Equilibrium Play in Matches: Binary Markov Games

We study two-person extensive form games, or "matches," in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in "point games." We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game (thus, playing minimax at each point assures a player that the probability he will win the match is at least as great as under equilibrium play, no matter how his opponent plays); and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. The minimax result provides a rationale for the players to play minimax (and therefore Nash equilibrium) in every point game, even if the behavioral assumptions typically used to justify Nash equilibrium are not satisfied. An application to tennis is provided.