The fair allocation of scarce resources is a central problem in mathematics, computer science, operations research, and economics. While much of the fair-division literature assumes that individuals have underlying cardinal preferences, eliciting exact numerical values is often cognitively burdensome and prone to inaccuracies. A growing body of work in fair division addresses this challenge by assuming access only to ordinal preferences. However, the restricted expressiveness of ordinal preferences makes it challenging to quantify and optimize cardinal fairness objectives such as envy. In this paper, we explore the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy. We consider various settings based on whether the true preferences of the agents are stochastic or worst-case, and whether the inaccuracies, modeled as additive noise, are stochastic or worst-case. When the true preferences are stochastic, we show that envy-free allocations can be computed with high probability; this is achieved both in the setting with stochastic and worst-case noise. This generalizes a notable result in stochastic fair division, which establishes a similar guarantee, albeit in the absence of any noise. When the true preferences are worst-case, and the noise is bounded, we analyze the maximum envy achieved by the Round-Robin algorithm. This bound is shown to be tight for deterministic algorithms, and applications of this bound are provided. Lastly, we consider a setting with worst-case preferences and noise, where the true preferences for each item are revealed upon its allocation. Here, we give an efficient online algorithm that guarantees logarithmic maximum envy with high probability. This result generalizes a known result from algorithmic discrepancy to a setting with noisy input data.

翻译：稀缺资源的公平分配是数学、计算机科学、运筹学和经济学的核心问题。尽管公平分配领域的大量研究假设个体具有潜在的基数偏好，但获取精确的数值偏好通常认知负担较重且易于产生误差。公平分配领域日益增长的研究工作通过仅假设可获得序数偏好来应对这一挑战。然而，序数偏好有限的表达能力使得量化和优化基数公平目标（如嫉妒）变得困难。本文探讨了在给定不精确基数偏好的情况下不可分割物品公平分配的广泛图景，重点关注最小化嫉妒。我们考虑了基于智能体真实偏好是随机的还是最坏情况的，以及建模为加性噪声的不精确性是随机的还是最坏情况的各种设定。当真实偏好是随机时，我们证明可以高概率计算无嫉妒分配；这一结果在具有随机噪声和最坏情况噪声的设定中均能实现。这推广了随机公平分配中的一个显著结果，该结果建立了类似的保证，尽管是在没有任何噪声的情况下。当真实偏好是最坏情况的且噪声有界时，我们分析了轮询算法（Round-Robin）所实现的最大嫉妒。我们证明该界限对于确定性算法是紧的，并提供了该界限的应用。最后，我们考虑了一个具有最坏情况偏好和噪声的设定，其中每件物品的真实偏好在其被分配时揭示。在此，我们给出了一种高效的在线算法，该算法能以高概率保证对数级最大嫉妒。这一结果将算法差异理论中的一个已知结果推广到了具有噪声输入数据的设定。