Properties
The Shapley value has the following desirable properties:
1. Efficiency: The total gain is distributed:
2. Symmetry: If i and j are two actors who are equivalent in the sense that
for every subset S of N which contains neither i nor j, then φi(v) = φj(v).
3. Additivity: if we combine two coalition games described by gain functions v and w, then the distributed gains should correspond to the gains derived from v and the gains derived from w:
for every i in N.
4. Zero Player (Null player): The Shapley value of a null player i in a game v is zero. A player is null in if for all coalitions .
In fact, given a player set N, the Shapley value is the only map from the set of all games to payoff vectors that satisfies all four properties 1, 2, 3, and 4 from above.
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