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.

翻译：我们研究策略性代理具有可加估值时不可分割物品的公平分配问题。受确定性真实机制不可能性结果的驱动，我们聚焦于满足"期望真实性"(Truthful-in-Expectation, TIE)的随机化机制。从公平性视角出发，我们致力于为每个代理事后(ex-post)保证其"最大最小份额"(Maximin Share, MMS)的较大比例。在诸多研究成果中，Bu与Tao[FOCS 2025]提出了一种能保证事后$\frac{1}{n}$-MMS的TIE机制。首先，我们提出一种序数型TIE机制，可保证事后$\frac{1}{H_n + 2}$-MMS，其中$H_n$为第$n$个调和数($H_n \simeq \ln n$)。这对序数型机制而言近乎最优——即使非真实性的序数分配算法也无法获得优于$\frac{1}{H_n}$的逼近比。随后我们证明，仅需引入少量额外基数信息，即可将事后保证提升至$Ω(\frac{1}{\log\log n})$-MMS，代价是将激励要求松弛为可忽略项$\varepsilon(n)$对应的$(1-\varepsilon(n))$-TIE。最后针对两个代理情形，我们提出一种能实现事后$\frac{2}{3}$-MMS的TIE机制。所有机制均满足事前比例性(ex-ante proportional)（从而同时实现"两全其美"结果），且运行时间为多项式复杂度。此外，所有结论均可推广至截断比例份额(Truncated Proportional Share, TPS)——该份额至少与MMS相等。针对两个代理的$\frac{2}{3}$-TPS结果在TPS框架下已为最优。