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Existence and Computation of Fair Allocations under Constraints

Siddharth Barman, Ioannis Caragiannis, Sudarshan Shyam

TL;DR

A novel guarantee that FEF can always be achieved with Pareto-optimality is established, and it is shown that truthfulness is compatible with each of FEF and Pareto-optimality, individually.

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.

Existence and Computation of Fair Allocations under Constraints

TL;DR

A novel guarantee that FEF can always be achieved with Pareto-optimality is established, and it is shown that truthfulness is compatible with each of FEF and Pareto-optimality, individually.

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.
Paper Structure (10 sections, 10 theorems, 6 equations, 1 table, 3 algorithms)

This paper contains 10 sections, 10 theorems, 6 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Contributions
  3. Additional Related Work
  4. Notation and Preliminaries
  5. Existence of FEF Allocations
  6. Non-Convexity of FEF Allocations
  7. Existence of $\mathsf{FEF}$ and PO Allocations
  8. Are Fairness, Efficiency, and Truthfulness Compatible?
  9. Truthfulness
  10. Conclusion

Key Result

Lemma 3.3

For any fair-division instance, Algorithm algorithm:fefeps outputs a $\mathsf{FEF}$-$\varepsilon$ allocation.

Theorems & Definitions (26)

  • Definition 2.1: Feasible envy-freeness ($\mathsf{FEF}$)
  • Definition 2.2: Feasible envy-freeness up to any good ($\mathsf{FEF}_x$)
  • Definition 3.1
  • Definition 3.2: Approximate feasible envy-freeness $\mathsf{FEF}$-$\varepsilon$
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • proof
  • ...and 16 more