On the uniqueness of optimal strategies in symmetric matrix games

This note presents a characterization for strictly complementary optimal strategies in an extended neighborhood of any given non-unique strictly complementary optimal strategy for a symmetric matrix game. Specifically, we use the given strategy and the game payoff matrix to construct a test matrix which (i) establishes uniqueness of the strategy if the matrix is non-singular, and (ii) provides the algebraic foundation for characterization of alternate optimal strategies within a neighborhood of the original optimal strategy when the test matrix is singular, which is particularly significant if the given original strategy is known to be the analytic center of the region of all strictly complementary strategies. We also discuss the implications of our results to questions of uniqueness in general linear programs.