Past exam of the computer science course of the University of Oxford / 2018 / Computational game theory / 3 / b

Take three players and define the simple cooperative gameIt is a superadditive coalitional game: two disjoint coalitions cannot both have value one because each would need player 1. If one has value one, their union does too; if neither does, the desired inequality is immediate.

The vector $(1,0,0)$ belongs to the core of a cooperative game. More strongly, it is the unique core vector: singleton constraints give nonnegative coordinates, efficiency gives sum one, and the value-one coalitions $\{1,2\}$ and $\{1,3\}$ force $x_3=x_2=0$.

In a random player ordering, player 1 is pivotal unless it comes first, giving Shapley value $2/3$. Player 2 is pivotal only in order $1,2,3$, and player 3 only in order $1,3,2$. ThusThe coalition $\{1,2\}$ receives only $5/6<1$, so the Shapley value is outside the core. This proves that superadditivity and nonempty core do not imply Shapley stability.