Refined best reply correspondence and dynamics
AbstractWe call a correspondence, defined on the set of mixed strategy profiles, a generalized best reply correspondence if it has (1) a product structure, is (2) upper semi-continuous, (3) always includes a best reply to any mixed strategy profile, and is (4) convex- and closed-valued. For each generalized best reply correspondence we define a generalized best reply dynamics as a differential inclusion based on it. We call a face of the set of mixed strategy profiles a minimally asymptotically stable face (MASF) if it is asymptotically stable under some such dynamics and no subface of it is asymptotically stable under any such dynamics. The set of such correspondences (and dynamics) is endowed with the partial order of point-wise set-inclusion and, under a mild condition on the normal form of the game at hand, forms a complete lattice with meets based on point-wise intersections. The refined best reply correspondence is then defined as the smallest element of the set of all generalized best reply correspondences. We ultimately find that every Kalai and Samet's (1984) persistent retract, which coincide with Basu and Weibull's (1991) CURB sets based, however, on the refined best reply correspondence, contains a MASF. Conversely, every MASF must be a Voorneveld's (2004) prep set, again, however, based on the refined best reply correspondence.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 451.
Length: 28 pages
Date of creation: Jul 2011
Date of revision:
Evolutionary game theory; best response dynamics; CURB sets; persistent retracts; asymptotic stability; Nash equilibrium refinements; learning;
Find related papers by JEL classification:
- C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
- C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
- C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2011-07-27 (All new papers)
- NEP-GTH-2011-07-27 (Game Theory)
- NEP-MIC-2011-07-27 (Microeconomics)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Matsui, Akihiko, 1992. "Best response dynamics and socially stable strategies," Journal of Economic Theory, Elsevier, vol. 57(2), pages 343-362, August.
- K. Ritzberger & J. Weibull, 2010.
"Evolutionary Selection in Normal-Form Games,"
Levine's Working Paper Archive
452, David K. Levine.
- Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2012. "The refined best-response correspondence in normal form games," Working Papers 466, Bielefeld University, Center for Mathematical Economics.
- E. Kohlberg & J.-F. Mertens, 1998.
"On the Strategic Stability of Equilibria,"
Levine's Working Paper Archive
445, David K. Levine.
- Van Damme, E. & Hurkens, S., 1993.
"Commitment Robust Equilibria and Endogenous Timing,"
9356, Tilburg - Center for Economic Research.
- van Damme, Eric & Hurkens, Sjaak, 1996. "Commitment Robust Equilibria and Endogenous Timing," Games and Economic Behavior, Elsevier, vol. 15(2), pages 290-311, August.
- Damme, E.E.C. van & Hurkens, J.P.M., 1993. "Commitment robust equilibria and endogenous timing," Discussion Paper 1993-56, Tilburg University, Center for Economic Research.
- Damme, E.E.C. van & Hurkens, J.P.M., 1996. "Commitment robust equilibria and endogenous timing," Open Access publications from Tilburg University urn:nbn:nl:ui:12-73412, Tilburg University.
- Blume, A., 1992.
"Equilibrium Refinements in Sender-Receiver Games,"
92-12, University of Iowa, Department of Economics.
- Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
- Voorneveld, Mark, 2004. "Preparation," Games and Economic Behavior, Elsevier, vol. 48(2), pages 403-414, August.
- Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
- Ehud Kalai & Dov Samet, 1982. "Persistent Equilibria in Strategic Games," Discussion Papers 515, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
- Balkenborg, D. & Jansen, M. & Vermeulen, D., 1998.
"Invariance properties of persistent equilibria and related solution concepts,"
Discussion Paper Series In Economics And Econometrics
9806, Economics Division, School of Social Sciences, University of Southampton.
- Balkenborg, Dieter & Jansen, Mathijs & Vermeulen, Dries, 2001. "Invariance properties of persistent equilibria and related solution concepts," Mathematical Social Sciences, Elsevier, vol. 41(1), pages 111-130, January.
- Balkenborg, Dieter & Schlag, Karl H., 2007. "On the evolutionary selection of sets of Nash equilibria," Journal of Economic Theory, Elsevier, vol. 133(1), pages 295-315, March.
- I. Gilboa & A. Matsui, 2010.
"Social Stability and Equilibrium,"
Levine's Working Paper Archive
534, David K. Levine.
- Hurkens Sjaak, 1995. "Learning by Forgetful Players," Games and Economic Behavior, Elsevier, vol. 11(2), pages 304-329, November.
- Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868, October.
- Kets, Willemien & Voorneveld, Mark, 2005.
"Learning to be prepared,"
Working Paper Series in Economics and Finance
590, Stockholm School of Economics.
- Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
- repec:hal:wpaper:hal-00713871 is not listed on IDEAS
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dr. Frederik Herzberg).
If references are entirely missing, you can add them using this form.