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Finding all Nash equilibria of a finite game using polynomial algebra

机译：使用多项式代数找到有限博弈的所有纳什均衡

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The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article, we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Gröbner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game. Keywords Nash equilibrium - Normal form game - Algebraic variety JEL Classification C72 Our earlier paper (Datta 2003c) contains much of the material which is surveyed more expansively here. We would like to express our gratitude to the following for generously taking the time to personally discuss with us the use of their software packages: Andrew McLennan and Ted Turocy (Gambit McKelvey et al. 2006), Gert-Martin Greuel (Singular Greuel et al. 2001), and Jan Verschelde (PHC Verschelde 1999). We would also like to thank Gabriela Jeronimo for sending us a preprint of her paper with Daniel Perrucci and Juan Sabia, and Andrew McLennan for suggesting she do so. We would like to thank Richard Fateman and Bernd Sturmfels for supervising the research leading up to that paper, during which the author was partially supported by NSF grant DMS 0138323. We would also like to acknowledge our debt to Bernd Sturmfels, especially for teaching us about the application of polynomial algebra to Nash equilibria, in the lectures leading to Sturmfels (2002).
机译：有限博弈的纳什均衡集是多项式方程组的非负解集。在这篇调查文章中，我们描述了如何构造某些特殊博弈，并解释了如何找到相应多项式系统的所有复根，包括所有纳什均衡。然后，我们解释如何通过从特殊构造游戏的解开始的多面同性连续性，找到任意泛型游戏的多项式系统的所有复杂根。我们描述了使用Gröbner基来求解这些多项式系统并学习有关解集如何随收益函数变化的几何信息。最后，我们回顾了使用Gambit软件包来查找有限博弈的所有纳什均衡的情况。关键字Nash均衡-正规形式博弈-代数形式JEL分类C72我们先前的论文（Datta 2003c）包含许多在此进行更广泛调查的材料。对于以下人员，我们非常感谢他们抽出宝贵的时间与我们亲自讨论他们的软件包的使用：Andrew McLennan和Ted Turocy（Gambit McKelvey等，2006），Gert-Martin Greuel（Singular Greuel等） （2001年）和扬·范思凯德（PHC Verschelde 1999）。我们还要感谢Gabriela Jeronimo向我们发送了她的论文的预印本，其中包括Daniel Perrucci和Juan Sabia，以及Andrew McLennan建议她这样做。我们要感谢Richard Fateman和Bernd Sturmfels指导了这篇论文的研究，在此期间，作者获得了NSF授予DMS 0138323的部分支持。我们还要感谢我们对Bernd Sturmfels的欠债，特别是因为我们对我们的了解在导致Sturmfels（2002）的讲座中，将多项式代数应用于Nash均衡。

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