In many situations, several agents need to make a sequence of decisions. For example, a group of workers that needs to decide where their weekly meeting should take place. In such situations, a decision-making mechanism must consider fairness notions. In this paper, we analyze the fairness of three known mechanisms: round-robin, maximum Nash welfare, and leximin. We consider both offline and online settings, and concentrate on the fairness notion of proportionality and its relaxations. Specifically, in the offline setting, we show that the three mechanisms fail to find a proportional or approximate-proportional outcome, even if such an outcome exists. We thus introduce a new fairness property that captures this requirement, and show that a variant of the leximin mechanism satisfies the new fairness property. In the online setting, we show that it is impossible to guarantee proportionality or its relaxations. We thus consider a natural restriction on the agents' preferences, and show that the leximin mechanism guarantees the best possible additive approximation to proportionality and satisfies all the relaxations of proportionality.

翻译：在许多情境中，多个决策主体需要就一系列决策做出选择。例如，一组工作人员需要决定每周会议的地点。在此类情境下，决策机制必须兼顾公平性概念。本文分析了三种已知机制的公平性：循环轮询、最大纳什福利和leximin（词典序最大最小原则）。我们同时考虑离线与在线场景，重点关注比例公平性概念及其松弛形式。具体而言，在离线场景中，我们证明即使存在比例性或近似比例性结果，上述三种机制也无法找到此类结果。为此我们引入一种能刻画该需求的新型公平属性，并证明改进后的leximin机制满足该属性。在在线场景中，我们证明不可能保证比例性或其松弛形式。因此我们考虑对主体偏好施加自然限制，并证明leximin机制可保证比例性的最佳可加近似，同时满足比例性的所有松弛条件。