In cooperative game theory, the Shapley value is a method (solution concept) for fairly distributing the total gains or costs among a group of players who have collaborated. For example, in a team project where each member contributed differently, the Shapley value provides a way to determine how much credit or blame each member deserves. 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.

The Shapley value determines each player's contribution by considering how much the overall outcome changes when they join each possible combination of other players, and then averaging those changes. In essence, it calculates each player's average marginal contribution across all possible coalitions. It is the only solution that satisfies four fundamental properties: efficiency, symmetry, additivity, and the dummy player (or null player) property, which are widely accepted as defining a fair distribution.

This method is used in many fields, from dividing profits in business partnerships to understanding feature importance in machine learning.

Suppose we have a situation where players can win certain rewards by cooperating (forming a coalition) to accomplish a task; such situations are often called coalitional games. For a coalition (set of players) ${\displaystyle S}$, we define the payoff or value function ${\displaystyle v(S)}$ as the total sum of payoffs that the members of ${\displaystyle S}$ can obtain by cooperating.

The Shapley value is one way to divide up the value created by a coalition between its members. It is a "fair" distribution in the sense that it is the only distribution with certain desirable properties (listed below). According to the Shapley value, the amount that player ${\displaystyle i}$ is given in a coalitional game ${\displaystyle (v,N)}$ is

where ${\displaystyle n}$ is the total number of players and the sum extends over all subsets ${\displaystyle S}$ of ${\displaystyle N}$ not containing player ${\displaystyle i}$, including the empty set. Also note that ${\displaystyle {n \choose k}}$ is the binomial coefficient. The formula can be interpreted as follows: imagine the coalition is formed one actor at a time, with each actor demanding their contribution ${\displaystyle v(S\cup \{i\})-v(S)}$ as a fair compensation, and then for each actor take the average of this contribution over the possible different permutations in which the coalition can be formed.

An alternative, equivalent formula for the Shapley value is:

where the sum ranges over all ${\displaystyle n!}$ orders ${\displaystyle R}$ of the players and ${\displaystyle P_{i}^{R}}$ is the set of players in ${\displaystyle N}$ which precede ${\displaystyle i}$ in the order ${\displaystyle R}$.