lecture4_Computational-Aspects_of_GameTheory.pdf - Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecture 4 Nash Equilibria Lecturer

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Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecture 4: Nash Equilibria Lecturer: Bruno Codenotti The concept of Nash Equilibrium embodies the fundamental idea of economics that people act according to their incentives , and thus Nash equilibria turn out to be relevant in every area where incentives are important, i.e., all of Economic Theory. 4.1 Introduction In this lecture, we first analyze two-player zero sum games, and then move on to consider two-player non- constant sum games. For zero-sum games, we first look at the cases where equilibria in the pure strategies do exist. These cases are characterized by the presence of a saddle point in the payoff matrix. Since saddle points do not exist in general, we then consider the case where players can use mixed strategies. In this scenario, we show the equivalence of zero-sum games and linear programming. More precisely, we prove that: Given a zero-sum game, the optimal strategies of the players can be represented by two linear programs, which are the dual of one another; we can then use duality theory to show the existence of equilibria. Given a pair of primal-dual linear programs, we construct a zero sum game, and show that the optimal solutions to the linear programs can be easily extracted from the mixed strategies which give a NE for the game. In the second part of the lecture, we first show that NE for two-player non-constant sum games can be described as the solution to a computational problem known as linear complementarity , and then prove the Nash Theorem, which shows the existence of NE. Before starting with our analysis, we recall some basic definitions. We consider two-player games in normal form . These games are described by a pair of matrices ( A, B ), whose entries are the payoffs of the two players, called row and column player. A = ( a ij ) is the payoff matrix of the row player, and B = ( b ij ) is the payoff matrix of the column player. The rows (resp. columns) of A and B are indexed by the row (resp. column) player’s pure strategies . The entry a ij is the payoff to the row player, when she plays her i -th pure strategy and the opponent plays his j -th pure strategy. Similarly, b ij is the payoff to the column player, when he plays his j -th pure strategy and the opponent plays her i -th pure strategy. A mixed strategy is a probability distribution over the set of pure strategies which indicates how likely it is that each pure strategy is played. More precisely, in a mixed strategy a player associates to her i -th pure strategy a quantity p i between 0 and 1, such that i p i = 1, where the sum ranges over all pure strategies. Let us consider the game ( A, B ), where A and B are m × n matrices. In such a game the row player has m pure strategies, while the column player has n pure strategies. Let x be a mixed strategy of player R (the row player), and y a mixed strategy of player C (the column player). Strategy x is the m -tuple 4-1
4-2 Lecture 4: Nash Equilibria x = ( x 1 , x 2 , . . . , x m ), where x i 0, and m i =1 x i = 1. Similarly, y = ( y 1 , y 2 , . . . , y n ), where y