Last Updated on 28. April 2023 by Martin Schuster

The Shapley value is a solution concept in cooperative game theory. With the value, one can determine what pay-off a participant in a coalition should receive. It tries to take the individual contribution of each participant into account.

Let’s consider the following case. There are three participants who work on a project.

First, we need to find out all possible coalitions with their value.

Coalition	Value of the coalition
v({A})	12
v({B})	6
v({C})	9
v({A, B})	24
v({A, C})	27
v({B, C})	15
v({A, B, C})	36
v({o})	0

Algorithm

1. First, we make a list of all orders of participants.
2. For every order, we determine the value of the coalition consisting of the participants that come before participant i. This value is V_-i.
3. For every order, we determine the value of the coalition consisting of the participants, including i. This value is V_i+.
4. Calculate marginal contribution M_i = W_+I – W_-i
5. Calculate average over all M_i.

Contribution of player B

Order	v({participant before B})	v({participants including B})	Marginal contribution participant B
A, B, C	v({A}) = 12	v({A, B}) = 24	24-12 = 12
A, C, B	v({A, C}) = 27	v({A, C, B}) = 36	36-27 = 9
B, A, C	v({o}) = 0	v({B}) = 6	6-0 = 6
B, C, A	v({o}) = 0	v({B}) = 6	6-0 = 6
C, A, B	v({A, C}) = 27	v({A, C, B}) = 36	36-27= 9
C, B, A	v({C}) = 15	v({C, B}) = 15	15-9 = 6

Shapley Value player B: S_B = (12+9+6+6+9+6)/6 = 8

We do the same for players A and C.

S_A = 17

S_C = 11