Efficiency Loss in Market Mechanisms for Resource Allocation

249 Pages·2004·1.98 MB·English

Efficiency Loss in
Market Mechanisms for
Resource Allocation
by
Ramesh Johari
A.B.,Mathematics,HarvardUniversity(1998)
CertificateofAdvancedStudyinMathematics,UniversityofCambridge(1999)
Submittedtothe
DepartmentofElectricalEngineeringandComputerScience
inPartialFulfillmentoftheRequirementsfortheDegreeof
DoctorofPhilosophy
inElectricalEngineeringandComputerScience
atthe
MassachusettsInstituteofTechnology
June2004
c 2004MassachusettsInstituteofTechnology. Allrightsreserved.
(cid:13)
Signature of Author...................................................................
DepartmentofElectricalEngineeringandComputerScience
May18,2004
Certified by...........................................................................
JohnN.Tsitsiklis
ProfessorofElectricalEngineeringandComputerScience
ThesisSupervisor
Accepted by..........................................................................
ArthurC.Smith
Chairman,CommitteeonGraduateStudents
DepartmentofElectricalEngineeringandComputerScience

Efficiency Loss in Market Mechanisms
for Resource Allocation
by
Ramesh Johari
SubmittedtotheDepartmentofElectricalEngineering
andComputerScienceonMay18,2004,
inpartialfulfillmentoftherequirementsforthedegreeof
DoctorofPhilosophyinElectricalEngineeringandComputerScience
Abstract
This thesis addresses a problem at the nexus of engineering, computer science, and
economics: inlargescale,decentralizedsystems,howcanweefficientlyallocatescarce
resources among competing interests? On one hand, constraints are imposed on the
system designer by the inherent architecture of any large scale system. These con-
straints are counterbalanced by the need to design mechanisms that efficiently allo-
cate resources, even when the system is being used by participants who have only
theirownbestinterestsatstake.
We consider the design of resource allocation mechanisms in such environments.
The analytic approach we pursue is characterized by four salient features. First, the
monetary value of resource allocation is measured by the aggregate surplus (aggre-
gateutilitylessaggregatecost)achievedatagivenallocation. Anefficientallocationis
onewhichmaximizesaggregatesurplus. Second, wefocusonmarket-clearingmech-
anisms,whichsetasinglepricetoensuredemandequalssupply. Third,allthemech-
anisms we consider ensure a fully efficient allocation if market participants do not
anticipate the effects of their actions on market-clearing prices. Finally, when market
participants are price anticipating, full efficiency is generally not achieved, and we
quantifytheefficiencyloss.
Wemaketwomaincontributions. First,forthreeeconomicenvironments,wecon-
siderspecificmarketmechanismsandexactlyquantifytheefficiencylossintheseenvi-
ronmentswhenmarketparticipantsarepriceanticipating. Thefirsttwoenvironments
address settings where multiple consumers compete to acquire a share of a resource
in either fixed or elastic supply; these models are motivated by resource allocation
in communication networks. The third environment addresses competition between
4
multipleproducerstosatisfyaninelasticdemand; thismodelismotivatedbymarket
designinpowersystems.
Oursecondcontributionistoestablishthat,underreasonableconditions,themech-
anisms we consider minimize efficiency loss when market participants anticipate the
effects of their actions on market-clearing prices. Formally, we show that in a class of
market-clearingmechanismssatisfyingcertainsimplemathematicalassumptionsand
forwhichthereexistfullyefficientcompetitiveequilibria,themechanismsweconsider
uniquelyminimizeefficiencylosswhenmarketparticipantsarepriceanticipating.
Thesis Supervisor: JohnN.Tsitsiklis
Title: ProfessorofElectricalEngineering
Acknowledgments
Firstandforemost, Iamdeeplyindebtedtomyadvisor, ProfessorJohnTsitsiklis. His
encouragement,support,andadvicehavebeenimmenselyvaluable,bothinpersonal
and professional terms. I am particularly grateful for his emphasis on simplicity and
elegance in research, and for the genuine concern he has shown for my development
asanacademic.
I am also grateful to Professor Frank Kelly for his enthusiasm and counsel, a con-
stant throughout my graduate career. I first became familiar with network pricing
problems in conversations with him, and this thesis bears the hallmark of lessons
learnedduringmystayinCambridgeunderhistutelage.
