Minimal and locally minimal games and game forms |
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Authors: | Endre Boros Kazuhisa Makino |
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Institution: | a RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, United States b Department of Mathematical Informatics, Graduate School of Information and Technology, University of Tokyo, Tokyo, 113-8656, Japan |
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Abstract: | By Shapley’s (1964) theorem, a matrix game has a saddle point whenever each of its 2×2 subgames has one. In other words, all minimal saddle point free (SP-free) matrices are of size 2×2. We strengthen this result and show that all locally minimal SP-free matrices also are of size 2×2. In other words, if A is a SP-free matrix in which a saddle point appears after deleting an arbitrary row or column then A is of size 2×2. Furthermore, we generalize this result and characterize the locally minimal Nash equilibrium free (NE-free) bimatrix games.Let us recall that a two-person game form is Nash-solvable if and only if it is tight V. Gurvich, Solution of positional games in pure strategies, USSR Comput. Math. and Math. Phys. 15 (2) (1975) 74-87]. We show that all (locally) minimal non-tight game forms are of size 2×2. In contrast, it seems difficult to characterize the locally minimal tight game forms (while all minimal ones are just trivial); we only obtain some necessary and some sufficient conditions. We also recall an example from cooperative game theory: a maximal stable effectivity function that is not self-dual and not convex. |
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Keywords: | Game Game form Saddle point Nash equilibrium Effectivity function Minimal Locally minimal Monotone Weakly monotone |
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