The Controversy of Linearity in Shapley's Axioms

The Shapley value is a unique function which obeys three axioms, which are anonymous, pareto optimal, and linear. This is a solution concept for cooperative games with transferable utility. But when the set of games which is considered is the set of simple games, the condition of linearity has no sense. In this paper, we introduce the concept of monotonicity to answer the controversy in linearity.