[Paper Review] Budget-feasible Maximum Nash Social Welfare Allocation is Almost Envy-free

Xiaowei Wu, Bo Li+1 more|arXiv (Cornell University)|Dec 7, 2020

Game Theory and Voting Systems22 references4 citations

TL;DR

This paper studies the fairness properties of maximum Nash social welfare (NSW) allocations under budget constraints, where each item has a cost and agents have individual budgets. It proves that a Max-NSW allocation is always a 1/4-approximation of envy-freeness up to one good (EF1), and this ratio improves to 1/2 as the budget-to-cost ratio increases, with tightness shown via constructed examples.

ABSTRACT

The Nash social welfare (NSW) is a well-known social welfare measurement that balances individual utilities and the overall efficiency. In the context of fair allocation of indivisible goods, it has been shown by Caragiannis et al. (EC 2016 and TEAC 2019) that an allocation maximizing the NSW is envy-free up to one good (EF1). In this paper, we are interested in the fairness of the NSW in a budget-feasible allocation problem, in which each item has a cost that will be incurred to the agent it is allocated to, and each agent has a budget constraint on the total cost of items she receives. We show that a budget-feasible allocation that maximizes the NSW achieves a 1/4-approximation of EF1 and the approximation ratio is tight. The approximation ratio improves gracefully when the items have small costs compared with the agents' budgets; it converges to 1/2 when the budget-cost ratio approaches infinity.

Motivation & Objective

• To analyze the fairness of maximum Nash social welfare (NSW) allocations in budget-feasible settings where agents have limited budgets for item costs.
• To determine whether Max-NSW allocations maintain fairness guarantees such as envy-freeness up to one good (EF1) under budget constraints.
• To quantify the approximation ratio of Max-NSW allocations to EF1 fairness, and to study how this ratio depends on the budget-cost ratio.
• To establish tight bounds on the fairness approximation, showing that 1/4 is the best possible ratio in the worst case.
• To explore how fairness improves as the budget-cost ratio increases, and to derive a smooth convergence to 1/2-EF1 in the large-budget limit.

Proposed method

• Formalizing a budget-feasible allocation problem where each item has a cost and each agent has a budget constraint on total cost of allocated items.
• Defining a new fairness notion, EF1, adapted to budget constraints: agent i envies agent j only if no subset of j’s bundle (within i’s budget) gives i strictly higher value even after removing the most valuable item.
• Proving that a Max-NSW allocation is always 1/4-EF1 using a contradiction-based argument, constructing a hypothetical allocation that improves NSW to derive a contradiction.
• Constructing a tight example with budget-cost ratio 1 to show that 1/4 is the best possible approximation ratio, demonstrating the bound is tight.
• Analyzing the large-budget regime by deriving a smooth approximation ratio of (1/2 - 5/κ^{1/4})-EF1, where κ is the minimum budget-to-cost ratio.
• Using a fractional item partitioning technique to balance cost and value across subsets, enabling the construction of intermediate allocations to compare NSW values.

Experimental results

Research questions

• RQ1 Does a Max-NSW allocation remain EF1 when budget constraints are introduced?
• RQ2 What is the best possible approximation ratio of Max-NSW allocations to EF1 fairness in the budget-feasible setting?
• RQ3 How does the fairness approximation ratio change as the budget-to-cost ratio increases?
• RQ4 Can the approximation ratio be improved beyond 1/4 in any regime, and if so, under what conditions?
• RQ5 Is there a smooth convergence to 1/2-EF1 as the budget-cost ratio tends to infinity?

Key findings

• A Max-NSW allocation is always 1/4-EF1, meaning no agent envies another by more than a factor equivalent to one item’s value in the worst case.
• The 1/4-approximation ratio is tight, as demonstrated by a constructed instance where Max-NSW achieves exactly 1/4-EF1.
• In the large-budget regime, where the budget-cost ratio κ approaches infinity, the approximation ratio converges to 1/2.
• The fairness approximation improves gracefully with increasing κ, following the bound (1/2 - 5/κ^{1/4})-EF1.
• The result shows that Max-NSW remains a strong fairness approximation even under budget constraints, with performance improving as budgets grow relative to item costs.
• The paper establishes that a 1/4-EF1 allocation always exists and is Pareto optimal in this setting, though exact EF1 allocations may not always exist.