In the literature on simultaneous non-cooperative games, it is a widely used fact that a positive affine (linear) transformation of the utility payoffs neither changes the best response sets nor the Nash equilibrium set. We investigate which other game transformations also possess one of these two properties when being applied to an arbitrary N-player game (N >= 2): (i) The Nash equilibrium set stays the same. (ii) The best response sets stay the same. For game transformations that operate player-wise and strategy-wise, we prove that (i) implies (ii) and that transformations with property (ii) must be positive affine. The resulting equivalence chain gives an explicit description of all those game transformations that always preserve the Nash equilibrium set (or, respectively, the best response sets). Simultaneously, we obtain two new characterizations of the class of positive affine transformations.

翻译：在关于同时非合作博弈的文献中，一个广泛使用的事实是对效用收益进行正仿射（线性）变换既不会改变最优反应策略集，也不会改变纳什均衡集。我们研究了当应用于任意N人博弈（N ≥ 2）时，哪些其他博弈变换也具备这两种性质之一：（i）纳什均衡集保持不变；（ii）最优反应策略集保持不变。对于按玩家和按策略操作的博弈变换，我们证明了（i）蕴含（ii），且具有性质（ii）的变换必须是正仿射的。由此产生的等价链给出了所有那些始终保持纳什均衡集（或分别地，最优反应策略集）的博弈变换的显式描述。同时，我们获得了正仿射变换类的两个新刻画。