Continuous Envy-Free Allocation via Reciprocal Preference Matrices: An Alternative to Sperner-Based Discretization

L.A. Calvo, S. Merchán Rubira, D. de Pereda, J. Rodrigo

Summary:

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.

Spanish layman's summary:

Este trabajo propone un nuevo enfoque continuo para problemas de asignación libre de envidia con tres agentes y tres alternativas. Frente a los métodos tradicionales basados en discretizaciones y etiquetados de Sperner, las preferencias se representan mediante matrices recíprocas de comparación por pares. Esto permite formular el problema de manera exacta y continua, caracterizar matemáticamente las regiones de asignación libre de envidia y reducir el número de parámetros necesarios para representar las preferencias.

English layman's summary:

This work proposes a new continuous approach to envy-free allocation problems with three agents and three alternatives. In contrast to traditional methods based on discretization and Sperner labelings, preferences are represented through reciprocal pairwise-comparison matrices. This provides an exact continuous formulation, allows the mathematical characterization of envy-free allocation regions, and reduces the number of parameters required to represent agents’ preferences.

Keywords: fair division; envy-free allocation; reciprocal pairwise-comparison matrices; preference elicitation; Sperner lemma

Registration date: 03-nov-2026