Cooperative game
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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.
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 - from law to legal contract.
Mathematical treatment
A cooperative game is given by specifying a value for every (nonempty) coalition. Mathematically speaking, the game is a function
[ \nu \; : \; 2^N \; \to \Re ]
from the set of coalitions to a set of payments. 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.
Useful axioms of cooperative game theory
- Super Additivity
- If A and B are any two coalitions then their joint value is no less than the sum of their values: [ \nu (A \cup B) \; \ge \; \nu (A) \; + \; \nu (B) ].
- Monotonicity
- Larger coalitions gain more: [ A \subseteq B \Rightarrow \nu (A) \le \nu (B) ].
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 [ \nu (A) \; = \; 1 \; - \; \nu (N \setminus A) ]. That is, a coalition is winning if and only if its complement (opposition) is 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 'guarranteeing', 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 guarranteeing', 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 core
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.
Shapley's value
The Kernel
Is a vector allocating payoffs to players which is:- Efficient
- Personally reasonable
Some examples of recreational cooperative games
One example is "Stand Up", where a number of individuals sit down, link arms (all facing away from each other) and attempt to stand up. This objective becomes more difficult as the number of players increases.
Another is the counting game, where the players, as a group, attempt to count to 20 with no two participants saying the same number twice. In a cooperative version of volleyball, the emphasis is on keeping the ball in the air for as long as possible.
Role-playing games are the most common form of recreational cooperative game, where coalition forming (and sometimes coalition disintegration) are an intrinsic part of the game. In such games, the players, who act through persons called "characters", usually strive toward intertwined goals. However, each character has his or her own ambitions, and ultimately, individual goals. Hence conflict between groups of characters often occurs in these games.
Many multiplayer card games and board games exhibit cooperative behaviour, usually where a group of weak players join forces to halt the progress of a leading player. An interesting example is Diplomacy, a board game in which the forming and breaking of coalitions is the main strategic aspect of play.
References
- Osborne, M.J. and Rubinstein, A. (1994) A Course in Game Theory, MIT Press (see Chapters 13,14,15)
- Luce, R.D. and Raiffa, H. (1957) Games and Decisions: An Introduction and Critical Survey, Wiley & Sons. (see Chapter 8).
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