Budget-feasible Maximum Nash Social Welfare is Almost Envy-free

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.

Approximately Envy-Free Budget-Feasible Allocation

“…We also investigate the extent to which an NSW maximizing allocation approximates EF1 in the large budget setting. It has been shown that (for nonidentical valuation functions) such an allocation is 1/2-approximate EF1 [25]. We prove that when agents have identical valuations, this approximation ratio improves and approaches 1, which coincides with the result of Caragiannis et al [5] for the unconstrained setting.…”

“…Finally, we consider the large budget setting, in which the agents' budgets are much larger than the sizes of items [11,8,22,18,25]. We show that under this setting, our polynomial-time algorithm computes an allocation whose approximation ratio of EF1 is close to 1.…”

“…Later on, Biswas and Barman [3] provided a polynomial time algorithm to compute an EF1 allocation. More recently, progress was also made on the setting with non-identical valuations [9,25].…”

“…As mentioned in [25], the above questions appear to be non-trivial. In this paper, we answer both questions affirmatively.…”

“…A natural model for such applications is recently proposed by Wu et al [25]. In their model, each item j ∈ M has a size s j , and each agent i ∈ N has a budget B i > 0, which puts an upper bound on the total size of items this agent can receive.…”