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

Give each permanent member weight seven, each nonpermanent member weight one, and use the strict threshold $t=38$. If a coalition omits a permanent member, its weight is at most $4\cdot 7+10=38$, so it loses. If it includes all five permanent members and $r$ others, its weight is $35+r$, which exceeds 38 exactly when $r\ge 4$. These are precisely the required winning coalitions. Thus a weighted voting game representation isThe strict-threshold convention is important: in the alternative convention requiring weight at least the quota, the quota is 39.

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 $\alpha$.

False. Take three players of weight one and strict threshold one, so a coalition wins exactly when it contains at least two players. Set $S=\{1,2\}$ and $T=\{2,3\}$. ThenThe convex cooperative game inequality would require $1\ge 1+1-0=2$. Thus even this elementary majority weighted voting game is not convex.

True, with the usual normalization $v(\varnothing )=0$. For a convex cooperative game, the supermodular inequality implies increasing marginal contributions: if $A\subseteq B$ and $i\notin B$, apply it to $A\cup \{i\}$ and $B$ to obtainFix an ordering $\pi$ and let $P_i$ be the set of players before $i$. Its marginal contribution vector is $m_i^{\pi }=v(P_i\cup \{i\})-v(P_i)$. Summing in order telescopes to ${\sum }_i{m}_i^{\pi }=v(N)$. For any coalition $S$, $S\cap P_i\subseteq P_i$, so increasing marginals giveThese are exactly the efficiency and coalition constraints of the core of a cooperative game. Thus every marginal contribution vector is in the core. The core is a convex set, being an intersection of linear half-spaces and an efficiency hyperplane. The Shapley value is the average of the marginal contribution vectors over all orderings, so it too lies in the core. This proves Shapley value belongs to the core of a convex game, without needing a separate existence theorem for the core.