Simple cooperative game ($v(S)\in \{0,1\}$)

A simple cooperative game is a monotone transferable utility game in which each coalition either loses, with value zero, or wins, with value one. In the usual nontrivial convention the empty coalition loses and the grand coalition wins. The Shapley value is then the probability that a player changes a losing coalition into a winning one when players enter in uniformly random order.

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.