University of Southern California Dissertations and Theses

Taxi driver learns dynamic multi-market equilibrium

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Taxi driver learns dynamic multi-market equilibrium

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Content T axi Driver Learns
D yn amic Mul ti-market Equilibrium
b y
R uda Zhang
A Thesis Presented t o t he
F A CUL TY OF THE USC GRADU A TE SC HOOL
UNIVERSITY OF SOUTHERN C ALIF ORNIA
In P artial F ulfillment of t he
R equirements f or t he Deg ree
MAS TER OF AR TS
(EC ON OMICS)
Ma y 2018
abs tra ct
T axi is a v aluable part of urban tr ansportation and indus trial or g anization, where taxi
driv ers a re bo t h ride ser vice pro viders and independent contr act ors. This paper s tudies
t h e s treet-hail taxi indus tr y , where each taxi in ser vice is seen as a multi-mar k et fir m, e v -
er y s treet segment is a dis tinct mar k et, and fir ms allocate ser vice time across t he s treet
netw or k. W e g ener alize t he model int o a g ame of multi-mar k et com petition among fir ms
of equal capacity , and pro v e t hat t he g ame has pure-s tr ategy N ash equilibrium (PSNE),
which is (1) symmetric, (2) essentiall y unique in t hat mar ginal pla y er pa y offs are unif or m
across all in v es ted mar k ets, and (3) globall y asym p t o ticall y s table under g r adient adjus t-
ment process and imitativ e lear ning. The agg reg ate s tr ategy at equilibrium maximizes a
“po tential function”, and it differs from t he social op timal s tr ategy f or mos t f or ms of pro-
duction functions. W it h 868 million trip recor ds of all 13,237 Medallion taxis (y ello w cab)
in N e w Y or k City from 2009 t o 2013, w e v alidate t hat taxi driv ers ’ beha vior conf or m t o t his
equilibrium under fix ed tr affic speeds and taxi demand. Consis tent wit h t he equilibrium,
mar ginal segment income is cons tant o v er time-of-da y in each shif t. W it h t he launch of
S t reet Hail Liv er y (g reen cab) in late 2013, which increases s treet-hail v ehicles out of core
Manhattan, w e obser v e a decrease in y ello w cab pickups be y ond Eas t 96t h S treet and W es t
110t h S treet.
i
A c kn o wledgments
The aut hor t hank Henr y S. F arber of Princet on U niv ersity and A bhishek N ag ar a j of UC
Ber k ele y f or sharing NY C taxi trip recor ds. W e also t hank OpenS treetMap contribut ors f or
NY C map data, and Open Source R outing Machine contribut ors f or g r aph prepar ation and
map matching modules. R esearch is funded b y N ational Science F oundation under Gr ant
N o. 14-524 R esilient Inter dependent Infr as tr ucture Processes and Sys tems.
ii
C ontents
abs tra ct i
A c kn o wledgments ii
1 I ntr oduction 1
2 T axi driver decision makin g 4
3 M ul ti-market oligopol y 12
4 Em piric al resul ts 21
4.1 Spatial equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Dynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Driv er lear ning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 P olicy im pact on equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Di scussion 29
5.1 Efficiency : t he problem of social cos t . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Macroscopic inter pretation: t her modynamic s . . . . . . . . . . . . . . . . . . 31
6 M a terials and Methods 33
6.1 N e w Y or k City taxi trip recor ds . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Bibliography 34
iii
C h apter 1
Intr oduction
Cities are places wit h t he denses t human inter action, and it is t hus im portant t o under -
s tand t he regularity of urban agglomer ations. W it hin t he emer ging field of urban science,
f e w s tudies ha v e manag ed t o build com prehensiv e models t hat explain specific aspects of
cities. T o t his end, recent efforts ha v e used entrop y models (W ilson 2010 ), r andom w alk
models (Gonzalez, Hidalgo, and Bar abasi 2008 ), scaling la w s (Bettencourt 2013 ), and spa-
tial economic equilibrium (Glaeser 2008 ).
Despite t he dis tinct met hodologies applied t o urban science, one pre v alent perspectiv e
is t o treat a city as a sys tem. Meier 1968 sees t he metropolis as a tr ansaction-maximizing
sys tem where population density s timulates tr ansactions betw een individual act ors. F or -
res ter 1969 defines urban sys tem as t he built en vironment and t he inter acting population
wit hin an urban area, where t he built en vironment consis ts of buildings and infr as tr uc-
ture sys tems such as tr ansportation, ener gy dis tribution, and w ater suppl y and treatment.
A modeling met hodology f or urban sys tems is sociodynamics (W eidlich and Haag 1983 ),
which models t he e v olution of city configur ation as a s t ochas tic process, e v ol ving in ac-
cor dance wit h a mas ter equation. W eidlich 1999 used t his fr ame w or k t o s tudy t he im pact
of tr ansportation sys tem on regional de v elopment around cities. Based on models from
non-equilibrium s tatis tical ph ysics, ho w e v er , t his g ener al met hodology dismisses t he pos-
1
C hapter 1 Intr oduction
sibility of urban sys tems being in equilibrium.
As urban tr ansportation exhibits dail y , w eekl y and seasonal periodic beha vior , w e be-
lie v e it is reasonable t o s tudy cities as sys tems in dynamic equilibrium, at leas t f or ur -
ban tr ansportation sys tems. In f act, equilibrium met hods ha v e been applied t o tr ansporta-
tion since t he discussion o v er t oll roads in Pigou 1920 and Knight 1924 . Gener alizing t he
economis ts ’ exam ple, W ar drop 1952 described tw o beha vior al principles of route choice: a
N ash equilibrium and a P aret o op timal. Beckmann, McGuire, and W ins ten 1956 pro vided
t h e firs t mat hematical f or mulation of t he problem and pro v ed t he exis tence and unique-
ness of tr affic equilibrium in pure s tr ategies. F or a re vie w of de v elopment in t he f ollo wing
50 y ears, see f or exam ple Bo y ce, Mahmassani, and N agur ne y 2005 . Ho w e v er , a k e y issue of
t h e af orementioned tr affic equilibrium liter ature is its narro w assum p tion on individual
incentiv es: t hat individuals minimize t heir tr a v el time, or t hat social planners minimize
t h e t o tal tr a v el time of t he society . P eople mo v e f or v arious reasons and seldom beha v e
as time minimizers. T o tr ul y unders tand human mobility and successfull y tackle tr affic
cong es tion and related inefficiencies, an equilibrium model of urban tr ansportation has t o
account f or different incentiv e s tr uctures of t he population.
W e reg ar d t he regularity in urban tr ansportation as t he equilibrium outcome of indi-
vidual decision making in response t o tr ansportation demand and ser vices. This paper ,
in particular , pro vides equilibrium models f or taxi tr ansportation. T axi driv ers are bo t h
ride ser vice pro viders and independent contr act ors. W it h ex og enous tr affic speeds and
passeng er demand, income maximization of taxi driv ers lead t o an economic equilibrium.
W e f or malize taxi driv er decision making as a non-cooper ativ e g ame, sol v e its N ash equi-
librium, and s ho w its s tability under non-equilibrium dynamics. W e also pro vide a macro-
scopic inter pretation of t his equilibrium as a t her modynamic equilibrium, and discuss t he
economic efficiency of s treet-hail taxis. W it h fiv e y ears of N e w Y or k City (NY C) taxi trip
recor ds, w e v alidate t he equi librium in space, o v er time, and under policy chang e, and
2
C hapter 1 Intr oduction
examine t he lear ning process of taxi driv ers.
3
C h apter 2
T axi driver decision makin g
Each taxi in ser vice can be seen as a multi-mar k et fir m, where e v er y s treet segment is
a dis tinct mar k et, and t he fir m allocates its ser vice time among v arious s treet segments.
Com petition among taxis could lead t o specific choices of driv er s tr ategies, called equilib-
rium.
The tr ansportation decision of a taxi driv er can be sim pl y expressed as: taxi driv ers max-
imize t heir income b y choosing t heir driving s tr ategy . W e ignore t he exit decision of taxi
driv ers, and assume t hat individuals who driv e a taxi can ear n at leas t as much income
as t heir cos t, i.e. t heir alter nativ e income. When t his condition does no t hold, r ational in-
dividuals w ould no t be driving a taxi. W e sho w in t he f ollo wing t hat driv er’ s objectiv e
function is s tr ategicall y equiv alent t o trip re v enue, and f or malize driv er’ s decision as an
op t imization problem.
Income s tr ucture of a taxi driv er differs b y t he property r ights of t he taxi in use. Owner -
driv ers are Medallion o wners who also driv e t heir taxis, so t he y ha v e no lease t o pa y .
Driv ers of driv er -o wned v ehicle (DO V) lease a Medallion from fleets, ag ents, or Medal-
lion o wners, and eit her o wn or finance t he purchase of t he v ehicle, at different lease cos ts.
Ot her driv ers lease bo t h t he Medallion and t he v ehicle. In an y kind of such leases, t he
driv er pa ys a fix ed amount of mone y eit her per shif t which las ts 12 hours, or per w eek
4
C hapter 2 T axi driver decision makin g
in long er -ter m leases.
1
The lease ma y op tionall y include g asoline surchar g e, also a fix ed
amount, since 2012-09-30.
2
T axi lease type can be inf erred from d riv er names on t he taxi’ s
r ate car d: if a taxi has named driv ers, its o wner typicall y uses long-ter m lease; if it has un-
specified driv er , its o wner typicall y uses shif t lease. T able 2.1 sho w s t he number of NY C
taxis in 2005 b y t heir manag er and driv er types, deriv ed from Schaller 2006 .
