On $$\alpha $$ α -constant-sum games

Given any $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , an $$\alpha $$ α -constant-sum game (abbreviated as $$\alpha $$ α -CS game) on a finite set of players, N, is a function that assigns a real number to any coalition $$S\subseteq N$$ S ⊆ N , such that the sum of the worth of the coalition S and the worth of its complementary coalition $$N\backslash S$$ N \ S is $$\alpha $$ α times the worth of the grand coalition. This class contains the constant-sum games of Khmelnitskaya (Int J Game Theory 32:223–227, 2003) (for $$\alpha = 1$$ α = 1 ) and games of threats of (Kohlberg and Neyman, Games Econ Behav 108:139–145, 2018) (for $$\alpha = 0$$ α = 0 ) as special cases. An $$\alpha $$ α -CS game may not be a classical TU cooperative game as it may fail to satisfy the condition that the worth of the empty set is 0, except when $$\alpha =1$$ α = 1 . In this paper, we (i) extend the $$\alpha $$ α -quasi-Shapley value giving the Shapley value for constant-sum games and quasi-Shapley-value for threat games to any class of $$\alpha $$ α -CS games, (ii) extend the axiomatizations of Khmelnitskaya (2003) and Kohlberg and Neyman (2018) to any class of $$\alpha $$ α -CS games, and (iii) introduce a new efficiency axiom which, together with other classical axioms, characterizes a solution that is defined by exactly the Shapley value formula for any class of $$\alpha $$ α -CS games.