Strategic complementarity and substitutability without transitive indifference

Kukushkin, Nikolai S. (2011): Strategic complementarity and substitutability without transitive indifference.

Abstract

We study what useful implications strategic complementarity or substitutability may have when the indifference relation(s) need not be transitive. Two results are obtained about the existence of a monotone selection from the best response correspondence when both strategies and parameters form chains. Two more results are obtained about the existence of a Nash equilibrium in games with strategic complementarities where strategy sets are chains, but monotone selections from the best response correspondences need not exist.

• Item Type: MPRA Paper
• Language: English
• Keywords: strong acyclicity; interval order; single crossing; monotone selection; Nash equilibrium
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Aizerman, M.A., and F.T. Aleskerov, 1995. Theory of Choice. North-Holland, Amsterdam.

Athey, S., 2001. Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69, 861-889.

Aumann, R.J., 1962. Utility theory without the completeness axiom. Econometrica 30, 445-462.

Aumann, R.J., 1964. Utility theory without the completeness axiom: A correction. Econometrica 32, 210-212.

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

Fishburn, P.C., 1973. The Theory of Social Choice. Princeton University Press, Princeton.

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

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

Kukushkin, N.S., 2003. Acyclicity of Monotonic Endomorphisms. Russian Academy of Sciences, Dorodnicyn Computing Center, Moscow. A version available at http://www.ccas.ru/mmes/mmeda/ququ/MonoEndo.pdf

Kukushkin, N.S., 2004. Best response dynamics in finite games with additive aggregation. Games and Economic Behavior 48, 94-110.

Kukushkin, N.S., 2005. "Strategic supplements" in games with polylinear interactions. EconWPA Paper \#0411008. Available at http://ideas.repec.org/p/wpa/wuwpga/0411008.html

Kukushkin, N.S., 2008. Maximizing an interval order on compact subsets of its domain. Mathematical Social Sciences 56, 195-206.

LiCalzi, M., and A.F. Veinott, Jr., 1992. Subextremal functions and lattice programming. Unpublished manuscript. Available at http://ideas.repec.org/p/wpa/wuwpge/0509001.html

Milgrom, P., and J. Roberts, 1990. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58, 1255-1277.

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

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

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

Quah, J., and B. Strulovici, 2009. Comparative statics, informativeness, and the interval dominance order. Econometrica 77, 1949-1992.

Reny, P.J., 2011. On the existence of monotone pure strategy equilibria in Bayesian games. Econometrica 79, 499-553.

Sen, A.K., 1984. Collective Choice and Social Welfare. North-Holland, Amsterdam.

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

Smith, T.E., 1974. On the existence of most-preferred alternatives. International Economic Review 15, 184-194.

Topkis, D.M., 1979. Equilibrium points in nonzero-sum $n$-person submodular games. SIAM Journal on Control and Optimization 17, 773-787.

Veinott, A.F.,Jr., 1989. Lattice Programming. Unpublished lectures.

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