Specialthanksgotomythesiscommitteemembers,ProfessorAbhijitBanerjeeand
Professor Robert Gallager. Each devoted significant time and effort to my thesis, and
theirsuggestionsandcommentsledtosubstantialimprovementinthefinalproduct.
The Laboratory for Information and Decision Systems (LIDS) provided an ideal
workenvironmentfortheinterdisciplinaryworkofthisthesis. Iamparticularlygrate-
ful to Professor Dimitri Bertsekas, for helping me navigate the subtleties of convex
analysis; to Professor Vincent Chan, for his steady hand in directing the lab; and to
ProfessorSanjoyMitter,forhisintellectualstewardshipofmytimeatMIT.Iaminthe
debtofthesefacultyandmanyothersatMITfortheirsupportoverthelastfouryears.
I have also benefited from time spent with colleagues at MIT. I would especially
like to thank Shie Mannor, whose dedication to research helped keep me disciplined;
and Emin Martinian, a frequent counterpart for lunch conversations, both academic
andotherwise. Finally,IwasparticularlyfortunatetohavehadConstantineCarama-
nis as an officemate, with whom I’ve shared many interesting discussions since my
daysasanundergraduate.
Last, I owe a great debt to Hsin Chau, for her support, encouragement, and sense
ofhumor;hercompanionshiphasbeeninvaluableinallfacetsoflife.
ThisresearchwassupportedbyaNationalScienceFoundationGraduateResearchFellow-
ship, bytheDefenseAdvancedResearchProjectsAgencyundertheNextGenerationInternet
Initiative,andbytheArmyResearchOfficeundergrantDAAD10-00-1-0466.
5

Preliminaries
Notation
We use R to denote the real numbers, and R+ to denote [0, ). Italics will be used to
∞
denote scalars, e.g., x. Boldface will be used to denote vectors, e.g., x = (x ,...,x ).
1 n
When x is a scalar, the notation (x)+ will be used to denote the positive part of x;
i.e., (x)+ = x if x 0, and (x)+ = 0 if x 0. If x ,...,x Rm, and x =
1 n
≥ ≤ ∈
(x ,...,x ), we will use x to denote the components of x other than x ; that is,
1 n −i i
x = (x ,...,x ,x ,...,x ). Throughout the thesis, if f : (Rm)n R is a real-
−i 1 i−1 i+1 n
→
valued function of n vectors x ,...,x Rm, we let f(x ;x ) denote the function f
1 n i −i
∈
asafunctionofx whilekeepingthecomponentsx fixed.
i −i
Convex analytic methods play a key role in this thesis, and we collect some re-
quired notions here [14, 103]. An extended real-valued function is a function f : Rn
→
[ , ]; such a function is called proper if f(x) > for all x, and f(x) < for at
−∞ ∞ −∞ ∞
least one x. We say that a vector γ Rn is a subgradient of an extended real-valued
∈
functionf atxifforallx Rn,wehave:
∈
f(x) f(x)+γ>(x x).
≥ −
The subdifferential of f at x, denoted ∂f(x), is the set of all subgradients of f at x.
We say that f is subdifferentiable at x if ∂f(x) = . We will typically be interested in
6 ∅
subgradients of a convex function f, and supergradients of a concave function f. A
vector γ is a supergradient of f if γ is a subgradient of f; thus we denote the
− −
superdifferentialoff atxby ∂[ f(x)].
− −
For extended real-valued functions f : R [ , ], we will require some addi-
→ −∞ ∞
tionalconcepts. Wedenotetherightdirectionalderivativeoff atxby∂+f(x)/∂xand
left directional derivative of f at x by ∂−f(x)/∂x (if these exist). If f is convex, then
∂f(x) = [∂−f(x)/∂x,∂+f(x)/∂x],providedthedirectionalderivativesexist.
7
8
Prerequisites
The main prerequisites for this thesis are a background in real analysis at the level of
Rudin[110],aswellassomefacilitywithconvexoptimizationandelementaryconvex
analysis. SourcesforbackgroundonconvexoptimizationincludethebooksbyWhittle
[145], Bertsekas [13], and Boyd and Vandenberghe [17], while background on convex
analysismaybefoundinthetextsbyBertsekasetal. [14]andRockafellar[103].
