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Every nontrivial simple cooperative game that is a convex cooperative game is a unanimity game. If two coalitions win, the supermodular inequality forces their intersection to win as well. Intersecting the finite nonempty family of winning coalitions gives a winning set $K\ne \varnothing$. Every winning coalition contains $K$; conversely monotonicity makes every superset of $K$ win. Thus the game is $u_K$. Conversely, every unanimity game is a supermodular set function, as a direct check of the winning and losing cases shows.

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.

A simple cooperative game is a transferable utility game $(N,v)$ withA coalition of value $1$ is a winning coalition; a value-$0$ coalition loses. The displayed definition uses the usual monotone, nontrivial convention: the empty coalition loses and the grand coalition wins.

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.

There is exactly one such nontrivial game for every $n\ge 1$: the unanimity game requiring every player. PutIt is a simple cooperative game, and permuting players preserves every coalition value. Thus it has the required symmetry of players in a cooperative game. It is also a convex cooperative game: if both $S,T$ are winning, all four terms in the supermodular inequality have value $1$; if exactly one is winning, their union is $N$ and their intersection loses, giving equality $1=1$; if neither wins, the right-hand side is zero and the inequality follows from nonnegativity. For $n=1$ the same definition gives the sole player value $1$.

To prove uniqueness, first establish that convex simple games are unanimity games. If $S,T$ are winning coalitions, the supermodular inequality givesso their intersection is also winning. The family of winning coalitions is finite and nonempty. Its intersection $K$ is therefore winning and nonempty, since the empty coalition loses. Every winning coalition contains $K$, and monotonicity makes every superset of $K$ winning. Hence $v=u_K$.

If $K$ were a proper nonempty subset of $N$, take $i\in K$ and $j\notin K$, and let $T=N\setminus \{i,j\}$. Then $T\cup \{i\}=N\setminus \{j\}$ contains $K$ and wins, whereas $T\cup \{j\}=N\setminus \{i\}$ loses. This violates symmetry of players in a cooperative game, which requires those two coalition values to agree. Thus $K=N$, proving the claimed count. The “nontrivial” convention excludes the identically zero game; if that degenerate game were also called simple, it would supply one additional symmetric convex example.

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.

A coalition of value $1$ in a simple cooperative game. Monotonicity makes every superset of a winning coalition winning. An inclusion-minimal winning coalition has no winning proper subset.