Editing Shapley value (section)

=== Glove game ===
The glove game is a coalitional game where the players have left- and right-hand gloves and the goal is to form pairs. Let
:<math>N = \{1, 2, 3\},</math>
where players 1 and 2 have right-hand gloves and player 3 has a left-hand glove.

The value function for this coalitional game is
:<math>
v(S) = 
\begin{cases}
  1  & \text{if }S \in \left\{ \{1,3\},\{2,3\},\{1,2,3\} \right\};\\
  0 & \text{otherwise}.\\
\end{cases}
</math>

The formula for calculating the Shapley value is

:<math>\varphi_i(v)= \frac{1}{|N|!}\sum_R\left [ v(P_i^R \cup \left \{ i \right \}) - v(P_i^R) \right ],</math>

where {{mvar|R}} is an ordering of the players and <math>P_i^R</math> is the set of players in {{mvar|N}} which precede {{mvar|i}} in the order {{mvar|R}}.

The following table displays the marginal contributions of Player 1.

:<math>
\begin{array}{|c|r|}
\text{Order }R\,\! & MC_1
\\
\hline
{1,2,3}
&v(\{1\}) - v(\varnothing) = 0 - 0 = 0
\\
{1,3,2}
&v(\{1\}) - v(\varnothing) = 0 - 0 = 0
\\
{2,1,3}
&v(\{1,2\}) - v(\{2\}) = 0 - 0 = 0
\\
{2,3,1}
&v(\{1,2,3\}) - v(\{2,3\}) = 1 - 1 = 0
\\
{3,1,2}
&v(\{1,3\}) - v(\{3\}) = 1 - 0 =1
\\
{3,2,1}
&v(\{1,3,2\}) - v(\{3,2\}) = 1 - 1 = 0
\end{array}
</math>

Observe
:<math>\varphi_1(v)= \!\left(\frac{1}{6}\right)(1)=\frac{1}{6}.</math>

By a symmetry argument it can be shown that
:<math>\varphi_2(v)=\varphi_1(v)=\frac{1}{6}.</math>

Due to the efficiency axiom, the sum of all the Shapley values is equal to 1, which means that

:<math>\varphi_3(v) = \frac{4}{6} = \frac{2}{3}.</math>