The Leximin Approach for a Sequence of Collective Decisions

TL;DR

This paper examines fairness in sequential collective decision-making, showing limitations of existing mechanisms and proposing a leximin-based approach that guarantees fairness under certain conditions.

Contribution

It introduces a new fairness property for sequential decisions and demonstrates that a leximin variant satisfies this property, improving fairness guarantees.

Findings

Existing mechanisms fail to ensure proportionality in offline settings.

A new fairness property is proposed for sequential decisions.

A leximin-based mechanism guarantees the best possible fairness approximation online.

Abstract

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.