On Pareto-Optimal and Fair Allocations with Personalized Bi-Valued Utilities

TL;DR

This work investigates fair division of $m$ indivisible goods among $n$ agents under personalized bi-valued utilities, focusing on Pareto-optimality and EFX fairness. It introduces a structural approach based on the minimum Pareto-improvement graph to characterize PO when the per-agent value ratio $r_i=a_i/b_i$ is an integer, enabling a polynomial-time PO decision and dominance results; for fractional $r_i$, PO testing becomes coNP-complete. The authors also prove that EFX allocations always exist in this setting and provide a polynomial-time algorithm (Match&Modify&Freeze) to compute one, extending prior results to personalized utilities. They experimentally and theoretically delineate the tractability frontier by contrasting integer versus fractional ratios and pose open questions about the coexistence of EFX and Pareto-optimality. The results advance the understanding of fair and efficient allocations with heterogeneous, personalized valuations and offer practical algorithms for EFX in this broader utility class.

Abstract

We study the fair division problem of allocating $m$ indivisible goods to $n$ agents with additive personalized bi-valued utilities. Specifically, each agent $i$ assigns one of two positive values $a_i > b_i > 0$ to each good, indicating that agent $i$'s valuation of any good is either $a_i$ or $b_i$. For convenience, we denote the value ratio of agent $i$ as $r_i = a_i / b_i$. We give a characterization to all the Pareto-optimal allocations. Our characterization implies a polynomial-time algorithm to decide if a given allocation is Pareto-optimal in the case each $r_i$ is an integer. For the general case (where $r_i$ may be fractional), we show that this decision problem is coNP-complete. Our result complements the existing results: this decision problem is coNP-complete for tri-valued utilities (where each agent's value for each good belongs to $\{a,b,c\}$ for some prescribed $a>b>c\geq0$), and this decision problem belongs to P for bi-valued utilities (where $r_i$ in our model is the same for each agent). We further show that an EFX allocation always exists and can be computed in polynomial time under the personalized bi-valued utilities setting, which extends the previous result on bi-valued utilities. We propose the open problem of whether an EFX and Pareto-optimal allocation always exists (and can be computed in polynomial time).

Theorems & Definitions (34)

• Lemma 1: Equivalence under Ratio-Based Utilities
• proof
• Definition 1: Item-Exchange Graph
• Definition 2: Minimum Pareto-improvement Graph
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof