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Approximate Equilibria in Large Games a Dissertation APPROXIMATE EQUILIBRIA IN LARGE GAMES A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MANAGEMENT SCIENCE AND ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Yu Wu August 2012 © 2012 by Yu Wu. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/jv163hr1839 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ramesh Johari, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Nicholas Bambos I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Christina Aperjis Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract The complexity of studying Nash equilibrium in large games often scales with the size of the system: as it increases, computing the exact Nash equilibrium can soon become intractable. However, when the number of players in the system approaches infinity and no individual player has a significant impact on the system, we can approximate the system by considering each single player no longer playing against other individual players but a single aggregation of all other players. In this paper, we apply this idea to study and approximate Nash equilibria in two large scale games. In part I, we consider a model of priced resource sharing that combines both queueing behavior and strategic behavior. We study a priority service model where a single server allocates its capacity to agents in proportion to their payment to the system, and users from different classes act to minimize the sum of their cost for processing delay and payment. As the exact processing time of this system is hard to compute and cannot be characterized in closed form, we introduce the concept of aggregate equilibrium to approximate the exact Nash equilibrium, by assuming each individual player plays against a common aggregate priority that characterizes the large system. We then introduce the notion of heavy traffic equilibrium as an alter- native approximation of the Nash equilibrium, derived by considering the asymptotic regime where the system load approaches capacity. We show that both aggregate equilibrium and heavy traffic equilibrium are asymptotically exact in heavy traffic. We present some numerical results for both approximate equilibria, and discuss effi- ciency and revenue, and in particular provide a bound for the price of anarchy of the heavy traffic equilibrium. In part II, we study the reputation system of large scale online marketplace. We iv develop a large market model to study reputation mechanisms in online marketplaces. We consider two types of sellers: commitment sellers, who are intrinsically honest but may be unable to accurately describe items because of limited expertise; and strategic sellers, who are driven by a profit maximization motive. We focus on stationary equi- libria for this dynamic market, in particular, on separating equilibria where strategic sellers are incentivized to describe the items they have for sale truthfully, and char- acterize the conditions under which such equilibria exist. We then complement our theoretical results with computational analysis and provide insights on the features of markets that may incentivize truthfulness in equilibrium. v Acknowledgements First, I would like to give my deepest gratitude to my advisor Ramesh Johari for his endless support and constant encouragement in my education at Stanford. His passion, optimism, attitude towards work and life has inspired and will continue to inspire me. I am very grateful to Christina Aperjis and Loc Bui for their insightful suggestions in collaborating in several areas of this thesis. I would like to thank Nick Bambos, Ashish Goel and Balaji Prabhakar for serving my dissertation committee and provid- ing great advice to the development of this work. I want to thank all the faculty and staff in the Department of Management Science and Engineering, for their generous help that makes my stay at Stanford a very enjoyable one. I would like to gratefully acknowledge support from a Larry Yung Stanford Grad- uate Fellowship, and the National Science Foundation under grant CMMI-0948434. In these five years at Stanford, I benefited tremendously from fruitful discus- sions with fellow students, especially Krishnamurthy Iyer, Xiangrui Meng and Zizhuo Wang. I enjoyed very much my life at Stanford, which has been filled with joyful memories with my friends here: Ming Chen, Hao Chen, Pei He, Yi Liu, Zhen Qian, Xu Tan and many others. I would also like to express my gratitude to Shanshan Xu, Weipeng Zhang and Yali Zhang for giving me constant emotional support despite the geographical distance between us. I have a special thank you to Beichen Wang, who embraces me with heartfelt friendship and influences me with his kindness and integrity. Finally, I would like to thank my parents Qingping Wu and Yuan Wang for their unconditional love, support and encouragement. I owe the most thanks to them. vi Contents Abstract iv Acknowledgements vi 1 Part I: Resource Sharing Game 1 1.1 Introduction . .1 1.2 Resource Sharing Game . .4 1.2.1 Characterizing Processing Times . .6 1.2.2 Nash Equilibrium . .8 1.3 Job Level Game: Aggregate Approximation . 10 1.3.1 Aggregate Priority . 11 1.3.2 Aggregate Processing Time . 12 1.3.3 Aggregate Equilibrium . 16 1.4 Job Level Game: Heavy Traffic Approximation . 20 1.4.1 Heavy Traffic Processing Time . 20 1.4.2 Heavy Traffic Equilibrium . 21 1.4.3 Sensitivity . 24 1.4.4 Efficiency . 25 1.4.5 Revenue . 28 1.5 Class Level Game . 30 1.5.1 Aggregate Equilibrium . 33 1.5.2 Heavy Traffic Equilibrium . 36 1.6 Numerics . 37 vii 1.6.1 Approximation Errors: AE vs. HTE . 38 1.6.2 Approximation Sensitivity . 38 1.6.3 Price of Anarchy . 41 1.7 Extensions . 41 1.7.1 A Multi-server Model . 44 1.8 Conclusion . 47 2 Part II: Reputation System 49 2.1 Introduction . 49 2.2 Model . 53 2.2.1 Preliminaries . 53 2.2.2 Market Dynamics and Stationary Equilibrium . 56 2.2.3 Separating Equilibrium . 58 2.3 Existence of Separating Equilibria . 60 2.4 Approximating Separating Equilibria . 65 2.5 Computational Analysis . 68 2.6 Conclusion . 72 Bibliography 73 viii List of Tables 1.1 Fairness-efficiency tradeoff on α...................... 26 ix List of Figures 1.1 Relative error of AE and HTE with different system loads. 39 1.2 Relative error of HTE under different parameters. 40 1.3 Price of anarchy of HTE under different parameters. 42 2.1 Density functions and their logarithmic ratio. 71 2.2 The function B(r) with different class weights. 72 x Chapter 1 Part I: Resource Sharing Game 1.1 Introduction A range of resource sharing systems, such as computing or communication services, exhibit two distinct characteristics: queueing behavior and strategic behavior. Queue- ing behavior arises because jobs or flows are served with the limited capacity of system resources. Strategic behavior arises because these jobs or flows are typically generated by self-interested, payoff-maximizing users. Analysis of strategic behavior in queueing systems has a long history, dating to the seminal work of Naor [47]; see the book by Hassin and Haviv [22] for a comprehensive survey. The interaction of queueing and strategic behaviors has become especially important recently, with the rise of paid re- source sharing systems such as cloud computing platforms. For example, [2] and [12] discussed systems with multiple service providers, modeled as first-come-first-serve queues, that compete in both price and response time for potential buyers. In this chapter we consider a particular queueing model where a single server is shared among multiple jobs, and the service capacity allocated to each job depends on its priority level. The particular scheduling policy we consider is known in the liter- ature as the discriminatory processor sharing (DPS) policy, introduced by Kleinrock [38]. In the DPS model, the server shares its capacity in proportion to the priority level of all jobs currently in the system. This service allocation rule is a special case of a more general scheduling policy for queueing networks known as proportionally 1 CHAPTER 1. PART I: RESOURCE SHARING GAME 2 fair resource sharing [34, 42]; such scheduling policies have been studied extensively in the context of networked resource sharing (see [32, 54] and references therein). A survey of the DPS literature can also be found in [3]. We consider a DPS system in steady state, and study a job level game where every individual job is a single strategic user. Each user chooses a payment β, which corresponds to the priority level of that user. The user also incurs a cost proportional to total processing time. The users' goal is to choose priority levels to minimize the sum of expected processing cost and payment. (We also briefly discuss a class level game, where every class is a single user.) This game is inspired by resource sharing in real services.
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