Cooperative game theory

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The cooperative game theory is a branch of the mathematical game theory , in which, in contrast to the non-cooperative game theory, the players have no actions or strategies with which they strive for advantageous conditions. In contrast, players in cooperative game theory receive payouts that are based on two pillars. On the one hand, the payouts depend on the coalition functions, and on the other hand, on the solution concept used.

Players and coalitions

The players in cooperative game theory are often summarized in a (finite) set N and the players themselves are numbered from 1 to n. Subsets of players are also called coalitions, with N being the grand coalition.

Coalition functions

Coalition functions (often also called characteristic functions) are used to describe the economic, political or social possibilities that are open to all coalitions. A distinction is made between coalition functions with and coalition functions without transferable benefits; accordingly, a distinction is made between games with and without side payments.

Coalition functions with transferable benefits

In the case of transferable benefits, each coalition is assigned a real number by the coalition function, which is called the worth. In the simplest case, the transferable benefit is a cash payment. For example, in the glove game, there are left-gloved players and right-gloved players. The respective sets L and R are disjoint and their union gives N. It is assumed that only pairs of gloves have a value (of 1 monetary unit ). The value of a coalition K (the functional value of the coalition function at K) is equal to the number of pairs of gloves that the players can form from K, and thus the number of monetary units that they can generate with them.

Coalition functions without transferable benefits

In the case of non-transferable benefits, each coalition is assigned a set of payout vectors by the coalition function. One example is the exchange economy. By exchanging bundles of goods, players can realize different utility vectors. Non-transferable benefits are e.g. This also happens, for example, when a coalition gains or loses intangible goods such as fame, honor, health, freedom, etc. through its cooperation.

Cooperative game theory as an axiomatic theory of coalition functions

The cooperative game theory is the axiomatic theory of coalition functions. The coalition functions should describe the economic, political or social possibilities open to the coalitions. There are a variety of solution concepts. A solution concept assigns payouts for the players to each coalition function. The assignment can be made using a formula (an algorithm) or by specifying general principles of division (axioms).

Solutions for cooperative games

Solution concepts have been developed for cooperative games , including the Nash negotiation solution , Kalai-Smorodinski solution , the Shapley value , the Gauthier solution , the Kalai-Rosenthal solution , the imputation set , the nucleolus or the mean-voter Solution .

The Zeuthen-Harsanyi model can thus be viewed as a non-cooperative implementation of the cooperative Nash solution .

As important representatives of cooperative game theory, Robert Aumann received the Nobel Prize in Economics in 2005 and Lloyd S. Shapley in 2012 .

criticism

The negative attitude that is often expressed towards cooperative game theory can be briefly summarized as follows: Cooperative game theory is not non-cooperative game theory. In fact, actions, goals, knowledge of the actions of the other players do not appear in the basic concepts of cooperative game theory. The plus point of cooperative game theory is that it can make statements about payouts even if it is not entirely clear which actions are available to the players in which order and what they know about previous actions.

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