Exclusion Dilation Operator

• The exclusion dilation operator is an extension of bilateral claims rules that incorporates lower and upper thresholds to exclude initial gains and residual losses.
• It employs a piecewise construction, allocating resources in three segments: below the lower threshold, within the exclusion band via a dilation of a benchmark rule, and above the upper threshold.
• The operator preserves key properties like endowment- and claim-monotonicity while intentionally violating equal-treatment and order-preservation to capture policy-driven asymmetries.

The exclusion dilation operator is an operator-based extension of bilateral claims rules for environments with lower and upper exclusion thresholds, where each threshold determines whether an individual is excluded from initial gains or residual losses. In the formulation examined in "The exclusion dilation operator for bilateral claims problems" (Calo-Blanco, 16 Mar 2026), the operator transforms a standard two-agent rule into an extended rule by first allocating with respect to exclusion thresholds and then distributing the remaining resources through a dilation transformation of an underlying benchmark rule. The construction is explicitly asymmetric: it preserves homogeneity and monotonicity, but it is designed to violate some classical symmetry-based axioms, most notably order-preservation, in order to reflect legal and policy-induced asymmetries.

1. Extended bilateral claims framework

The underlying setting is a standard two-agent claims, or rationing, problem. Agents $1$ and $2$ have nonnegative claims $c=(c_1,c_2)\in \mathbb{R}_+^2$ on a divisible endowment $E\in \mathbb{R}_+$. An allocation is a vector $x=(x_1,x_2)$ satisfying $0\le x_i\le c_i$ for each $i$, together with the balance condition $x_1+x_2=E$. A standard allocation rule is a function

$$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$

subject to the same feasibility constraints.

The extended model adds two threshold vectors, $\ell=(\ell_1,\ell_2)\in [0,1)^2$ and $2$0, with $2$1 for each claimant. The parameter $2$2 is interpreted as the fraction of claimant $2$3's claim below which that claimant is excluded from initial gains, while $2$4 is the fraction at which claimant $2$5 is excluded from residual losses. The corresponding aggregate thresholds are

$2$6

so that $2$7. An extended problem is therefore a quadruple $2$8, and an extended rule maps such quadruples to feasible allocations (Calo-Blanco, 16 Mar 2026).

2. Piecewise construction of the operator

Fix a benchmark rule $2$9. The exclusion dilation operator $c=(c_1,c_2)\in \mathbb{R}_+^2$0 is constructed in three stages. The first stage handles endowments below the aggregate lower threshold $c=(c_1,c_2)\in \mathbb{R}_+^2$1; the third stage handles endowments above the aggregate upper threshold $c=(c_1,c_2)\in \mathbb{R}_+^2$2; the middle stage applies a dilation of $c=(c_1,c_2)\in \mathbb{R}_+^2$3 within an exclusion-adjusted resource band.

Using the abbreviations

$c=(c_1,c_2)\in \mathbb{R}_+^2$4

the exposition defines

$c=(c_1,c_2)\in \mathbb{R}_+^2$5

The middle segment is the distinctive part of the construction. When $c=(c_1,c_2)\in \mathbb{R}_+^2$6, the problem is shrunk to the exclusion-space

$c=(c_1,c_2)\in \mathbb{R}_+^2$7

and the residual endowment

$c=(c_1,c_2)\in \mathbb{R}_+^2$8

is allocated by a dilation-transformed version of the benchmark rule. The exposition states that $c=(c_1,c_2)\in \mathbb{R}_+^2$9 satisfies $E\in \mathbb{R}_+$0, with each coordinate obtained by evaluating the original rule at a scaled-down endowment and then stretching coordinate $E\in \mathbb{R}_+$1 by factor $E\in \mathbb{R}_+$2. Equivalently, one may determine one coordinate and recover the other from balance.

The verbal interpretation of the three cases is direct. If $E\in \mathbb{R}_+$3, claimant $E\in \mathbb{R}_+$4 receives $E\in \mathbb{R}_+$5. If $E\in \mathbb{R}_+$6, claimant $E\in \mathbb{R}_+$7 receives the lower-exclusion baseline $E\in \mathbb{R}_+$8 plus the dilated award $E\in \mathbb{R}_+$9. If $x=(x_1,x_2)$0, each claimant is first honored up to the upper exclusion $x=(x_1,x_2)$1, and the surplus $x=(x_1,x_2)$2 is then handed entirely to the claimant who still has room, so that the total equals $x=(x_1,x_2)$3 (Calo-Blanco, 16 Mar 2026).

For the proportional benchmark rule,

$x=(x_1,x_2)$4

the exposition gives an explicit middle-segment expression: $x=(x_1,x_2)$5 This makes the geometry of the dilation transparent: the baseline $x=(x_1,x_2)$6 fixes the lower threshold allocation, while the increment is apportioned according to the widths $x=(x_1,x_2)$7 of the exclusion band.

3. Axiomatic profile

The paper organizes the operator’s normative behavior around preservation and non-preservation of familiar claims-allocation axioms. Preservation is understood conditionally: if the benchmark rule $x=(x_1,x_2)$8 satisfies a property, then the extended rule $x=(x_1,x_2)$9 does so as well.

Endowment monotonicity is preserved because each piece of the definition is nondecreasing in $0\le x_i\le c_i$0, and the dilation of a nondecreasing rule remains nondecreasing. Claim-monotonicity is also preserved: when claimant $0\le x_i\le c_i$1's own claim rises, the thresholds and dilations move proportionally, and the path of awards does not cross downward. Homogeneity is preserved because scaling $0\le x_i\le c_i$2 by $0\le x_i\le c_i$3 scales thresholds and segments proportionally.

