Computational Learning Theory Spring Semester , 2003 / 4 Lecture 4 : 2 - Player Zero Sum Games

In this lecture we will discuss 2-player zero sum games. Such games are completely competitive , where whatever one player wins, the other must lose. Examples of such games include chess, checkers, backgammon, etc. We will show that in such games: • An equilibrium always exists; • All equilibrium points yield the same payoff for all players; • The set of equilibrium points is actually the cartesian product of independent sets of equilibrium strategies per player. We will also show applications of this theory. Definition Let G be the game defined by N, (A i) , (u i) where N is the number of players, A i is the set of possible pure strategies for player i, and u i is the payoff function for player i. Let A be the cartesian product A = n i=1 A i. Then G is a zero sum game if and only if: ∀ a ∈ A, n i=1 u i (a) = 0 (4.1) In other words, a zero sum game is a game in which, for any outcome (any combination of pure strategies, one per player), the sum of payoffs for all players is zero. We naturally extend the definition of u i to any probability distribution p over A by u i (p) = E a ∼ p (u i (a)). The following is immediate due to the linearity of the expectation and the zero sum constraint: Corollary 4.1 Let G be a zero sum game, and ∆ the set of probability distributions over A. Then ∀ p ∈ ∆, n i=1 u i (p) = 0 (4.2)