Existence of envy-free allocations with zero-demand goods

Determine whether envy-free allocations always exist under Leontief preferences when some agents are permitted to have zero demand for some goods.

Background

The main results assume strictly positive demand for every good by every agent. The paper notes that the setting in which agents may have zero demand for some goods remains to be resolved in general. It observes only the special case that an envy-free allocation exists when every agent positively demands at least two goods, via a zero-welfare construction, and notes that some of the presented algorithms extend to this broader setting.

References

This work opens several promising directions for future research: (i) studying the complexity and existence of other fairness notions, such as equitability; (ii) investigating whether envy-freeness and equitability can be achieved simultaneously; and (iii) determining whether envy-free allocations always exist when some agents may have zero demand for some goods.

— Envy-Free Allocation of Indivisible Goods under Leontief Preferences  (2609.28308 - Inamdar et al., 23 Sep 2026) in Section Outlook