We would like to thank Carlos Alos-Ferrer, Pierpaolo Battigalli, Eddie Dekel, Amanda Friedenberg, Drew Fudenberg, Klaus Ritzberger, Karl Schlag, Mark Voorneveld, Jörgen Weibull, and editor Martin Osborne as well as four anonymous referees for helpful comments and suggestions.
Refined best reply correspondence and dynamics
Article first published online: 22 JAN 2013
Copyright © 2013 Dieter Balkenborg, Josef Hofbauer, and Christoph Kuzmics
Volume 8, Issue 1, pages 165–192, January 2013
How to Cite
Balkenborg, D., Hofbauer, J. and Kuzmics, C. (2013), Refined best reply correspondence and dynamics. Theoretical Economics, 8: 165–192. doi: 10.3982/TE652
- Issue published online: 22 JAN 2013
- Article first published online: 22 JAN 2013
- Submitted 2009-10-9. Final version accepted 2012-1-27. Available online 2012-1-27.
- Evolutionary game theory;
- best response dynamics;
- CURB sets;
- persistent retracts;
- asymptotic stability;
- Nash equilibrium refinements;
We call a correspondence, defined on the set of mixed strategy profiles, a generalized best reply correspondence if it (i) has a product structure, (ii) is upper hemicontinuous, (iii) always includes a best reply to any mixed strategy profile, and (iv) is 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 pointwise set inclusion and, under a mild condition on the normal form of the game at hand, forms a complete lattice with meets based on pointwise intersections. The refined best reply correspondence is then defined as the smallest element of the set of all generalized best reply correspondences. We find that every persistent retract (Kalai and Samet 1984) contains a MASF. Furthermore, persistent retracts are minimal CURB (closed under rational behavior) sets (Basu and Weibull 1991) based on the refined best reply correspondence. Conversely, every MASF must be a prep set (Voorneveld 2004), based again, however, on the refined best reply correspondence.