Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 37 / 5 / c / Solution

False. Take three players of weight one and strict threshold one, so a coalition wins exactly when it contains at least two players. Set $S=\{1,2\}$ and $T=\{2,3\}$. ThenThe convex cooperative game inequality would require $1\ge 1+1-0=2$. Thus even this elementary majority weighted voting game is not convex.