T able 2.1: NY C taxis b y manag er -driv er type, 2005
Owner -driv er N amed driv er U nspecified
Owner 3730 1210 -
Fleet - 1481 635
A g ent - 1435 4305
T axi driv ers also pa y f or fuel usag e, which depends on v ehicle model, v ehicle speed and
acceler ation, air tem per ature, and air conditioning. As of v ehicle model, af ter t he 2008-
05-02 TL C auction, 275 of t he 13237 Medallions are res tricted t o alter nativ e fuel v ehicles,
but man y unres tricted Medallion o wners v oluntaril y con v erted t o clean-fuel v ehicles
3
(see
T able 2.2 ). F or g asoline/h ybrid light passeng er v ehicles oper ating at urban tr affic speed
(16-40 km/h, or 10-25 m ph), fuel consum p tion per hour is almos t cons tant, see S. C. Da vis,
Dieg el, and Boundy 2017 . This means t hat fuel cos t per ser vice time can be seen as a con-
s tant f or each taxicab reg ar dless of speed — w e do no t consider taxis par k ed b y t he curb
wit h engine off activ el y in ser vice. Ev en wit hout t his obser v ation, fuel cos t per ser vice
time w ould s till be appro ximatel y cons tant f or a driv er in one shif t, as long as t he driv er
has consis tent driving speeds and acceler ation patter ns.
A taxi driv er ear ns t he remaining f are and tips af ter pa ying f or lease, fuel, or bo t h. F or -
mall y , t he hour l y income 𝑢 𝑖 of driv er 𝑖 deriv es from hour l y trip re v enue 𝜋 𝑖 , minus hour l y
1
TL C R ules §58-21: Leasing a T axicab or Medallion.
2
TL C lease cap r ules chang e. http://www.nyc.gov/html/tlc/downloads/pdf/lease_cap_
rules_passed.pdf
3
TL C 2008-2013 Annual R eports. http://www.nyc.gov/html/tlc/html/archive/annual.
shtml
5
C hapter 2 T axi driver decision makin g
T able 2.2: NY C taxis b y v ehicle fuel type
End of y ear Gasoline Hybrid-electric Diesel CN G
*
2007 12422 728 0 -
2008 11394 1843 0 -
2009 10177 3043 17 -
2010 9029 4185 19 4
2011 7540 5681 14 2
2012 6455 6769 10 3
2013 5320 7905 9 3
*
CN G: com pressed natur al g as
fuel cos t 𝑓 𝑖 , minus amortized hour l y lease pa yment 𝑟 𝑖 :
𝑢 𝑖 = 𝜋 𝑖 − 𝑓 𝑖 − 𝑟 𝑖 (2.1)
The amortized hour l y lease pa yment b y t he driv er is 𝑟 𝑖 = 𝑅 𝑖 /𝑇
𝑖 , where 𝑇 𝑖 deno tes driv er
t o tal ser vice time during t he lease ter m, and 𝑅 𝑖 deno tes lease pa yment, i.e. rent of t he
Medallion taxicab. Depending on t he lease, 𝑓 𝑖 or 𝑟 𝑖 ma y be zero. Since 𝑓 𝑖 and 𝑟 𝑖 are con-
s tant f or driv er 𝑖 in an y giv en shif t, t he y do no t affect t he driv er’ s driving s tr ategy . Thus,
driv er’ s objectiv e is s tr ategicall y equiv alent t o trip re v enue 𝜋 𝑖 . W e no te t hat alt hough v e-
hicle maintenance is ano t her cos t t o driv ers who o wn t he v ehicle, it is no t rele v ant t o t he
driv er s tr ategy of our interes t.
T o define taxi driv ers ’ driving s tr ategy , w e firs t anal yze taxi tr ansportation. T axis in ser -
vice are eit her v acant or occupied: when v acant, driv ers search t he s treets f or hailers; when
occupied, driv ers tak e t he passeng ers t o t heir des tination. T axi driv ers can freel y choose
ho w t he y spend t heir search time o v er t he s treet netw or k. Once t he y find hailers, driv ers
will s t op searching and pick t hem up.
4
T axi f are r ate is set b y t he city go v er nment, which
ma y be metered or has a flat r ate, depending on t he des tination. U nder flat r ate, driv ers
4
In real lif e, no t all taxi driv ers pick up e v er y hailer t he y meet. The y ma y discriminate hailers based on
t he des tination, r ace, or o t her f act ors, due t o profitability , security , or end-of-shif t concer ns. See NY C 311
recor ds f or com plaints about taxis ser vice denial.
6
C hapter 2 T axi driver decision makin g
are bes t off taking t he f as tes t pat h. Metered r ates char g e b y dis tance or dur ation, based on
a speed t hreshold, which are typicall y set such t hat driv ers ha v e no incentiv e t o driv e slo w .
Alt hough driv ers do ha v e an incentiv e t o tak e routes long er t han t he f as tes t pat h, passen-
g ers typicall y are mo tiv ated t o super vise trip dur ation. In case of driv er fr aud, det ouring is
no t a common s tr ategy (Balaf outas, K erschbamer , and Sutter 2015 ). Thus, w e assume t hat
taxi driv er’ s deliv er y s tr ategy is t o tak e passeng ers t o t heir des tination via t he f as tes t pat h,
so trip dur ation betw een tw o specific locations onl y depend on tr affic speed. W e can see
t h at t he onl y s tr ategic element f or taxi driv ers is ho w t he y allocate t heir search time.
N o w w e f or malize driv ers ’ driving s tr ategy . Let 𝑁 be t he set of taxi driv ers currentl y in
ser vice. Let 𝐺 = (𝑉 , 𝐸) be t he road netw or k wit hin t he urban area being s tudied, where 𝑉 is t he set of intersections and dead ends, and 𝐸 is t he set of s treet segments. S treet segment
𝑥 ∈ 𝐸 has lengt h 𝑙 𝑥 , wit h tr affic speed 𝑣 𝑥 and taxi search speed ̃ 𝑣 𝑥 . Define demand r ate 𝜇 𝑑𝑥𝑦 as t he frequency of hailers s tart hailing on segment 𝑥 who are going t o segment 𝑦 ; such
a g roup of hailers ha v e im patience 𝜇 𝑡𝑥𝑦 = 1/𝔼𝑇
𝑥𝑦 , t he reciprocal of hailer mean patience.
W it hin a short time inter v al, en vironment condition ℰ = ( v , 𝜇 𝑑 , 𝜇 𝑡 ) can be considered as
cons tant, where v is t he v ect or of tr affic speeds, and 𝜇 𝑑 and 𝜇 𝑡 are matrices of hailer demand
and im patience. S tr ategy f or driv er 𝑖 can be defined as t he spatial dis tribution of suppl y
r ates 𝜇 𝑠𝑖 , where 𝜇 𝑠𝑖 𝑥 = (𝜇
𝑠𝑖 )
𝑥 is t he frequency at which driv er 𝑖 enters segment 𝑥 as a v acant
taxi. Equiv alentl y , driv er s tr ategy can be defined as t he dis tribution of driv er’ s search time
per un it time:
𝑡 𝑠𝑖 𝑥 𝑡 =
𝑙 𝑥 ̃ 𝑣 𝑥 𝜇 𝑠𝑖 𝑥 (2.2)
This sho w s t hat on each segment, driv er search time is linear l y related t o driv er suppl y
r ate. Define pickup r ate 𝜇 𝑝𝑖 𝑥𝑦 as t he frequency at which driv er 𝑖 pick s up passeng ers on
𝑥 going t o 𝑦 . These attributes natur all y agg reg ates on each segment: 𝜇 𝑝𝑥 = ∑
𝑖 ∑
𝑦 𝜇 𝑝𝑖 𝑥𝑦 ,
𝜇 𝑠𝑥 = ∑
𝑖 𝜇 𝑠𝑖 𝑥 , 𝜇 𝑑𝑥 = ∑
𝑦 𝜇 𝑑𝑥𝑦 , and 𝜇 𝑡𝑥 = 1/𝔼𝑇
𝑥 . Pickup r ate can t hus be expressed as a
7
C hapter 2 T axi driver decision makin g
function of suppl y r ate, demand r ate and hailer im patience: 𝜇 𝑝𝑥 (𝜇
𝑠𝑥 , 𝜇 𝑑𝑥 , 𝜇 𝑡𝑥 ) . Zhang and
Ghanem 2018 proposed a class of pickup models and pro v ed t hat t he pickup r ate func-
tions are increasing, s trictl y conca v e, and arbitr aril y differentiable, wit h respect t o suppl y
r ate; f or t hree representativ e models, anal ytical f or ms of t he pickup r ate functions are also
pro vided.
W e no w relate driv er s tr ategy wit h driv er re v enue. Let Π
𝑥𝑦 be t he re v enue of a single trip
from 𝑥 t o 𝑦 , which onl y depends on tr affic speeds v . W e can write hour l y re v enue originated
on 𝑥 as 𝜋 𝑥 = ∑
𝑦 Π
𝑥𝑦 𝜇 𝑝𝑥𝑦 and a v er ag e re v enue of a trip originated on 𝑥 as Π
𝑥 = 𝜋 𝑥 /𝜇
𝑝𝑥 .
