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Mechanism Design and Analysis Using Simulation-Based Game Models by Yevgeniy Vorobeychik A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Science and Engineering) in The University of Michigan 2008 Doctoral Committee: Professor Michael P. Wellman, Chair Professor Edmund H. Durfee Associate Professor Satinder Singh Baveja Associate Professor Emre Ozdenoren c Yevgeniy Vorobeychik 2008 All Rights Reserved I dedicate this work to my wife, Polina, and my daughter, Avital. ii ACKNOWLEDGMENTS There is little doubt that my dissertation would never have been completed, had it not been for the support of a great number of people, including family, friends, and faculty mentors. I am very greatful to my thesis advisor, Michael Wellman, for having guided me along these past six years with timely advice, encouragement, and invaluable ideas that have had a direct and profound influence on my work. I also want to thank my committee members, Satinder Singh, Ed Durfee, and Emre Ozdenoren. My numerous adventures into each of their offices had invariably left me rejuvenated and brimming with new ideas. All of my published work to date is co-authored, and I am grateful to all of the co- contributors. Prominent among these are Chris Kiekintveld, my perennial officemate— before and after the move to our wonderful new building—who has been both a great collegue and a great friend, Daniel Reeves, who has been not only a collegue and a friend, but also a mentor, particularly in my days as a fledgling researcher, and Patrick Jordan, whose intellectual insight is often sharp and always invaluable. I am grateful to my col- legues at Yahoo! Research, Jenn Wortman, John Langford, Lihong Li, Yiling Chen, and David Pennock for the fantastic time I spend working with them during my internship, and to Isaac Porche for his tutelage during my summer tenure at RAND Corporation. Finally, I am grateful to my family for having supported me in my lengthy doctoral endeavor, particularly to my wife, Polina, who had been with me through the highs and the lows, and who gave me a most precious gift—a beautiful daughter Avital. iii TABLE OF CONTENTS DEDICATION .................................... ii ACKNOWLEDGMENTS .............................. iii LIST OF TABLES .................................. x LIST OF FIGURES ................................. xi LIST OF APPENDICES .............................. xv ABSTRACT .....................................xvi CHAPTER 1 Introduction . 1 1.1 A High-Level Model . 4 1.2 Contributions . 6 1.3 Overview of Thesis . 7 2 Game-Theoretic Preliminaries . 9 2.1 Games in Normal Form . 9 2.1.1 Notation . 9 2.1.2 Basic Normal-Form Solution Concepts . 11 2.2 Games of Incomplete Information and Auctions . 13 2.3 Games in Extensive Form . 15 2.3.1 A Formal Description . 15 2.3.2 Infinitely Repeated Games with Complete and Perfect Information . 17 Part I Will the Optimal Mechanism Please Stand Up? 19 3 Mechanism Design: Classical and Computational Approaches . 19 3.1 Social Choice Theory . 19 3.2 Mechanism Design Under Incomplete Information . 23 iv 3.2.1 Setup . 23 3.2.2 Implementation in Dominant Strategies . 24 3.2.3 Implementation in Bayes-Nash Equilibrium . 28 3.2.4 Participation Constraints . 30 3.2.5 Bilateral Trade . 31 3.2.6 Weak vs. Strong Implementation . 31 3.3 Implementation vs. Optimality . 32 3.4 Auction Design . 33 3.4.1 Single Item Auctions . 34 3.4.2 Multi-Unit Auctions . 38 3.4.3 Combinatorial Auctions . 39 3.5 Computational Approaches to Mechanism Design . 41 3.5.1 Automated Mechanism Design (AMD) . 41 3.5.2 Partial Revelation of Preferences . 45 3.5.3 Incremental Mechanism Design . 46 3.5.4 Evolutionary Approaches . 47 3.5.5 The Metalearning Approach . 47 4 Simulation-Based Mechanism Design . 49 4.1 Trading Agent Competition and the Supply-Chain Game . 50 4.1.1 The Story of Day-0 Procurement in TAC/SCM . 51 4.1.2 The TAC/SCM Design Problem . 52 4.2 What is Simulation-Based Mechanism Design? . 54 4.3 A General Framework for Simulation-Based Mechanism Design 55 4.4 Formalization of the TAC/SCM Design Problem . 57 4.5 Simulation-Based Design Analysis . 58 4.5.1 Estimating Nash Equilibria . 58 4.5.2 Data Generation . 62 4.5.3 Results . 63 4.6 “Threshold” Design Problems . 70 4.7 General Objective Functions . 72 4.7.1 Learning the Objective Function . 72 4.7.2 Stochastic Search . 74 4.7.3 Constraints . 75 4.8 Solving Games Induced by Mechanism Choices . 77 4.9 Sensitivity Analysis . 