應用短有理生成函數求解奈許均衡問題之多重數值實驗

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研究生：	吳定群 Wu, Ding-Chiun
論文名稱：	應用短有理生成函數求解奈許均衡問題之多重數值實驗 Numerical Experiments of Nash Equilibrium Problem using Short Rational Generating Function
指導教授：	李雨青 Lee, Yu-Ching
口試委員:	郭佳瑋 Kuo, Chia-Wei 吳浩庠 Wu, Hao-Hsiang 王俊涵 Wang, Chun-Han
學位類別：	碩士 Master
系所名稱：	工學院 - 工業工程與工程管理學系 Department of Industrial Engineering and Engineering Management
論文出版年：	2024
畢業學年度：	112
語文別：	英文
論文頁數：	54
中文關鍵詞：	生成函數、整數規劃、奈許均衡、賽局理論
外文關鍵詞：	generating function, integer programming, Nash equilibrium, game theory
相關次數：	點閱：48 下載：0
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這篇論文主要在針對我們的演算法進行多項數值實驗，進而驗證理論與演算法的正確性。我們的演算法充分利用 short rational generating function的特性，使我們能在特定條件之下，像是策略的維度固定還有目標為線性，在多項式時間內找到奈許均衡點。我們使用 python作為我們演算法的程式語言，並且在第四章，我們跑了三個經典的整數規劃賽局問題，分別是旅行者問題 Traveler Dilemma背包問題 Knapsack Problem還有 Normal form game。我們不僅跑不同的賽局，同時也針對單一賽局使用不同的參數設定，這有助於了解演算法在不同策略及大小與玩家數之下，時間的變化。我們在演算法中的多個步驟設置時間計時器，這有助於我們找到演算法最花時間的部分是在第一階段做交集聯集的地方，同時演算法的快慢也受到策略集合的大小還有玩家數目所影響。這篇論文的主要貢獻除了展示一個可以找到所有均衡解的演算法的細節也透過大量的數值實驗，讓讀者更了解此演算法的效能與暴力解演算法的效能差距，並且詳細探討影響演算法效能的潛在因素，提供未來學者進行賽局均衡點尋找的範例。

This paper primarily focuses on conducting numerous numerical experiments for our algorithm, thereby validating the correctness of both theory and algorithm. Our algorithm fully exploits the characteristics of short rational generating functions, enabling us to find Nash equilibrium points in polynomial time under specific conditions, such as fixed dimensions of the profile and linear objectives. Python serves as the programming language for our algorithm. In Chapter Four, we ran three classic integer programming game problems: the normal form game, the traveling salesman problem, and the knapsack problem. Not only did we run various games, but we also adjusted parameters for individual games. This aids in understanding how the algorithm's runtime varies with different strategies, sizes, and numbers of players.
Throughout our algorithm, we implemented timers at various steps, helping us identify that the most time-consuming part occurs in the first phase during the intersection and union operations. The speed of the algorithm is also influenced by the size of the strategy set and the number of players. The main contributions of this paper include not only detailing an algorithm capable of finding all equilibrium solutions but also providing extensive numerical experiments to help readers better understand the performance of this algorithm compared to brute-force algorithms. Additionally, the paper thoroughly explores potential factors affecting algorithm performance, offering a valuable example for future researchers in the field of game theory.

摘要
I
ABSTRACT II
CONTENTS IV
FIGURE 1
TABLE 1
CHAPTER1 INTRODUCTION 3
CHAPTER2 LITERATURE REVIEW 5
CHAPTER3 METHODOLOGY 8
3.1 NOTATIONS AND DEFINITIONS 8
3.2 EXTENDED GAME 8
3.3 NORMAL FORM GAME 10
3.4 RATIONAL GENERATING FUNCTION 11
3.5 HADAMARD PRODUCT 12
3.6 ALGORITHM OVERVIEW 14
3.7 ALGORITHM PHASE 1 15
3.8 ALGORITHM PHASE 2 16
v
CHAPTER4 NUMERICAL RESULT  18
4.1 TRAVELER’S DILEMMA 18
4.1.1 Traveler’s Dilemma with 2 Players 21
4.1.2 Traveler’s Dilemma with 3 Players 22
4.2 KNAPSACK PROBLEM  23
4.2.1 Knapsack Problem with 2 Players and 2 Items  24
4.2.2 Knapsack Problem with 2 Players and 3 Items  26
4.2.3 Knapsack Problem with 3 Players and 2 Items  28
4.2.4 Knapsack Problem with 3 Players and 3 Items  30
4.2.5 Knapsack Problem with 4 Players and 2 Items  31
4.2.6 Knapsack Problem with 4 Players and 3 Items  32
4.2.7 Knapsack Problem with 4 Players and 5 Items  34
4.3 NORMAL FORM GAME  34
4.3.1 Two-Player Normal Form Game 36
4.3.2 Three-Player Normal Form Game  37
4.4 SUMMARIZE 39
CHAPTER5 CONCLUSION 42
REFERENCES 44
APPENDIX 46

