Game Transformations that preserve Nash Equilibrium sets and/or Best Response sets

In the literature on simultaneous non-cooperative games, it is well-known that a positive affine (linear) transformation (PAT) of the utility payoffs do not change the best response sets and the Nash equilibrium set. PATs have been successfully used to expand the classes of 2-player games for which we can compute a Nash equilibrium in polynomial time. We investigate which game transformations other than PATs also possess one of the following properties: (i) The game transformation shall not change the Nash equilibrium set when being applied on an arbitrary game. (ii) The game transformation shall not change the best response sets when being applied on an arbitrary game. First, we prove that property (i) implies property (ii). Over a series of further results, we derive that game transformations with property (ii) must be positive affine. Therefore, we obtained two new and equivalent characterisations with game theoretic meaning for what it means to be a positive affine transformation. All our results in particular hold for the 2-player case of bimatrix games.