We study the problem of fair sequential decision making given voter preferences. In each round, a decision rule must choose a decision from a set of alternatives where each voter reports which of these alternatives they approve. Instead of going with the most popular choice in each round, we aim for proportional representation across rounds, using axioms inspired by the multi-winner voting literature. The axioms require that every group of $\alpha\%$ of the voters that agrees in every round (i.e., approves a common alternative), must approve at least $\alpha\%$ of the decisions. A stronger version of the axioms requires that every group of $\alpha\%$ of the voters that agrees in a $\beta$ fraction of rounds must approve $\beta\cdot\alpha\%$ of the decisions. We show that three attractive voting rules satisfy axioms of this style. One of them (Sequential Phragm\'en) makes its decisions online, and the other two satisfy strengthened versions of the axioms but make decisions semi-online (Method of Equal Shares) or fully offline (Proportional Approval Voting). We present empirical results for these rules based on synthetic data and U.S. political elections. We also run experiments using the moral machine dataset about ethical dilemmas: We train preference models on user responses from different countries and let the models cast votes. We find that aggregating these votes using our rules leads to a more equal utility distribution across demographics than making decisions using a single global preference model.

翻译：我们研究在给定选民偏好情况下的公平序贯决策问题。在每一轮中，决策规则必须从备选方案集中选择一个决策，每位选民需报告其认可哪些备选方案。不同于每轮直接选择最受欢迎的方案，我们借鉴多席位投票文献中的公理思想，追求跨轮次的比例代表性。该公理要求：在每一轮中均达成一致（即认可共同备选方案）的任意α%选民群体，其认可的决策比例不得低于α%。该公理的强化版本进一步要求：在β比例轮次中达成一致的任意α%选民群体，其认可的决策比例不得低于β·α%。我们证明三种具有吸引力的投票规则满足此类公理。其中一种规则（Sequential Phragmén）采用在线决策方式，另外两种规则（Method of Equal Shares与Proportional Approval Voting）分别以半在线和完全离线方式决策，且满足强化版公理要求。基于合成数据与美国政治选举数据，我们展示了这些规则的实证结果。此外，我们利用道德机器数据集中关于伦理困境的实验数据：通过训练不同国家用户反馈的偏好模型，并让模型进行投票。研究发现，相较于使用单一全局偏好模型进行决策，采用我们的规则聚合这些投票能实现更平等的跨人口统计学效用分布。