APPENDIX A. PARAMETER AND RESULTS TABLE FOR THE TWO-PLAYER TRAVELER’S DILEMMA.  46
APPENDIX B. PARAMETER TABLE FOR THE THREE-PLAYER TRAVELER’S DILEMMA.  46
APPENDIX C. EXAMPLE DATA FOR THE TWO-PLAYER, TWO-ITEM KNAPSACK PROBLEM 47
APPENDIX D. PARAMETER TABLE FOR THE TWO-PLAYER, TWO-ITEM KNAPSACK PROBLEM.  47
APPENDIX E. EXAMPLE DATA FOR THE TWO-PLAYER, THREE-ITEM KNAPSACK PROBLEM 48
APPENDIX F. PARAMETER TABLE FOR THE TWO-PLAYER, THREE-ITEM KNAPSACK PROBLEM. 48
APPENDIX G. EXAMPLE DATA FOR THE THREE-PLAYER, TWO-ITEM KNAPSACK PROBLEM 49
APPENDIX H. PARAMETER TABLE FOR THE THREE-PLAYER, TWO-ITEM KNAPSACK PROBLEM.  49
APPENDIX I. PARAMETER TABLE FOR THE THREE-PLAYER, THREE-ITEM KNAPSACK PROBLEM.  50
APPENDIX J. PARAMETER TABLE FOR THE FOUR-PLAYER, TWO-ITEM KNAPSACK PROBLEM.  51
vii
APPENDIX K. PARAMETER TABLE FOR THE FOUR-PLAYER, THREE-ITEM KNAPSACK PROBLEM.  51
APPENDIX L. PARAMETER TABLE FOR THE FAILED EXAMPLE OF FOUR-PLAYER, THREE-ITEM KNAPSACK PROBLEM.  52
APPENDIX M. PARAMETER TABLE FOR THE FOUR-PLAYER, FIVE-ITEM KNAPSACK PROBLEM.  52
APPENDIX N. PARAMETER TABLE FOR THE TWO-PLAYER NORMAL FORM GAME  53
APPENDIX O. PARAMETER TABLE FOR THE TWO-PLAYER NORMAL FORM GAME  53
TABLE 1 NOTATIONS AND DEFINITIONS 8
TABLE 2 EXPERIMENTAL RESULTS FOR THE TWO-PLAYER, TWO-ITEM KNAPSACK PROBLEM. 25
TABLE 3 EXPERIMENTAL RESULTS FOR THE TWO-PLAYER, THREE-ITEM KNAPSACK PROBLEM. 27
TABLE 4 EXPERIMENTAL RESULTS FOR THE THREE-PLAYER, TWO-ITEM KNAPSACK PROBLEM. 29
TABLE 5 EXPERIMENTAL RESULTS FOR THE THREE-PLAYER, THREE-ITEM KNAPSACK PROBLEM. 30
TABLE 6 EXPERIMENTAL RESULTS FOR THE TOUR-PLAYER, TWO-ITEM KNAPSACK PROBLEM. 31
TABLE 7 EXPERIMENTAL RESULTS FOR THE TOUR-PLAYER, THREE-ITEM KNAPSACK PROBLEM. 33
TABLE 8 EXPERIMENTAL RESULTS FOR THE TWO-PLAYER NORMAL FORM PROBLEM 36
TABLE 9 EXPERIMENTAL RESULTS FOR THE THREE-PLAYER NORMAL FORM PROBLEM  37
TABLE 10 THE TIME VARIATIONS OF OUR ALGORITHM AND THE ENUMERATION ALGORITHM  39
TABLE 11 PARAMETERS AFFECTING THE PERFORMANCE OF THE ALGORITHM 39
                                

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