EFX Exists for Three Agents

We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation.

💡 Research Summary

This paper resolves a long-standing open question in the fair division of indivisible goods: it proves that an allocation satisfying Envy-Freeness up to any item (EFX) always exists for three agents with additive valuations. Furthermore, it disproves a conjecture by Caragiannis et al. regarding the relationship between EFX allocations and Nash welfare.

The fair division problem involves distributing a set M of m indivisible items among n agents. Each agent i has an additive valuation function v_i, assigning a non-negative value to each item. An allocation X = (X_1, …, X_n) assigns a bundle X_i to each agent i. EFX is a compelling fairness relaxation where no agent i envies another agent j after the removal of any single item from j’s bundle (i.e., v_i(X_i) ≥ v_i(X_j \ {g}) for all g in X_j). While an EFX allocation is known to exist for two agents, its existence for three or more agents has remained enigmatic despite significant effort.

The paper’s first major contribution is a constructive, algorithmic proof that an EFX allocation always exists for three agents (n=3). The proof handles the general case where agents have distinct additive valuations. The authors begin by reducing the problem to “non-degenerate” instances, where no agent values two different subsets equally. This is achieved by adding vanishingly small, lexicographically structured perturbations to the valuations, ensuring that an EFX allocation for the perturbed instance is also EFX for the original.

The core algorithmic innovation lies in overcoming two key technical barriers. The first is the “Bundle Splitting” technique. The algorithm maintains a partial EFX allocation. For each agent’s bundle X_i, it identifies a specific item (a “pin”) that splits the bundle into an “upper” and “lower” half (U_i and L_i) based on the agent’s own valuation ordering. The algorithm then meticulously analyzes the envy relationships between agents concerning these half-bundles. By systematically moving these upper or lower halves between agents under precise rules, it improves the allocation while preserving the EFX property.

The second barrier concerns progress measurement. Prior approaches relied on showing Pareto improvements (where no agent’s utility decreases and at least one increases). However, the authors demonstrate in Section 5 an instance where a certain partial EFX allocation is not Pareto-dominated by any complete EFX allocation, rendering a pure Pareto-improvement approach insufficient. To overcome this, they introduce a novel lexicographic potential function. They fix an arbitrary ordering of the three agents (a, b, c) and define the potential of allocation X as φ(X) = (v_a(X_a), v_b(X_b), v_c(X_c)). The algorithm’s update rules are designed to guarantee that this lexicographic potential strictly increases with each reallocation. Crucially, the utility of agent a never decreases, and when Pareto improvements are impossible, the rules force an increase in v_a(X_a). Since valuations are discrete in the perturbed space, this guarantees convergence to a complete EFX allocation in finite steps.

The second major contribution is falsifying a conjecture by Caragiannis et al. They had conjectured that adding an item to a problem that has an EFX allocation yields a new problem that also has an EFX allocation with at least as high Nash welfare (the geometric mean of agents’ utilities). This conjecture, if true, would have implied the existence of complete EFX allocations with good efficiency properties. The authors construct a specific three-agent instance where there exists a partial EFX allocation (with some items unallocated) whose Nash welfare is strictly higher than the Nash welfare of any possible complete EFX allocation for that instance. This result is significant because it reveals an inherent limitation of iterative approaches that start with an inefficient partial allocation and try to allocate leftover items without harming fairness. It underscores the distinct and sometimes competing priorities of fairness (EFX) and efficiency (Nash welfare).

In summary, this paper achieves a landmark result by settling the existence of EFX allocations for three agents. It introduces powerful new technical tools—bundle splitting and a lexicographic potential function—that are likely to influence future research on fair division beyond this specific result. By also disproving a key conjecture, it deepens the understanding of the complex interplay between fairness and efficiency in resource allocation.