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6.896 Topics in Algorithmic Game Theory February 10, 2010 Lecture 3

By Lecturer Constantinos Daskalakis, Scribe Pablo Azar and Anthony Kim


In the previous lecture we saw that there always exists a Nash equilibrium in two-player zero-sum games. Moreover, the equilibrium enjoys several attractive properties such as polynomial-time tractability, convexity of the equilibrium set, and uniqueness of players ’ payoffs in all equilibria. In the present and the following lecture we investigate whether simple and natural distributed protocols can find the value/equilibrium strategies of a zero-sum game. We have in mind very generic settings in which the players may be oblivious to the exact specifications of the zero-sum game they are playing. We only require that they know what strategies are available to them, and can observe how well each of their strategies performs against the choices of their opponent. 1 Fictitious Play Let (R, C)m×n be a two player zero-sum game. Suppose that the game is played repeatedly by its two players. Say that, at time t = 0, the row player plays strategy i0 and the column player strategy j0. For any row-player strategy i, define V 0 (i) = Ri,j0 to represent the payoff achieved by strategy i, given the current history of play by the other player (in this case the history has length 1). Similarly, for any column-player strategy j, define U 0 (j) = Ri0,j to represent the loss incurred by strategy j given the history of play of the row player

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