We call a correspondence, defined on the set of mixed strategy profiles, a generalized

best reply correspondence if it has a product structure, is upper

hemi-continuous, always includes a best reply to any mixed strategy profile,

and 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 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.