Assume patience and des tination are appro ximatel y uncorrelated f or hailers wit h t he same
origin, which means ∀𝑥 , 𝑦 ∈ 𝐸, 𝜇 𝑡𝑥 ≈ 𝜇 𝑡𝑥𝑦 . Then hailers on t he same segment ha v e an equal
chance of being pick ed up reg ar dless of t heir des tination:
∀𝑥 ∈ 𝐸, 𝜇 𝑝𝑥𝑦 ∝ 𝜇 𝑑𝑥𝑦 , ∀𝑦 ∈ 𝐸 Thus, t he a v er ag e re v enue f or a trip originated on 𝑥 onl y depends on tr affic speeds and de-
mand r ates: Π
𝑥 ( v , 𝜇 𝑑𝑥 ) = ∑
𝑦 Π
𝑥𝑦 𝜇 𝑑𝑥𝑦 /𝜇
𝑑𝑥 . Since driv ers are assumed no t t o discriminate
hailers:
∀𝑖 ∈ 𝑁 , ∀𝑥 ∈ 𝐸, 𝜇 𝑝𝑖 𝑥𝑦 ∝ 𝜇 𝑝𝑥𝑦 , ∀𝑦 ∈ 𝐸 Driv er re v enue originated on a segment 𝜋 𝑖 𝑥 = ∑
𝑦 Π
𝑥𝑦 𝜇 𝑝𝑖 𝑥𝑦 can t hus be written as 𝜋 𝑖 𝑥 =
∑
𝑦 Π
𝑥𝑦 𝜇 𝑝𝑥𝑦 𝜇 𝑝𝑖 𝑥 /𝜇
𝑝𝑥 = Π
𝑥 𝜇 𝑝𝑖 𝑥 . Since each pass of a v acant taxi has an equal chance of
picking up a hailer reg ar dless of t he driv er :
∀𝑥 ∈ 𝐸, 𝜇 𝑝𝑖 𝑥 ∝ 𝜇 𝑠𝑖 𝑥 , ∀𝑖 ∈ 𝑁 W e ha v e 𝜋 𝑖 𝑥 = Π
𝑥 𝜇 𝑝𝑖 𝑥 = Π
𝑥 𝜇 𝑝𝑥 𝜇 𝑠𝑖 𝑥 /𝜇
𝑠𝑥 . Driv er hour l y trip re v enue can t hus be expressed
8
C hapter 2 T axi driver decision makin g
wit h explicit function de pendency as:
𝜋 𝑖 = ∑
𝑥∈𝐸
𝜋 𝑖 𝑥 = ∑
𝑥∈𝐸
Π
𝑥 ( v , 𝜇 𝑑𝑥 )𝜇
𝑝𝑥 (𝜇
𝑠𝑥 , 𝜇 𝑑𝑥 , 𝜇 𝑡𝑥 )
𝜇 𝑠𝑖 𝑥 𝜇 𝑠𝑥 (2.3)
A more anal yticall y con v enient definition of driv er s tr ategy is driv er’ s allocation of ser -
vice time. Ser vice time 𝑡 𝑖 𝑥 = 𝑡 𝑠𝑖 𝑥 + 𝑡 𝑝𝑖 𝑥 is t he t o tal time driv er 𝑖 spends searching and
deliv ering trips originated on 𝑥 during a period of time 𝑡 . The r ationale of using ser vice
time dis tribution as driv er s tr ategy ins tead of suppl y r ate or search time is t hat: ser vice
time is a conser v ed quantity and identical f or all driv ers; mean while, ser vice time is mono-
t onic in suppl y r ate and preser v es properties of t he pickup r ate function. Let 𝑡 𝑥𝑦 be t he
trip dur ation from 𝑥 t o 𝑦 , which onl y depends on tr affic speeds v . The a v er ag e dur ation
of a trip originated on 𝑥 is 𝑡 𝑥 ( v , 𝜇 𝑑𝑥 ) = ∑
𝑦 𝑡 𝑥𝑦 𝜇 𝑑𝑥𝑦 /𝜇
𝑑𝑥 = ∑
𝑦 𝑡 𝑥𝑦 𝜇 𝑝𝑥𝑦 /𝜇
𝑝𝑥 , wit h reasoning
similar t o a v er ag e trip re v enue Π
𝑥 . The proportion of time driv er 𝑖 spends deliv ering trips
originated on 𝑥 is t hus 𝑡 𝑝𝑖 𝑥 /𝑡 = ∑
𝑦 𝑡 𝑥𝑦 𝜇 𝑝𝑖 𝑥𝑦 = ∑
𝑦 𝑡 𝑥𝑦 𝜇 𝑝𝑥𝑦 𝜇 𝑝𝑖 𝑥 /𝜇
𝑝𝑥 = 𝑡 𝑥 𝜇 𝑝𝑖 𝑥 = 𝑡 𝑥 𝜇 𝑝𝑥 𝜇 𝑠𝑖 𝑥 /𝜇
𝑠𝑥 ,
wit h reasoning similar t o 𝜋 𝑖 𝑥 . T og et her wit h Equation 2.2 , t he proportion of ser vice time
driv er 𝑖 allocates on 𝑥 can t hus be written as:
𝑠 𝑖 𝑥 =
𝑡 𝑠𝑖 𝑥 + 𝑡 𝑝𝑖 𝑥 𝑡 = (
𝑙 𝑥 ̃ 𝑣 𝑥 + 𝑡 𝑥 𝜇 𝑝𝑥 𝜇 𝑠𝑥 ) 𝜇 𝑠𝑖 𝑥 (2.4)
This sho w s t hat on each segment, driv er ser vice time is also linear l y related t o driv er suppl y
r ate: ∀𝑥 ∈ 𝐸, 𝑠 𝑖 𝑥 ∝ 𝜇 𝑠𝑖 𝑥 , ∀𝑖 ∈ 𝑁 . F rom Equation 2.4 , ser vice time on a segment 𝑠 𝑥 =
𝜇 𝑠𝑥 𝑙 𝑥 / ̃ 𝑣 𝑥 + 𝜇 𝑝𝑥 𝑡 𝑥 . W it h pickup r ate function 𝜇 𝑝𝑥 (𝜇
𝑠𝑥 , 𝜇 𝑑𝑥 , 𝜇 𝑡𝑥 ) and cons tant en vironment
condition ℰ , pickup r ate is im plicitl y a function of ser vice time: 𝜇 𝑝𝑥 (𝑠
𝑥 , ℰ ) . Each t axi driv er
mus t allocate all t he ser vice time among t he s treet segments: ∑
𝑥 𝑡 𝑖 𝑥 = 𝑡 , or equiv alentl y
∑
𝑥 𝑠 𝑖 𝑥 = 1 . The driving s tr ategy of taxi driv er 𝑖 is t hus s
𝑖 ∈ 𝑆 𝑖 , where t he s tr ategy space
𝑆 𝑖 = Δ
|𝐸|−1
, a sim plex of dimension one less t han t he number of segments. N o w w e can
9
C hapter 2 T axi driver decision makin g
f or mall y write t he op timization problem of a taxi driv er :
maximize ∑
𝑥∈𝐸
Π
𝑥 ( v , 𝜇 𝑑𝑥 )𝜇
𝑝𝑥 (𝑠
𝑥 , ℰ )
𝑠 𝑖 𝑥 𝑠 𝑥 subject t o s
𝑖 ≥ 0
s
𝑖 ⋅ 1 = 1
(2.5)
N o w w e pro v e t hat pickup r ate 𝜇 𝑝𝑥 (𝑠
𝑥 , ℰ ) is also increasing, s trictl y conca v e, and arbitr aril y
differentiable wit h respect t o 𝑠 𝑥 . W it h cons tant en vironment condition ℰ , t he im plicit func-
tion can be abs tr acted t o 𝑧 = 𝑎𝑥 + 𝑏𝑦 , where 𝑧 = 𝑠 𝑥 , 𝑥 = 𝜇 𝑠𝑥 , 𝑦 = 𝜇 𝑝𝑥 , 𝑎 = 𝑙 𝑥 / ̃ 𝑣 𝑥 , and 𝑏 = 𝑡 𝑥 ;
𝑦 (𝑥 ) is increasing, s trictl y conca v e, and arbitr aril y differentiable, while 𝑎, 𝑏 > 0 are con-
s tants. Our proposition is t hus equiv alent t o: 𝑦 (𝑧) is also increasing, s trictl y conca v e, and
arbitr aril y differentiable. Differentiablity is sim pl y preser v ed b y t he linear relation. Since
𝑧(𝑥 ) = 𝑎𝑥 + 𝑏𝑦 (𝑥 ) is increasing, its in v erse 𝑥 (𝑧) is t hus also increasing; b y com position,
𝑦 (𝑧) = 𝑦 (𝑥 (𝑧)) is also increasing. By im plicit differentiation, d 𝑦 / d 𝑧 = 𝑦 ′
(𝑥 )/(𝑎 + 𝑏𝑦 ′
(𝑥 )) ,
and t hus d
2
𝑦 / d 𝑧 2
= 𝑎𝑦 ″
(𝑥 )/(𝑎 + 𝑏𝑦 ′
(𝑥 ))
3
. Since 𝑦 ′
(𝑥 ) > 0 and 𝑦 ″
(𝑥 ) < 0 , 𝑦 ″
(𝑧) < 0 , which
means 𝑦 (𝑧) is also s trictl y conca v e.
Despite t he v arious alter nativ es w e proposed as f or mal driv er s tr ategy , here w e point
out ho w taxi driv ers w ould im plement such a s tr ategy . Picture a taxi driv er 𝑖 who is f amil-
iar wit h t he tr affic of t he city , and hailer and taxi dis tributions t hroughout a da y . T o ear n
more mone y , t he driv er has a plan on ho w much time t o spend searching different places
f or hailers; t he plan v aries f or different time of da y . A t t he beginning of 𝑖 ’ s shif t, t he driv er
heads t o t he region where t he plan allocates t he mos t search time. Af ter deliv ering t he
firs t pickup, t he driv er is lik el y t o be in a region wit h less planned search time. T o a v oid
o v er -searching t he current region, 𝑖 driv es back t o t he pref erred region. If 𝑖 goes t hrough
t he pref erred region wit hout a pickup, t he driv er w ould circle around and continue t he
search, as long as t he t o tal search time wit hin t he region is no t t oo long com pared wit h
10
C hapter 2 T axi driver decision makin g
t h e plan. The driv er does no t alw a ys search or immediatel y go back t o t he region wit h t he
highes t pla nned search time, but w ould balance t he allocation of realized search time t o
appro ximate t he plan. But when 𝑖 drops off at a location wit h v er y little planned search
time, t he driv er w ould directl y head t o a place nearb y where t he plan giv es more search
time, since a single pass w ould typicall y suffice f or t he drop-off location. Because t he t o tal
search time is limited f or an y giv en shif t, t he driv er w ould no t be able t o perf ectl y im-
plement t he s tr ategy in one shif t. But agg reg ated o v er time, t he dis tribution of realized
search time could reasonabl y appro ximate an intended s tr ategy . In later sections, w e tak e
t he dis tribution of realized search time as a driv er’ s s tr ategy .
11
C h apter 3
Mul ti-market oligopol y
In t his section w e f or malize t he g ame of multi-mar k et com petition among fir ms of equal
capacity , and pro v e t hat t he g ame has pure-s tr ategy N ash equilibrium (PSNE), which is
symmetric and essentiall y unique in t hat mar ginal pla y er pa y offs are unif or m across all
in v es ted mar k ets.
W e use subscrip t 𝑥 t o deno te a mar k et, or product ; subscrip t 𝑖 f or a fir m, or pla y er ; sub-
scrip t −𝑖 f or opponents of fir m 𝑖 . Boldf ace deno tes a v ect or ; single subscrip t indicates sum-
mation. Conditions in parent heses are op tional.
