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 = {c 1, ..., c 15} depending on the preference pre-orders 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.
The second author gratefully acknowledges the partial support of DIMACS, the NSF Center for Discrete Mathematics and Theoretical Computer Science.
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Boros, E., Elbassioni, K., Gurvich, V., Makino, K., Oudalov, V. (2008). A Complete Characterization of Nash-Solvability of Bimatrix Games in Terms of the Exclusion of Certain 2×2 Subgames . In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_13
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DOI: https://doi.org/10.1007/978-3-540-79709-8_13
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