Abstract
We consider a finite symmetric game where the set of strategies for each player is a one-dimensional integer interval. We show that a pure strategy equilibrium exists if the payoff function is concave with respect to the own strategy and satisfies a pair of symmetrical conditions near the symmetric strategy profiles. As an application, we consider a symmetric Cournot game in which each firm chooses an integer quantity of product. It is shown, among other things, that if the industry revenue function is concave, the inverse demand function is nonincreasing, and the cost function is convex, then the payoff function of the firm satisfies the conditions and this symmetric game has a pure strategy equilibrium.
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Notes
- 1.
Every symmetric game with two strategies has a pure strategy equilibrium as shown in Cheng et al. (2004).
- 2.
- 3.
The authors owe this example to Federico Quartieri.
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 25380233. The authors thank valuable comments from the referees and editors. Especially, we thank Federico Quartieri for providing a good example. Of course all remaining errors are our responsibility.
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Iimura, T., Watanabe, T. (2016). Pure Strategy Equilibria in Finite Symmetric Concave Games and an Application to Symmetric Discrete Cournot Games. In: von Mouche, P., Quartieri, F. (eds) Equilibrium Theory for Cournot Oligopolies and Related Games. Springer Series in Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-29254-0_7
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DOI: https://doi.org/10.1007/978-3-319-29254-0_7
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