Efficiency Loss in Resource Allocation Games Yunjian Xu

246 Pages·2012·0.78 MB·English

Efficiency Loss in Resource Allocation Games
by
Yunjian Xu
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2012
⃝c Massachusetts Institute of Technology 2012. All rights reserved.
Author ..............................................................
Department of Aeronautics and Astronautics
July 20, 2012
Certified by..........................................................
Munther A. Dahleh
Professor of EECS
Thesis Committee Member
Certified by..........................................................
John N. Tsitsiklis
Clarence J. Lebel Professor of Electrical Engineering
Thesis Supervisor
Accepted by.........................................................
Eytan H. Modiano
Chairman, Department Committee on Graduate Students
Chair, Thesis Committee
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Efficiency Loss in Resource Allocation Games
by
Yunjian Xu
Submitted to the Department of Aeronautics and Astronautics
on July 20, 2012, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
The overarching goals of this thesis are to quantify the efficiency loss due to market
participant strategic behavior, and to design proper pricing mechanisms that reduce
the efficiency loss. The concept of efficiency loss is intimately related to the concept
of “price of anarchy,” which was advanced by Koutsoupias and Papadimitriou, and
comparesthemaximumsocialwelfarewiththatachievedataworstNashequilibrium.
This thesis focuses on the following two topics: (i) For a market with an arbitrary
number of participants, how much is the Nash equilibrium close, in the sense of price
of anarchy, to a social optimum? (ii) For a resource allocation/pricing mechanism,
is the social welfare achieved at an economic equilibrium asymptotically optimal, as
the number of market participants goes to infinity?
Regarding the first topic, we quantify the efficiency loss in classical Cournot
oligopoly games, where multiple oligopolists compete by choosing quantities. We also
compare the total profit earned at a Cournot equilibrium to the maximum possible
total profit that would be obtained if the suppliers were to collude.
For the second topic, related to the efficiency in large economics, we analyze the
efficiency of Kelly’s proportional allocation mechanism in large-scale wireless commu-
nication systems. We study a corresponding Bayesian game in which each user has
incomplete information on the state or type of the other users, and show that the
social welfare achieved at a Bayes-Nash equilibrium is asymptotically optimal, as the
number of users increases to infinity.
Finally, for electricity delivery systems, we propose a new dynamic pricing mecha-
nism that explicitly encourages consumers to adapt their consumption so as to offset
the variability of demand on conventional units. Through a dynamic game-theoretic
formulation, we show that the proposed pricing mechanism achieves social optimality
asymptotically, as the number of consumers increases to infinity.
Thesis Supervisor: John N. Tsitsiklis
Title: Clarence J. Lebel Professor of Electrical Engineering
Thesis Supervisor
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Acknowledgments
Foremost, I would like to express my profoundest gratitude to my advisor, John
Tsitsiklis. I am sincerely grateful for his encouragement, motivation, and especially
for his constantly insightful suggestions that have been immensely valuable during
my entire Ph.D study.
My sincere thanks goes to the members of my thesis committee, Munther Dahleh
and Eytan Modiano, for their significant effort devoted to my dissertation research.
Also thank you to Asu Ozdaglar, Mardavij Roozbehani and Minghui Zhu for being
part of my general committee. Their valuable feedback considerably improved the
quality of this thesis.
I would also like to express my deep appreciation to Dimitri Bertsekas, Michael
Caramanis,RameshJohari,MihalisMarkakis,JagdishRamakrishnan,DevavratShah,
Mengdi Wang, Moe Win, Kuang Xu, Yuan Zhong, and Spyros Zoumpoulis, with
whom I have had conversations that helped me in my Ph.D study and research.
I am extremely grateful to my family, for the support they have given me over the
years.
This research was sponsored by National Science Foundation under grants ECCS-
0701623 and CMMI-0856063, and by a Graduate Fellowship from Shell.