Game setup of multi-mar k et oligopol y . F or fir ms 𝑖 ∈ 𝑁 , |𝑁 | = 𝑛 , each dis tributing a unit
of resources o v er mar k ets 𝑥 ∈ 𝐸 , |𝐸| = 𝑚 :
1. T o tal pa y off in a mar k et 𝑢 𝑥 (𝑠
𝑥 ) , 𝑠 𝑥 ≥ 0 , 𝑢 𝑥 (0) = 0 , is (increasing) non-decreasing,
differentiable, and (s trictl y) conca v e;
2. P a y off per in v es tment in a mar k et 𝑝 𝑥 (𝑠
𝑥 ) = 𝑢 𝑥 /𝑠
𝑥 , 𝑠 𝑥 > 0 , is (decreasing) non-increasing;
no t necessaril y con v ex;
3. Pla y er pa y off in a mar k et 𝑢 𝑖 𝑥 (𝑠
𝑖 𝑥 ; 𝑠 −𝑖 𝑥 ) = 𝑝 𝑥 (𝑠
𝑥 )𝑠
𝑖 𝑥 , 𝑠 𝑖 𝑥 ∈ [0, 1] , 𝑠 −𝑖 𝑥 = ∑
𝑗 ≠𝑖
𝑠 𝑗 𝑥 ∈
[0, 𝑛 − 1] ;
12
C hapter 3 Mul ti-market oligopol y
4. Pla y er pa y off 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) = ∑
𝑥 𝑢 𝑖 𝑥 (𝑠
𝑖 𝑥 ; 𝑠 −𝑖 𝑥 ) , s
𝑖 ∈ 𝑆 𝑖 = Δ
𝑚−1
, s
−𝑖
= ∑
𝑗 ≠𝑖
s
𝑗 ∈ 𝑆 −𝑖
=
(𝑛 − 1)Δ
𝑚−1
; Here Δ
𝑚−1
= { v ∈ ℝ
𝑚 ∣ v ≥ 0, v ⋅ 1 = 1} is t he (𝑚 − 1) -dimensional
sim plex.
5. Mar ginal pla y er pa y off in a mar k et at equilibrium 𝜙 𝑥 (𝑠
𝑥 ) = 𝑝 𝑥 (𝑠
𝑥 ) + 𝑝 ′
𝑥 (𝑠
𝑥 )𝑠
𝑥 /𝑛 , or
equiv alentl y 𝜙 𝑥 (𝑠
𝑥 ) = 𝑢 ′
𝑥 (𝑠
𝑥 )/𝑛 + (1 − 1/𝑛)𝑢
𝑥 (𝑠
𝑥 )/𝑠
𝑥 , is (positiv e) non-neg ativ e, (de-
creasing) non-increasing;
6. P o tential function Φ( s ) = ∑
𝑥 ∫
𝑠 𝑥 0
𝜙 𝑥 (𝑡 ) d 𝑡 , s ∈ 𝑛Δ
𝑚−1
, which means t hat Φ( s ) =
∑
𝑥 [𝑢
𝑥 (𝑠
𝑥 )/𝑛 + (1 − 1/𝑛) ∫
𝑠 𝑥 0
𝑢 𝑥 (𝑡 )/𝑡 d 𝑡 ] ;
Multi-mar k et oligopol y is similar t o Cour no t oligopol y (Cour no t 1838 ), but differs in
significant w a ys. In Cour no t oligopol y , each pla y er chooses a production le v el of t he same
product, whose mar ginal retur n decreases wit h t o tal production; while t he multi-mar k et
oligopol y can be seen as a multi-product Cour no t g ame, where all pla y ers ha v e t he same
t o tal productivity . F or mall y , Cour no t oligopol y can be written as: 𝐺 𝑐 = {𝑁 , 𝑄, u } , where
pla y er s tr ategy 𝑞 𝑖 ∈ 𝑄 𝑖 = ℝ
≥0
, and pla y er pa y off function 𝑢 𝑖 (𝑞
𝑖 , 𝑞 −𝑖
) = 𝑝(𝑞)𝑞
𝑖 − 𝑐 𝑞 𝑖 ; price
𝑝(𝑞) is a decreasing function on t o tal productivity 𝑞 = ∑
𝑖 𝑞 𝑖 , and mar ginal cos t 𝑐 is as-
sumed t o be cons tant. The multi-mar k et oligopol y ins tead has 𝑚 products, and each pla y er
dis tributes one unit of resource 𝑠 among t he products, ear ning pa y off from all products in-
v es ted.
T o pro v e t hat multi-mar k et fir ms of t he same capacity ha v e a unique and symmetric
PSNE, w e f ollo w a lis t of propositions sho wn belo w . Bef ore g etting int o t he details, w e
point out t he k e ys t o t he proof: con v ex g ame guar antees PSNE exis ts; equal capacity leads
t o symmetr y ; and mono t onic mar ginal pa y offs pro vide a unique solution.
Propositions:
1. Φ( s ) is (s trictl y) conca v e.
13
C hapter 3 Mul ti-market oligopol y
2. 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is (s trictl y) conca v e , ∀𝑖 , ∀ s
−𝑖
∈ 𝑆 −𝑖
.
3. Multi-mar k et oligopol y is a con v ex g ame.
4. Multi-mar k et oligopol y has PSNE, (all s trict).
5. Multi-mar k et oligopol y can onl y ha v e symmetric P SNE.
6. Multi-mar k et oligopol y ha v e a (unique) esse ntiall y unique PSNE, in t hat mar ginal
pla y er pa y offs are unif or m across all in v es ted mar k ets.
7. The equilibrium of multi-mar k et oligopol y is globall y asym p t o ticall y s table under
g r adient adjus tment process.
Let 𝑃 𝑥 (𝑠
𝑥 ) = ∫
𝑠 𝑥 0
𝑝 𝑥 (𝑡 ) d 𝑡 . Since 𝑃 𝑥 (𝑠
𝑥 ) is a differentiable real function wit h a con v ex
domain, it is (s trictl y) conca v e if and onl y if it is globall y (s trictl y) dominated b y its linear
expansions: ∀𝑠
0
> 0, ∀𝑠
𝑥 ≥ 0, 𝑠 𝑥 ≠ 𝑠 0
,
𝑃 𝑥 (𝑠
𝑥 ) − [𝑃
𝑥 (𝑠
0
) + 𝑝 𝑥 (𝑠
0
)(𝑠
𝑥 − 𝑠 0
)]
= ∫
𝑠 𝑥 𝑠 0
𝑝 𝑥 (𝑡 ) d 𝑡 − 𝑝 𝑥 (𝑠
0
)(𝑠
𝑥 − 𝑠 0
)
= ∫
𝑠 𝑥 𝑠 0
𝑝 𝑥 (𝑡 ) − 𝑝 𝑥 (𝑠
0
) d 𝑡 ≤ 0
This is tr ue because 𝑝 𝑥 (𝑠
𝑥 ) is (decreasing) non-increasing. Because Φ( s ) is a positiv e linear
tr ansf or mation of 𝑃 𝑥 (𝑠
𝑥 ) and 𝑢 𝑥 (𝑠
𝑥 ) which is also (s trictl y) conca v e, it im plies t hat Φ( s )
is (s trictl y) conca v e on t he non-neg ativ e cone ℝ
𝑚 ≥0
. Because sim plex 𝑛Δ
𝑚−1
is a con v ex
subset of t he non-neg ativ e cone ℝ
𝑚 ≥0
, it im plies t hat Φ( s ) is (s trictl y) conca v e on t he sim plex
𝑛Δ
𝑚−1
. This pro v es Proposition 1 .
Because sim plex 𝑆 𝑖 is a con v ex subset of t he non-neg ativ e cone ℝ
𝑚 ≥0
, if 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is
(s trictl y) conca v e on ℝ
𝑚 ≥0
, ∀𝑖 , ∀ s
−𝑖
∈ 𝑆 −𝑖
, t hen 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is also (s trictl y) conca v e on 𝑆 𝑖 ,
∀𝑖 , ∀ s
−𝑖
∈ 𝑆 −𝑖
. It suffices t o pro v e t he f or mer s tatement wit hout cons tr aints on opponent
14
C hapter 3 Mul ti-market oligopol y
s tr ategies: 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is (s trictl y) conca v e on ℝ
𝑚 ≥0
, ∀𝑖 , ∀ s
−𝑖
∈ ℝ
𝑚 ≥0
. Because 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is a
positiv e linear tr ansf or mation of 𝑢 𝑖 𝑥 (𝑠
𝑖 𝑥 ; 𝑠 −𝑖 𝑥 ) , 𝑥 ∈ 𝐸 , it suffices if 𝑢 𝑖 𝑥 (𝑠
𝑖 𝑥 ; 𝑠 −𝑖 𝑥 ) is (s trictl y)
conca v e on ℝ
≥0
, ∀𝑥 , ∀𝑖 , ∀𝑠
−𝑖 𝑥 ≥ 0 . T o sim plify no tations, t his is equiv alent t o 𝑢 𝑖 𝑥 (𝑠; 𝑐 ) =
𝑝 𝑥 (𝑠 + 𝑐 )𝑠 (s trictl y) conca v e on ℝ
≥0
, ∀𝑥 , ∀𝑐 ≥ 0 . This can be pro v ed b y definition, and w e
do no t include t he proof here because it is s tr aightf or w ar d but tedious. The k e y t o t his
proof is t hat 𝑢 𝑥 (𝑠
𝑥 ) is (s trictl y) conca v e and 𝑝 𝑥 (𝑠
𝑥 ) is (decreasing) non-increasing; eit her of
t he o p tional conditions can guar antee s trict conca vity . This pro v es Proposition 2.
A con v ex g ame is a g ame where each pla y er has a con v ex s tr ategy space and a conca v e
pa y off function 𝑢 𝑖 (𝑠
𝑖 ; 𝑠 −𝑖
) f or all opponent s tr ategies. In t his g ame, pla y er s tr ategy space is
t he same sim plex 𝑆 𝑖 = Δ
𝑚−1
f or all pla y ers, which is con v ex. T og et her wit h Proposition 2,
t his is pro v es Proposition 3.
