A Purification Theorem for Perfect-Information Games

Kalmar [2, 1928-9] proved that Chess is strictly determined. Von Neumann-Morgenstern [5, 1944] proved the same for any finite two-person zero-sum perfect-information (PI) game. The latter result yields a minimax theorem for (finite) non-zero-sum PI games. Fix a PI, and a player, Ann. Convert this game to a two-person zero-sum game between Ann and the other players (considered as one player), in which Ann gets the same payoffs as in the initial game. In the new game, the minmax of Ann’s payoffs is equal to the maxmin of Ann’s payoffs. Since this statement involves only Ann’s payoffs, it must also hold in the initial game. In this note we first prove a generalization of this fact: The minmax and maxmin of Ann’s payoffs are still equal, when they are taken over subsets of strategies, provided the subsets for the players other than Ann are rectangular. We then use this generalization to prove a purification result: Fix a PI game, a strategy s for Ann, and rectangular subsets of strategies for the players other than Ann. Suppose s is weakly dominated with respect to these subsets. That is, suppose there is a mixed strategy for Ann that gives her at least as high a payoff for each profile of pure strategies for the other players, taken from the given subsets, and a strictly higher payoff against at least one such profile. Then there is a pure strategy for Ann that weakly dominates s in the same sense. Kalmar’s [2, 1928-9] proof was forward looking. Another forward-looking argument is Kuhn’s [3, 1950], [4, 1953] well-known proof that every finite PI game has a pure-strategy Nash equilibrium.” Our arguments, too, are forward looking, proceeding by induction on the length of the tree.