Refined best reply correspondence and dynamics

Abstract

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.