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 $ρ_n(\varepsilon)$ satisfying $ρ_n(\varepsilon) = 1-O(\varepsilon^2)$ as $\varepsilon\to 0$ for fixed $n$. We further consider the stronger sequential requirement that $\operatorname{EF1}$ be maintained after every item assignment. For this setting, we introduce \emph{PriorityNet}, a deep reinforcement learning framework trained with Proximal Policy Optimization (PPO) and equipped with prospective $\operatorname{EF1}$ action masking, which guarantees prefix-wise $\operatorname{EF1}$ by construction. Across 3,000 test instances in each of the offline full-information and random-order online regimes ($n\in[2,20]$, $m\in[5,100]$), PriorityNet achieves mean normalized $\operatorname{NSW}$ 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 $0.9701$.