Cooperative Game - Mathematical Definition

Mathematical Definition

A cooperative game is given by specifying a value for every coalition. Formally, the game (coalitional game) consists of a finite set of players, called the grand coalition, and a characteristic function from the set of all possible coalitions of players to a set of payments that satisfies . The function describes how much collective payoff a set of players can gain by forming a coalition, and the game is sometimes called a value game or a profit game. The players are assumed to choose which coalitions to form, according to their estimate of the way the payment will be divided among coalition members.

Conversely, a cooperative game can also be defined with a characteristic cost function satisfying . In this setting, players must accomplish some task, and the characteristic function represents the cost of a set of players accomplishing the task together. A game of this kind is known as a cost game. Although most cooperative game theory deals with profit games, all concepts can easily be translated to the cost setting.