A con v ex g ame has PSNE if it has a com pact s tr ategy space and continuous pa y off func-
tions, see Nikaido and Isoda 1955 . Because t he product space of sim plices is com pact, t his
g ame has a com pact s tr ategy space 𝑆 ≡ ∏
𝑖 𝑆 𝑖 = ∏
𝑖 Δ
𝑚−1
. Because 𝑢 𝑥 (𝑠
𝑥 ) is continuous
∀𝑥 , pla y er pa y off 𝑢 𝑖 ( s ) is t hus cont inuous ∀𝑖 . This g ame t hus has PSNE. If 𝑢 𝑖 ( s
𝑖 ; s
−𝑖
) is
s trictl y conca v e, all PSNEs are s trict. This pro v es Proposition 4.
Giv en a P SNE s
∗
, f or all pla y er 𝑖 , equilibrium s tr ategy s
∗
𝑖 sol v es t he con v ex op timization
problem:
maximize 𝑢 𝑖 ( s
𝑖 ; s
∗
−𝑖
)
subject t o s
𝑖 ≥ 0
s
𝑖 ⋅ 1 = 1
(3.1)
Since t his con v ex op timization problem is s trictl y f easible, b y Slater’ s t heorem, it has s trong
duality . Since t he objectiv e function 𝑢 𝑖 ( s
𝑖 ; s
∗
−𝑖
) is differentiable, t he Kar ush-K uhn- T uck er
(KKT) t heorem s tates t hat op timal points of t he op timization problem is t he same wit h t he
15
C hapter 3 Mul ti-market oligopol y
solutions of t he KKT conditions:
∇ 𝑢 𝑖 + 𝜆 𝑖 − 𝜈 𝑖 1 = 0 (saddle point)
s
𝑖 ≥ 0 (primal cons tr aint 1)
𝜆 𝑖 ≥ 0 (dual cons tr aint)
s
𝑖 ∘ 𝜆 𝑖 = 0 (com plementar y slackness)
s
𝑖 ⋅ 1 = 1 (primal cons tr aint 2)
(3.2)
Here oper at or ∘ deno tes t he Hadamar d product: (𝑥 ∘ 𝑦 )
𝑖 = 𝑥 𝑖 𝑦 𝑖 . Giv en t he saddle point
conditions 𝜕 𝑢 𝑖 /𝜕 𝑠 𝑖 𝑥 + 𝜆 𝑖 𝑥 − 𝜈 𝑖 = 0, ∀𝑥 , t he dual cons tr aint im plies t hat t he mar ginal pa y off
f or pla y er 𝑖 in mar k et 𝑥 is bounded abo v e: 𝜕 𝑢 𝑖 /𝜕 𝑠 𝑖 𝑥 ≤ 𝜈 𝑖 , ∀𝑥 . If pla y er 𝑖 in v es ts in mar k et 𝑥 ,
𝑠 𝑖 𝑥 > 0 , b y com plementar y slackness t he upper bound is tight, which means t he mar ginal
pa y offs f or pla y er 𝑖 are unif or m in all mar k ets 𝑖 in v es ts. T og et her , mar ginal pla y er pa y offs
at equilibrium ha v e relation:
𝜕 𝑢 𝑖 𝜕 𝑠 𝑖 𝑦 ≤
𝜕 𝑢 𝑖 𝜕 𝑠 𝑖 𝑥 = 𝜈 𝑖 , ∀𝑖 , ∀𝑥 , 𝑦 , 𝑠 ∗
𝑖 𝑥 > 𝑠 ∗
𝑖 𝑦 ≥ 0 (3.3)
Since 𝜕 𝑢 𝑖 /𝜕 𝑠 𝑖 𝑥 = 𝑝 𝑥 (𝑠
𝑥 )+𝑝
′
𝑥 (𝑠
𝑥 )𝑠
𝑖 𝑥 , and 𝑝 ′
𝑥 < 0 , t his is equiv alent t o 𝑝 𝑥 (𝑠
∗
𝑥 ) ≤ 𝜈 𝑖 +|𝑝
′
𝑥 (𝑠
∗
𝑥 )|𝑠
∗
𝑖 𝑥 , ∀𝑥 ,
wit h equality if 𝑠 ∗
𝑖 𝑥 > 0 . If pla y er 𝑖 in v es ts more in mar k et 𝑥 t han pla y er 𝑗 does, 𝑠 ∗
𝑖 𝑥 > 𝑠 ∗
𝑗 𝑥 ≥ 0 ,
t his im plies 𝜈 𝑖 + |𝑝
′
𝑥 (𝑠
∗
𝑥 )|𝑠
∗
𝑖 𝑥 ≤ 𝜈 𝑗 + |𝑝
′
𝑥 (𝑠
∗
𝑥 )|𝑠
∗
𝑗 𝑥 . But because t he pla y ers ha v e t he same capacity ,
pla y er 𝑖 mus t ha v e in v es ted less in some mar k et 𝑦 t han pla y er 𝑗 does: 𝑠 ∗
𝑗 𝑦 > 𝑠 ∗
𝑖 𝑦 ≥ 0 , which
im plies 𝜈 𝑗 + |𝑝
′
𝑦 (𝑠
∗
𝑦 )|𝑠
∗
𝑗 𝑦 ≤ 𝜈 𝑖 + |𝑝
′
𝑦 (𝑠
∗
𝑦 )|𝑠
∗
𝑖 𝑦 . T og et her , t hese inequalities im pl y |𝑝
′
𝑥 (𝑠
∗
𝑥 )|(𝑠
∗
𝑖 𝑥 −
𝑠 ∗
𝑗 𝑥 ) + |𝑝
′
𝑦 (𝑠
∗
𝑦 )|(𝑠
∗
𝑗 𝑦 − 𝑠 ∗
𝑖 𝑦 ) ≤ 0 . This contr adicts our assum p tion on pla y er resource allocation,
t hus all pla y ers mus t ha v e t he same s tr ategy in equilibrium. This pro v es Proposition 5.
F rom Propos ition 4 and 5, multi-mar k et fir ms ha v e PSNE, which are symmetric. F or a
symmetric PSNE s
∗
, pla y er s tr ategy s
∗
𝑖 = s
∗
/𝑛 , and mar ginal pla y er pa y offs in in v es ted mar -
k ets are t he same f or all pla y ers: 𝜈 𝑖 = 𝜈 , ∀𝑖 . N o w t he relation among equilibrium mar ginal
16
C hapter 3 Mul ti-market oligopol y
pla y er pa y offs can be re written as: 𝑝 𝑥 (𝑠
∗
𝑥 ) + 𝑝 ′
𝑥 (𝑠
∗
𝑥 )𝑠
∗
𝑥 /𝑛 ≤ 𝜈 , ∀𝑥 , wit h equality in in v es ted
mar k ets, 𝑠 ∗
𝑥 > 0 . Because 𝜙 𝑥 (𝑠
𝑥 ) = 𝑝 𝑥 (𝑠
𝑥 ) + 𝑝 ′
𝑥 (𝑠
𝑥 )𝑠
𝑥 /𝑛 , t his is equiv alent t o
𝜙 𝑥 (𝑠
∗
𝑥 ) ≤ 𝜈 , ∀𝑥 (3.4)
wit h equality in in v es ted mar k ets. Since 𝑢 𝑥 (𝑠
𝑥 ) is a uni-v ariate differentiable (s trictl y) con-
ca v e function, 𝑢 ′
𝑥 (𝑠
𝑥 ) is (decreasing) non-increasing. Because 𝑝 𝑥 (𝑠
𝑥 ) is also (decreasing)
non-increasing, 𝜙 𝑥 (𝑠
𝑥 ) = 𝑢 ′
𝑥 (𝑠
𝑥 )/𝑛 + (1 − 1/𝑛)𝑝
𝑥 (𝑠
𝑥 ) is (decreasing) non-increasing. Define
in v erse function 𝜙 −1
𝑥 ∶ ℝ
≥0
→ ℝ
≥0
, so t hat 𝜙 −1
𝑥 (𝜈 ) = 0 f or 𝜈 > 𝜙 −1
𝑥 (0) . The function is
non-increasing (decreasing f or 𝜈 ≤ 𝜙 −1
𝑥 (0) ) and t he equilibrium satisfies:
𝑠 ∗
𝑥 = 𝜙 −1
𝑥 (𝜈 ), ∀𝑥 (3.5)
Since t o tal in v es tment equals t he number of pla y ers, ∑
𝑥 𝑠 𝑥 = 𝑛 , mar ginal pla y er pa y off in
in v es ted m ar k ets 𝜈 is deter mined b y :
∑
𝑥 𝜙 −1
𝑥 (𝜈 ) = 𝑛 (3.6)
Because t he lef t-hand side of Equation 3.6 is decreasing f or 𝜈 ≤ 𝑚𝑎𝑥 𝑥 𝜙 −1
𝑥 (0) where t he
lef t-hand side is positiv e, t he equation giv es a unique solution 𝜈 . Thus Equation 3.5 giv es
a (unique) essentiall y unique s
∗
. F igure 3.1 sho w s t his process g r aphicall y . This pro v es
Proposition 6.
