Fair and Efficient Balanced Allocation for Indivisible Goods

TL;DR

This paper presents polynomial-time algorithms for fair and efficient allocation of indivisible goods under balanced constraints, specifically for agents with bivalued valuations or at most two valuation types, combining EF1 fairness and fractional Pareto optimality.

Contribution

It establishes the existence and efficient computation of EF1 and fPO allocations under balanced constraints for specific valuation cases, using novel graph matching and duality techniques.

Findings

Polynomial-time algorithms for EF1 and fPO allocations under balanced constraints.

First solutions for these fairness and efficiency criteria in the specified cases.

New insights into constrained fair division problems.

Abstract

We study the problem of allocating indivisible goods among agents with additive valuation functions to achieve both fairness and efficiency under the constraint that each agent receives exactly the same number of goods (the \emph{balanced constraint}). While this constraint is common in real-world scenarios such as team drafts or asset division, it significantly complicates the search for allocations that are both fair and efficient. Envy-freeness up to one good (EF1) is a well-established fairness notion for indivisible goods. Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely accepted efficiency criteria. Our main contribution establishes both the existence and polynomial-time computability of allocations that are simultaneously EF1 and fPO under balanced constraints in two fundamental cases: (1) when each agent has a personalized bivalued valuation, and (2) when agents have at most two distinct valuation types,. Our algorithms leverage novel applications of maximum-weight matching in bipartite graphs and duality theory, providing the first polynomial-time solutions for these cases and offering new insights for constrained fair division problems.