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This article is about a part of a game theory. For video gaming , see Cooperative gameplay.

A cooperative game is a game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. It is like 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 behaviour on the members of the coalition. Such mechanisms, however, are abundant in real life situations, such as contract law.

Mathematical treatment

A cooperative game is given by specifying a value for every coalition. Mathematically speaking, the game is a function $\nu \; : \; \mathcal{P}(N) \; \to \mathbb{R}$ from the set of coalitions to a set of payments (the characteristic function). The function describes how much collective payoff a set of players can gain by forming a coalition. 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. It is assumed that the empty coalition gains nil.

Properties for characterization

Superadditivity

Characteristic functions are often assumed to be superadditive.(Owen 1995, p. 213) This means that the joint value of disjoint coalitions is no less than the sum of their values:

\nu (A \cup B) \; \ge \; \nu (A) \; + \; \nu (B)

A\cap B = \emptyset

Monotonicity

Larger coalitions gain more:

A \subseteq B \Rightarrow \nu (A) \le \nu (B)

. This follows from superadditivity if payoffs are normalized so singleton coalitions have value zero.

Simple games

A simple game is a special kind of cooperative game, where the payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".

A simple game is called proper, if the complement (opposition) of any winning coalition is losing. It is called strong, if $\nu (A) \; = \; 1 \; - \; \nu (N \setminus A)$ ; that is, a coalition is winning if and only if its complement is losing.
A veto player in a simple game is a player who is included in all winning coalitions. That is, all coalitions not containing the veto player are losing.

Relation with non-cooperative theory

Let G be a strategic (non-cooperative) game. Then, assuming that coalitions have the ability to enforce coordinated behaviour, there are several cooperative games associated with G. These games are often referred to as representations of G.

The α-effective game associates with each coalition the sum of gains its members can \'guarantee\' by joining forces. By \'guaranteeing\', it is meant that the value is the max-min, e.g. the maximal value of the minimum taken over the opposition\'s strategies.
The β-effective game associates with each coalition the sum of gains its members can \'strategically guarantee\' by joining forces. By \'strategically guaranteeing\', it is meant that the value is the min-max, e.g. the minimal value of the maximum taken over the opposition\'s strategies.

Solution concepts for cooperative theory

A cooperative game describes payoffs given for coalitions. Players will only join a coalition if they expect to gain from it. So, in order to find what coalitions will actually be created, one needs to estimate both the relative power of different coalitions, as well as the strength of the different players within each coalition.

The stable set

The stable set of a game (also known as the von Neumann-Morgenstern solution (von Neumann & Morgenstern 1944)) is the first solution proposed for games with more than 2 players. A stable set satisfies two properties:

Internal stability: No alternative in the stable set is dominated by another alternative in the set.
External stability: All alternatives outside the set are dominated by at least one alternative in the set.

Von Neumann and Morgenstern saw the stable set as the collection of acceptable behaviours in a society: None is clearly preferred to any other, but for each unacceptable behaviour there is a preferred alternative. The definition is very general allowing the concept to be used in a wide variety of game formats.

Properties

A stable set may or may not exist (Lucas 1969), and if it exists it is typically not unique (Lucas 1992). Stable sets are usually difficult to find. This and other difficulties have led to the development of many other solution concepts.

A positive fraction of cooperative games have unique stable sets consisting of the core (Owen 1995, p. 240.).

A positive fraction of cooperative games have stable sets which discriminate $n-2$ players. In such sets at least $n-3$ of the discriminated players are excluded (Owen 1995, p. 240.).

The core

Main article: Core (economics)

The core of a game is a set of vectors allocating payoffs to players, which preserve the following conditions:

Efficiency: it is assumed that the players form the grand coalition (a coalition containing all players), and so the sum of individual payoffs should equal the value of the grand coalition.
Strategic stability or balance: no coalition can earn more by defecting from the grand coalition. E.g. no coalition has a value greater than the sum of its members\' payoffs.

Note that the core of a game may be empty (see Bondareva-Shapley theorem).

Shapley\'s value

Main article: Shapley value

The kernel

Is a vector allocating payoffs to players which is:

Efficient
Personally reasonable

References

Lucas, William F. (1969). "The Proof That a Game May Not Have a Solution". Transactions of the American Mathematical Society 136: 219-229.
Lucas, William F. (1992), "Von Neumann-Morgenstern Stable Sets", Handbook of Game Theory, Volume I., Amsterdam: Elsevier, pp. 543-590
Luce, R.D. and Raiffa, H. (1957) Games and Decisions: An Introduction and Critical Survey, Wiley & Sons. (see Chapter 8).
Osborne, M.J. and Rubinstein, A. (1994) A Course in Game Theory, MIT Press (see Chapters 13,14,15)
Owen, Guillermo (1995), Game Theory (3rd ed.), San Diego: Academic Press, ISBN 0-12-531151-6
von Neumann, John & Morgenstern, Oskar (1944), Theory of Games and Economic Behavior, Princeton: Princeton University Press

External links

v • d • e Topics in game theory
Definitions	Normal form game · Extensive form game · Cooperative game · Information set · Preference
Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayesian-Nash · Perfect Bayesian · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy · Risk dominance · Pareto efficiency
Strategies	Dominant strategies · Pure strategy · Mixed strategy · Tit for tat · Grim trigger · Collusion
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design · Stochastic game · Nontransitive game
Games	Prisoner\'s dilemma · Traveler\'s dilemma · Coordination game · Chicken · Volunteer\'s dilemma · Dollar auction · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game · Public goods game · Nash bargaining game · Blotto games · War of attrition
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow\'s theorem

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