Better response dynamics and Nash equilibrium in discontinuous games

Kukushkin, Nikolai S. (2017): Better response dynamics and Nash equilibrium in discontinuous games.

Abstract

Philip Reny's approach to games with discontinuous utility functions can work outside its original context. The existence of Nash equilibrium and the possibility to approach the equilibrium set with a finite number of individual improvements are established, under conditions weaker than the better reply security, for three classes of strategic games: potential games, games with strategic complements, and aggregative games with appropriate monotonicity conditions.

• Item Type: MPRA Paper
• Language: English
• Keywords: discontinuous game; potential game; Bertrand competition; strategic complements; aggregative game
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Bergstrom, T., 1975. Maximal elements of acyclic relations on compact sets. Journal of Economic Theory 10, 403-404.

Bich, P., 2009. Existence of pure Nash equilibria in discontinuous and non quasiconcave games. International Journal of Game Theory 38, 395-410.

Dubey, P., O. Haimanko, and A. Zapechelnyuk, 2006. Strategic complements and substitutes, and potential games. Games and Economic Behavior 54, 77-94.

Huang, Z., 2002. Fictitious play in games with a continuum of strategies. Ph.D. Thesis, State University of New York at Stony Brook, Department of Economics.

Jensen, M.K., 2010. Aggregative games and best-reply potentials. Economic Theory 43, 45-66.

Jensen, M.K., 2012. Aggregative games and best-reply potentials: Erratum. https://sites.google.com/site/mkaaejensen/research

Kukushkin, N.S., 1994. A fixed-point theorem for decreasing mappings. Economics Letters 46, 23-26.

Kukushkin, N.S., 2005. "Strategic supplements" in games with polylinear interactions. EconWPA Paper #0411008

Kukushkin, N.S., 2011. Nash equilibrium in compact-continuous games with a potential. International Journal of Game Theory 40, 387-392.

Kukushkin, N.S., 2013a. Increasing selections from increasing multifunctions. Order 30, 541-555.

Kukushkin, N.S., 2013b. Monotone comparative statics: changes in preferences versus changes in the feasible set. Economic Theory 52, 1039-1060.

Kukushkin, N.S., 2016. Cournot tatonnement in aggregative games with monotone best responses. In: Contributions on Equilibrium Theory for Cournot and Related Games: Essays in Honour of Koji Okuguchi. Von Mouche, P., and F. Quartieri, eds. Springer, 45-59.

Kukushkin, N.S., S. Takahashi, and T. Yamamori, 2005. Improvement dynamics in games with strategic complementarities. International Journal of Game Theory 33, 229-238.

LiCalzi, M., and A.F. Veinott, Jr., 1992. Subextremal functions and lattice programming. EconWPA Paper #0509001

McLennan, A., P.K. Monteiro, and R. Tourky, 2011. Games with discontinuous payoffs: A strengthening of Reny's existence theorem. Econometrica 79, 1643-1664.

Milgrom, P., and C. Shannon, 1994. Monotone comparative statics. Econometrica 62, 157-180.

Monderer, D., and L.S. Shapley, 1996. Potential games. Games and Economic Behavior 14, 124-143.

Novshek, W., 1985. On the existence of Cournot equilibrium. Review of Economic Studies 52, 85-98.

Prokopovych, P., 2013. The single deviation property in games with discontinuous payoffs. Economic Theory 53, 383-402.

Prokopovych, P., and N.C. Yannelis, 2017. On strategic complementarities in discontinuous games with totally ordered strategies. Journal of Mathematical Economics 70, 147-153.

Quah, J.K.-H., 2007. The comparative statics of constrained optimization problems. Econometrica 75, 401-431.

Quah, J.K.-H., and B. Strulovici, 2009. Comparative statics, informativeness, and the interval dominance order. Econometrica 77, 1949-1999.

Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029-1056.

Reny, P.J., 2011. Strategic approximations of discontinuous games. Economic Theory 48, 17-29.

Reny, P.J., 2016. Nash equilibrium in discontinuous games. Economic Theory 61, 553-569.

Shannon, C., 1995. Weak and strong monotone comparative statics. Economic Theory 5, 209-227.

Topkis, D.M., 1978. Minimizing a submodular function on a lattice. Operations Research 26, 305-321.

Veinott, A.F., Jr., 1989. Lattice Programming. Unpublished notes from lectures delivered at Johns Hopkins University.

Vives, X., 1990. Nash equilibrium with strategic complementarities. Journal of Mathematical Economics 19, 305-321.

Wakker, P., 1988. Continuity of preference relations for separable topologies. International Economic Review 29, 105-110.

Walker, M., 1977. On the existence of maximal elements. Journal of Economic Theory 16, 470-474.