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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.

Yes; without self-loops, such a nontrivial simple cooperative game has exactly one positive edge, of weight $1$. For each pair,because singleton and loop contributions are zero. Summing all edge weights gives $v(N)=1$, so precisely one unordered pair $\{a,b\}$ has weight $1$, and all other pairs have weight zero.

Thus the winning coalitions are exactly those containing both $a$ and $b$. For any $0\le \lambda \le 1$, setThis payoff vector is efficient, since ${\sum }_i{x}_i=v(N)=1$. Every winning coalition receives $1$ and every losing coalition receives a nonnegative total, so all inequalities $x(S)\ge v(S)$ hold. Hence it belongs to the core of a cooperative game.

In fact these are all core allocations. Singleton conditions require $x_i\ge 0$, and the winning pair requires $x_a+x_b\ge 1$; efficiency then forces equality and zero payoffs outside the pair. The endpoints are both veto players, consistent with the general characterization of the core of a simple cooperative game.

No. Two positive self-loops and one negative edge give a simple cooperative game with an empty core of a cooperative game. Give players $1,2$ loop weights ${\ell }_1={\ell }_2=1$, put $w_{12}=-1$, and give every other loop and edge weight zero. For a coalition $S$,Thus a coalition wins exactly when it contains player $1$ or player $2$. This is monotone, has values in $\{0,1\}$, and satisfies $v(\varnothing )=0,v(N)=1$. Extra players for any $n\ge 3$ do not alter any coalition value.

The core singleton inequalities would require $x_1\ge 1$, $x_2\ge 1$ and $x_i\ge 0$ for every other player, while efficiency requires ${\sum }_i{x}_i=1$. These requirements are inconsistent. Equivalently, the two disjoint winning singleton coalitions leave no veto player. Signed weights are essential to this particular representation and are permitted in the induced subgraph game model.

For a fixed nonempty coalition $K$, the simple cooperative game in which a coalition wins exactly when it contains $K$. Members of $K$ are exactly the veto players. It is a convex cooperative game: intersections of winning coalitions still contain $K$, and checking the remaining zero-one cases establishes the supermodular inequality. Full symmetry of players in a cooperative game forces $K=N$.

A player belonging to every winning coalition. Giving all of the grand-coalition value to a veto player satisfies every core constraint in a nontrivial simple cooperative game. If there are no veto players, its core of a simple cooperative game is empty.