A complete characterization of Nash-solvability of bimatrix games in terms of the exclusion of certain 2 × 2 subgames

Authors:
Endre Boros;Khaled Elbassioni;Vladimir Gurvich;Kazuhisa Makino;Vladimir Oudalov
Affiliations:
RUTCOR, Rutgers University, Piscataway, NJ;Max-Planck-Institut für Informatik, Saarbrücken, Germany;RUTCOR, Rutgers University, Piscataway, NJ;Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan;RUTCOR, Rutgers University, Piscataway, NJ
Venue:
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Year:
2008

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Abstract

In 1964 Shapley observed that a matrix has a saddle point whenever every 2×2 submatrix of it has one. In contrast, a bimatrix game may have no Nash equilibrium (NE) even when every 2 × 2 subgame of it has one. Nevertheless, Shapley's claim can be generalized for bimatrix games in many ways as follows. We partition all 2×2 bimatrix games into fifteen classes C = {c1, ..., c15} depending on the preference preorders of the two players. A subset t ⊆ C is called a NE-theorem if a bimatrix game has a NE whenever it contains no subgame from t. We suggest a general method for getting all minimal (that is, strongest) NE-theorems based on the procedure of joint generation of transversal hypergraphs given by a special oracle. By this method we obtain all (six) minimal NE-theorems.