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arXiv · 2608.10326

Resolving Envy by Adding Goods with Bounded Supply: A Type-Count Dichotomy and Two-Agent Hardness

Chuang-Chieh Lin·Guillaume Fertin·Po-An Chen·Stéphane Vialette·Géraldine Jean·Emile Benoist·Colin Cleveland
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Abstract

We study envy elimination by adding goods (EEAG) when the additional pool has bounded supply and no separate budget bound. We establish a sharp type-count dichotomy for binary additive valuations. With one additional item type, EEAG is polynomial-time solvable for any number of agents. More generally, our algorithm permits arbitrary nonnegative integer per-copy values. The envy constraints form a system of difference constraints, and Bellman--Ford returns the componentwise least feasible extension. In contrast, with exactly two additional item types, EEAG is \textsf{NP}-complete even when both types have positive finite supply and the approvers of one type form a subset of the approvers of the other. This closes the two-type case left open by Bentert et al. Separately, we prove weak \textsf{NP}-completeness even for two agents with identical additive valuations, one initially endowed good, and a growing number of unit-supply item types. Thus, bounded-supply hardness appears both with two item-types and many agents and with two agents and many item-types.

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Chuang-Chieh Lin, Guillaume Fertin, Po-An Chen, Stéphane Vialette, Géraldine Jean, Emile Benoist, Colin Cleveland. 2026-08-10. Resolving Envy by Adding Goods with Bounded Supply: A Type-Count Dichotomy and Two-Agent Hardness. https://arxiv.org/abs/2608.10326

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