Games of threats

A game of threats on a finite set of players, N, is a function d that assigns a real number to any coalition, S⊆N, such that d(S)=−d(N∖S). A game of threats is not necessarily a coalitional game as it may fail to satisfy the condition d(∅)=0. We show that analogs of the classic Shapley axioms for coalitional games determine a unique value for games of threats. This value assigns to each player an average of d(S) across all the coalitions that include the player. Games of threats arise naturally in value theory for strategic games, and may have applications in other branches of game theory.