Abstract
In a recent but well known paper, Reny has proved the existence of Nash equilibria for compact and quasiconcave games, with possibly discontinuous payoff functions. In this paper, we prove that the quasiconcavity assumption in Reny’s theorem can be weakened: we introduce a measure allowing to localize the lack of quasiconcavity, which allows to refine the analysis of equilibrium existence (I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper).
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References
Bagh A, Jofre A (2006) Reciprocal upper semicontinuity and better reply secure games: a comment. Econometrica 74(6): 1715–1721
Baye MR, Tian G, Zhou J (1993) Characterization of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev Econ Stud 60: 935–948
Carmona G (2009) An existence result for discontinuous games. J Econ Theory 144(3): 1333–1340
Dasgupta P, Maskin E (1986) The existence of an equilibrium in discontinuous economics games I: theory. Rev Econ Stud 53: 1–26
Frenk JBG, Kassay G et al (2005) Introduction to convex and quasiconvex analysis, Chapter 1. In: Hadjisavvas N et al. (eds) Handbook of generalized convexity and generalized monotonicity, Nonconvex optimization and its applications, vol 76. Springer, Heidelberg, pp 121–149
Friedman J, Nishimura K (1981) Existence of Nash equilibrium in n person games without quasi-concavity. Int Econ Rev 22: 637–648
Kostreva MM (1989) Nonconvexity in noncooperative game theory. Int J Game Theory 18: 247–259
McClendon JF (2005) Existence of solutions of games with some non-convexity. Int J Game Theory 15: 155–162
McLennan A (1989) Fixed points of contractible valued correspondences. Int J Game Theory 18: 175–184
Reny PJ (1999) On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67: 1029–1056
Starr RM (1969) Quasi-equilibria in markets with non-convex preferences. Econometrica 37: 25–38
Topkis DM (1979) Equilibrium points in nonzero-sum n-person submodular games. SIAM J Control Optim 17: 773–787
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I wish to thank P. J. Reny, two anonymous referees and the associated editor for corrections, suggestions and remarks which led to improvements in the paper.
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Philippe, B. Existence of pure Nash equilibria in discontinuous and non quasiconcave games. Int J Game Theory 38, 395–410 (2009). https://doi.org/10.1007/s00182-009-0160-y
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DOI: https://doi.org/10.1007/s00182-009-0160-y