Microeconomics(particularlymarkettheory)andgametheoryalsoplayakeyrole
inthisthesis,andsomebasicknowledgeofthetwofieldsishelpful. ThetextbyVarian
providesaconciseintroductiontomicroeconomictheory[137],whilethetextbookby
Mas-Colell et al. provides deeper coverage [82]. As to game theory, in this thesis we
will only use elementary concepts from game theory, particularly Nash equilibrium;
however,someunderstandingofthemodelingissuesishelpful. Forthispurpose,see
the books by Fudenberg and Tirole [43], Myerson [89], and Osborne and Rubinstein
[96](wherethelastreferenceisaconciseintroductionfortheuninitiatedreader).
Bibliographic Note
Portions of the content of Chapter 2 appear in the paper by Johari and Tsitsiklis [60];
exceptions are Sections 2.1.3, 2.3, 2.4.3, and 2.5.1. Sections 3.1, 3.2, 3.3, and 3.4 will
appearinthepaperbyJoharietal. [58].
Contents
Abstract 3
Acknowledgments 5
Preliminaries 7
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
BibliographicNote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ListofFigures 13
1 Introduction 15
1.1 Consumers,Producers,andAggregateSurplus . . . . . . . . . . . . . . . 16
1.2 Market-ClearingMechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 PriceTakingBehaviorandCompetitiveEquilibrium . . . . . . . . . . . . 22
1.4 PriceAnticipatingBehaviorandNashEquilibrium . . . . . . . . . . . . . 23
1.5 AnExample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.1 AMarket-ClearingMechanism . . . . . . . . . . . . . . . . . . . . 26
1.5.2 PriceTakingConsumers . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.3 PriceAnticipatingConsumers . . . . . . . . . . . . . . . . . . . . 28
1.5.4 EfficiencyLoss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.6 ContributionsofThisThesis . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 MultipleConsumers,InelasticSupply 33
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 PriceTakingUsersandCompetitiveEquilibrium . . . . . . . . . 40
2.1.2 PriceAnticipatingUsersandNashEquilibrium . . . . . . . . . . 42
2.1.3 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 EfficiencyLoss: TheSingleLinkCase. . . . . . . . . . . . . . . . . . . . . 50
2.3 ProfitMaximizingLinkManagers . . . . . . . . . . . . . . . . . . . . . . 57
9
10 CONTENTS
2.4 GeneralNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.1 AnExtendedGame . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.2 EfficiencyLoss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.4.3 AComparisontoProportionallyFairPricing . . . . . . . . . . . . 81
2.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.1 StochasticCapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.2 AGeneralResourceAllocationGame . . . . . . . . . . . . . . . . 87
2.6 ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3 MultipleConsumers,ElasticSupply 91
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.1.1 PriceTakingUsersandCompetitiveEquilibrium . . . . . . . . . 97
3.1.2 PriceAnticipatingUsersandNashEquilibrium . . . . . . . . . . 99
3.1.3 NondecreasingElasticityPriceFunctions . . . . . . . . . . . . . . 106
3.2 EfficiencyLoss: TheSingleLinkCase. . . . . . . . . . . . . . . . . . . . . 108
3.3 InelasticSupplyvs. ElasticSupply . . . . . . . . . . . . . . . . . . . . . . 119
3.4 GeneralNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.5 CournotCompetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.5.1 ModelswithBoundedEfficiencyLoss . . . . . . . . . . . . . . . . 142
3.5.2 GeneralNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.6 CournotCompetitionwithLatency . . . . . . . . . . . . . . . . . . . . . . 155
3.7 ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4 MultipleProducers,InelasticDemand 163
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.1.1 PriceTakingFirmsandCompetitiveEquilibrium . . . . . . . . . 170
4.1.2 PriceAnticipatingFirmsandNashEquilibrium . . . . . . . . . . 173
4.2 EfficiencyLoss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.3 NegativeSupply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.4 StochasticDemand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.5 ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5 CharacterizationTheorems 189
5.1 MultipleConsumers,InelasticSupply . . . . . . . . . . . . . . . . . . . . 191
5.1.1 AFirstCharacterizationTheorem . . . . . . . . . . . . . . . . . . 192
5.1.2 ASecondCharacterizationTheorem . . . . . . . . . . . . . . . . . 208
5.1.3 ATwoUserMechanismwithArbitrarilyLowEfficiencyLoss . . 211
5.2 MultipleProducers,InelasticDemand . . . . . . . . . . . . . . . . . . . . 214
5.3 ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229