Computation of Completely Mixed Equilibrium Payoffs in Bimatrix Games

Computing the (Nash) equilibrium payoffs in a given bimatrix game (i.e., a finite two-person game in strategic form) is a problem of considerable practical importance. One algorithm that can be used for this purpose is the Lemke-Howson algorithm (Lemke and Howson (1964); von Stengel (2002)), which is guaranteed to find one equilibrium. Another, more elementary, approach is to compute the equilibrium payoffs by: (1) “guessing” the support of the equilibrium, i.e., the set of pure strategies each player uses with positive probability (which can always be done by systematically checking all possibilities, if necessary); (2) finding a pair of mixed strategies with this support, such that all the pure strategies designated to either player give that player the same payoff against the other player’s mixed strategy; (3) computing the corresponding payoffs; and, finally, (4) checking that none of the other pure strategies of either player gives that player a higher payoff. For an equilibrium with known support, the equilibrium payoffs are easier to compute. Indeed, it is shown below that, in the case of known support, steps (1), (2), and (4) can essentially be dispensed with. Examples of bimatrix games having equilibria with known supports include generalized rock–scissors–paper games, such as the (symmetric) game with payoff matrix