Researchers prove fair+optimal allocation impossible for submodular valuations

In a new arXiv paper (2607.18220), researchers Harish Chandramouleeswaran and Prajakta Nimbhorkar resolve a classic open problem in fair division: the nonexistence of allocations that are both envy-free up to one item (EF1) and Pareto optimal (PO) when agents have monotone submodular valuations. Their counterexample uses just two agents with unweighted coverage valuations—a strict subclass of submodular functions. This proves that even in simple, well-behaved settings, we cannot guarantee both fairness (no agent envies another after removing one item) and efficiency (no reallocation can make someone better off without harming another).

Beyond the negative result, the authors show that checking whether any such allocation exists is NP-hard for monotone submodular valuations. This moves the frontier of impossibility from subadditive valuations (previously known) down to the more restrictive coverage valuations, which are common in resource allocation and mechanism design. The paper also extends the result to chores (disutility) with unweighted coverage costs, showing symmetry between goods and bads. This has implications for algorithmic fairness, market design, and AI systems that allocate resources among self-interested agents.