Shapley value

The Shapley value is a solution concept in cooperative game theory. It was named in honour of Lloyd Shapley, who introduced it in 1953. To each cooperative game it assigns a unique distribution (among the players) of a total surplus generated by the coalition of all players. The Shapley value is characterized by a collection of desirable properties. Hart (1989) provides a survey of the subject.

The setup is as follows: a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game? Or phrased differently: how important is each player to the overall cooperation, and what payoff can he or she reasonably expect? The Shapley value provides one possible answer to this question.

Formally, a coalitional game is defined as: There is a set N (of n players) and a function ${\displaystyle v}$ that maps subsets of players to the real numbers: ${\displaystyle v\;:\;2^{N}\to \mathbb {R} }$, with ${\displaystyle v(\emptyset )=0}$, where ${\displaystyle \emptyset }$ denotes the empty set. The function ${\displaystyle v}$ is called a characteristic function.

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