[Paper Review] On Best-of-Both-Worlds Fairness via Sum-of-Variances Minimization

Moshe Babaioff, Yuval Grofman|arXiv (Cornell University)|Jan 23, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper analyzes minimizing the sum of variances of agents’ values under ex-ante proportionality to achieve BoBW fairness, showing strengths for identical valuations (two agents) but strong negatives for non-identical valuations and for n≥3 identical agents.

ABSTRACT

We consider the problem of fairly allocating a set of indivisible goods among agents with additive valuations. Ex-ante fairness (proportionality) can trivially be obtained by giving all goods to a random agent. Yet, such an allocation is very unfair ex-post. This has motivated the Best-of-Both-Worlds (BoBW) approach, seeking a randomized allocation that is ex-ante proportional and is supported only on ex-post fair allocations (e.g., on allocations that are envy-free-up-to-one-good (EF1), or give some constant fraction of the maximin share (MMS)). It is commonly pointed out that the distribution that allocates all goods to one agent at random fails to be ex-post fair as it ignores the variances of the values of the agents. We examine the approach of trying to mitigate this problem by minimizing the sum-of-variances of the values of the agents, subject to ex-ante proportionality. We study the ex-post fairness properties of the resulting distributions. In support of this approach, observe that such an optimization will indeed deterministically output a proportional allocation if such exists. We show that when valuations are identical, this approach indeed guarantees fairness ex-post: all allocations in the support are envy-free-up-to-any-good (EFX), and thus guarantee every agent at least 4/7 of her maximin share (but not her full MMS). On the negative side, we show that this approach completely fails when valuations are not identical: even in the simplest setting of only two agents and two goods, when the additive valuations are not identical, there is positive probability of allocating both goods to the same agent. Thus, the supporting ex-post allocation might not even be EF1, and might not give an agent any constant fraction of her MMS. Finally, we present similar negative results for other natural minimization objectives that are based on variances.

Motivation & Objective

• Motivate and formalize the sum-of-variances (SoV) objective under ex-ante proportionality as a BoBW approach.
• Characterize ex-post fairness properties of distributions minimizing SoV when valuations are identical.
• Investigate how SoV minimization fares for non-identical valuations and for more than two agents.
• Show computational hardness and foundational limitations of SoV across variants and objectives.

Proposed method

• Define ex-ante proportional distributions and the SoV objective across allocations.
• Characterize the support of SoV-minimizing distributions as distance-minimizing to the ex-ante proportional share vector.
• Prove that with identical valuations and n=2, SoV-minimizers are MMS-fair ex-post (hence EFX) and identify NP-hardness for computing such distributions.
• Extend to n≥3 identical valuations to show ex-post MMS-fairness fails in some cases but ex-post EFX holds, implying constant MMS approximation.
• Provide negative results for non-identical valuations, showing EF1/EFX may fail in the support, even for 2 agents and 2 goods, and analyze other variance-based objectives.

Experimental results

Research questions

• RQ1 Does minimizing the sum of variances under ex-ante proportionality guarantee ex-post fairness (EF1/EFX or MMS) for identical valuations?
• RQ2 What ex-post fairness properties do SoV-minimizing distributions exhibit when valuations are identical with more than two agents?
• RQ3 Do non-identical valuations undermine ex-post fairness for SoV-minimizing distributions, even in the simplest two-agent, two-good setting?
• RQ4 Are there computational or complexity barriers to finding SoV-minimizing distributions under ex-ante proportionality?
• RQ5 Do alternative variance-based objectives similarly fail to guarantee ex-post fairness?

Key findings

• For two agents with identical valuations, every SoV-minimizing distribution is MMS-fair ex-post (and thus EFX ex-post).
• For n≥3 identical valuations, SoV-minimizers are EFX ex-post, yielding a 2/3 MMS approximation for n=3 and 4/7 MMS for n≥4; the 3-agent MMS approximation bound is at most 275/304 ≈ 90.4%.
• Computing an SoV-minimizing distribution under ex-ante proportionality for two identical agents is NP-hard.
• With non-identical valuations, there exists a two-agent, two-good instance where every SoV-minimizer places positive probability on an allocation giving both goods to one agent, breaking EF1 (and thus EFX) ex-post and not guaranteeing constant MMS.
• Similar negative results hold for other natural variance-based objectives (max variance, max std dev, sum of std devs, variance of variances, etc.).
• The SoV approach does not ensure ex-post fairness when valuations differ, despite ex-ante proportionality.