and all of the subjects performed the experiment before learning supply
chain coordination. We selected six class sections in total, across two consecutive
semesters. In order to limit the noise arising from different sections’ instructors, we
limited the number of different instructors to three. Among the six sections, four were
from the same instructor, and the other two were from two other instructors. Each
participant was randomly assigned one of the eight treatments.
To run the experiment, we scheduled with the instructors to attend a certain class
period, and we went into the classes in the last 20 minutes. We brought in refreshments as
a fixed payment across subjects. The subjects were told to be in the role play in the story
and to give their personal responses, with no discussion allowed. All replies were
supposed to be based on their personal opinions, and they were told that there were no
right or wrong answers. Participants were also permitted to provide analysis or
explanation if they wished to do so.
64
2.4.Result
2.4.1. Descriptive statistics
We collected a total of 206 answers, and 5 of them were removed due to unreadable
or ambiguous handwriting. We coded the data by the following rules. For role, the buyer
was coded as “1” and the sellers’ alliance was coded as “0”. Low initial profit margin
was coded as “0” and “1” otherwise. For the size effect, we coded it based on the
different roles. For the seller, small size was coded as “0” and large size “1”. For the
buyer who faced the coalition, the small size was coded as “1” and large size “0”. That is,
we coded the size factor as “1” when it had a positive impact.
Finally, for the perceived change in profit margin, which was our dependent variable,
we coded it “3” if the change was more than 0.5, “1” if less than -0.5, and “2” otherwise.
We considered the change to be minor if it was within the range +0.5 to -0.5, and this
represented the middle category. However, when we strictly required the middle category
to be exactly 0, which meant no change at all, the result was almost the same. Table 8
below presents a summary of the outcomes in different categories.
Table 8: Data Summary
N Percentage
Level 1 51 25.4%
2 30 14.9%
3 120 59.7%
Total 201 100%
65
2.4.2. Ordinal regression model
In this study, we use the ordinal regression method to model the relationship between
the ordinal outcome variable (e.g., different levels of change in the profit margin) and the
explanatory variables (e.g., factors concerning the characteristics of both parties in the
negotiation). Unlike linear and logistic regression models, ordinal regression does not
assume normality and homogeneity of variance, but it does require the assumption of
parallel lines across all levels of the categorical outcome. Thus, it is preferable in this
study because the ordered categorical outcome contains only a small number of discrete
categories that make it implausible to assume normality and constant variance.
The ordinal regression model (or proportion odds model) is written in the following
form:
SIZE MARGIN ROLE f
i i 3 2 1
) ( β β β α θ − − − = 2 , 1 = i ,
where () f is the link function that connects the random component on the left side of the
equation and the systematic component on the right. Two commonly used link functions
are logit and complementary log-log, with the latter being appropriate when a higher
category is more probable. In our case, nearly 60% of the outcomes fall in category 3, the
highest one, and therefore we chose complementary log-log as the link function. Feeding
in this link function, we have the following model:
SIZE MARGIN ROLE f
S M R i i i
β β β α θ θ − − − = − − = )) 1 ln( ln( ) ( 2 , 1 = i ,
where:
66
i
θ ( 2 , 1 = i ) are the odds for category i . Specifically,
=
1
θ Pr(outcome of 1) / Pr(outcome of 2 or 3),
=
2
θ Pr(outcome of 1 or 2) / Pr(outcome of 3).
Notice that the last category does not have odds associated with it because the
probability of choosing up to and including the last category is 1. Each odd has its own
) 2 , 1 ( = i
i
α that can be viewed as intercepts in the linear regression (these terms are
called the threshold values and usually are not of much interest). All odds have the same
coefficients
R
β ,
M
β , and
S
β , which means that the effects of the three independent
variables are the same for different odds’ link functions. This is an assumption of
parallelism that needed to be checked to see if it could be satisfied.
