Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 31 / 6 / b

The core of a cooperative game is the set of efficient allocations for which every coalition receives at least its own value: $x(S)\ge v(S)$ for every $S\subseteq N$. The singleton constraints give individual rationality, and the remaining proper-coalition inequalities areEfficiency turns the second into $x_2\le 2$; together with $x_2\ge 2$ this forces $x_2=2$. Put $x_1=t$ and $x_3=10-t$. The remaining constraints give $t\ge 1$, $t\le 6$, and $t\le 7$ from $x_3\ge 3$. The first pair constraint adds only $t\ge 1$. HenceEach point in this segment satisfies all singleton, pair, and grand-coalition constraints, so the description is sufficient as well as necessary. Its endpoints are $(1,2,9)$ and $(6,2,4)$.