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.

本文分析了三种决策机制：轮流执行、最大 Nash 福利和 leximin，并考虑了比例公正性及其放宽的公正性概念。研究表明，在离线设置中，三种机制都不能保证比例或近似比例结果，因此引入了一种新的公正标准，并表明 leximin 机制的变体满足新的公正标准。在在线设置中，不可能保证比例或其放宽条件，因此考虑了对应代理偏好的自然限制，并表明 leximin 机制保证最佳可能的加性逼近比例性并满足比例性的所有放宽条件。