The g r adient adjus tment process (Arro w and Hur wicz 1960 ) is a heuris tic lear ning r ule
where pla y ers adjus t t heir s tr ategies accor ding t o t he local g r adient of t heir pa y off func-
tions, projected ont o t he tang ent cone of pla y er s tr ategy space. F or mall y , g r adient adjus t-
17
C hapter 3 Mul ti-market oligopol y
F igure 3.1: Deter mination of equilibrium. Mar ginal pla y er pa y off 𝜈 at equilibrium can be
deter mined from ∑
𝑥 𝑠 𝑥 (𝜈 ) = 𝑠 . Equilibrium allocation in each mar k et can t hen
be deter mined b y 𝑠 ∗
𝑥 = 𝑠 𝑥 (𝜈 ) .
ment process is a dynamical sys tem:
d s
𝑖 d 𝑡 = 𝑃 𝑇 (𝑠
𝑖 )
∇
𝑖 𝑢 𝑖 ( s ), ∀𝑖 (3.7)
Here ∇
𝑖 deno tes t he g r adient wit h respect t o pla y er s tr ategy 𝑠 𝑖 , 𝑇 (𝑠
𝑖 ) is t he tang ent cone of
pla y er s tr ategy space 𝑆 𝑖 at point 𝑠 𝑖 , and 𝑃 is t he projection oper at or . F or all interior points
of t he pla y er s tr ategy space, 𝑃 𝑇 (𝑠
𝑖 )
is sim pl y t he centering matrix, 𝑀 1
= 𝐼 − 11
T
/𝑚 . T o
pro v e t hat t he dynamical sys tem is globall y asym p t o ticall y s table, w e sho w t hat 𝑉 ( s ) =
Φ( s
∗
) − Φ( s ) is a global L y apuno v function: a function t hat is positiv e-definite, contin-
uousl y differentiable, and has neg ativ e-definite time deriv ativ e. F rom Proposition 1 and
KKT t heorem, t he maximal points of t he po tential function Φ( s ) is deter mined t he same
w a y as Equation 3.4 . That is, t he maximal set of t he po tential function is identical t o t he
PSNE of multi-mar k et oligopol y . This means 𝑉 ( s ) > 0 at non-equilibrium points, so 𝑉 ( s )
18
C hapter 3 Mul ti-market oligopol y
is positiv e-definite. 𝑉 ( s ) is clea r l y continuousl y differentiable, and its time deriv ativ e
d 𝑉 ( s )
d 𝑡 = −∇ Φ( s ) ⋅
d s
d 𝑡 = −
𝑚 ∑
𝑥=1
𝑛 ∑
𝑖 =1
⎛
⎜
⎜
⎝
𝜕 𝑢 𝑖 𝑥 𝜕 𝑠 𝑖 𝑥 −
1
𝑚 𝑚 ∑
𝑦 =1
𝜕 𝑢 𝑖 𝑦 𝜕 𝑠 𝑖 𝑦 ⎞
⎟
⎟
⎠
𝜙 𝑥 = −𝑛
𝑚 ∑
𝑥=1
⎛
⎜
⎜
⎝
𝜙 𝑥 −
1
𝑚 𝑚 ∑
𝑦 =1
𝜙 𝑦 ⎞
⎟
⎟
⎠
𝜙 𝑥 = −𝑚𝑛 ( 𝜙 2
− 𝜙 2
)
Here 𝜙 = ∑
𝑚 𝑥=1
𝜙 𝑥 /𝑚 and 𝜙 2
= ∑
𝑚 𝑥=1
𝜙 2
𝑥 /𝑚 . Thus, 𝜙 2
− 𝜙 2
≥ 0 , wit h equality if and onl y
if 𝜙 𝑥 are all equal. F rom Equation 3.4 , w e can see t hat d 𝑉 / d 𝑡 ≤ 0 , wit h equality onl y at
equilibrium points s
∗
. W e ha v e t hus sho wn t hat 𝑉 ( s ) is a global L y apuno v function of t he
dynamical sys tem, which immediatel y im plies Proposition 7.
The s tability result in Proposition 7 is onl y intended t o sho w t hat under a sim ple and
plausible lear ning r ule, global asym p t o tic s tability of N ash equilibrium is possible in multi-
mar k et oligopol y so t hat t he equilibrium can be em piricall y obser v ed. The g r adient adjus t-
ment process adop ted in t his paper is no t meant t o be t he exact lear ning r ule used in real
lif e, which is har d t o deter mine. But com pared wit h Ba y esian or bes t-response lear ning
r u les, it is less demanding on t he pla y ers as it does no t require com plete inf or mation of
t h e g ame or long-ter m memor y of t he pla y ers. And e v en if some pla y ers adop t alter nativ e,
non-economic lear ning r ules, t he s tability of t he equilibrium ma y w ell be preser v ed. F or
exam ple, ne w driv ers ma y sim pl y choose imitativ e lear ning (R o t h and Ere v 1995 ; F uden-
ber g and Le vine 2009 ), or in o t her w or ds “f ollo w t he older driv ers”. In t his case t he N ash
equilibrium is s till t he s table f ocus as all pla y ers adop t t he same s tr ategy and t he r ational
pa y off-im p ro ving pla y ers adjus t t o t he equilibrium. By imitativ e lear ning, ne w driv ers sa v e
t h e possibl y long process of s tr ategy adjus tment and quickl y con v er g e t o t he equilibrium
s tr ategy . This allo w s t he equilibrium remain s table under an e v ol ving set of driv ers.
19
C hapter 3 Mul ti-market oligopol y
W e no te t hat t he pa y off in multi-mar k et oligopol y is no t s trictl y diagonall y conca v e,
so t he uniqueness and s tability results canno t f ollo w R osen 1965 . But t he eig en v alues o f
t h e Jacobian ∇ ( d s / d 𝑡 ) are alw a ys neg ativ e, so local asym p t o tic s tability at t he equilib-
rium is guar anteed under g r adient dynamics wit h individual-specific adjus tment speeds.
W e also no te t hat unlik e t he Cour no t g ame, multi-mar k et oligopol y is no t an agg reg ate
g a me defined in Selten 1970 or later g ener alizations, because pla y er s tr ategies are multi-
dimensional. Thus it does no t inherit t he s tability under discrete-time bes t-response dy -
namics. Multi-mar k et oligopol y is also no t a po tential g ame and t hus does no t inherit t he
g ener al dynamic s tability properties in Monderer and Shaple y 1996 . Ins tead, w e pro vided
a “po tential function” t hat is a global L y apuno v function f or t he g r adient dynamics.
20
C h apter 4
Empiric al resul ts
4.1 Sp a tial equilibrium
T o v alidate t hat taxi driv ers f ollo w t he t heoretical equilibrium, w e proceed in tw o parts.
F irs t , t he s tr ategy profile at equilibrium is symmetric, which means t hat all driv ers use
t h e same s tr ategy . Second, giv en driv ers use t he same s tr ategy , unif or m mar ginal driv er
re v e nue across s treet segments means t hat 𝜙 𝑥 is t he same on all segments.
Alt hough driv er s tr ategy , t he spatial dis tribution of ser vice time, is no t directl y obser v ed,
it is proportional t o driv er pickup probability on each segment. If all driv ers use t he same
s tr ategy , t he pickup probability dis tribution across segments of each driv er shall be t he
same as t hat of t he o v er all dis tribution. Then each driv er’ s actual pickups shall be a sam-
ple of t he corresponding categorical r andom v ariable. Since t here are t housands of s treet
segments, pickup recor ds of each driv er is no t enough t o tes t t he probability model. W e
partition t he segments int o 10 equi-probable g roups, so pickup counts in t hese g roups
shall be a multinomial r andom v ariable wit h t he same probability f or each g roup. Driv ers ’
pickup counts in t hese g roups can be tes ted b y a corrected log lik elihood r atio of multino-
mial dis tributions (Smit h et al. 1981 ). F or each driv er , t he pickup counts are nor malized
int o a probability v ect or x . And w e consider t he de viation from t he equi-probable v ect or ,
21
C hapter 4 Empiric al resul ts
x
′
= x − 1 /10 , lar g e if its 𝐿1 -nor m ex ceeds 0.3. Onl y a small fr action of driv ers ha v e s ta-
tis ticall y significant lar g e de viations, see F ig. 4.1 A . The t hreshold f or lar g e de viation is
arbitr ar y , but t he result sho w s t hat mos t driv ers use similar s tr ategies.
T o sho w t hat mar ginal driv er re v enue on all s treet segments are equilibr ated, w e no te
t h at 𝜙 𝑥 ≈ 𝑢 𝑥 /𝑠
𝑥 , because at an y moment t he number of taxi driv ers in ser vice in Manhattan
is in t he t housands. So it suffices t o sho w t hat re v enue originated on each segment 𝑢 𝑥 is
proportional t o t o tal driv er ser vice time attributable t o t he segment 𝑠 𝑥 , which is t he sum
of search time 𝑡 𝑠𝑥 and trip time 𝑡 𝑝𝑥 per unit time. Because t he ma jority of taxi trips are
metered, which is calculated from trip dis tance and time in slo w tr affic, driv er re v enue
from each trip is highl y correlated t o trip dur ation reg ar dless of driv er s tr ategy , especiall y
when tr affic speeds are hold s tationar y . T o a v oid t he influence of t his f act, consider trip
time is a linear function of trip re v enue, t hen 𝑢 𝑥 ∝ 𝑠 𝑥 is equiv alent t o 𝑢 𝑥 ∝ 𝑡 𝑠𝑥 , and w e tr y
t o sho w t he latter . Because search routes are no t reco r ded in t he trip recor ds, w e tak e trip
recor ds betw een 6pm and 7pm on w eekda ys in spring, and es timate taxi routes betw een
trips b y shortes t dis tance routing. W e consider t his approach accep table because during
t he selected hours, tr affic is roughl y at a unif or m cong es ted speed wh ile a v er ag e taxi search
time is t he shortes t, so route de viation from t he shortes t pat h is unlik el y . The correlation
betw een re v enue and es timated search time on s treet segments are reasonabl y high, wit h
𝑅 2
= 0.85 , see F ig. 4.1 B. Because shortes t pat h routing pro vides a single route f or trips
wit h t he same origin and des tination, es timated search time ma y be concentr ated t o a f e w
s treet segments, which w eak ens t he actual correlation.
4.2 D yn amic equilibrium
As en vironment condition ℰ v aries o v er times of a da y , taxi equilibrium will also v ar y . If
taxi driv ers are free t o choose when t o w or k and are indifferent about w or king at different
22
C hapter 4 Empiric al resul ts
times of a da y , taxi suppl y will adjus t so t hat at equilibrium t he a v er ag e driv er income
is t he same t hroughout a da y . T o v erify t his, w e examine t he tr a ject or y of a v er ag e driv er
re v e nue and number of driv ers t hroughout a typical w eekda y , sho wn in F ig. 4.2 . A v er ag e
driv er re v enues during 8am-4pm and 6pm-3am center around $35/hour and $30/hour
respectiv el y , and are cons tant in t he sense t hat its o v er all v ariation is about t he same as its
short-ter m v ariation. The difference betw een a v er ag e driv er re v enue f or t hese tw o periods
can be explained b y tw o f act ors. F irs t, t he t o tal number of taxis is limited and no t all is
a v ailable f or t he night shif t, so no t all driv ers who w ould lik e t o w or k at night can g et a
taxi. Second, t he lease r ate f or da y shif ts is less t han t hose of night shif ts, so t he difference in
a v er a g e driv er income betw een t he tw o periods is less t han t hat of a v er ag e driv er re v enue.