2.4.3. Ordinal regression results
The proportional odds model is fitted using PLUM in SPSS, and Table 9 contains the
estimated coefficients for the model. The estimates labeled “Threshold” are the
i
α ’s (the
intercept-equivalent terms), and the estimates labeled “Location” are the coefficients for
the predictor variables. The Wald statistics for Role, Margin, and Size are 21.113; 33.169;
and 12.796, respectively. They are all significant at the 0.05 level, which means that all
three factors are related to the dependent variable.
When initial profit margin is 0, it receives a positive coefficient of 1.597, which
means that those with a lower initial profit share of the fixed pie are more likely to
67
perceive a profit margin increase than those with a higher initial profit margin. This
supports our hypothesis H1.
The coefficient of Role is 1.208, indicating that a sellers’ alliance (Role = 0) is more
likely to choose the upper category—i.e., perceive a profit margin increase—than is the
buyer. This supports our second hypothesis.
For the size effect, the coefficient is positive when size is “0”. This means that
without a positive size impact the buyer or the sellers’ delegate is more likely to perceive
a profit margin increase, which supports our third hypothesis. Specifically, the result
shows that sellers who coalesce with a smaller size (size = 0 for seller) are more likely to
perceive a profit margin increase than with a larger size. A buyer who faces a larger
alliance (size = 0 for buyer) is more likely to perceive a profit margin increase than is a
buyer facing a smaller alliance.
With the predicted parameters, the expected probability of falling into a specific
outcome category is calculated by the ordinal regression model in the previous section.
We cross tabulate the actual outcome and the predictions to obtain the confusion matrix
in Table 10. The prediction accuracy is (33 + 109)/201 = 70.65%, which is not bad.
Table 9: Parameter Estimate
Estimate Std. Error Wald Sig.
Threshold [Level=1] .132 .176 .562 .453
[Level=2] .877 .174 25.274 .000***
Location [Role=0] 1.208 .263 21.113 .000***
[Margin=0] 1.597 .277 33.169 .000***
[Size=0] .884 .247 12.796 .000***
***: Significant at 0.05.
68
Table 10: Confusion Matrix
Predicted
Actual 1 3 Total
1 33 18 51
2 15 15 30
3 11 109 120
Totals 59 142 201
The overall model test result in Table 11 has a significance level far less than 0.05, so
we can reject the null hypothesis that the location coefficients for all of the variables in
the model are 0. The goodness-of-fit statistics in Table 12 are not significant, which
supports the similarity of the observed and expected cell counts. Table 13 shows that the
test of parallel lines has a large significance level, which supports the model assumption
that the location parameters are the same for all categories. Notice that this is an
important assumption for ordinal regression modeling that must be satisfied in order to
validate the use of the model. We also tried to break the outcome into more categories,
but then this assumption will not be satisfied.
Table 11: Model Fitting
Model -2 Log Likelihood Chi-Square df Sig.
Intercept Only
Final
141.427
53.373
88.054
3
.000***
Table 12: Goodness-of-Fit
Chi-Square df Sig.
Pearson
Deviance
10.898
12.123
11
11
.452
.354
Table 13: Test of Parallelism
Model -2 Log Likelihood Chi-Square df Sig.
Null Hypothesis
General
53.373
52.614
.759
3
.859
69
To summarize, all three hypotheses are supported, and we have the following main
results from the ordinal regression. By the Nash bargaining solution, the perceived
increase in profit margin is translated into the expected increase in negotiation power.
• • • • The players who have a lower initial profit margin (i.e., a smaller share of the
fixed profit pie) are more likely to perceive a profit margin increase from the
expected negotiation result than are those with a higher initial share of the fixed
profit pie.
• • • • Sellers who join forces are more likely to perceive a profit margin increase than
the buyer who faces the coalition.
Finally, a sellers’ alliance with fewer members is more likely to perceive a profit
margin increase than a coalition with more members. Conversely, a buyer who faces a
larger-size coalition is more likely to perceive a profit margin increase than a buyer
facing a smaller-size coalition.
