Constant-sum cooperative game ($v(S)+v(N\setminus S)=v(N)$)

A transferable utility game satisfying the displayed equality for every coalition. This forces $v(\varnothing )=0$. It does not by itself guarantee a nonempty core: core rigidity in a constant-sum cooperative game shows that nonemptiness requires the characteristic function of a coalitional game to be additive.

For a core allocation $x$, the inequalities for $S$ and its complement, together with efficiency, give $x(S)\ge v(S)$ and $x(S)=v(N)-x(N\setminus S)\le v(N)-v(N\setminus S)=v(S)$. Thus equality holds for every coalition; singletons force $x_i=v(\{i\})$ and all other values must be their sums. Conversely, if the characteristic function of a coalitional game is additive, that unique allocation satisfies every core inequality with equality. Hence a constant-sum cooperative game has either an empty core or this one-point core.