During 4pm-6pm mos t double-shif ted taxis chang e driv ers, which means suppl y decisions
during t his period is no t up t o t he driv ers, so t he a v er ag e driv er re v enue is no t cons tant.
During 3am- 6am v er y f e w driv ers are at w or k , and t he high a v er ag e driv er re v enue jus tifies
t h e cos t of w or king when mos t people pref er t o be sleeping. During 6am-8am mos t da y
shif t driv ers s tart w or king, and alt hough t he a v er ag e driv er re v enue is no t cons tant, it
s tabilizes as more driv ers become activ e.
In contr as t t o t he equilibr ation of a v er ag e driv er income o v er time, giv en en vironment
condition ℰ , equilibrium mar ginal segment re v enue is a decreasing function of t he number
of driv ers: 𝜙(𝑠) ∣ ℰ . This relation is har d t o measure wit hout controlled experiment, but
it is reflected in t he obser v ational data if t he number of driv ers is f orced t o chang e much
f as t er t han t he en vironment does, such as during shif t tr ansition. N o te t hat equilibrium
mar ginal segment re v enue and a v er ag e driv er re v enue are appro ximatel y t he same: 𝜙 ≈
∑
𝑥 𝑢 𝑥 / ∑
𝑥 𝑠 𝑥 , because 𝜙 𝑥 ≈ 𝑢 𝑥 /𝑠
𝑥 . R eusing F ig. 4.2 , t he do wn w ar d trend in 5pm-6pm
reflects 𝜙(𝑠) f or t hat time of da y , when people lea v e w or k and taxis retur n f or t he night
shif t.
23
C hapter 4 Empiric al resul ts
4.3 Driver learnin g
It is natur al t o ask if driv ers lear n t o use t he same s tr ategy resulting in spatiall y unif or m
mar ginal re v enue. W e use driv ers ’ firs t appear ance in trip recor ds t o measure t heir o v er -
all experience, and t he number of trips driv ers made during a chosen time slo t t o measure
t h eir experience wit h t he specific situation. F ig. 4.3 sho w s t hat s tr ategy de viation decreases
wit h driv er experience. A v er ag e driv er de viation con v er g es wit hin about one y ear of driv -
ing, while decreases mono t onicall y wit h situation-specific experience.
4.4 Policy imp a ct on equilibrium
Chang e in taxi regulation affects t he equilibrium. On 2013-08-08, NY C TL C launched S treet
Hail Liv er y , kno wn as g reen cabs, which are allo w ed t o pick up s treet-hail passeng ers out-
side core Manhattan, defined as sou t h of W es t 110t h S treet and Eas t 96t h S treet. This g r ad-
uall y increased suppl y of s treet-hail ser vice be y ond core Manhattan, decreasing mar ginal
driv er re v enue on segments t herein. Equilibrium demands t hat mar ginal driv er re v enue
on segments wit hin core Manhattan should also decrease t o t he same le v el, which im plies
more suppl y of y ello w cabs in core Manhattan where t he y ha v e ex clusiv e rights t o ser vice.
W e com pare t he time-series of pickup percentag e in t he region bor dering core Manhattan
in 2012 and 2013, see F ig. 4.4 . Ex cluding irregularity due t o Hurricane Sandy and holi-
da ys, t he percentag e is s table in t he later mont hs of bo t h y ears, reflecting a robus t decline
in y e llo w c ab suppl y be y ond core Manhattan.
24
C hapter 4 Empiric al resul ts
F igure 4.1: V alidation of equilibrium, using trip recor ds in S pring 2011 and Spring 2012.
(U pper) Size and s tatis tical significance of driv er de viation from a v er ag e s tr at-
egy in T ue- Thu PM peak s, 6pm-10pm. 3.66% of all driv ers ha v e s tatis ticall y
significant ( 𝑝 > 0.05 ) lar g e de viations ( || x
′
||
1
> 0.3 ). (Lo w er) Correlation be-
tw een search time and income on s treet segments ( 𝑅 2
= 0.85 ), bo t h in log scale
and s tandar dized, f or Mon-F ri 6pm-7pm. If t he tw o quantities are proportional,
mar ginal income on s treet segments are unif or m.
25
C hapter 4 Empiric al resul ts
5 10 15 17 19 24 5
25
30
35
40
45
50
Average driver revenue, $/hour
Hour
Average number of taxis
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Average driver revenue, $/hour
Number of drivers
0 2000 4000 6000 8000 10000
25
30
35
40
45
50
F igure 4.2: N umber of driv ers and a v er ag e driv er income t hroughout a typical W ednesda y
in spring, 2011-2012. (U pper) T ime series. (Lo w er) T r a ject or y . Each do t repre-
sents one minute, a v er ag ed o v er all obser v ations and colored b y t he hour . T ext
labels mar k hours of a da y , from 0 t o 23, positioned at t he middle of t he hour .
26
C hapter 4 Empiric al resul ts
F igure 4.3: Driv er lear ns equilibrium. Using t he same trip recor ds as F ig. 2A . (U pper) A v -
er ag e driv er de viation decreases wit h t heir y ears of driving. Driv ers s tarted be-
f ore data collection are clus tered near 2009. R ed cur v e sho w s local reg ression.
Dashed line sho w s a v er ag e de viation of driv ers s tarted betw een mid-2009 and
2011. (Lo w er) A v er ag e driv er de v iation decreases wit h number of trips made.
27
C hapter 4 Empiric al resul ts
0 50 100 150
1.5
2.0
2.5
3.0
Percent of pickups bordering core Manhattan
Days since July 1
2012
2013
F igure 4.4: Equilibrium shif t af ter g reen cab launch. Solid cur v es are 7-da y rolling pickup
probability in t he region bor dering core Manhattan, wit h 90% propability in-
ter v al. This probability slightl y reduced af ter 2012 f are r aise (red line), g reatl y
increased during Hurricane Sandy (shaded area), and moder atel y increased
during Thank sgiving (v ertical lines) and Chris tmas.
28
C h apter 5
Discussion
5.1 Efficien cy : the pr oblem of social c os t
W e no tice t hat in multi-mar k et oligopol y t he t o tal pa y off 𝑢( s ) = ∑
𝑖 ∈𝑁
𝑢 𝑖 ≠ Φ( s ) , which
means t o tal pa y off is g ener all y no t maximized in N ash equilibrium, t hus no t sociall y op-
timal. In f act, if t o tal pa y off is maximized, t hen 𝜕 𝑢 𝑥 /𝜕 𝑠 𝑥 ≥ 𝜕 𝑢 𝑦 /𝜕 𝑠 𝑦 , ∀𝑥 , 𝑦 ∈ 𝐸, 𝑠 𝑥 > 0 ,
which means mar ginal pa y off are t he same f or all in v es ted mar k ets. Com pare wit h Equa-
tion 3.4 and t he definition of 𝜙 𝑥 (𝑠
𝑥 ) , a w eighted a v er ag e of mar ginal and a v er ag e pa y off
is balanced ins tead. W it h 𝑛 ≫ 1 in t he case of NY C taxi sys tem, w e ha v e 𝜙 ≈ 𝑢 𝑥 /𝑠
𝑥 . So at
equilibrium t he a v er ag e segment income per ser vice time are effectiv el y t he same f or all
searched segments.
This is similar t o t he Cour no t g ame. The t o tal pa y off in t he Cour no t g ame is 𝑢(𝑞) =
∑
𝑖 ∈𝑁
𝑢 𝑖 = 𝑝(𝑞)𝑞 − 𝑐 𝑞 . Assuming 𝑝(𝑞) differentiable, t he social op timum is 𝑢 ∗
= (𝑝(𝑞
∗
) − 𝑐 )𝑞
∗
,
where 𝑞 ∗
satisfies 𝑝 ′
(𝑞
∗
)𝑞
∗
+ 𝑝(𝑞
∗
) = 𝑐 . The N ash equilibrium is 𝑞 𝑖 = 𝑞 †
/𝑛, ∀𝑖 ∈ 𝑁 , where
𝑞 †
satisfies 𝑝 ′
(𝑞
†
)𝑞
†
/𝑛 + 𝑝(𝑞
†
) = 𝑐 . This mak es 𝑞 †
> 𝑞 ∗
and 𝑢(𝑞
†
) < 𝑢(𝑞
∗
) , so t he N ash
equilibrium is no t social op timal, and decreases furt her as t he number of pla y er increases.
Ho w e v er , t he N ash equilibrium of multi-mar k et oligopol y is sociall y op timal if mar k et
pa y offs are po w er functions of t he same or der : 𝑢 𝑥 (𝑠
𝑥 ) = 𝑎 𝑥 𝑠 𝑝 𝑥 , 𝑎 𝑥 > 0, 𝑝 ∈ (0, 1) . In t his case
29
C hapter 5 Discussion
t h e t o tal pa y off 𝑢 = ∑
𝑥∈𝐸
𝑎 𝑥 𝑠 𝑝 𝑥 and pla y er pa y off 𝑢 𝑖 = ∑
𝑥∈𝐸
𝑎 𝑥 𝑠 𝑝−1
𝑥 𝑠 𝑖 𝑥 . A t social equilibrium,
𝜕 𝑢 𝑥 /𝜕 𝑠 𝑥 = 𝑎 𝑥 𝑝𝑠 𝑝−1
𝑥 is a cons tant f or all mar k ets 𝑥 ∈ 𝐸 . Since ∑
𝑥∈𝐸
𝑠 𝑥 = ∑
𝑖 ∈𝑁
∑
𝑥∈𝐸
𝑠 𝑖 𝑥 = 𝑛 ,
social op timal s tr ategy is 𝑠 ∗
𝑥 = 𝑛𝑎 1/(1−𝑝)
𝑥 / ∑
𝑦 ∈𝐸
𝑎 1/(1−𝑝)
𝑦 , ∀𝑥 ∈ 𝐸 . A t N ash equilibrium, f or
all pla y ers 𝑖 ∈ 𝑁 , let 𝑢 𝑖 𝑥 = 𝑢 𝑥 𝑠 𝑖 𝑥 /𝑠
𝑥 , t hen 𝜕 𝑢 𝑖 𝑥 /𝜕 𝑠 𝑖 𝑥 = 𝑎 𝑥 ((𝑝 − 1)𝑠
𝑝−2
𝑥 𝑠 𝑖 𝑥 + 𝑠 𝑝−1
𝑥 ) is a con-
s tant f or all mar k ets 𝑥 ∈ 𝐸 . This means ∑
𝑖 ∈𝐸
𝜕 𝑢 𝑖 𝑥 /𝜕 𝑠 𝑖 𝑥 = (𝑝 − 1 + 𝑛)𝑎
𝑥 𝑠 𝑝−1
𝑥 is a cons tant f or all
mar k ets 𝑥 ∈ 𝐸 , which giv es t he same agg reg ate s tr ategy 𝑠 †
𝑥 = 𝑛𝑎 1/(1−𝑝)
𝑥 / ∑
𝑦 ∈𝐸
𝑎 1/(1−𝑝)
𝑦 , ∀𝑥 ∈
𝐸 , so t he N ash equilibrium is social op timal. Use t he condition ag ain, w e find N ash equi-
librium 𝑠 †
𝑖 𝑥 = 𝑠 †
𝑥 /𝑛, ∀𝑖 ∈ 𝑁 , 𝑥 ∈ 𝐸 .
