賽局理論是一個理論框架，用於設想競爭者之間的社會情境，在給定已知的payoff或可量化的結果，我們能得到一個讓所有玩家在自身最佳獲利下不會單方面改變自身決策的結果。而在近年來相關的研究中，賽局理論也運用在廣泛的課題，像是經濟、生物和工程學等等。而本篇論文所探討的generalized Nash equilibrium problem，能描述更一般化的情況，過往在解決這類問題，我們會採取混和策略的方式，給予每位玩家的決策一個機率值，並能確保必定會有一個混和納許均衡點。然而，這單一結果並不夠實際且無法全面的預測對手的決策。因此，在這篇論文中，我們設計了一套演算法，目標是取得一個賽局中所有的整數均衡點並列舉出來，這是在過往尚未有人實現過的。而在我們的數據實驗中，我們在三個範例下分別獲得了1、13、3個整數納許均衡點。此外，我們對演算法進行了運行時間的測試，並發現我們能在兩小時內解決一個14維度以下的賽局遊戲。

Game theory is a theoretical framework for conceiving social situations among competing players. With known payoffs or quantifiable consequences, we can get an outcome that no player can increase their payoff by changing strategy unilaterally. In recent research, game theory has applied to many issues, e.g., economics, biology, engineering. Furthermore, the generalized Nash equilibrium problem is mentioned more often because of the general conditions. For solving this problem, we usually use the concept of mixed strategy, where every players’ strategies are assigned a probability respectively, and this method will guarantee there is a mixed Nash equilibrium point. However, the unique solution is not practical enough and it can’t predict rivals’ strategies completely. Thus, in this thesis, we design an algorithm in order to find all the integer Nash equilibria points and enumerate them, and this has never been done before. In our numerical experiments, we get 1, 13 and 3 integer Nash equilibria for examples 1, 2, and 3 respectively. Moreover, we examine our algorithm’s running time and find that we can enumerate the integer Nash equilibria of a game with up to 14 variables in 2 hours.

List of Tables...vi
List of Figures...vii
Chapter 1 Introduction...1
Chapter 2 Literature Review...3
2.1 Generalized Nash Equilibrium Problem...3
2.1.1 GNEP formulation...3
2.1.2 Solution of GNEP...4
2.2 Short rational generating function...5
2.3 Integer programming in game theory...7
Chapter 3 Model...9
3.1 Environment...9
3.2 Algorithm...13
Chapter 4 Numerical Experiment...16
4.1 Numerical results of examples...16
4.2 Comparisons with other algorithms...18
4.2.1 Gambit...19
4.2.2 Newton method for GNEP...20
4.2.3 Comparisons...22
4.3 Running time of algorithm...25
Chapter 5 Conclusions and the Future Work...28
References...30
Appendix...32
Appendix A. example 1’s reformulation...32
Appendix B. example 2’s reformulation...33

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