CMSC 858 F : Algorithmic Game Theory Fall 2010

In game theory as we have studied it so far this semester, games are assumed to be interactions between completely independent and selfish parties. For example, Nash equilibria were assumed to be stable, since no single player could alter their strategy and benefit. However, there could, for example, be a situation where there are two Nash equilibria. In the current equilibrium state, no player benefits from changing their strategy, but two players together could be enough to switch the equilibrium to a different, more beneficial equilibrium. This mean that the first equilibrium is not in fact stable in the case of cooperating adversaries. When cooperating groups are fixed, we deal with this by simply considering cooperating players to be a single player that controls all members of the cooperating group. However, this does not deal with questions of how players choose to cooperate or not with each other. We study this group formation through cooperative games. A cooperative game has two main components. The first is a set of agents, N = {1, 2, . . . , n}. The second is a value function. We use 2 to denote the power set P(N). The value function maps subsets of N to non-negative real numbers, V : 2 → R ∪ {0}. Intuitively, one should think of the value function as representing the payout that a subset of players can achieve as a result of cooperating. The players in this game are making no choices other than who to cooperate with. The outcome of the game is a subset of cooperating players, along with a payment vector ~x = (x1, . . . , xn) that specifies what portion of the payout is given to each player in the game.