Pure-Strategy Equilibrium for Bi-matrix Games and Coordination Games

We consider pure-strategy equilibrium for bi-matrix games and prove a version of the seminal “Equivalence Theorem” of Mangasarian and Stone, when it is restricted to pure-strategy profiles. We then consider bi-matrix games where both players have the same pay-off matrix. The existence of a pure strategy equilibrium for such games, that we refer to as “two-person identical pay-off matrix” (TPIPM) games, is established easily. We subsequently define coordination games as the class of all symmetric bi-matrix games, each such game having at least two pure-strategy equilibria and the set of pure-strategy equilibria for each of which is identical to the set of pure-strategy equilibria of the corresponding TPIPM game where the pay-off matrix is the average of the two pay-off matrices. We discuss two bi-matrix games well-known for problems with regard to coordination in choice of strategy, to show that they agree with our definition of coordination games.