A note on the per capita Shapley support levels value

Abstract

The per capita Shapley support levels value extends the Shapley value to cooperative games with a level structure. This value prevents symmetrical groups of players of different sizes from being treated equally. We use efficiency, additivity, the null player property, and two new properties to give an axiomatic characterization. The first property, called joint productivity, is a fairness property within components and makes the difference to the Shapley levels value. If all players of two components are only jointly productive, they should receive the same payoff. Our second axiom, called neutral collusions, is a fairness axiom for players outside a component. Regardless of how players of a component organize their power, as long as the power of the coalitions that include all players of the component remains the same, the payoff to players outside the component does not change.