EFX Exists for Three Types of Agents

Fair division of indivisible resources is a well-researched problem at the intersection of theoretical computer science and economics. The problem arises in a variety of practical settings, from allocating slots or assets to distributing aid or shared goods. One of the most intuitive notions of fairness is envy-freeness (EF) [Fol67], where each individual is content with their share compared to others. However, when the resources are indivisible - such as physical objects, housing units, or assets like artwork -- achieving true envy-freeness becomes impossible in many cases. While EF provides a natural measure of fairness, the combinatorial nature of indivisible goods often renders EF allocations unattainable, highlighting the necessity for more nuanced fairness criteria.