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On pure equilibria for bimatrix games. (English) Zbl 0795.90087

Summary: L. S. Shapley [Ann. Math. Stud. 52, 1-28 (1964; Zbl 0126.162)] gave several conditions for the existence of pure saddlepoints for a matrix game. We show that only a few of these conditions, when translated to the situation of a bimatrix game guarantee the existence of pure equilibria. Further, we associate with a bimatrix game a directed graph as well as a so-called ‘binary game’. If this graph has no cycles, then the bimatrix game in question has a pure equilibrium. It is shown that the binary game for a bimatrix game without a pure equilibrium possesses a ‘fundamental’ subgame, which can be characterized by means of ‘minimal’ cycles.

MSC:

91A05 2-person games
91A43 Games involving graphs

Citations:

Zbl 0126.162
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Full Text: DOI

References:

[1] Shapley LS (1964) Some topics in two-person games. Ann of Math Stud 52:1-28 · Zbl 0126.16204
[2] Smadici C (1979) Generalized saddle points for bimatrix games. Proceedings of the Third Colloquium on Operations Research (Cluy-Napoca, 1978) Univ. ?Babes-Bolyai?, Cluy-Napoca 249-256
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