Correlated Strategies and Correlated Equilibrium

Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to [email protected]. A more thorough version would be available soon in this space. We commence our study of cooperative game theory with a discussion of games and solution concepts which enable us to capture cooperation among players. In this chapter, we first study games with contracts followed by games with communication. We introduce the key concept of correlated strategies and correlated equilibrium. The motivation to look at games with cooperation is the fact that in many games (for example, the Prisoner's Dilemma), Nash equilibria yield non-optimal payoffs compared to certain non-equilibrium outcomes. Let us consider the following modified version of the prisoner's dilemma problem whose payoff matrix is shown in Figure 1. 2 1 x 2 y 2 x 1 2,2 0,6 y 1 6,0 1,1 Figure 1: Payoff matrix of a modified version of prisoner's dilemma game Note in the above that the unique equilibrium (which happens to be a strictly dominant strategy equilibrium) is (y 1 , y 2) which yields a payoff profile (1,1). The non-equilibrium outcome (x 1 , x 2) yields higher payoffs (2,2). In situations like these, the players may like to transform the game to extend the set of equilibria to include better outcomes. There could be several ways of achieving this transformation: • The players communicate among themselves to coordinate their moves • The players formulate contractual agreements 1