playing T. Thus, a mixed-strategy Nash equilibrium in this game is for each player to randomly choose H or T with and . Oddness of equilibrium points In 1971, Robert Wilson came up with the Oddness Theorem, which says that "almost all" finite games have a finite and odd number of Nash equilibria. In 1993, Harsanyi published an alternative proof of the result. "Almost all" here means that any game with an infinite or even number of equilibria is very special in the sense that if its payoffs were even slightly randomly perturbed, with probability one it would have an odd number of equilibria instead. The prisoner's dilemma, for example, has one equilibrium, while the battle of the sexes has three-- two pure and one mixed, and this remains true even if the payoffs change slightly. The free money game is an example of a "special" game with an even number of equilibria. In it, two players have to both vote "yes" rather than "no" to get a reward and the votes are simultaneous. There are two pure-strategy Nash equilibria, (yes, yes) and (no, no), and no mixed strategy equilibria, because the strategy "yes" weakly dominates "no". "Yes" is as good as "no" regardless of the other player's action, but if there is any chance the other player chooses "yes" then "yes" is the best reply. Under a small random perturbation of the payoffs, however, the probability that any two payoffs would remain tied, whether at 0 or some other number, is vanishingly small, and the game would have either one or three equilibria instead. See also Notes References Game theory textbooks . Dixit, Avinash, Susan Skeath and David Reiley. Games of Strategy. W.W. Norton & Company. (Third edition in 2009.) An undergraduate text. . Suitable for undergraduate and business students. Fudenberg, Drew and Jean Tirole (1991) Game Theory MIT Press. . Lucid and detailed introduction to game theory in an explicitly economic context. Morgenstern, Oskar and John von Neumann (1947) The Theory of Games and Economic Behavior Princeton University Press. . . A modern introduction at the graduate level. . A comprehensive reference from a computational perspective; see Chapter 3. Downloadable free online. Original Nash papers Nash, John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences 36(1):48-49. Nash, John (1951) "Non-Cooperative Games" The Annals of Mathematics 54(2):286-295. Other references Mehlmann, A. (2000) The Game's