The alpha-constant-sum games

Given any alpha in [0,1], an alpha-constant-sum game on a finite set of players, N, is a function that assigns a real number to any coalition S (being a subset of the player set N), such that the sum of the worth of the coalition S and the worth of its complementary coalition N\S is alpha times of the worth of the grand coalition N. This class contains the constant-sum games of Khmelnitskaya (2003) (for alpha = 1) and games of threats of Kohlberg and Neyman (2018) (for alpha = 0) as special cases. An alpha-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 alpha = 1. In this paper, we will build a value theory for the class of alpha-constant-sum games, and mainly introduce the alpha-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 alpha-constant-sum games.