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