[Paper Review] Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods

Koichi Nishimura, Hanna Sumita|arXiv (Cornell University)|Feb 26, 2023

Experimental Behavioral Economics Studies4 citations

TL;DR

This paper establishes that for mixed divisible and indivisible goods with binary and linear valuations, maximum Nash welfare (MNW) allocations satisfy envy-freeness up to any good for mixed goods (EFXM), a stronger fairness notion than existing EF1M and EFM. The result generalizes prior MNW fairness guarantees for purely divisible or indivisible goods and holds for any symmetric strictly convex function minimization, proving MNW allocations are both Pareto optimal and fair under binary valuations.

ABSTRACT

We study fair allocation of resources consisting of both divisible and indivisible goods to agents with additive valuations. When only divisible or indivisible goods exist, it is known that an allocation that achieves the maximum Nash welfare (MNW) satisfies the classic fairness notions based on envy. Moreover, the literature shows the structures and characterizations of MNW allocations when valuations are binary and linear (i.e., divisible goods are homogeneous). In this paper, we show that when all agents' valuations are binary linear, an MNW allocation for mixed goods satisfies the envy-freeness up to any good for mixed goods (EFXM). This notion is stronger than an existing one called envy-freeness for mixed goods (EFM), and our result generalizes the existing results for the case when only divisible or indivisible goods exist. When all agents' valuations are binary over indivisible goods and identical over divisible goods (e.g., the divisible good is money), we extend the known characterization of an MNW allocation for indivisible goods to mixed goods, and also show that an MNW allocation satisfies EFXM. For the general additive valuations, we also provide a formal proof that an MNW allocation satisfies a weaker notion than EFM.

Motivation & Objective

• To investigate the fairness properties of maximum Nash welfare (MNW) allocations in settings with both divisible and indivisible goods.
• To determine whether MNW allocations satisfy stronger fairness notions than existing approximations like EF1M or EFM in mixed good settings.
• To extend known results on MNW fairness from purely divisible or indivisible goods to the mixed case under binary and linear valuations.
• To establish that MNW allocations minimize any symmetric strictly convex function of utilities under binary valuations, generalizing prior results.
• To provide a formal proof of the existence and fairness properties of MNW allocations in the mixed goods setting.

Proposed method

• Prove that any MNW allocation is Pareto optimal (PO), which is essential for deriving fairness guarantees.
• Use a contradiction argument: assume an MNW allocation violates EFXM, then construct a new allocation with higher Nash welfare, contradicting MNW optimality.
• Apply a geometric transfer argument using a piece of divisible good with proportional utility ratios to show welfare improvement.
• Define and analyze two cases: when the envy arises from a divisible good component and when it arises from an indivisible good, using minimal ratio goods to ensure improvement.
• Leverage the linearity of valuations and binary utility values (0 or 1) to ensure that utility ratios are preserved under transfers.
• Use the fact that Nash welfare is maximized when the product of positive utilities is maximized, and any reallocation increasing this product contradicts optimality.

Experimental results

Research questions

• RQ1 Does an MNW allocation for mixed divisible and indivisible goods satisfy envy-freeness up to any good for mixed goods (EFXM) when all agents have binary and linear valuations?
• RQ2 How does the MNW allocation compare to existing fairness notions like EF1M and EFM in the mixed goods setting?
• RQ3 Can the fairness guarantees of MNW allocations in purely divisible or indivisible settings be extended to the mixed case under binary valuations?
• RQ4 Is there a general fairness principle—such as minimizing a symmetric strictly convex function—under which MNW allocations remain optimal and fair in mixed settings?
• RQ5 Can EFXM allocations be computed in polynomial time for binary and linear valuations in the mixed goods model?

Key findings

• For binary and linear valuations, every MNW allocation for mixed divisible and indivisible goods satisfies EFXM, a fairness notion stronger than EF1M and EFM.
• The result generalizes known MNW fairness guarantees: in purely divisible goods, MNW implies envy-freeness; in purely indivisible goods with binary valuations, MNW implies EFX; this paper unifies both under EFXM.
• MNW allocations are Pareto optimal (PO), and this property is essential in proving that any envy-based deviation can be used to construct a welfare-improving reallocation.
• The proof constructs a welfare-improving allocation by transferring a carefully chosen piece of divisible good or an indivisible good with minimal utility ratio, leading to a contradiction with MNW optimality.
• An EFXM allocation can be computed in polynomial time using existing algorithms: EFM algorithms for linear valuations and EFX algorithms for binary indivisible goods.
• The fairness guarantee holds not only for Nash welfare but also for any symmetric strictly convex function of utilities, generalizing the result beyond just the geometric mean.