Game Transformations That Preserve Nash Equilibria or Best Response Sets

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.