Game theory | Fair division | Cooperative games
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. 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. For cost-sharing games with concave cost functions, the optimal cost-sharing rule that optimizes the price of anarchy, followed by the price of stability, is precisely the Shapley value cost-sharing rule. (A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.) In mechanism design, this means that the Shapley value solution concept is optimal for these sets of games. (Wikipedia).
