Pseudo-Shapley value for weak games of threats

For a real number $$\omega $$ ω , a weak game of threats (N, v) consists of a set N of n players and a function $$v:2^N\rightarrow \mathbb {R}$$ v : 2 N → R such that $$\omega v(\emptyset )+(1-\omega )v(N)=0$$ ω v ( ∅ ) + ( 1 - ω ) v ( N ) = 0 , where $$v(\emptyset )\ne 0$$ v ( ∅ ) ≠ 0 possibly. It is shown that there exists a unique value with respect to $$\omega $$ ω for weak games of threats that satisfies efficiency, linearity, symmetry and the null player property.