Econ 618: Topic 11 Introduction to Coalitional Games

Consider a game with a finite set of players. A coalition is a nonempty subset of the set of players. A coalitional game with transferable payoff consists of (1) a finite set of players, N (2) a function v : S ⊆ N −→ R. The value of the function for a given S, v(S) is known as the worth of the coalition S. v(S) is the total payoff that is available for division among the members of the coalition S. The coalition can choose to divide v(S) in any way amongst its members. The set of joint actions that the coalition can take consists of all possible divisions of v(S) amongst the members of S. Thus, an action by a coalition is an allocation of v(S) amongst its members. In many situations, the payoff that a coalition can achieve depends on the actions taken by players outside the coalition. We assume this is not the case here. The phrase transferable payoff captures the notion that every action of S generates a distribution of payoffs amongst the members of S that has the same sum or in other words every allocation has the same total value. A coalitional game in which this is not the case, has non-transferable payoff. The following two examples illustrate the difference.