Past exam of the MSc in advanced computer science course of the University of Oxford / 2021 / Computational game theory / 4 / c / ii / Solution

The core is empty. Efficiency would require $x_1+x_2+x_3=v(N)=1$. Each two-player coalition has value $1$, so core membership would also requireAdding gives $2(x_1+x_2+x_3)\ge 3$, contradicting efficiency, which makes the left side $2$. Thus no allocation can meet all coalition demands. The obstruction uses only pair coalitions and the grand coalition; imposing individual rationality cannot remove it. The equal allocation $(1/3,1/3,1/3)$ is efficient but gives each pair only $2/3<1$, so symmetry does not supply a core allocation.