Cooperative game

The cooperative game theory is a branch of mathematical game theory, in which, in contrast to the non-cooperative game theory, the players no actions or strategies are available, with which they seek advantageous conditions. Players in the cooperative game theory, however, receive payments, which are based on two pillars. For one, the payoffs depend on the coalition functions, on the other hand by the solution concept applied.

Players and coalitions

The players in the cooperative game theory are often summarized in a set N and numbered by the players themselves from 1 to n. Subsets of the players is also called coalitions, where N is called the grand coalition.

Coalition functions

Coalition functions ( often called characteristic functions ) are used to describe the economic, political and social opportunities that are open to all coalitions. A distinction coalition functions with and without transferable utility coalition functions.

Coalition functions with transferierbarem benefits

In transferierbarem benefits of each coalition is assigned by the coalition function is a real number, the value that you (English: worth ) is called. For example, it is in the glove game player with left-hand gloves, and those with the right gloves. The respective amounts of L and R are disjoint and their union results in N. It is assumed that only pairs of gloves, a value ( of 1) have. The value of a coalition K ( the function value of the coalition function in K) is equal to the number of pairs of gloves, which can form the players from K.

Coalition functions without transferable utility

In nichttransferierbarem benefits of each coalition is assigned by the coalition function a lot of payoffs. An example is the exchange economy. Players can be realized by the exchange of goods bundling different utility vectors.

Cooperative game theory as an axiomatic theory of coalition functions

The cooperative game theory is the axiomatic theory of coalition functions. The coalition functions are intended to describe the economic, political and social opportunities that are open to the coalitions. There are a number of solutions. A solution concept assigns to each coalition function payouts to players. The assignment can be done by a formula ( an algorithm ), or an indication of the general layout principles ( axioms ).

Solutions for cooperative games

For cooperative games has its own solution concepts developed, including Imputationsmenge, nucleolus, Nash bargaining solution, Kalai - Smorodinski solution, the Shapley value or the mean- voter solution.

Criticism

The cooperative game theory often have placed negative attitude can be summed up briefly as follows: Cooperative game theory is not non-cooperative game theory. In fact, actions, goals, knowledge come over the actions of the other players do not occur in the basic concepts of cooperative game theory. As a plus point can post the cooperative game theory, that they also may make statements about payments when not quite clear in what order are open to what actions the players and what they know about previous actions.