2.5 Discussion
Recall that we base our study on a Nash bargaining solution in order to measure the
change in negotiation power via the perceived contract price. With all cost parameters
unchanged, the outcome should be the same as the original wholesale price. However,
only 14.9% of the subjects give this result, and the rest think that it should be changed.
70
The ordinal regression model predicts only two results, either “Down” or “Up,” but not
remaining the same. Those who predict “Down” are the subjects in the buyer role with a
high initial profit margin, regardless of the size of the sellers’ coalition. All of the other
subjects perceive “Up” in the wholesale price. Because this is a validated model, we
conclude that the concept of joining forces alone will influence the negotiation power of
either the group or the party facing the group under these circumstances.
In this experiment, the influence on wholesale price is related to three factors, shown
by the ordinal regression result in the previous section. In what follows, we discuss these
main effects in more detail.
1. Holding everything else equal, a sellers’ alliance is more likely to perceive a
profit margin increase in the negotiation than is the buyer.
In this setting, the sellers can be considered active because they initiate the joining of
forces. Comparatively, the buyer is placed in a somewhat passive position and forced to
face the alliance of sellers. This difference probably leads to the different perceptions of
the buyer and the sellers’ alliance. Notice, however, that this does not mean that the buyer
does not expect a profit increase. Indeed, the predictions for a buyer with a low initial
profit margin, regardless of coalition size, are all “Up.” We shall discuss this along with
the factor of initial profit margin later.
The regression model shows that a buyer who started with a low initial profit margin
tends to concede and expect “Down” in the profit margin from the negotiation. Besides
71
the argument about passive position, this may also be caused by the issue of fairness. As
the suppliers form a coalition, the buyer, who already has a large percentage of the fixed
profit pie, may consider it fair to give up part of the profit to the alliance. This argument
is used by some subjects as the reasoning for their answers. However, the sellers’ alliance
does not behave in the same manner, even when it starts with a high initial profit margin.
That is, the sellers’ alliance expects an increase in negotiation power no matter the initial
profit share. There are several possible explanations for this. One is that the active
position in the joining of forces dominates the perception with regard to change in the
profit margin, and thus subjects are less likely to engage in concession. Another is that
the perceived increase in negotiation power represents an argument as to why the parties
join forces. Thus, even with a high initial profit share, the sellers’ alliance may consider
the increase of negotiation power is a purpose of joining forces. The last reason may be
that the buyer is more risk-averse than the sellers’ alliance. By the Nash bargaining
solution, the sellers’ alliance is more likely to perceive an increase in negotiation power
than is the buyer.
2. Holding everything else equal, the party with the lower initial profit margin is
more likely to perceive a profit margin increase than the party with the higher
initial profit margin.
As discussed earlier, negotiation power cannot be measured directly. However, based
on the Nash bargaining solution, the change in the profit margin, or the share of the fixed
72
profit pie, indicates a change in negotiation power. Thus, we consider profit margin an
indirect measure of negotiation power, holding everything else equal. To begin with, it is
natural for each seller and each buyer to have his or her negotiation power before the
sellers form a coalition. This is reflected in the given original wholesale contract price, or
the initial profit margin.
The regression result shows that this original status affects the perception with regard
to the negotiation outcome of a new wholesale price, which indicates that the expected
bargaining power shifts. Specifically, the higher the original profit margin, the less likely
the subject will be to perceive a profit margin in the “Up” category. One possible reason
is that parties with a higher initial profit share tend to be more risk averse than a party
with a low initial profit share. By the Nash bargaining solution, these parties are less
powerful and thus expect lower gains in negotiation power than do parties with lower
initial profit margins.
