Past exam of the computer science course of the University of Oxford / 2020 / Computational game theory / 3 / c / Solution

No representation exists, even allowing arbitrary signed weights and self-loops. Every singleton coalition loses, so each loop weight must be ${\ell }_i=v(\{i\})=0$. Every two-player coalition wins, which then forcesThe represented grand-coalition value would consequently becontradicting its required value $1$. The obstruction is the additive edge representation, not a restriction that weights be nonnegative: the singleton and pair values already determine every relevant weight.