6.853 Topics in Algorithmic Game Theory September 13, 2011 Lecture 2

Two-Player (Normal-Form) Games. A two-player normal-form game is specified via a pair (R,C) of m × n payoff matrices. The two players of the game, called the row player and the column player, have respectively m and n pure strategies. As the players’ names imply, the pure strategies of the row player are in one-to-one correspondence with the rows of the payoff matrices, while the strategies of the column player correspond to columns. So if the row player plays strategy i and the column player strategy j, then their respective payoffs are given by R[i, j] and C[i, j]. Players may also randomize over their strategies, leading to mixed—as opposed to pure—strategies. We represent the simplex of mixed strategies available to the row player by ∆m and those available to the column player by ∆n. Thus, if x ∈ ∆m, then x is an m-dimensional vector 〈x1, x2, . . . , xm〉 such that 0 ≤ xi ≤ 1 for each i ∈ [m] := {1, . . . ,m} and ∑m i=1 xi = 1; similarly for y ∈ ∆n. For a pair of mixed strategies x ∈ ∆m and y ∈ ∆n, the expected payoff of the row and column player are respectively x Ry and x Cy.