We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree.

翻译：我们利用向量丛研究标准形式$n$人博弈的全混合纳什均衡轨迹，称之为纳什均衡概形。当收益张量格式平衡时，我们研究纳什判别簇，即纳什均衡概形非既约或具有正维分量的博弈代数簇。我们证明该簇的余维为一。我们对二进制三人博弈纳什均衡概形的所有可能分量进行了分类。我们证明若收益张量呈边界格式，则纳什判别簇具有两个分量：一个不可约超曲面与一个更高余维分量。具有非平衡收益张量格式的一般博弈不存在全混合纳什均衡。我们定义了存在正数个全混合纳什均衡的博弈的纳什结式簇，证明其不可约性并确定了其余维与次数。