The exclusion dilation operator for bilateral claims problems

TL;DR

This paper introduces the exclusion dilation operator for bilateral claims problems, enabling resource allocation with exclusion thresholds that reflect legal and policy asymmetries, while analyzing its properties.

Contribution

It proposes a novel operator that extends standard allocation rules to incorporate exclusion thresholds and characterizes its axiomatic properties.

Findings

Maintains homogeneity and monotonicity.

Violates order preservation to reflect asymmetries.

Provides a formal method for resource allocation with exclusions.

Abstract

This paper examines bilateral claims problems with lower and upper exclusion thresholds that determine whether an individual is excluded from initial gains or losses. We introduce the exclusion dilation operator, a method that transforms standard rules into extended rules incorporating exclusion thresholds. The operator first allocates gains and losses with respect to these thresholds and then distributes the remaining resources through a dilation transformation of an underlying standard rule. We axiomatically characterize this operator and examine which standard properties of the theory of fair allocation it preserves. While the operator maintains key properties such as homogeneity and monotonicity, it intentionally violates others, most notably order preservation, to reflect the asymmetries induced by exclusion thresholds. Our approach provides a formal methodology for resource allocation in contexts where symmetry is not appropriate due to legal and policy considerations.