On the set of proper equilibria of a bimatrix game
The paper is devoted to the analysis of the set of proper equilibria of a bimatrix game. It discusses the well-known problem which started from the structure of the set of Nash equilibria of a matrix game. This set is the Cartesian product of two polytopes in the case of a matrix game. The main result of this paper is that the set of proper equilibria of a bimatrix game is the finite union of polytopes. The mathematical technique for proving this result is based on the representation of the set of strategies (the strategy space) of each player into a finite number of equivalence classes. A Myerson set of proper equilibria is considered. The main results are the following: Theorem 1. The set of proper equilibria of a bimatrix game is the union of a finite number of polytopes. Theorem 2. The intersection of two maximal Myerson sets is empty on a face of both maximal sets.
scientific article; zbMATH DE number 429105