Axiomatic characterizations of the family of Weighted priority values

We introduce a new family of values for TU-games with a priority structure, which both containsthe Priority value recently introduced by B´eal et al. (2022) and the Weighted Shapley values (Kalaiand Samet, 1987). Each value of this family is called a Weighted priority value and is constructedas follows. A strictly positive weight is associated with each agent and the agents are partiallyordered according to a binary relation. An agent is a priority agent with respect to a coalitionif it is maximal in this coalition with respect to the partial order. A Weighted priority valuedistributes the dividend of each coalition among the priority agents of this coalition in proportionto their weights. We provide an axiomatic characterization of the family of the Weighted Shapleyvalues without the additivity axiom. To this end, we borrow the Priority agent out axiom fromB´eal et al. (2022), which is used to axiomatize the Priority value. We also reuse, in our domain,the principle of Superweak differential marginality introduced by Casajus (2018) to axiomatizethe Positively weighted Shapley values (Shapley, 1953). We add a new axiom of Independence ofnull agent position which indicates that the position of a null agent in the partial order does notaffect the payoff of the other agents. Together with Efficiency, the above axioms characterize theWeighted Shapley values. We show that this axiomatic characterization holds on the subdomainwhere the partial order is structured by levels. This entails an alternative characterization of theWeighted Shapley values. Two alternative characterizations are obtained by replacing our principleof Superweak differential marginality by Additivity and invoking other axioms.