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Solving Eight Treasures of Game Theory SOLVING EIGHT TREASURES OF GAME THEORY PROBLEMS USING BI-CRITERIA METHOD by ZHINENG YE Submitted in partial fulfillment of the requirements For the degree of Master of Science Thesis Advisor: Professor Behnam Malakooti Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY May, 2016 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Zhineng Ye candidate for the degree of Master of Science Committee Chair Prof. Behnam Malakooti Committee Member Prof. Vira Chankong Committee Member Prof. Mingguo Hong Date of Defense Jan. 13, 2016 *We also certify that written approval has been obtained for any proprietary material contained therein. Table of Contents Chapter 1 Introduction ............................................................................................................. 1 1.1 History of Game Theory ................................................................................................. 1 1.2 Nash equilibrium ............................................................................................................. 2 1.3 Utility and the definition of rationality ........................................................................... 4 1.4 Bi-Criteria and its application in Game theory .............................................................. 5 1.4.1 Concept of Bi-criteria and its application in Game theory ...................................... 5 1.4.2 An example using bi-criteria method in game ......................................................... 8 1.5 Research object and thesis structure .............................................................................. 9 Chapter 2 ................................................................................................................................. 11 Solving Eight Treasures of Game Theory Problems .............................................................. 11 Using Bi-Criteria Method ....................................................................................................... 11 2.1 Traveler’s dilemma ....................................................................................................... 11 2.2 A Matching Pennies Game ............................................................................................ 14 2.3 Coordination game with a secure option ...................................................................... 15 2.4 Minimum effort coordination game .............................................................................. 16 2.5 The Kreps Game ........................................................................................................... 19 2.6 Dynamic Game 1 with complete information ............................................................... 21 2.7 Dynamic Game 2 with complete information ............................................................... 24 2.8 Two-stage bargaining Game ......................................................................................... 26 Chapter 3 ................................................................................................................................. 29 Reasons and Discussions ......................................................................................................... 29 3.1 Reasons why Nash equilibrium fails ............................................................................. 29 3.2 Why use bi-criteria to solve ........................................................................................... 31 3.3 Is Nash equilibrium a rational choice? ......................................................................... 32 APPENDIX ............................................................................................................................. 34 Reference ................................................................................................................................. 41 List of Figures Figure 1- Claim Frequency in a Traveler's Dilemma for R=180(Dark bars) and R=5(Light bars) 12 Figure 2- Effort choice frequencies for a minimum-effort coordination game with high effort cost (Dark bars) and low effort cost (Light bars) ............................................................................... 17 Figure 3- payoff matrix for row player in a minimum effort game ............................................. 18 Figure 4- payoff matrix for column player in a minimum effort game ........................................ 18 Figure 5- weighted payoff matrix for column player in a minimum effort game ......................... 19 Figure 6- Graph solution to the basic Kreps Game ..................................................................... 20 Figure 7- Payoff tree for Dynamic Game 1, Experiment 1 ......................................................... 22 Figure 8- Payoff tree for Dynamic Game 1, Experiment 2 ......................................................... 23 Figure 9- Payoff tree for Dynamic Game 2, Experiment 1 ......................................................... 24 Figure 10- Payoff tree for Dynamic Game 2, Experiment 2 ....................................................... 25 List of Tables Table 1-Payoff Matrix example of Pure Nash Equilibrium .......................................................... 3 Table 2- Payoff matrix example of Mixed Nash Equilibrium ....................................................... 3 Table 3- Payoff matrix of prisoner’s dilemma ............................................................................. 8 Table 4-Weighted payoff matrix of Prisoner’s dilemma ............................................................... 9 Table 5- Payoff matrix of traveler’s dilemma for R=5 ............................................................... 13 Table 6- Weighted payoff matrix of traveler’s dilemma for R=5 ................................................ 13 Table 7- Payoff matrix of traveler’s dilemma for R=180............................................................ 13 Table 8- Weighted payoff matrix of traveler’s dilemma for R=180 ............................................ 13 Table 9- Payoff Matrix of a matching pennies game .................................................................. 14 Table 10- Weighted payoff matrix for Matching Pennies asymmetric game ............................... 15 Table 11- Weighted payoff matrix for Matching Pennies reversed game .................................... 15 Table 12- Payoff Matrix of Coordination Game......................................................................... 15 Table 13- Weighted payoff matrix of coordination game ........................................................... 16 Table 14- Payoff Matrix for the basic Kreps Game .................................................................... 19 Table 15- Payoff Matrix for Positive payoff Kreps Game .......................................................... 21 Acknowledgments Throughout this research, I’m glad to have the opportunity to work with Professor Behnam Malakooti, his excellent student Mohammad Komaki and other students as well. I learned not only research skill from them, but also the personalities we need to go through research. I would like to thank them all, but here I would love to acknowledge some that have been particularly helpful and supportive. I would like to express my greatest gratitude to my advisor, Prof. Behnam Malakooti, for his persistent support and tireless teaching. I appreciated his passion, patience and understanding throughout my graduate study. I would like to extend my thanks to Mohammad Komaki, who gave a great helping hand on this project. I am impressed by his masterful skill in using different kinds of software to solve problems in a short period of time. And I always have my questions solved when I come to him for help. Finally, I would like to thank my parents for their love and my friends for their support throughout this project. Solving Eight Treasures of Game Theory Problems Using Bi-Criteria Method Abstract by Zhineng Ye Game theory is a strategic mathematics model of how we make decisions. It is widely applied in fields like economics and psychology to make our decisions more competitive and favorable. Nash equilibrium, the foundation of game theory, is always the first method attempted to solve a problem, especially in a two-person game. In Goeree and Holt’s 2001 paper, “Ten little treasures of game theory and ten intuitive contradictions”, they did ten experiments on different kinds of two-person games, each game associates with a basic version and a contradictive version. From the experiment result, the basic version matches our assumption and intuition for most of the games, while the contradictive version disagrees with that, which the Nash equilibrium can’t explain the latter. But if certain weights are assigned to the payoffs of the two players and the additive weighted payoff is used as the new payoff for one player, then it may become possible to solve the “new” game and explain the result using Nash equilibrium. This is how the bi-criteria method works in game theory. Logically, the weight is assigned to the payoffs because the other player’s payoff is important to a decision maker. Using this method, almost all of the games in Goeree and Holt’s paper (2001) can be solved. Chapter 1 Introduction 1.1 History of Game Theory Game theory was first discussed in 1713 in a letter by James Waldegrave when he was playing a two-person card game. The first
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