= Core of a simple cooperative game
{title2=$\operatorname{Core}(v)=\{x\ge0:\sum_i x_i=1,\ x_i=0\text{ for non-veto players}\}$}
For a monotone <simple cooperative game> with empty value $0$ and grand-<coalition> value $1$, the <core of a cooperative game> consists exactly of the nonnegative unit-total <payoff> <vector>s supported on <veto players>.
For the proof, singleton <core> conditions give $x_i\ge0$. Every <winning coalition> must receive at least $1$, the full available total, so all players outside it receive zero. A positively paid player therefore belongs to every <winning coalition> and is a <veto player>. Conversely, a unit-total nonnegative allocation supported on <veto players> gives every <winning coalition> exactly $1$ and every losing <coalition> at least $0$. These are precisely the <core> inequalities. The <core> is thus nonempty if and only if a <veto player> exists.
Back to article page
