Skip to main content

Pure strategy equilibrium in finite weakly unilaterally competitive games


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  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).


  1. Davey BA, Priestly HA (1990) Introduction to lattices and order. Cambridge University Press, Cambridge

    Google Scholar 

  2. Duersch P, Oechssler J, Schipper BC (2012a) Pure strategy equilibria in symmetric two-player zero-sum games. Int J Game Theory 41:553–564

  3. Duersch P, Oechssler J, Schipper BC (2012b) Unbeatable imitation. Games Econ Behav 76:88–96

  4. Friedman JW (1983) On characterizing equilibrium points in two person strictly competitive games. Int J Game Theory 12:245–247

    Article  Google Scholar 

  5. Friedman JW (1991) Game theory with applications to economics, 2nd edn. Oxford University Press, Oxford

    Google Scholar 

  6. Hehenkamp B, Possajennikov A, Guse T (2010) On the equivalence of Nash and evolutionary equilibrium in finite populations. J Econ Behav Organ 73:254–258

    Article  Google Scholar 

  7. Kats A, Thisse J-F (1992) Unilaterally competitive games. Int J Game Theory 21:291–299

    Article  Google Scholar 

  8. Milgrom P, Roberts J (1990) Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58:1255–1277

    Article  Google Scholar 

  9. Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14:124–143

    Article  Google Scholar 

  10. Nash J (1951) Non-cooperative games. Ann Math 54:286–295

    Article  Google Scholar 

  11. Schaffer ME (1988) Evolutionarily stable strategies for a finite population and a variable contest size. J Theor Biol 132:469–478

    Article  Google Scholar 

  12. Schaffer ME (1989) Are profit-maximizers the best survivors?—A Darwinian model of economic natural selection. J Econ Behav Organ 12:29–45

    Article  Google Scholar 

Download references


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.

Author information



Corresponding author

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”.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Iimura, T., Watanabe, T. Pure strategy equilibrium in finite weakly unilaterally competitive games. Int J Game Theory 45, 719–729 (2016).

Download citation


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

JEL Classification

  • C72 (Noncooperative game)