# Cooperative Game Theory

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## Cooperative Game Theory

a branch of the theory of games in which games are considered without due regard for the strategic opportunities of the players (thus, cooperative game theory studies a certain class of models of general games). In particular, it includes the investigation of nonstrategic (cooperative) games, from the very start devoid of a strategic aspect. In a cooperative game the opportunities and preferences of different groups of players (coalitions) are given and from these are derived situations that are optimal (stable, fair) for the players, including the distribution of the total payoffs among them; the principles of optimality themselves are established and their realizability in various classes of games demonstrated and concrete realizations are found. Many economic and sociological phenomena lend themselves to a description in terms of cooperative games.

Simplest of all is the description of the classical cooperative game, which consists in indicating (1) a set of players J, (2) a family Rn of subsets of J (coalitions of interest), and (3) a function ν defined on Rn and assuming real values. [ν(K) may be understood (sometimes with certain stipulations) as the sum that the coalition K can distribute among its own members.] Usually, but not always, the function v is considered to be superadditive: ν (K ∪ L) ≥ ν (K) + ν (L) for K ∩ L = ø This reflects the additional opportunities that groups acquire through union. A characteristic of the classical cooperative theory of games is the possibility of unlimited transfers of payoffs from some players to others without altering the utility (value) of the payoffs in the process. A more general type of game is one without side payments, where certain limitations are imposed on such transfers.

Let J = {1, . . . , n} ; a vector x = (x1, . . . , xn) for which

and Xi v( {i}) for all i ∊ J is called an imputation. An imputation x is said to dominate over an imputation y = (y1 . . . , yn) if there exists a coalition K (favoring it) such that

and Xi > yi for all iK. The optimal behavior of the partici-pants in a cooperative game may tend toward a set of imputations that do not dominate over other imputations (c-core), or toward a set of imputations that do not dominate over each other but that together dominate over all the remaining imputations (the Neumann-Morgenstern solution), or toward a set of imputations in which the “dissatisfaction” of the coalition is minimized in a certain sense (n-core), and so forth. Some of these principles of optimality cannot always be achieved; the realization of others is sometimes not unique. Finding realizations is often difficult. Thus, the mathematical problem of establishing the optimal behavior in cooperative games is very complex both in principle and in practice.

### REFERENCES

Neumann, J. von, and O. Morgenstern. Teoriia igr i ekonomicheskoe povedenie. Moscow, 1970. (Translated from English.)
Vorob’ev, N. N. “Sovremennoe sostoianie teorii igr.” Uspekhi matematicheskikh nauk, 1970, vol. 25, no. 2.
Owen, G. Teoriia igr. Moscow, 1971. (Translated from English.)
Rosenmiiller, J. Kooperative Spiele und Märkte. Berlin-Heidelberg-New York, 1971.

N. N. VOROB’EV

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