Another possible explanation is the cognition of fairness. The background setting puts
the buyer and the sellers’ alliance in a monopoly-to-monopoly situation. If fairness is the
only cause driving the perception of negotiation outcome, subjects are expected to predict
the outcome that gives them 50% of the fixed profit pie. Clearly, this cannot be the only
cause since few subjects expect this result. However, the perception of fairness may give
some explanation for the results. For example, subjects in the buyer role perceive the
result in the “Down” category when they have a high initial profit share and in the “Up”
73
category when they have a low initial profit share. In other words, their expectations
move toward the middle point, where each party obtains 50% of the profit. However,
subjects in the sellers’ alliance role always expect profit margin go up, as predicted by the
regression model, which contradicts the fairness concept. Interestingly though, some
subjects in this role with a low initial profit margin did claim fairness as their argument
for their perception of profit increase. Because this is a preliminary study, it is hard to tell
which reasoning is more appropriate than the others in explaining subjects’ behaviors.
3. Holding everything else equal, the party without a positive size impact is more
likely to perceive a profit margin increase than the party with a positive size
impact.
This result indicates that a sellers’ coalition with fewer members is more likely to
perceive an increase in negotiation power than is a coalition with more members. The
buyer facing a larger joint force is more likely to expect an increase in his or her
negotiation power than is a buyer encountering a smaller joint force. This may look
counter-intuitive at a glance. However, one of the main causes may be risk-averseness.
For sellers, the larger the coalition size, the riskier the alliance may be perceived to be in
the negotiation because more sellers will be out of business if an agreement cannot be
reached. Thus, the alliance tends to become more risk-averse, or less aggressive toward
the monopoly buyer. This means that it will be less likely to perceive an increase in
negotiation power than a coalition with fewer members. On the other side, the buyer
74
knows that more sellers are stakeholders with the larger coalition size, which means that
the buyer can take better advantage of his or her monopoly status in the negotiation.
Therefore, the buyer is more likely to perceive a profit increase, or more negotiation
power, than in the case in which he or she faces a smaller-size alliance.
2.6 Managerial insight
This study is a very first step in trying to understand the behavior of negotiators in
the situation in which sellers join forces. Although the results are based on perceptions of
the final negotiation outcome, they reveal the reasoning of people with regard to joining
forces in the negotiation process. Sellers with lower negotiation power might be at better
advantage to form a coalition because the buyer is more likely to yield in this situation.
Comparatively, the party who already has a big share of the profit pie may have less
incentive to join the coalition because the room for increase is small and the probability
that the other party will perceive more power is larger, making it harder to achieve the
desired negotiation outcome. When facing a monopoly, a large-size alliance may be
inferior to a small-size one because the opponent is more likely to show more strength
against the coalition with more members.
75
2.7 Further research
As a new topic, there are many findings to be observed. In this study, we have only
two levels for the factors of initial profit margin and coalition size. It will be valuable if
the study can be repeated with an added middle level in these two factors. With more
levels in the factors, the results might be analyzed using other regression models that help
to interpret the numeric change in the outcome. Another possible venue is to introduce
new factors such as alternatives. As found by Ellison and Snyder (2001), the benefit from
a large-size coalition can be completely wiped out when an alternative is available. It will
be interesting to see when the value of an alternative can dominate other factors such as
size. Ideally, we would conduct an unstructured bargaining experiment mimicking reality,
but this is very difficult because most subjects feel it is hard to get into real role playing
when facing strangers. In the near future, we will set up negotiation experiments on
structured bargaining, which gives us more control over the procedures.
76
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Abstract (if available)
Abstract
This dissertation is on two topics. Chapter 1, substitution and variety, focuses two major issues in retail inventory management: (1) the assortment problem
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Asset Metadata
Creator
Chen, Feng
(author)
Core Title
Substitution and variety, group power in negotiation
School
Marshall School of Business
Degree
Doctor of Philosophy
Degree Program
Business Administration
Publication Date
02/03/2009
Defense Date
12/15/2008
Publisher
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Tag
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), Rajagopalan, Sampath (
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), Ross, Sheldon M. (
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), Sriram, Dasu (
committee member
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Creator Email
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