Proportional Aggregation of Preferences for Sequential Decision Making

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, using axioms inspired by the multi-winner voting literature. 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 center dot alpha% of the decisions. We show that three attractive voting rules satisfy axioms of this style. One of them (Sequential Phragmen) 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.