  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.
