Existence and Computation of Fair Allocations under Constraints

2026-02-27

AI Summary

This study addresses the fair allocation of divisible goods under agents’ budget constraints and generalized assignment restrictions, introducing a charitable mechanism to handle unallocated surplus. By leveraging fixed-point theory, mechanism design, and set-based feasibility modeling, the work establishes the existence of feasible envy-free (FEF) allocations and, for the first time, demonstrates that FEF can be simultaneously achieved with Pareto optimality. It further delineates the compatibility boundaries among truthfulness, fairness, and efficiency: while all three cannot be jointly satisfied, truthfulness is compatible with either FEF or Pareto optimality individually. This research extends the theoretical foundations of FEF, reveals the non-convex structure of its solution space, and offers a novel paradigm for constrained resource allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

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

Abstract

We study fair division of divisible goods under generalized assignment constraints. Here, each good has an agent-specific value and size, and every agent has a budget constraint that limits the total size of the goods she can receive. Since it may not always be feasible to assign all goods to the agents while respecting the budget constraints, we use the construct of charity to accommodate the unassigned goods. In this constrained setting with charity, we obtain several new existential and computational results for feasible envy-freeness (FEF); this fairness notion requires that agents are envy-free, considering only budget-feasible subsets. First, we simplify and extend known existential results for FEF allocations. Then, we show that the space of FEF allocations has a non-convex structure. Next, using a fixed-point argument, we establish a novel guarantee that FEF can always be achieved with Pareto-optimality. Furthermore, we give an alternative proof of the fact that one cannot additionally obtain truthfulness in this context: There does not exist a mechanism that is simultaneously truthful, fair, and Pareto-optimal. On the positive side, we show that truthfulness is compatible with each of FEF and Pareto-optimality, individually.

Problem

• fair division
• budget constraints
• envy-freeness
• feasible allocation
• charity

Innovation

• fair division
• feasible envy-freeness
• budget constraints
• Pareto-optimality
• truthfulness