On α-constant-sum games

Given any α∈ [0 , 1] , an α-constant-sum game (abbreviated as α-CS game) on a finite set of players, N, is a function that assigns a real number to any coalition S⊆ N, such that the sum of the worth of the coalition S and the worth of its complementary coalition N\ S is 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 α= 1) and games of threats of (Kohlberg and Neyman, Games Econ Behav 108:139–145, 2018) (for α= 0) as special cases. An 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 α= 1. In this paper, we (i) extend the α-quasi-Shapley value giving the Shapley value for constant-sum games and quasi-Shapley-value for threat games to any class of α-CS games, (ii) extend the axiomatizations of Khmelnitskaya (2003) and Kohlberg and Neyman (2018) to any class of 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 α-CS games.