求解整數賽局純奈許均衡的演算法在決策科學領域一直是不斷被探討的議題。很多研究都提出了有效率的方法來求解，像是搜尋、逼近或是窮舉法演算法。但是，這些演算法都存在限制，例如搜尋和逼近演算法通常只找尋一組奈許均衡點，而窮舉法則是太過於耗時。因此，我們實現了一個基於生成函數的演算法。
根據過去的文獻顯示，在整數賽局中，玩家的擴展策略可以被表示成有理生成函數的形式，我們便能利用生成函數的特性來有效的求解純奈許均衡點。這個方法是將非奈許均衡點以限制式的方式表達出來，再利用生成函數的特性計算出純奈許均衡點的集合。接下來運用二元搜尋找出所有純奈許均衡點。計算過程中需使用Residue去計算有理生成函數所包含整數點個數的近似值，自然會產生一些數值誤差。因此我們在演算法最後建立驗證的步驟，將找出的奈許均衡點代入原問題確認。
相較過去提出的演算法通常限制於解決正則形式賽局問題 (Sandholm et al., 2005; Porter et al., 2008; Wu et al., 2014) 且每個玩家的策略集數目通常不超過500 (Sagratella, 2016; Carvalho et al., 2021) ， 我們實現的演算法可以 1) 應用在任何以多面體策略集表示的賽局問題上，包含GENP；2) 有能力解決玩家策略數較多的賽局問題；3) 在可接受的時間範圍內，可以列出所有的純奈許均衡點。除此之外，我們也在研究中提出四種不同的實例來驗證演算法的可行性。實證的過程中我們發現演算法的計算與搜尋速度取決於玩家策略集數目及問題的維度，當玩家的擴展策略數越多或是維度越大時，所需時間就比較長。我們的實證結果可以歸類當玩家策略集數目不超過560時，我們的演算法可以在5秒內找到所有純奈許均衡點；當玩家策略集數目不超過7650時，我們的演算法可以在12分鐘內找出所有純奈許均衡點。

Finding Pure Nash Equilibrium (PNE) in Integer Programming Games (IPG) has been a long-standing issue. Searching, heuristic, and enumerating algorithms have been developed to find PNE, however, these algorithms usually have some limitations. For example, searching and heuristic algorithms can only find one PNE, and the enumeration algorithm is time-consuming. Thus, we propose an algorithm that can find all PNE using the short rational generating function.
Based on existing studies, the extended deviated profile of each player can be encoded into the short rational generating function. This algorithm computes the set containing PNE by subtracting all extended profiles from the extended deviated profile of each player. Next, we enumerate all PNE via binary search. In the enumeration process, the ordinary method - residue approach - to counting the number of integer points encoded in the short rational generating function naturally causes numerical deviation. Therefore, we construct a validation process to check the output of the algorithm.
Compared to existing algorithms that can only solve problems in normal form (Sandholm et al., 2005; Porter et al., 2008; Wu et al., 2014) and median-and-small problems in finite time with the strategies size of each player under 500 (Sagratella, 2016; Carvalho et al., 2021), our proposed algorithm can: i) apply to games with polyhedron strategies set; ii) have the capability to solve large-size strategies sets of each player; iii) find entire PNE in an acceptable running time. Moreover, we implemented our algorithm on different kinds of examples: traveler's dilemma, normal form, and knapsack game. The results show that the running time for finding PNE is determined by the size of the player's strategies set, and the dimension of the problem usually affects the strategies size. We can conclude that if the problem's strategies size of each player is under 560, we can solve it within 5 seconds. For the large-size problems where the strategies size of each player is 7650, we can deal with it within 12 minutes.

1. Introduction 7
2. Literature Review 10
2.1 Algorithm for Finding Mixed Nash Equilibrium of integer-valued strategies set 11
2.2 Algorithm for Finding Pure Nash Equilibrium on IPG 12
3. Methodology 14
3.1 Nash Equilibria 15
3.1.1 Extended Game 15
3.1.2 The extended deviated profile of each player 16
3.1.3 Normal-form Game 17
3.2 Short Rational Generating Function 18
3.2.1 Residue approach 18
3.2.2 Hadamard product 21
3.3 Algorithm 22
4. Implementation 33
5. Conclusion 40
6. Reference 42

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