A characterization of the family of Weighted priority values

We introduce a new family of values for TU-games with a priority structure. This family both contains the Priority value recently introduced by Béal et al. (2021) and the Weighted Shapley values (Kalai and Samet, 1987). Each value of this family is called a Weighted priority value and is constructed as follows. A strictly positive weight is associated with each agent and the agents are partially ordered according to a binary relation. An agent is a priority agent with respect to a coalition if it is maximal in this coalition with respect to the partial order. A Weighted priority value distributes the dividend of each coalition among the priority agents of this coalition in proportion to their weights. We provide an axiomatic characterization of the family of the Weighted Shapley values without the additivity axiom. To this end, we borrow the Priority agent out axiom from Béal et al. (2021), which is used to axiomatize the Priority value. We also reuse, in our domain, the principle of Super weak differential marginality introduced by Casajus (2018) to axiomatize the Positively weighted Shapley values (Shapley, 1953a). We add a new axiom of Independence of null agent position which indicates that the position of a null agent in the partial order does not affect the payoff of the other agents. Together with Efficiency, the above axioms characterize the Weighted Shapley values. Finally, we show that this axiomatic characterization holds on the subdomain where the partial order is structured by levels. This entails an alternative characterization of the Weighted Shapley values.