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Oddness of the number of equilibrium points: A new proof

Abstract

A new proof is offered for the theorem that, in “almost all” finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those “exceptional” games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.

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This research has been supported by Grant GS-3222 of the National Science Foundation, through the Center for Research in Management Science, University of California, Berkeley.

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Harsanyi, J.C. Oddness of the number of equilibrium points: A new proof. Int J Game Theory 2, 235–250 (1973). https://doi.org/10.1007/BF01737572

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Keywords

  • Equilibrium Point
  • Economic Theory
  • Game Theory
  • Payoff Function
  • Topological Property