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. We formalize this aim using axioms based on Proportional Justified Representation (PJR), which were proposed in the literature on multi-winner voting and were recently adapted to multi-issue decision making. The axioms require that every group of $\alpha\%$ of the voters, if it agrees in every round (i.e., approves a common alternative), then those voters 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). The first two are polynomial-time computable, and the latter is based on an NP-hard optimization, but it admits a polynomial-time local search algorithm that satisfies the same axiomatic properties. We present empirical results about the performance of these rules based on synthetic data and U.S. political elections. We also run experiments where votes are cast by preference models trained on user responses from the moral machine dataset about ethical dilemmas.

翻译：我们研究了给定选民偏好下的公平序贯决策问题。在每一轮中，决策规则必须从一组备选方案中选择一个决策，每个选民报告他们认可哪些备选方案。我们并非在每个回合中选择最受欢迎的方案，而是追求比例代表性。我们基于比例公正代表权（PJR）的公理来形式化这一目标，该公理最初在多人投票文献中提出，最近被应用于多议题决策。这些公理要求：若每一轮中任意α%的选民群体都同意（即认可同一个备选方案），则这些选民必须认可至少α%的决策。更强版本的公理要求：若任意α%的选民群体在β比例轮次中达成一致，则他们必须认可β·α%的决策。我们证明三种有吸引力的投票规则满足此类公理。其中一种（序贯Phragmén规则）在线做出决策，另外两种满足加强版公理但采用半在线（等分法）或完全离线（比例批准投票）方式。前两种规则可在多项式时间内计算，后者基于NP难优化问题，但存在满足相同公理性质的多项式时间局部搜索算法。我们基于合成数据和美国政治选举展示了这些规则的实证结果，同时开展了利用道德困境数据集——道德机器（Moral Machine）中用户反馈训练出的偏好模型进行投票的实验。