Partial-implementation invariance and claims problems

In the context of claims problems, we formulate an invariance axiom of a rule with respect to its “partial implementation”: having applied the rule to a problem and distributed their awards to some of the claimants, we consider the problem obtained by setting the claims of these claimants equal to zero and decreasing the endowment by the sum of their awards; we require that in this problem the rule assigns to each remaining claimant the same amount as it did initially. We formulate several variants of this requirement of “partial-implementation invariance” and a “converse” of it. We investigate how it relates to known axioms and ask whether it is preserved under certain “operators” that have been defined on the space of rules. Our main result is a fixed-population characterization of a family of rules introduced and characterized by Young (1987) in a variable-population framework, known as the “parametric rules”.