Past exam of the computer science course of the University of Oxford / 2019 / Computational game theory / 3 / b

Use the inclusive quota: a coalition wins at weight at least $25$. For each player, count losing predecessor coalitions that become winning when that player joins. Group these swing counts by predecessor cardinality $t=0,1,2,3$:

For example, player one is pivotal for predecessor coalitions $\{2\}$, $\{2,3\}$, $\{2,4\}$, $\{3,4\}$ and $\{2,3,4\}$. Player three is pivotal only for $\{1,4\}$; player four has the symmetric counterpart.

Each swing contributes $1/8$ to the probabilistic Banzhaf index. For the Shapley value, a swing with $t$ predecessors contributes $t!(3-t)!/4!$, which is $1/4,1/12,1/12,1/4$ for the four possible sizes. Applying these weights to the table givesThe Shapley values sum to one. The unnormalized Banzhaf indices sum to $5/4$; they must not be replaced by normalized shares in this part.