EF1-Constrained Nash Social Welfare with Identical Additive Valuations: Complexity, Guarantees, and Experiments

We study the allocation of indivisible goods among agents with identical additive valuations, focusing on envy-freeness up to one good (EF1) and Nash social welfare (NSW). Since every maximum-NSW allocation is EF1 under additive valuations, the associated threshold problem inherits the known strong NP-hardness of NSW maximization under identical additive valuations and is strongly NP-complete. We therefore focus on welfare guarantees satisfied by arbitrary EF1 allocations. Although every such allocation is known to achieve an e-1/e-approximation to the unrestricted optimal NSW, we identify conditions yielding stronger guarantees. Under uniform valuations, every EF1 allocation is NSW-optimal. Under an varepsilon-small-item condition, every EF1 allocation achieves an explicit approximation ratio ρn(varepsilon) satisfying ρn(varepsilon) = 1-O(varepsilon2) as varepsilon→ 0 for fixed n. We further consider the stronger sequential requirement that operatornameEF1 be maintained after every item assignment. For this setting, we introduce PriorityNet, a deep reinforcement learning framework trained with Proximal Policy Optimization (PPO) and equipped with prospective operatornameEF1 action masking, which guarantees prefix-wise operatornameEF1 by construction. Across 3,000 test instances in each of the offline full-information and random-order online regimes (n∈[2,20], m∈[5,100]), PriorityNet achieves mean normalized operatornameNSW values of 0.9911 and 0.9701, respectively. Relative to the offline Longest Processing Time (LPT) heuristic and the online least-valued-bundle rule, it attains instance-wise win-minus-loss rates of +27.10% and +17.87%. Its aggregate welfare matches the offline LPT baseline to four decimal places and modestly improves upon the online baseline, from 0.9694 to 0.9701.