Cross invariance, the Shapley value, and the Shapley–Shubik power index
提出一个简单公理“交叉不变性”，结合可加性和哑元公理，刻画了TU博弈上的沙普利值（简单博弈上的沙普利-舒比克权力指数），并展示了可加性可被强单调性替代或完全去掉，效率是推导而非假设的。
Abstract In this paper we propose a simple axiom which, along with the axioms of additivity (transfer) and dummy player, characterizes the Shapley value (the Shapley–Shubik power index) on the domain of TU (simple) games. The new axiom, cross invariance, demands payoff invariance on symmetric players across “quasi-symmetric games,” that is, games where excluding null players, all players are symmetric. Additionally, we demonstrate that the axiom of additivity can be replaced by a new axiom called strong monotonicity, or it can be completely dropped if a stronger version of cross invariance is employed. We also show that the weighted Shapley values can be characterized using a weighted variant of cross invariance. Efficiency is derived rather than assumed in our characterizations. This fresh perspective contributes to a deeper understanding of the Shapley value and its applicability.