# Pure Strategy Equilibria in Finite Symmetric Concave Games and an Application to Symmetric Discrete Cournot Games

## 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.

## Keywords

Payoff Function Strategy Profile Real Interval Inverse Demand Function Pure Strategy Equilibrium## Notes

### 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.

## References

- Cheng SF, Reeves DM, Vorobeychik Y, Wellman MP (2004) Notes on equilibria in symmetric games. In: Proceedings of the 6th international workshop on game theoretic and decision theoretic agents, New YorkGoogle Scholar
- Dasgupta P, Maskin E (1986) The existence of equilibrium in discontinuous economic games, I: theory. Rev Econ Stud 53(1):1–26CrossRefGoogle Scholar
- Dubey P, Haimanko O, Zapechelnyuk A (2006) Strategic complements and substitutes, and potential games. Games Econ Behav 54(1):77–94CrossRefGoogle Scholar
- Favati P, Tardella F (1990) Convexity in nonlinear integer programming. Ric Oper 53:3–44Google Scholar
- Iimura T, Watanabe T (2014) Existence of a pure strategy equilibrium in finite symmetric games where payoff functions are integrally concave. Discret Appl Math 166:26–33CrossRefGoogle Scholar
- Jensen M (2010) Aggregative games and best-reply potentials. Econ Theory 43(1):45–66CrossRefGoogle Scholar
- Kukushkin NS (1994) A fixed-point theorem for decreasing mappings. Econ Lett 46(1):23–26CrossRefGoogle Scholar
- McManus M (1964) Equilibrium, numbers and size in Cournot oligopoly. Yorks Bull Soc Econ Res 16(2):68–75CrossRefGoogle Scholar
- Moulin H (1986) Game theory for the social sciences. NYU Press, New YorkGoogle Scholar
- Murphy F, Sherali H, Soyster A (1982) A mathematical programming approach for determining oligopolistic market equilibrium. Math Programm 24(1):92–106CrossRefGoogle Scholar
- Nikaido H, Isoda K (1955) Note on noncooperative convex games. Pac J Math 5(1):807–815CrossRefGoogle Scholar
- Novshek W (1985) On the existence of Cournot equilibrium. Rev Econ Stud 52(1):85–98CrossRefGoogle Scholar
- Okuguchi K (1964) The stability of the Cournot oligopoly solution: a further generalization. Rev Econ Stud 31(2):143–146CrossRefGoogle Scholar
- Okuguchi K (1973) Quasi-competitiveness and Cournot oligopoly. Rev Econ Stud 40(1):145–148CrossRefGoogle Scholar
- Reny PJ (1999) On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67(5):1029–1056CrossRefGoogle Scholar
- Roberts J, Sonnenschein H (1976) On the existence of Cournot equilibrium without concave profit functions. J Econ Theory 13(1):112–117CrossRefGoogle Scholar