The exclusion dilation operator for bilateral claims problems

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.