The Uncertainty Principle of Game Theory

In 1928, von Neumann's minimax theorem [33] on two-player, zero-sum games showed that, in general, optimal strategies of players are random, even if the rules of the game do not involve chance. In this paper we investigate the degree of uncertainty or randomness possible in optimal strategies. We show that Heisenberg's uncertainty principle of physics corresponds to a similar principle for zero-sum games and derive an explicit lower bound for the uncertainty of a game. We also demonstrate that this lower bound is a constraint on the set of possible optimal strategies and apply it to develop a simple but effective method for finding an approximately optimal strategy. A two-player, finite, zero-sum game is traditionally described by a payoff matrix A = iatj), where the rows correspond to the strategies of the first player (P) and the columns correspond to the strategies of the second player (Q). (The reason for these particular player "names" will become apparent later.) The individual strategies / and j are called pure strategies. The real number ai} is simply the amount of money or payoff that the first player receives from the second player if the first player chooses strategy "/" and the second player chooses strategy "/'; the chosen strategy of the opponent is unknown to each player. If