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The random arrival construction of the Shapley value, determining average contributions to a coalition, is intuitively fair. However, the notion of fairness can be made rigorous by requiring the satisfaction of a number of axioms, some of which have been hinted at in earlier discussion. A number of desirable properties for a value can be advanced.

For group rationality, the total value of the players should be the value of the grand coalition. Thus, to assign a value to each player in X we require that

Σi=1:n Φi(X, v) = v(X)

A value should not a priori favour any particular player over another. That is, should the return to any coalition featuring Player i and not Player j be the same as the return to the coalition with Player i replaced by Player j, then the values of players i and j should be equal.

Any player whose presence in a coalition does not alter its payoff should receive a value of 0.

Given games (X, v) and (X,w), then we can define the game (X, v+w) as (v+w)(S) = v(S)+w(S). Logically, we should require that Φ(X, v + w) = Φ(X, v) + Φ(X,w), i.e., that the return of playing the sum of two games is the sum of the returns of each game.

These four properties can be summarised as follows.

Definition: The Shapley axioms for games in coalitional form are
  • Efficiency: Σi=1:n Φi(X, v) = v(X)
  • Symmetry: If v(S ∪ {i}) = v(S ∪ {j}) ∀ S ∈ P(X)\{i, j} then Φi(X, v) = Φj(X, v).
  • Dummy: If v(S ∪ {i}) - v(S) = 0 ∀ S ∈ P(X) then Φi(X, v) = 0
  • Additivity: Φ(X, v + w) = Φ(X, v) + Φ(X,w) for any games (X, v) and (X,w)

Shapley's theorem asserts the existence and uniqueness of a Shapley function for any given coalitional form game satisfying the Shapley axioms. For the proof, see the print version of this document (details on the project homenode).

Part of A survey of game theory- see project homenode for details and links to the print version.

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