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Pure strategy equilibrium in finite weakly unilaterally competitive games

Abstract

We consider the finite version of the weakly unilaterally competitive game (Kats and Thisse, in Int J Game Theory 21:291–299, 1992) and show that this game possesses a pure strategy Nash equilibrium if it is symmetric and quasiconcave (or single-peaked). The first implication of this result is that unilaterally competitive or two-person weakly unilaterally competitive finite games are solvable in the sense of Nash, in pure strategies. We also characterize the set of equilibria of these finite games. The second implication is that there exists a finite population evolutionarily stable pure strategy equilibrium in a finite game, if it is symmetric, quasiconcave, and weakly unilaterally competitive.

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Notes

  1. 1.

    This notion of solvability, which appears in Friedman (1983) and Kats and Thisse (1992), is a slight generalization of the solvability in Nash (1951). In Nash (1951) and Friedman (1983), the existence of an equilibrium was tacitly given; we explicitly include it in the definition of solvability.

  2. 2.

    In the earlier version of this paper whether or not our class of games is included in the class of generalized ordinal potential games was left as an open problem. The authors are greatly indebted to a reviewer for pointing out this counterexample that appeared in Duersch et al. (2012b, Example 2).

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Acknowledgments

The authors thank two anonymous reviewers for their valuable comments and suggestions. This work is supported by JSPS Grant-in-Aid for Scientific Research (C) (KAKENHI) 25380233.

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Correspondence to Takuya Iimura.

Additional information

The earlier version of this paper was presented at UECE Lisbon Meetings 2013 under the title “On the pure strategy equilibrium of finite weakly unilaterally competitive games”.

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Iimura, T., Watanabe, T. Pure strategy equilibrium in finite weakly unilaterally competitive games. Int J Game Theory 45, 719–729 (2016). https://doi.org/10.1007/s00182-015-0481-y

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Keywords

  • Weakly unilaterally competitive game
  • Finite game
  • Symmetric game
  • Quasiconcave game
  • Existence of a pure strategy equilibrium
  • Solvable game

JEL Classification

  • C72 (Noncooperative game)