Fast algorithms for rank-1 bimatrix games

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r − 1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component

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.Comment: 18 pages, 0 figure