77 4.10 Probabilistic Analysis in TAC/SCM . 79 4.11 Comparative Statics . 85 4.12 Conclusion . 86 5 A General Framework for Computational Mechanism Design on Con- strained Design Spaces . 87 5.1 Automated Mechanism Design for Bayesian Games . 87 5.1.1 Designer’s Optimization Problem . 88 5.1.2 Computing Nash Equilibria . 90 v 5.1.3 Dealing with Constraints . 91 5.1.4 Evaluating the Objective . 95 5.2 Extended Example: Shared-Good Auction (SGA) . 97 5.2.1 Setup . 97 5.2.2 Automated Design Problems . 99 5.3 Applications . 105 5.3.1 Myerson Auctions . 106 5.3.2 Vicious Auctions . 110 5.4 Conclusion . 117 Part II One Nash, Two Nash, Red Nash, Blue Nash 118 6 Related Work on Computational Game Theory . 118 6.1 Complexity of Computing Nash Equilibria . 118 6.2 Solving Finite Two-Person Games . 119 6.2.1 Zero-Sum Games . 120 6.2.2 General Two-Person Games with Complete Information . 121 6.3 Solving General Games . 125 6.3.1 Simplicial Subdivision . 125 6.3.2 The Liapunov Minimization Method . 125 6.3.3 The Govindan-Wilson Algorithm . 126 6.3.4 Logit Equilibrium Path Following Method . 129 6.3.5 Search in the Pure Strategy Support Space . 130 6.4 “Learning” Algorithms . 131 6.4.1 Iterative Best Response . 132 6.4.2 Fictitious Play . 132 6.4.3 Replicator Dynamics for Symmetric Games . 133 6.5 Solving Games of Incomplete Information . 134 6.5.1 Restricted Strategy Spaces and Convergence . 135 6.5.2 Empirical Game Theory . 135 6.6 Compact Representations of Games . 136 6.6.1 Graphical Games . 136 6.6.2 Multi-Agent Influence Diagrams . 137 6.6.3 Action-Graph Games . 137 7 Simulation-Based Game Modeling: Formal Notation and Finite-Game Methods . 138 7.1 Simulation-Based and Empirical Games . 139 7.2 A General Procedure for Solving Simulation-Based Games . 140 7.3 Simulation Control . 141 7.4 Approximating Nash Equilibria in Empirical Games . 147 7.4.1 Nash Equilibrium Approximation Metrics . 148 7.4.2 Estimating Nash Equilibria in (Small) Finite Games . 150 7.4.3 Estimating Nash Equilibria in Large and Infinite Games . 151 vi 7.5 Applications to TAC/SCM Analysis . 153 7.5.1 Analysis of Day-0 Procurement in TAC/SCM 2003 . 154 7.5.2 Analysis of Day-0 Procurement in TAC/SCM 2004 . 157 7.6 Consistency Results About Nash Equilibria in Empirical Games . 158 7.7 Probabilistic Bounds on Approximation Quality . 161 7.7.1 Distribution-Free Bounds . 162 7.7.2 Confidence Bounds for Finite Games with Normal Noise 164 7.7.3 Confidence Bounds for Infinite Games That Use Finite Game Approximations . 165 7.7.4 Experimental Evaluation of Bounds . 166 7.8 Maximum Likelihood Equilibrium Estimation . 167 7.9 Comparative Statics . 170 7.10 Conclusion . 171 8 Learning Payoff Functions in Infinite Games . 173 8.1 Payoff Function Approximation . 174 8.1.1 Problem Definition . 174 8.1.2 Polynomial Regression . 175 8.1.3 Local Regression . 176 8.1.4 Support Vector Machine Regression . 177 8.1.5 Finding Mixed-Strategy Equilibria . 177 8.1.6 Strategy Aggregation . 178 8.2 First-Price Sealed-Bid Auction . 179 8.3 Market-Based Scheduling Game . 183 8.4 Using in Model Selection . 192 8.5 Using as the Approximation Target . 194 8.6 Future Work . 198 8.6.1 Active Learning . 198 8.6.2 Learning Bayes-Nash Equilibria . 199 8.7 Conclusion . 199 9 Stochastic Search Methods for Approximating Equilibria in Infinite Games202 9.1 Best Response Approximation . 203 9.1.1 Continuous Stochastic Search for Black-Box Optimization203 9.1.2 Globally Convergent Best Response Approximation . 205 9.2 Nash Equilibrium Approximation . 207 9.2.1 Equilibrium Approximation via Iterated Best Response . 208 9.2.2 A Globally Convergent Algorithm for Equilibrium Ap- proximation . 208 9.3 Examples . 212 9.3.1 Mixed Strategy Equilibria in Finite Games . 212 9.3.2 Symmetric Games . 212 9.4 Infinite Games of Incomplete Information . 213 9.4.1 Best Response Approximation . 213 9.4.2 Strategy Set Restrictions . 213 vii 9.5 Experimental Evaluation of Best Response Quality . 214 9.5.1 Experimental Setup . 214 9.5.2 Two-Player One-Item Auctions . 216 9.5.3 Five-Player One-Item Auctions . 219 9.5.4 Sampling and Iteration Efficiency . 219 9.6 Experimental Evaluation of Equilibrium Quality . 221 9.6.1 Experimental Setup . 221 9.6.2 Two-Player One-Item Auctions . 221 9.6.3 Five-Player One-Item Auctions . 222 9.6.4 Two-Player Two-Item First-Price Combinatorial Auction 223 9.7 Experimental Comparison of Equilibrium Approximation Meth- ods . 224 9.8 Conclusion . ..
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