By contrast, equal-treatment-of-equals fails in general. If $0\le x_i\le c_i$4 but the claimants have different lower or upper exclusions, the initial and final segments may award them differently. Order-preservation also fails: a larger claim does not necessarily imply a weakly larger award or weakly larger loss. In the abstract, this is described not as an incidental defect but as an intentional violation, introduced to capture asymmetries induced by exclusion thresholds.

Midpoint and self-duality survive only under symmetric exclusions, namely when $0\le x_i\le c_i$5 for each claimant. Under that condition, the midpoint property—if $0\le x_i\le c_i$6 then $0\le x_i\le c_i$7—is preserved, and the self-duality relation

$0\le x_i\le c_i$8

is preserved as well. The exposition attributes this to the fact that, under symmetric exclusions, the three segments reflect each other.

Convexity and related comparative-shape properties are preserved only locally. Restricted endowment convexity is preserved within the exclusion band $0\le x_i\le c_i$9, but not over the entire endowment range. Progressivity or regressivity is preserved in the middle region when the exclusions respect the original claim order, that is, when $i$0. Concavity or convexity of the award path is likewise preserved only inside the exclusion-space (Calo-Blanco, 16 Mar 2026).

Axiom	Preserved by $i$1?	Condition
Equal-treatment-of-equals	No	—
Order-preservation	No	—
Endowment-monotonicity	Yes	—
Claim-monotonicity	Yes	—
Homogeneity	Yes	—
Midpoint	Yes	If $i$2
Self-duality	Yes	If $i$3
Restricted endowment convexity	Yes	On $i$4
Progressivity / Regressivity	Yes	On $i$5 if $i$6
Concavity / Convexity	Yes	On $i$7 if order-preserving

4. Breakdown of order-preservation

The failure of order-preservation is isolated in the exposition as a structural consequence of lower exclusions. The key lemma states that if $i$8 while $i$9, then for sufficiently small $x_1+x_2=E$0,

$x_1+x_2=E$1

even though claimant $x_1+x_2=E$2 has the larger claim. The proof sketch is immediate from the first stage of the piecewise definition: when the endowment is small, the split is governed only by the lower-exclusion fractions, not by the claims themselves.

The counterexample given is

$x_1+x_2=E$3

Then

$x_1+x_2=E$4

For the small endowment $x_1+x_2=E$5,

$x_1+x_2=E$6

Thus $x_1+x_2=E$7 but $x_1+x_2=E$8. In this framework, the anomaly is deliberate: the operator prioritizes exclusion structure over claim ranking at the relevant endowment levels. This suggests that the operator is not intended as a symmetric refinement of classical rationing rules, but as a mechanism for settings where claimant priority is externally specified by threshold parameters rather than inferred from claim magnitudes alone (Calo-Blanco, 16 Mar 2026).

5. Characterization by exclusion principles

The exposition states that the operator is uniquely pinned down by three principles. The first is Full exclusion: if $x_1+x_2=E$9, then the agent with the smaller lower threshold $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$0 receives zero. The text presents this as equivalent to the first line of the piecewise formula, with the clarification that if one $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$1, the formula forces the other share to zero.

The second is Null exclusion: if $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$2, then the agent with the larger upper threshold $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$3 is fully compensated, and the residual goes to the other claimant. This is stated to match the third line of the operator’s definition.

The third is Proportional exclusion-invariance. On the middle band $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$4, allocations must be a positive affine dilation of the benchmark rule $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$5. Formally, the exposition requires the existence of $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$6 with $$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$7 such that

$$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$8

where

$$R:\{(c,E): c_1,c_2\ge 0,\; 0\le E\le c_1+c_2\}\longrightarrow \mathbb{R}_+^2$$9

The text states that checking the three endowment regions shows that these axioms force the piecewise form and no alternative rule (Calo-Blanco, 16 Mar 2026).

6. Worked example and broader significance

The numerical illustration uses claims

$\ell=(\ell_1,\ell_2)\in [0,1)^2$0

so that

$\ell=(\ell_1,\ell_2)\in [0,1)^2$1

$\ell=(\ell_1,\ell_2)\in [0,1)^2$2

The benchmark rule is proportional: $\ell=(\ell_1,\ell_2)\in [0,1)^2$3

For $\ell=(\ell_1,\ell_2)\in [0,1)^2$4,

$\ell=(\ell_1,\ell_2)\in [0,1)^2$5

For $\ell=(\ell_1,\ell_2)\in [0,1)^2$6, the middle mass is

$\ell=(\ell_1,\ell_2)\in [0,1)^2$7

The dilated proportional rule gives

$\ell=(\ell_1,\ell_2)\in [0,1)^2$8

Hence

$\ell=(\ell_1,\ell_2)\in [0,1)^2$9

For $2$00, each claimant is honored up to the upper exclusion and the surplus is assigned as follows: $2$01 with $2$02. The exposition notes that, at small endowment levels, the example exhibits the same order failure emphasized earlier: the lower-exclusion structure can dominate raw claim size.

Within the scope stated in the paper, the operator provides a systematic way to construct asymmetric extensions of any standard bilateral rule whenever policy or legal constraints impose priority bands, minimum guarantees, or delayed compensation. The abstract explicitly places the contribution in contexts where symmetry is inappropriate because of legal and policy considerations, and the exposition names bankruptcy, tax credits, and divorce settlements as representative settings. Its broader significance lies in showing that one can preserve robustness properties such as monotonicity and homogeneity while formally identifying the classical axioms that must be relinquished once exclusion thresholds are treated as primitive features of the allocation problem (Calo-Blanco, 16 Mar 2026).