[Paper Review] Existence of EFX for Two Additive Valuations

Ryoga Mahara|arXiv (Cornell University)|Aug 20, 2020

Game Theory and Voting Systems4 citations

TL;DR

This paper proves that a complete EFX (envy-freeness up to any item) allocation always exists when agents have additive valuations and are divided into two distinct types, each with identical valuations. The authors provide a constructive algorithm that iteratively builds a Pareto-dominating EFX allocation from an initial partial EFX allocation, extending prior results for identical or two-agent settings to general numbers of agents and items.

ABSTRACT

Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.

Motivation & Objective

• To resolve the open problem of whether complete EFX allocations exist for general numbers of agents and items with non-identical additive valuations.
• To extend prior existence results—previously limited to two agents, identical valuations, or three agents—to settings with two distinct valuation types.
• To provide a constructive algorithm that generates a complete EFX allocation through iterative Pareto improvement from a partial EFX allocation.
• To establish a foundational step toward resolving the general EFX existence conjecture in fair division of indivisible items.

Proposed method

• The method begins with an initial partial EFX allocation and constructs a new allocation by reallocating items to improve Pareto dominance while preserving EFX fairness.
• It uses the concept of the 'minimum preferred set' to identify items that can be transferred without violating EFX conditions for envy from one agent to another.
• The algorithm selects agents who can be made better off via item swaps or reallocations, ensuring that envy relations are maintained up to any item removal.
• It applies a recursive construction where a new allocation is formed by replacing one agent's bundle with a preferred subset, maintaining EFX across all agent pairs.
• The proof relies on inductive reasoning over agent types and valuation structures, leveraging symmetry between agents of the same type.
• Key lemmas establish that the new allocation Pareto dominates the previous one and remains EFX under all envy-checking conditions.

Experimental results

Research questions

• RQ1 Does a complete EFX allocation exist when agents have additive valuations and belong to one of two distinct types?
• RQ2 Can a constructive algorithm be designed to generate such EFX allocations iteratively from a partial EFX allocation?
• RQ3 Is it possible to achieve Pareto dominance while preserving EFX fairness during reallocation steps?
• RQ4 How does the structure of two valuation types affect the feasibility of EFX beyond the two-agent or identical-valuation cases?

Key findings

• A complete EFX allocation always exists when agents are divided into two types, each with identical additive valuation functions.
• The existence result holds for any number of agents and items, without restriction on the number of distinct item values per agent.
• The proof is constructive: a Pareto-dominating EFX allocation can be systematically built from an initial partial EFX allocation.
• The algorithm ensures that envy relations are preserved under removal of any single item from the envied agent’s bundle, satisfying EFX for all agent pairs.
• The method fails for three-agent additive cases, indicating that more complex techniques are needed for generalizations beyond two types.
• The result represents a significant step toward resolving the long-standing open problem of EFX existence in the general case of additive valuations.