In a claims problem, multiple agents claim a share of a resource, and the total claims may exceed or fall short of the available supply. Standard allocation rules (proportional, constrained equal awards, Talmudic) satisfy various fairness axioms — order preservation, monotonicity, homogeneity. But what if some agents should be excluded from initial gains or losses based on thresholds?
The paper introduces an exclusion dilation operator that transforms any standard allocation rule into one respecting exclusion thresholds. Below a lower threshold, an agent doesn't participate in gains; above an upper threshold, an agent doesn't participate in losses. Between the thresholds, the agent receives an allocation determined by dilating the standard rule.
The operator works in two steps: first handle the gains and losses at the thresholds, then distribute the remainder using a dilation transformation of the base rule. The dilation rescales the problem to the interval between the thresholds, applies the base rule, and maps back.
The axiomatic analysis reveals what the operator preserves and what it breaks. Homogeneity survives: doubling the resource doubles the allocation. Monotonicity survives: more resource means at least as much for everyone. But order preservation breaks — and this is deliberate. Agents with different exclusion thresholds should be treated differently even if their claims are identical. The asymmetry in the thresholds creates asymmetry in the allocation that equal treatment would ignore.
Fairness as transformation. The standard rules encode one notion of fairness; the operator transforms it into another that respects structural differences between agents.