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

2025-07-24

AI Summary

This paper studies fair allocation of indivisible goods under personalized bivalued utilities—where each agent assigns to each item one of two positive values, possibly distinct across agents. Using combinatorial optimization and computational complexity analysis, we fully characterize the structure of all Pareto-optimal (PO) allocations. We prove that PO verification is polynomial-time solvable when value ratios are integers, but coNP-complete in general. Moreover, we establish for the first time that EFX (envy-freeness up to any good) allocations always exist under this model and can be constructed in polynomial time. These results collectively advance the state of the art on the decidability and constructibility of both PO and EFX allocations under heterogeneous preferences, providing fundamental structural insights into fair division theory and delivering efficient algorithmic guarantees.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 $a>b>cgeq0$), 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).

Problem

• Characterize Pareto-optimal allocations with bi-valued utilities
• Determine complexity of Pareto-optimality for fractional value ratios
• Examine existence and computation of EFX allocations

Innovation

• Characterizes Pareto-optimal allocations with bi-valued utilities
• Proves coNP-completeness for fractional value ratios
• Ensures EFX existence and polynomial-time computation