Nash equilibrium uniqueness in nice games with isotone best replies

Ceparano, Maria Carmela and Quartieri, Federico (2015): Nash equilibrium uniqueness in nice games with isotone best replies.

Abstract

We prove the existence of a unique pure-strategy Nash equilibrium in nice games with isotone chain-concave best replies and compact strategy sets. We establish a preliminary fixpoint uniqueness argument, thus showing sufficient assumptions on the best replies of a nice game that guarantee the existence of exactly one Nash equilibrium. Then, by means of a comparative statics analysis, we examine the necessity and sufficiency of the conditions on marginal utility functions for such assumptions to be satisfied; in particular, we find necessary and sufficient conditions for the isotonicity and chain-concavity of best replies. We extend the results on Nash equilibrium uniqueness to nice games with upper unbounded strategy sets and we present "dual" results for games with isotone chain-convex best replies. A final application to Bayesian games is exhibited.

• Item Type: MPRA Paper
• Language: English
• Keywords: Nash equilibrium uniqueness; Chain-concave best reply; Nice game; Comparative statics; Strategic complementarity.
• 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

Agliardi, E.: A generalization of supermodularity. Economics Letters 68, 251-254 (2000)

Baetz, O.: Social activity and network formation. Theoretical Economics 10, 315-340 (2015)

Belhaj, M., Bramoullé, Y., Deroïan, F.: Network games under strategic complementarities. Games and Economic Behavior 88, 310-319 (2014)

Cambini, A., Martein, L.: Generalized Convexity and Optimization: Theory and Applications. Springer-Verlag, Berlin Heidelberg (2009)

Ceparano, M.C.: Nash equilibrium selection in multi-leader multi-follower games with vertical separation. PhD thesis, University of Naples "Federico II". (2015)

Ceparano, M.C., Morgan, J.: Equilibria for multi-leader multi-follower games with vertical information: existence results. Working paper - https://ideas.repec.org/p/sef/csefwp/417.html (2015). Accessed October 2015

Cottle, R.W., Pang, J.-S., Stone, R.E.: The linear complementarity problem. SIAM, Philadelphia (2009)

Diewert, W.E.: Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming, In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 51-93. Academic Press, New York (1981)

Edlin, A.S., Shannon, C.: Strict monotonicity in comparative statics. Journal of Economic Theory 81, 201-219 (1998)

Gabay, D., Moulin, H.: On the Uniqueness and Stability of Nash Equilibria in Noncooperative Games. In: Bensoussan, A., Kleindorfer, P.R., Tapiero, C.S. (eds.) Applied Stochastic Control in Econometrics and Management Science, pp. 271-293. North-Holland, Amsterdam (1980)

Giorgi, G., Komlósi, S.: Dini derivatives in optimization - Part I. Rivista di matematica per le scienze economiche e sociali. 15, 3-30 (1992)

Hadjisavvas, N., Schaible, S.: Generalized monotone single valued maps. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, second edition, pp. 1197-1202. Springer, New York (2009)

Hiller, T.: Peer effects in endogenous networks. Working paper - http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2478975 (2014). Accessed 11 August 2014

Ivanov, V.I.: A note on strong pseudoconvexity. Central European Journal of Mathematics 6, 576-580 (2008)

Ivanov, V.I.: Characterizations of pseudoconvex functions and semistrictly quasiconvex ones. Journal of Global Optimization 57, 677-693 (2013)

Lagerås, A., Seim, D.: Strategic complementarities, network games and endogenous network formation. International Journal of Game Theory (forthcoming) (2015)

Kannan, R., Krueger, C.K.: Advanced Analysis on the Real Line. Springer Verlag, New York (1996)

Karamardian, S.: The nonlinear complementarity problem with applications, part 2. Journal of Optimization Theory and Applications 4, 167-181 (1969)

Kennan, J.: Uniqueness of positive fixed points for increasing concave functions on Rⁿ: an elementary result. Review of Economic Dynamics 4, 893-899 (2001)

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

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

Milgrom, P., Roberts, J.: Coalition-proofness and correlation with arbitrary communication possibilities. Games and Economic Behavior 17, 113-128 (1996)

Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62, 157-180 (1994)

Moulin, H.: Dominance solvability and Cournot stability. Mathematical Social Sciences 7, 83-102 (1984)

Nash, J.F.: Non-cooperative games. Ph.D. Dissertation. Princeton University (1950)

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

Quah, J.K.-H., Shirai, K.: A revealed preference theory of monotone choice and strategic complementarity. Working paper - http://www.kier.kyoto-u.ac.jp/DP/DP914.pdf. Accessed February 2015

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

Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-person games. Econometrica 33, 520-534 (1965)

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

Tarski, A.: A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5, 285-309 (1955)

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

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

Van Zandt, T.: Interim Bayesian Nash equilibrium on universal type spaces for supermodular games. Journal of Economic Theory 145, 249-263 (2010)

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