Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 40 / 4 / d / Solution

The core of a cooperative game consists of efficient allocations satisfying ${\sum }_{i\in S}{x}_i\ge v(S)$ for every coalition, so no coalition can obtain a greater total payoff by leaving. Efficiency has already been checked. For singletons,For pairs,The empty constraint is automatic and the grand constraint is equality. Thus every core inequality holds andAll four assertions about this allocation are true, although their definitions impose different requirements.