Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 35 / 5 / Solution

A coalitional game with transferable utility has a finite player set $N$ and a characteristic function of a coalitional game $v:2^N\to \mathbb{R}$, normalized by $v(\varnothing )=0$. The value $v(S)$ is the total payoff the coalition $S$ can secure and redistribute among its members.

An imputation is an efficient, individually rational payoff vector:The core of a cooperative game consists of imputations satisfying every coalition constraint:A coalition assigned less than its characteristic value can leave and make every member better off, by sharing the positive surplus. Conversely a coalition with no such surplus cannot improve every member's payoff simultaneously. This explains the stability represented by the core of a cooperative game.