The Shapley value without efficiency and additivity

• We characterize the Shapley value without efficiency and additivity.
• Besides the dummy player property, we employ another two axioms.
• Namely, the gain-loss property and fairness/differential marginality.
• Our characterization does not work within the class of simple games.

We provide a new characterization of the Shapley value neither using the efficiency axiom nor the additivity axiom. In this characterization, efficiency is replaced by the gain-loss axiom (Einy and Haimanko, 2011), i.e., whenever the total worth generated does not change, a player can only gain at the expense of another one. Additivity and the equal treatment axiom are substituted by fairness (van den Brink, 2001) or differential marginality (Casajus, 2011), where the latter requires equal productivity differentials of two players to translate into equal payoff differentials. The third axiom of our characterization is the standard dummy player axiom.