Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization
Abstract
Fair allocation problems on a continuous simplex are often approached through Sperner-type discretizations: agents label mesh vertices according to their preferences and a fully labelled elementary triangle gives a discrete localization of a fair outcome. We develop a continuous alternative for three agents and three alternatives by representing preferences with multiplicatively consistent reciprocal pairwise-comparison matrices. Two independent ratios per agent induce exact linear indifference boundaries and turn envy-free allocation into a finite-dimensional feasibility problem. For the six possible assignments, we introduce an assignment score Φ built from the agents' consistent priority vectors and prove that an incentive solution component is nonempty if and only if its assignment is the unique minimizer of Φ, whereas a disincentive solution component is nonempty if and only if its assignment is the unique maximizer. Consequently, the strict solution region is either empty or a unique convex preference region whose closure lies in the interior of the simplex. The weak solution set is never empty and degenerates exactly when Φ has ties. We also compare the exact continuous formulation with Sperner-based discretization, including the boundary convention required in the disincentive model. Under exact consistency, the model uses six independent pairwise thresholds, whereas a regular triangular mesh with resolution m contains (m+1)(m+2)/2 vertices. The method therefore exchanges a stronger parametric preference assumption for an exact continuous solution set and a resolution-independent elicitation stage.
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BibTeX
@misc{pascual2026continuous,
title = {{Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization}},
author = {Luis Ángel Calvo Pascual and Susana Merchán Rubira and Diego de Pereda Sebastián and Javier Rodrigo Hitos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38880},
eprint = {2610.38880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38880},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization
AU - Luis Ángel Calvo Pascual
AU - Susana Merchán Rubira
AU - Diego de Pereda Sebastián
AU - Javier Rodrigo Hitos
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38880
UR - https://arxiv.org/abs/2610.38880
AB - Fair allocation problems on a continuous simplex are often approached through Sperner-type discretizations: agents label mesh vertices according to their preferences and a fully labelled elementary triangle gives a discrete localization of a fair outcome. We develop a continuous alternative for three agents and three alternatives by representing preferences with multiplicatively consistent reciprocal pairwise-comparison matrices. Two independent ratios per agent induce exact linear indifference boundaries and turn envy-free allocation into a finite-dimensional feasibility problem. For the six possible assignments, we introduce an assignment score Φ built from the agents' consistent priority vectors and prove that an incentive solution component is nonempty if and only if its assignment is the unique minimizer of Φ, whereas a disincentive solution component is nonempty if and only if its assignment is the unique maximizer. Consequently, the strict solution region is either empty or a unique convex preference region whose closure lies in the interior of the simplex. The weak solution set is never empty and degenerates exactly when Φ has ties. We also compare the exact continuous formulation with Sperner-based discretization, including the boundary convention required in the disincentive model. Under exact consistency, the model uses six independent pairwise thresholds, whereas a regular triangular mesh with resolution m contains (m+1)(m+2)/2 vertices. The method therefore exchanges a stronger parametric preference assumption for an exact continuous solution set and a resolution-independent elicitation stage.
ER -