Cooperative game

A cooperative game is a game where groups of players (“coalitions”) may enforce cooperative behavior, hence the game is a competition between coalitions of players, rather than between individual players. An example is a coordination game, when players choose the strategies by a consensus decision-making process.

Recreational games are rarely cooperative, because they usually lack mechanisms by which coalitions may enforce coordinated behavior on the members of the coalition. Such mechanisms, however, are abundant in real life situations (e.g. Parliaments, social movements,…).

The basic elements of a cooperative game is:

• N players.
• ${\displaystyle v:2^{N}}$ possible coalitions.
• Characteristic function ${\displaystyle v:2^{N}\to \mathbb {R} }$ from the set of all possible coalitions of players to a set of payments that satisfies ${\displaystyle v(\emptyset )=0}$, (the value can be also represent a cost). It is possible to change from a cost-game to a profit-game. This games are called to be dual games.

There are important concepts related with cooperative games to know:

• The stable set of a game (also known as the von Neumann-Morgenstern solution).
• Core
• Shapley value
• Matroids

See also

Game Theory, Non-cooperative game

Papers

• Shapley, Lloyd S. (1971), Cores of convex games, International Journal of Game Theory 1 (1): 11-26
• Shapley, Lloyd S.; Shubik, M. (1966), Quasi-cores in a monetary economy with non-convex preferences, Econometrica (The Econometric Society) 34 (4): 805-827.