Core (game theory, $\{x:{\sum }_{i\in N}{x}_i=v(N),{\sum }_{i\in S}{x}_i\ge v(S)\forall S\}$)

The core consists of efficient payoff vectors for which each coalition receives at least its own attainable value. Thus no coalition can improve every member's payoff by leaving. It is a convex set defined by linear constraints, but can be empty. Every convex cooperative game has a nonempty core because each marginal contribution vector belongs to it.

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 vectors supported on veto players.

For the proof, singleton core conditions give $x_i\ge 0$. 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.

For $r\ge 2$, the core is zero when $n<r$, the nonnegative unit simplex when $n=r$, the single allocation $x_i=1/r$ when $n\ge 2r$ is divisible by $r$, and empty for all other $n\ge r$. To prove the obstruction, every $r$-person coalition requires payoff at least one. Summing these constraints counts each player $\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ times and gives $r\lfloor n/r\rfloor /n\ge 1$, impossible when $r\nmid n$. For a divisible $n$, all such coalition sums must equal one. Comparing coalitions with the same $r-1$ members makes all individual payoffs equal when $n>r$. The uniform allocation satisfies every coalition constraint because $|S|/r\ge \lfloor |S|/r\rfloor$. At $n=r$ all proper coalition values are zero, giving the whole simplex instead.