Shapley value: a standalone construct

Lloyd Shapley introduced this solution concept in 1953, in Contributions to the theory of games II, to answer how the total payoff a cooperating group can secure should be divided among its members. The Shapley value gives each player the average of the marginal contribution they add when they join a coalition, taken over every possible order in which the players could form the full group, so a member who tends to raise the group's payoff more receives more. Shapley proved that it is the only division of the total that satisfies four properties at once: efficiency (the whole payoff is handed out), symmetry (members who contribute identically get equal shares), the null-player property (a member who adds nothing to any coalition gets nothing), and additivity (the value of two games combined is the sum of their separate values). Later work sharpened this foundation, with Young (1985) deriving the same value from a single marginality axiom and Winter (2002) surveying its axiomatic and applied development in the Handbook of Game Theory. As a landmark, it is the canonical measure of fair allocation in cooperative settings, and its use reaches well beyond economics into cost sharing, the measurement of voting power, and, more recently, attributing a machine learning model's prediction to its individual input features.[1][2][3]

Unit

A payoff vector assigning one real-valued share to each player in a cooperative game

Recovery means

Not applicable

Measured by

Computed as each player's marginal contribution averaged over all orderings of the players, and approximated by Monte Carlo sampling of orderings when the number of players is large