Shapley Value

Definitions

Let N={1,2, …, n} be a set of players. A cooperative game v on N is expressed in characteristic function form if a function v: 2N → R (the characteristic function) is given, which assigns a value v(S) to each coalition S ⫅ N where v(Ø) = 0. A game is superadditive if v(S ∪ T) ≥ v (S) + v (T) for all disjoint coalitions S, T of N.

Axioms

The Shapley value ɸi for player i is based on four axioms. (In his original presentation, Shapley merged the first and third axioms into a single carrier axiom.)

• 1.
  Efficiency: all resources available to the whole coalition N are distributed among the players (i.e., Ʃ ɸi(v)=v(N)).
• 2.
  Symmetry: if players i and j have symmetrical roles in v, then φi = φj. (For example, if we transform the game v into the game v' where the only change is that player 1 assumes the label “2” and player 2 assumes the label “1,” then φ1(v) = φ2(v') and φ2(v) = φ1(v').)
• 3.
  Dummy player: if a player i does not bring any contribution to any coalition S ⫅ N (i.e., player i is a dummy player), then φi(v)= 0.
• 4.
  Additivity: if a game is the “sum” of two games, v' and v, defined on the same player set N, the win for each player i in the cumulative game is the sum of the winnings achieved by him or her in the two separate games:

φi(v' + v”)= φi(v)+ φi(v”) for all players i ɛ N.

Formula

It can be demonstrated that a unique function exists that respects the four axioms listed above. This function, the Shapley value, is:

for all i ɛ N, [Page 601]where s is the number of members in the coalition S.

The formula assigns each player i the sum of the contributions v(S) - v(S\{i}) made by him to all possible coalitions S ⫅ N, each multiplied by the coefficient , where n! = 1, 2, …, <>

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