Minimal and locally minimal games and game forms

By Shapley’s (1964) theorem, a matrix game has a saddle point whenever each of its 2 × 2 subgames have one. In other words, all minimal saddle point free (SP-free) matrices are of size 2 × 2. We strengthen this result and show that all locally minimal SP-free matrices are of size 2× 2. In other words, if A is a SP-free matrix in which a saddle point appears after deleting an arbitrary row or column, then A is of size 2× 2. Furthermore, we generalize this result and characterize the minimal and locally minimal Nash equilibrium free (NE-free) bimatrix games. Let us recall that a two-person game form is Nash-solvable if and only if it is tight (Gurvich, 1975). We show that all (locally) minimal non tight game forms are of size 2 × 2. In contrast, it seems difficult to characterize the locally minimal tight game forms (while all minimal ones are just trivial); we only obtain some necessary and some sufficient conditions. We also recall an example from cooperative game theory: a maximal stable effectivity function that is not self-dual and not convex.