Truthful-in-Expectation Mechanisms for MMS Approximation

2026-04-29

AI Summary

This work addresses the impossibility of simultaneously achieving strategyproofness and Maximin Share (MMS) fairness in deterministic mechanisms for indivisible goods allocation. The authors propose polynomial-time randomized mechanisms that are truthful-in-expectation (TIE), guaranteeing ex-ante proportionality and ex-post approximate MMS (or Threshold Proportional Share, TPS) fairness. Their key contributions include the first near-optimal ordinal TIE mechanism achieving a $1/(H_n + 2)$-approximation to MMS, where $H_n$ is the $n$-th harmonic number; an improved $\Omega(1/\log \log n)$-MMS guarantee using only limited cardinal information; and, for the two-agent setting, a TIE mechanism attaining the theoretically optimal $2/3$-MMS (which also equals $2/3$-TPS).

Abstract

We study fair allocation of indivisible goods among strategic agents with additive valuations. Motivated by impossibility results for deterministic truthful mechanisms, we focus on randomized mechanisms that are \emph{Truthful-in-Expectation (TIE)}. From a fairness perspective, we seek to guarantee every agent a large fraction of their \emph{Maximin Share (MMS)} ex-post. Among other results, Bu~and~Tao~[FOCS 2025] presented a TIE mechanism that guarantees $\frac{1}{n}$-MMS ex-post. First, we present an ordinal TIE mechanism that guarantees $\frac{1}{H_n + 2}$-MMS ex-post, where $H_n$ is the $n$-th harmonic number ($H_n \simeq \ln n$). This is nearly best possible for ordinal mechanisms, as even non-truthful ordinal allocation algorithms cannot obtain an approximation better than $\frac{1}{H_n}$. We then show that with just a small amount of additional cardinal information, the ex-post guarantee can be improved to $Ω(\frac{1}{\log\log n})$-MMS, at the cost of relaxing the incentive requirement to $(1-\varepsilon(n))$-TIE for negligible $\varepsilon(n)$. Finally, for two agents, we present a TIE mechanism that is $\frac{2}{3}$-MMS ex-post. All our mechanisms are ex-ante proportional (thus also providing ``Best-of-Both-Worlds'' results) and run in polynomial time. Moreover, all our results extend to the truncated proportional share (TPS), which is at least as large as the MMS. Our two-agent $\frac{2}{3}$-TPS result is best possible for the TPS.

Problem

• Truthful-in-Expectation
• Maximin Share
• Fair Allocation
• Indivisible Goods
• Mechanism Design

Innovation

• Truthful-in-Expectation
• Maximin Share
• Ordinal Mechanism
• Fair Division
• Truncated Proportional Share