[Paper Review] Approximately Envy-Free Budget-Feasible Allocation

Jiarui Gan, Bo Li+1 more|arXiv (Cornell University)|Jun 28, 2021

Game Theory and Voting Systems24 references4 citations

TL;DR

This paper presents a polynomial-time algorithm that computes a 1/2-approximate envy-free up to one item (EF1) allocation for agents with identical additive valuations under budget constraints. It introduces virtual budgets and feasible configurations to enable efficient, envy-free-like allocations, achieving a significantly improved approximation ratio over prior work.

ABSTRACT

In the budget-feasible allocation problem, a set of items with varied sizes and values are to be allocated to a group of agents. Each agent has a budget constraint on the total size of items she can receive. The goal is to compute a feasible allocation that is envy-free (EF), in which the agents do not envy each other for the items they receive, nor do they envy a charity, who is endowed with all the unallocated items. Since EF allocations barely exist even without budget constraints, we are interested in the relaxed notion of envy-freeness up to one item (EF1). The computation of both exact and approximate EF1 allocations remains largely open, despite a recent effort by Wu et al. (IJCAI 2021) in showing that any budget-feasible allocation that maximizes the Nash Social Welfare (NSW) is 1/4-approximate EF1. In this paper, we move one step forward by showing that for agents with identical additive valuations, a 1/2-approximate EF1 allocation can be computed in polynomial time. For the uniform-budget and two-agent cases, we propose efficient algorithms for computing an exact EF1 allocation. We also consider the large budget setting, i.e., when the item sizes are infinitesimal compared with the agents' budgets, and show that both the NSW maximizing allocation and the allocation our polynomial-time algorithm computes have an approximation close to 1 regarding EF1.

Motivation & Objective

• To address the lack of efficient algorithms for computing approximately envy-free allocations under budget constraints.
• To improve upon the 1/4-approximate EF1 ratio achieved by Nash Social Welfare-maximizing allocations.
• To design a polynomial-time algorithm that guarantees a better approximation ratio for identical-valuation agents.
• To explore exact EF1 computation in special cases such as two agents or uniform budgets.
• To investigate the performance of NSW-maximizing and proposed algorithms in the large-budget limit.

Proposed method

• Introduces the concept of a feasible configuration, representing a collection of item bundles that respect agents' budget constraints.
• Employs virtual budgets that are gradually increased and kept below actual agent budgets to allow bundle exchanges across agents with different real budgets.
• Uses a greedy assignment strategy that prioritizes items with the highest value-to-size ratio (density) to maintain efficiency.
• Applies a knapsack-style selection process with size constraints, removing low-density items to ensure budget feasibility.
• Relies on a potential function argument based on Nash Social Welfare to prove approximation guarantees.
• Uses contradiction-based proofs involving item subsets and value comparisons to establish the 1/2-approximation ratio.

Experimental results

Research questions

• RQ1 Can a better-than-1/4 approximation ratio for EF1 be achieved in budget-feasible allocation with identical additive valuations?
• RQ2 Is there a polynomial-time algorithm that computes a 1/2-approximate EF1 allocation under budget constraints?
• RQ3 Can exact EF1 allocations be computed efficiently in special cases such as two agents or uniform budgets?
• RQ4 How does the performance of NSW-maximizing and proposed algorithms behave in the large-budget limit?
• RQ5 What are the implications of introducing virtual budgets and feasible configurations in fair allocation under budget constraints?

Key findings

• For agents with identical additive valuations, a 1/2-approximate EF1 allocation can be computed in polynomial time.
• In the two-agent and uniform-budget cases, exact EF1 allocations can be computed efficiently.
• In the large-budget regime, both the NSW-maximizing allocation and the proposed algorithm achieve an approximation ratio approaching 1 for EF1.
• The proposed algorithm outperforms the prior 1/4-approximation guarantee of NSW-maximizing allocations.
• The use of virtual budgets and feasible configurations enables effective bundle exchange even when agents have different real budgets.
• Theoretical analysis confirms that the algorithm's approximation ratio is tight under the given assumptions.