Existence and Computation of Fair Allocations under Constraints

TL;DR

This paper investigates fair division of divisible goods with agent-specific constraints and charity, providing new existence, computational, and structural results for feasible envy-freeness, including Pareto-optimality and truthfulness limitations.

Contribution

It extends existential results for feasible envy-freeness under constraints, analyzes the structure of FEF allocations, and establishes Pareto-optimality guarantees using fixed-point methods.

Findings

FEF allocations exist under generalized constraints.

The space of FEF allocations is non-convex.

FEF can be achieved with Pareto-optimality using fixed-point arguments.

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.