This phenomenon of difference betw een cooper ativ e and com petitiv e decisions has been
s tudied f or a long time under different names. The concep t of economic inefficiency ref ers
t o a situation where t o tal income, or social w ealt h, is no t maximized, see Mill 1859 and
Sidgwick 1883 . The problem of social cos t is t he div er g ence betw een priv ate and social
cos ts or v alue, see t he discussion betw een Pigou 1920 and Knight 1924 . This discussion
br a nch int o t he concep t of exter nal effect, originated b y Meade 1952 . Independentl y , Gor -
don 1954 proposed rent dissipation, where maximum rent is no t realized at equilibrium.
Mar k et f ailure is ano t her ter minology of t he same phenomenon, see Bat or 1958 f or exam-
ple. Coase 1960 dismissed t he discussion of social cos t, calling t he reduced social income as
tr ansaction cos t. Exter nality e v ol v ed out of exter nal effect, whose proponents typicall y call
f or go v er nment regulation, see O. A . Da vis and Whins t on 1962 . Ins titution cos t is a g ener -
alization of tr ansaction cos t, see Cheung 1998 . A recent de v elopment in algorit hmic g ame
t h eor y uses tw o ter ms f or t his inefficiency of equilibria: price of a narch y (K outsoupias and
P apadimitriou 1999 ; R oughg ar den 2002 ), and price of s tability .
Despite t he v arious ter minologies, t he essence of t he problem is t he same: when individ-
uals do no t ha v e incentiv e in maximizing t he t o tal pa y off, equilibrium natur all y will differ
from t he op timum set, which b y definition results in less t o tal pa y off. Here I propose t he
main tak ea w a y f or t he case of multi-mar k et oligopol y :
30
C hapter 5 Discussion
Theorem If a property is heterog eneous in productivity , t he o wner canno t obtain t he op-
timal rent b y leasing t o multiple ten ants wit hout contr acting on t heir allocation of
effort.
5.2 Ma cr osc opic interpret a tion: thermod yn amics
In t his section, w e inter pret t he N ash equilibrium of taxi driv ers as a t her modynamic equi-
librium. This es tablishes a macroscopic equilibrium where agg reg ate beha vior is perceiv ed
as a tr ansport phenomenon built up from individual choices. This macroscopic vie w ig-
nores t he decision making and com petition of taxi driv ers, but helps unders tand t he out-
come of a social sys tem from t he perspectiv e of a ph ysical sys tem.
R eg ar d t o tal ser vice tim e 𝑠 , which equals t he number of driv ers in ser vice, as t o tal en-
er g y of t he taxi tr ansportation sys tem. R eg ar d t he po tential function Φ of multi-mar k et
oligopol y as entrop y . And reg ar d t he reciprocal of equilibrium mar ginal driv er re v enue,
𝜓 = 1/𝜙 , as tem per ature. 𝑠 , 𝜓 , and Φ are all s tate v ariables of t he taxi sys tem at equilibrium
giv en e n vironment condition ℰ .
Being a s tate v ariable and intensiv e property , 𝜓 is t he driving f orce of t he tr ansport of
ser v ice time 𝑠 o v er t he s treet segments. As w e ha v e pro v ed ear lier , t he g r adient dynamics of
multi-mar k et oligopol y alw a ys increases t he po tential function Φ( s ) , which is maximized
at equilibrium. When tw o taxi sys tems at equilibrium are put int o contact wit h an interf ace
per m eable t o t he tr ansf er of taxi ser vice time, 𝑠 will flo w from t he sys tem wit h higher 𝜓 t o
t h e one wit h lo w er 𝜓 . A t equilibrium, 𝜓 is homog eneous across all searched segments. In
summar y , w e can mak e t he f ollo wing s tatements of t her modynamics:
Zeroth Law T w o taxi sys tems in contact ha v e t he same equilibrium mar ginal driv er re v -
enue. Equiv alentl y ,
𝜓 1
= 𝜓 2
31
C hapter 5 Discussion
First Law T axi tr ansportation is t he tr ansf er process of t o tal ser vice time 𝑠 , which is a con-
ser v ed quantity . Equiv alentl y ,
d 𝑠 = ∑
𝑥∈𝐸
d 𝑠 𝑥 Second Law U nder fix ed taxi demand and tr affic s tate, a closed taxi sys tem maximizes its
entrop y Φ . Equiv alentl y ,
d Φ ≥
𝛿 𝑠 𝜓 The contact equilibrium defines equiv alent classes of taxi equilibrium, which are s trictl y
t o tall y or dered b y s tate v ariable 𝜓 . The manif old of taxi equilibrium is t hus one-dimensional,
par ameterized b y 𝜓 , and an y o t her s tate v ariable mus t depend on it. This means t hat s tate
space (Φ, 𝑠, 𝜓 ) ∣ ℰ has onl y one deg ree of freedom, and t his dependency is t he cons titutiv e
relation of t he taxi sys tem giv en en vironment condition ℰ . W riting t he cons titutiv e relation
explicitl y as 𝜓 (𝑠) ∣ ℰ , or equiv alentl y 𝜙(𝑠) ∣ ℰ , our discussion in Sectio n 4.2 has sho wn ho w
t h is cons titutiv e relation can be measured from obser v ational data under certain condi-
tions. R earr anging t he exact differential of 𝑠(Φ) ∣ ℰ giv es t he fundamental t her modynamic
relation of taxi equilibrium:
d 𝑠 = 𝜓 d Φ (5.1)
Our discussion in Section 4.4 can be seen as an exam ple of contact equilibrium, where
t h e ne w taxi sys tem (g reen cab) is g eog r aphicall y res tricted t o pickup outside core Man-
hattan. The increased taxi suppl y out of core Manhattan driv es do wn 𝜙 and t hus r aises
tem per ature 𝜓 , f orcing some y ello w cabs int o core Manhattan. The ne w equilibrium is
reached when tem per ature 𝜓 across t he boundar y are t he same.
32
C h apter 6
Ma terials and Methods
6.1 New Y ork City t axi trip rec ords
The N e w Y or k City (NY C) T axi and Limousine Commission (TL C) s tarted collecting trip
recor d of its Medallion T axis (aka y ello w cabs) using T axi P asseng er Enhancement Pro-
g r am (TPEP) de vice in Januar y 2009. Since t hen, t he TL C has been publishing t his digital
trip recor d continuousl y . R ecent s tudies ha v e used t hese trip recor ds t o s tudy labor suppl y
elas ticity of taxi driv ers F arber2014 , tax i pooling po tential Santi et al. 2014 , and tr ansporta-
tion sys tems resilience t o natur al hazar ds B. Dono v an and D. W or k 2015.
The trip recor d contains attributes, among o t hers: latitude, longitude and time s tam p of
bo t h pickups and drop-offs; trip dis tance and f are amount. Published trip recor ds till 2014
ma y also contain medallion ID (f or v ehicles) and hack license (f or driv ers), which w ere
remo v ed from published data later b y TL C due t o priv acy concer ns. W e ha v e g at hered
trip recor ds wit h t hese ID fields f or t he y ears from 2009 t o 2013, accounting f or 870 mil-
lion individual taxi trips. Brian Dono v an and D. B. W or k 2014 In t his s tudy , w e use t hese
data t o v alidate FHV s teady s tate and t o deriv e t he em pirical cons titutiv e relation f or NY C
taxicab trips. The ID fields are indispensable in t he v alidation, because t o es timate model
par a meters w e need t o tr ack a taxi betw een consecutiv e trips.
33
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36

Abstract (if available)

Abstract Taxi is a valuable part of urban transportation and industrial organization, where taxi drivers are both ride service providers and independent contractors. This paper studies the street-hail taxi industry, where each taxi in service is seen as a multi-market firm, every street segment is a distinct market, and firms allocate service time across the street network. We generalize the model into a game of multi-market competition among firms of equal capacity, and prove that the game has pure-strategy Nash equilibrium (PSNE), which is (1) symmetric, (2) essentially unique in that marginal player payoffs are uniform across all invested markets, and (3) globally asymptotically stable under gradient adjustment process and imitative learning. The aggregate strategy at equilibrium maximizes a ""potential function"", and it differs from the social optimal strategy for most forms of production functions. With 868 million trip records of all 13,237 Medallion taxis (yellow cab) in New York City from 2009 to 2013, we validate that taxi drivers' behavior conform to this equilibrium under fixed traffic speeds and taxi demand. Consistent with the equilibrium, marginal segment income is constant over time-of-day in each shift. With the launch of Street Hail Livery (green cab) in late 2013, which increases street-hail vehicles out of core Manhattan, we observe a decrease in yellow cab pickups beyond East 96th Street and West 110th Street.

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Creator Zhang, Ruda (author)

Core Title Taxi driver learns dynamic multi-market equilibrium

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Economics

Publication Date 04/11/2018

Defense Date 02/05/2018

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

Tag economic efficiency,equilibrium,game theory,Learning and Instruction,OAI-PMH Harvest,statistical physics,taxi driver,Urban transportation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kahn, Matthew (committee chair), Pesaran, Hashem (committee member), Ridder, Geert (committee member)

Creator Email rudazhan@gmail.com,rudazhan@usc.edu

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