The setup is as follows: a coalition of actors cooperates, and obtains a certain overall gain from that cooperation. Since some actors may contribute more to the coalition than others, the question arises how to fairly distribute the gains among the actors. Or phrased differently: how important is each actor to the overall operation, and what payoff can they reasonably expect?
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2 Example 3 Properties |
To formalize this situation, we use the notion of a coalition game:
we start out with a set N (of actors) and a (gain-) function v : 2N → R with the properties
Formal definition
whenever S and T are disjoint subsets of N.
The interpretation of the function v is as follows: if S is a coalition of actors which agree to cooperate, then v(S) describes the total expected gain from this cooperation, independent of what the actors outside of S do. The additivity condition expresses the fact that collaboration can only help but never hurt.
The Shapley value is one way to distribute the total gains to the actors, assuming that they all collaborate. It is a "fair" distribution in the sense that it is the only distribution with certain desirable properties to be listed below. The amount that actor i gets if the gain function v is being used is