COS 521: Advanced Algorithms∗ Game Theory and Linear Programming

A two player game (or more correctly, a two player normal-form game) is specified by two m × n payoff matrices R and C corresponding to the row and column player respectively. Each of these matrices has m rows corresponding to the m strategies of the row player and n columns corresponding to the n strategies of the column payer. The row player picks a row i ∈ [m], and the column player picks a column j ∈ [n]. We say that the outcome of the game is (i, j). The payoff to the row player is Ri,j and the payoff to the column player is Ci,j. The goal of each player is to maximize their own payoff. The strategies for the row and column player described so far are referred to as pure strategies. Both the row and column player may also pick a distribution over rows and columns respectively, referred to as mixed strategies. A mixed strategy for the row player can be specified by a vector x = (x1, . . . , xm) such that xi ≥ 0 and ∑ i xi = 1. Here the ith coordinate xi specifies the probability that the (pure) strategy i ∈ [m] is picked by the row player. The space of mixed strategies for the row player is denoted by ∆m. This is just the set of vectors x = (x1, . . . , xm) such that xi ≥ 0 and ∑ i xi = 1. Similarly, a mixed strategy for the column player is specified by a vector y = (y1, . . . , yn) such that yj ≥ 0 and ∑ j yj = 0. The space of mixed strategies for the column player is denoted by ∆n. If the row and column player play mixed strategies x,y respectively, then the interpretation is that both players pick independently from the probability distributions specified by x and y, i.e. the probability that the outcome is (i, j) is xiyj. The expected payoff to ∗ Princeton University, Spring 2013.