Multiagent Systems

of. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 402/731
5. Nets and coalitions 4. Payoff Division
Theorem 5.31 (Shapley-Value)
There is only one payoff division satisfying the above three
axioms. It is called the Shapley value of agent i and is
defined by
x i = 1
|A|!
∑
(|A| − |S| − 1)!|S|!(v S∪{i} − v S )
S⊆A
(|A| − S)! is the number of all possible joining orders of the
agents (to form a coalition).
(v S∪{i} − v S ) is i’s marginal contribution when added to set S.
There are |S|! ways for S to be built before i’s joining. There are
(|A| − |S| − 1)! ways for the remaining agents to form S (after i).
The Shapley value sums up the marginal contributions of agent i
averaged over all joining orders.
An expected gain can be computed by taking a random joining
order and computing the Shapley value.