The Shapley value

Abstract

Chapter summary This chapter presents the Shapley value, which is one of the two most important single-valued solution concepts for coalitional games. It assigns to every coalitional game an imputation, which represents the payoff that each player can expect to obtain from participating in the game. The Shapley value is defined by an axiomatic approach: it is the unique solution concept that satisfies the efficiency, symmetry, null player, and additivity properties. An explicit formula is provided for the Shapley value of a coalitional game, as a linear function of the worths of the various coalitions. A second characterization, due to Peyton Young, involves a marginality property that replaces the additivity and null player properties. The Shapley value of a convex game turns out to be an element of the core of the game, which implies in particular that the core of a convex game is nonempty. Similar to the core, the Shapley value is consistent: it satisfies a reduced game property, with respect to the Hart–Mas-Colell definition of the reduced game. When applied to simple games, the Shapley value is known as the Shapley–Shubik power index and it is widely used in political science as a measure of the power distribution in committees. This chapter studies the Shapley value , a single-valued solution concept for coalitional games first introduced in Shapley [1953]. Shapley's original goal was to answer the question “How much would a player be willing to pay for participating in a game?”