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Journal Article

On the Proof of the Existence of Undominated Strategies in Normal Form Games

Martin Kovár and Alena Chernikava
The American Mathematical Monthly
Vol. 121, No. 4 (April 2014), pp. 332-337
DOI: 10.4169/amer.math.monthly.121.04.332
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Page Count: 6
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On the Proof of the Existence of Undominated Strategies in
Normal Form Games
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Abstract In the game theory literature, there are two versions of the proof of the well-known fact that in a normal form game of n persons with compact spaces of strategies and continuous utility functions, the sets of undominated strategies are nonempty. The older one, stated in the first edition of the well-known book by Herve Moulin, depends on certain, relatively nontrivial results from measure theory, metric topology, and mathematical analysis. The proof is valid only for metrizable topological spaces. The second, revised edition of the same book contains a simplified proof, which is, however, incorrect. The author implicitly assumes that any linearly ordered set contains a cofinal subsequence, which is certainly not true. In this paper we correct, simplify, and generalize the second proof of Moulin by its reformulation in terms of topological convergence of nets. This modified technique also yields a slightly better result than is stated in the original. The assertion now holds for almost compact spaces. The argument used is elementary and easily understandable to non-experts.

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