Axiomatic characterizations of the family of Weighted priority values
Abstract
We introduce a new family of values for TU-games with a priority structure, which both contains
the Priority value recently introduced by B´eal et al. (2022) 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
B´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 axiomatize
the Positively weighted Shapley values (Shapley, 1953). 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. 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. Two alternative characterizations are obtained by replacing our principle
of Superweak differential marginality by Additivity and invoking other axioms.