Monotone and local potential maximizers in symmetric 3x3 supermodular games

Generalized notions of potential maximizer, monotone potential maximizer (MP-maximizer) and local potential maximizer (LP-maximizer), are studied. It is known that 2x2 coordination games generically have a potential maximizer, while symmetric 4x4 supermodular games may have no MP- or LP-maximizer. This note considers the case inbetween, namely the class of (generic) symmetric 3x3 supermodular coordination games. This class of games are shown to always have a unique MP-maximizer, and its complete characterization is given. A nondegenerate example demonstrates that own-action quasiconcave supermodular games may have more than one LP-maximizers.