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 S ⊆ N , such that d(S)=−d(N∖S) 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 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) 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.