Editing Shapley value (section)

=== Business example ===
Consider a simplified description of a business. An owner, ''o'', provides crucial capital in the sense that, without him/her, no gains can be obtained. There are ''m'' workers ''w''<sub>1</sub>,...,''w''<sub>''m''</sub>, each of whom contributes an amount ''p'' to the total profit. Let

:<math>N = \{o, w_1,\ldots,w_m\}.</math>

The value function for this coalitional game is
:<math>
v(S) = 
\begin{cases}
  (|S|-1)p  & \text{if }o \in S\;,\\
  0 & \text{otherwise}\;.\\
\end{cases}
</math>
Computing the Shapley value for this coalition game leads to a value of {{sfrac|''mp''|2}} for the owner and {{sfrac|''p''|2}} for each one of the ''m'' workers.

This can be understood from the perspective of synergy. The synergy function <math>w</math> is
:<math>
w(S) = 
\begin{cases}
  p,  & \text{if } S = \{ o, w_i \} \\
  0, & \text{otherwise}\\
\end{cases}
</math>

so the only coalitions that generate synergy are one-to-one between the owner and any individual worker.

Using the above formula for the Shapley value in terms of <math>w</math> we compute

: <math>\varphi_{w_i} = \frac{w(\{o, w_i \})}{2} = \frac{p }{2} </math>
and
: <math>\varphi_o = \sum_{i=1}^m \frac{w(\{o, w_i \})}{2} = \frac{m p }{2} </math>

The result can also be understood from the perspective of averaging over all orders. A given worker joins the coalition after the owner (and therefore contributes ''p'') in half of the orders and thus makes an average contribution of <math>\frac p2</math> upon joining. When the owner joins, on average half the workers have already joined, so the owner's average contribution upon joining is <math>\frac{mp}2</math>.