Abstract
For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis: a Hitchhiker’s Guide (3rd edn.). Springer, Berlin (2006)
Aliprantis C.D., Glycopantis D., Puzello D.: The joint continuity of the expected payoff functions. J Math Econ 42, 121–130 (2006)
Aubin J.-P., Cellina A.: Differential Inclusions. Springer, Berlin (1984)
Balder E.J.: A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J Control Optim 22, 570–598 (1984)
Balder E.J.: An equilibrium existence result for games with incomplete information and indeterminate outcomes. J Math Econ 40, 297–320 (2004)
Balder, E.J.: On Equilibria for Discontinuous Games: Nash Approximation Schemes. Preprint no. 1209. Department of Mathematics, University of Utrecht (2001)
Billingsley P.: Convergence of Probability Measures. Wiley, New York (1968)
Brown L.D., Purves R.: Measurable selections of extrema. Ann Stat 1, 902–912 (1973)
Carmona, G.: Understanding some recent existence results for discontinuous games. Econ Theory (to appear)
Dasgupta P., Maskin E.: The existence of equilibrium in discontinuous economic games I: theory. Rev Econ Stud 53, 1–26 (1986)
Dasgupta P., Maskin E.: The existence of equilibrium in discontinuous economic games II: applications. Rev Econ Stud 53, 28–42 (1986)
Glicksberg I.: A further generalization of Kakutani’s fixed point theorem with applications to Nash equilibrium points. Proc Natl Acad Sci USA 38, 170–172 (1952)
Hotelling H.: The stability of economic competition. Econ J 39, 41–57 (1929)
Jackson M.O., Simon L.K., Swinkels J.M., Zame W.R.: Communication and equilibrium in discontinuous games of incomplete information. Econometrica 70, 1711–1740 (2002)
Lebrun B.: Existence of an equilibrium in first price auctions. Econ Theory 7, 421–443 (1996)
Moore J.C.: Mathematical Methods for Economic Theory 2. Springer, New York (1999)
Myerson R.: Game Theory. Harvard University Press, Cambridge (1991)
Neveu J.: Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco (1965)
Pfanzagl J.: Convexity and conditional expectations. Ann Probab 2, 490–494 (1974)
Reny P.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056 (1999)
Simon L.K.: Games with discontinuous payoffs. Rev Econ Stud 54, 569–597 (1987)
Simon L.K., Zame W.R.: Discontinuous games and endogenous sharing rules. Econometrica 58, 861–872 (1990)
Valadier M.: Désintégration d’une mesure sur un produit. C.R Acad Sci Paris 267, 33–35 (1973)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
A question by an anonymous referee helped and stimulated the author to demarcate the present paper’s position with respect to Simon and Zame (1990). Also, remarks by Guilherme Carmona helped to improve the presentation.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Balder, E.J. An equilibrium closure result for discontinuous games. Econ Theory 48, 47–65 (2011). https://doi.org/10.1007/s00199-010-0574-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-010-0574-6
Keywords
- Nash equilibrium
- Discontinuous games
- Weak convergence of probability measures
- Endogenous sharing rule
- Kuratowski limes superior