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Contents
1 Introduction 15
1.1 Bayesian Proportional Resource Allocation Games . . . . . . . . . . . 16
1.2 Efficiency and Profit Loss in Cournot Oligopolies . . . . . . . . . . . 17
1.3 Pricing of Fluctuations in Electricity Markets . . . . . . . . . . . . . 18
2 Bayesian Proportional Resource Allocation Games 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 Background and related research . . . . . . . . . . . . . . . . 21
2.1.2 Summary and contributions of this chapter . . . . . . . . . . . 25
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Formal description of the model . . . . . . . . . . . . . . . . . 26
2.2.2 Resource allocation mechanisms . . . . . . . . . . . . . . . . . 27
2.3 The Value of Information - Non-strategic Formulations . . . . . . . . 32
2.4 Ex Ante and Ex Post Games with Kelly Mechanism . . . . . . . . . . 36
2.4.1 Ex Ante Game . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Ex post Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Bayesian Game: Existence of Equilibria . . . . . . . . . . . . . . . . . 38
2.6 Bayesian Game Examples . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Bayesian Games with a Large Number of Users . . . . . . . . . . . . 45
2.7.1 Assumptions, preliminaries, and approximate symmetry of BNEs 46
2.7.2 Asymptotic efficiency . . . . . . . . . . . . . . . . . . . . . . . 48
2.8 Efficiency Loss in Bayesian Games . . . . . . . . . . . . . . . . . . . . 51
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3 Efficiency Loss in a Cournot Oligopoly with Convex Market Demand 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1.2 Our contribution . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Formulation and Background . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Preliminaries on Cournot Equilibria . . . . . . . . . . . . . . . . . . . 61
3.3.1 Optimality and equilibrium conditions . . . . . . . . . . . . . 62
3.3.2 Efficiency of Cournot equilibria . . . . . . . . . . . . . . . . . 64
3.3.3 Restricting to linear cost functions . . . . . . . . . . . . . . . 66
3.3.4 Other properties of Cournot candidates . . . . . . . . . . . . . 70
3.3.5 Concave inverse demand functions . . . . . . . . . . . . . . . . 72
3.4 Affine Inverse Demand Functions . . . . . . . . . . . . . . . . . . . . 73
3.5 Convex Inverse Demand Functions . . . . . . . . . . . . . . . . . . . 75
3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7 Monopoly and Social Welfare . . . . . . . . . . . . . . . . . . . . . . 86
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Profit Loss in Cournot Oligopolies 91
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.2 Our contribution . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Formulation and Background . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Preliminaries on Cournot Equilibria . . . . . . . . . . . . . . . . . . . 94
4.3.1 Optimality and equilibrium conditions . . . . . . . . . . . . . 95
4.3.2 Profit ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.3 Restricting to linear cost functions . . . . . . . . . . . . . . . 98
4.3.4 Other properties of Cournot candidates . . . . . . . . . . . . . 101
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4.4 Profit ratio lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.5 Corollaries and Applications . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Efficiency Loss in a Class of Two-Sided Market Mechanisms 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Efficiency Loss of Nash Equilibria . . . . . . . . . . . . . . . . . . . . 119
5.4 Corollaries and Applications . . . . . . . . . . . . . . . . . . . . . . . 124
6 Pricing of Fluctuations in Electricity Markets 127
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . 131
6.1.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 The Pricing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.4 Dynamic Oblivious Equilibrium . . . . . . . . . . . . . . . . . . . . . 142
6.4.1 The sequence of prices induced by a dynamic oblivious strategy 143
6.4.2 Equilibrium strategies . . . . . . . . . . . . . . . . . . . . . . 144
6.4.3 Relation between a continuum model and a corresponding n-
consumer model . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5 Approximation in Large Games . . . . . . . . . . . . . . . . . . . . . 147
6.6 Asymptotic Social Optimality . . . . . . . . . . . . . . . . . . . . . . 150
6.6.1 Social welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.6.2 Asymptotic social optimality of a DOE . . . . . . . . . . . . . 151
6.7 Implementation of the proposed pricing mechanism . . . . . . . . . . 154
6.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.8.1 Equilibrium under Marginal Cost Pricing . . . . . . . . . . . . 156
6.8.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 157
6.9 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . 165
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A Proof of results in Chapter 2 177
A.1 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A.2 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.3 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
A.4 Proof of Theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.5 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 198
B Proof of results in Chapter 3 203
B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
B.2 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.3 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
C Proof of results in Chapter 4 215
C.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
C.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
D Proof of Theorem 5.1 225
E Proof of results in Chapter 6 231
E.1 Approximation of the supplier cost . . . . . . . . . . . . . . . . . . . 231
E.2 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
E.3 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
E.3.1 Proof of Part (a) . . . . . . . . . . . . . . . . . . . . . . . . . 239
E.3.2 Proof of Part (b) . . . . . . . . . . . . . . . . . . . . . . . . . 241
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