The alpha-constant-sum games

Given any α ∈ [0, 1], an α-constant-sum 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 of the worth of the grand coalition. This class contains the constant-sum games of Khmelnitskaya (2003, [3]) (for α = 1) and games of threats of Kohlberg and Neyman (2018, [4]) (for α = 0) as special cases. An α-constant-sum 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 will build a value theory for the class of α-constant-sum games, and mainly introduce the α-quasi-Shapley value. We characterize this value by classical axiomatizations for TU games. We show that axiomatizations of the equal division value do not work on these classes of α-constant-sum games.