Refined Best Reply Correspondence and Dynamics

28 Pages Posted: 29 Mar 2012 Last revised: 17 Feb 2015

See all articles by Dieter Balkenborg

Dieter Balkenborg

University of Exeter - Department of Economics

Josef Hofbauer

University of Vienna - Department of Mathematics

Christoph Kuzmics

University of Graz - Department of Economics

Date Written: July 15, 2011

Abstract

We 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.

Keywords: Evolutionary game theory, best response dynamics, CURB sets, persistent retracts, asymptotic stability, Nash equilibrium refinements, learning

JEL Classification: C62, C72, C73

Suggested Citation

Balkenborg, Dieter and Hofbauer, Josef and Kuzmics, Christoph, Refined Best Reply Correspondence and Dynamics (July 15, 2011). Institute of Mathematical Economics Working Paper No. 451, Available at SSRN: https://ssrn.com/abstract=2026243 or http://dx.doi.org/10.2139/ssrn.2026243

Dieter Balkenborg

University of Exeter - Department of Economics ( email )

Streatham Court
Exeter EX4 4PU
United Kingdom

Josef Hofbauer

University of Vienna - Department of Mathematics ( email )

Oskar-Morgenstern-Platz 1
A-1090 Vienna
Austria

Christoph Kuzmics (Contact Author)

University of Graz - Department of Economics ( email )

Universitaetsstrasse 15
RESOWI - F4
Graz, 8010
Austria

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