Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 37 / 5 / b / Solution

For a simple cooperative game, the Shapley value is the probability that a player is pivotal in a uniformly random ordering. Symmetry gives one value for permanent members and another for nonpermanent members.

A particular nonpermanent member is pivotal exactly when all five permanent members and exactly three of the other nine nonpermanent members precede them. The predecessor set then has size eight. There are $\left(\genfrac{}{}{0ex}{}{9}{3}\right)$ such sets, each giving $8!6!$ orderings. Their Shapley value is thereforeEvery ordering has exactly one pivotal member, since the empty coalition loses and the full coalition wins. This proves efficiency directly: $5\alpha +10\beta =1$, where $\alpha$ is the value of each permanent member. HenceThe vector has five entries $421/2145$ and ten entries $4/2145$. As a check, a permanent member is pivotal when they are last among the permanent members and occupy a position from nine to fifteen; counting those orderings gives the same $\alpha$.