When One Good Is Not Enough: EF1 and Pareto Optimality Are Not Compatible for Submodular Valuations

Abstract

One of the central questions in discrete fair division is whether fairness and efficiency can be achieved simultaneously. For indivisible goods, a canonical relaxation of envy-freeness is envy-freeness up to one good (EF1), while the standard efficiency benchmark is Pareto optimality (PO). In their seminal work, Caragiannis et al. showed that, for additive valuations, EF1 and PO are always compatible, and asked whether this compatibility extends to submodular valuations. This question has since become an important open problem in the study of fair division. In this paper, we settle the question in the negative. We construct an instance with two agents and eight goods, where both agents have submodular valuations, such that no EF1 allocation is even weakly Pareto optimal. Thus, the celebrated compatibility between EF1 and PO for additive valuations breaks down already for two agents under submodular valuations. We then map the boundary of this impossibility. On the negative side, we show that even for weighted matroid rank valuations, EF1 and fractional Pareto optimality (fPO) are incompatible. This rules out, in general, broad classes of weighted-welfare and Fisher-market-based approaches. On the positive side, we identify a common-envelope condition that restores compatibility. Under this condition, EF1+PO allocations exist for any number of agents. This yields new positive results showing that common-weight matroid-rank valuations always admit EF1+PO allocations. Finally, we quantify the efficiency loss that is unavoidable when insisting on EF1. Our submodular counterexample implies that there is a constant $\alpha<1$ such that no EF1 allocation is $\alpha$-\PO. For the broader class of subadditive valuations, we prove a tight two-agent bound: for any $\varepsilon>0$, there exists an instance in which no EF1 allocation is $\left(\frac{1}{\sqrt{2}}+\varepsilon\right)$-PO.