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 α% of the voters, if it agrees in every round (i.e., approves a common alternative), then those voters must approve at least α% of the decisions. A stronger version of the axioms requires that every group of α% of the voters that agrees in a β fraction of rounds must approve β⋅α% of the decisions. We show that three attractive voting rules satisfy axioms of this style. One of them (Sequential Phragmén) 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.

We interleave sampling based motion planning methods with pruning ideas from minimum spanning tree algorithms to develop a new approach for solving a Multi-Goal Path Finding (MGPF) problem in high dimensional spaces. The approach alternates between sampling points from selected regions in the search space and de-emphasizing regions that may not lead to good solutions for MGPF. Our approach provides an asymptotic, 2-approximation guarantee for MGPF. We also present extensive numerical results to illustrate the advantages of our proposed approach over uniform sampling in terms of the quality of the solutions found and computation speed. Note to Practitioners—MGPF is concerned with finding a collision-free, near-optimal path for a robot visiting a set of target configurations. This problem arises in applications that use robotic manipulators such as advanced manufacturing, surface inspection, package sorting, and in other logistical applications where the cost of the traveling between any two configurations of a robot cannot be readily determined a-priori. As robots are expected to perform a large number of tasks, the sequencing of these tasks become important specifically when the travel costs are challenging to estimate. This paper provides an approach to handle this problem in higher dimensions with theoretical guarantees as well as provides simulation results on a broad class of environments to corroborate its performance with respect to the state of the art.

We interleave sampling based motion planning methods with pruning ideas from minimum spanning tree algorithms to develop a new approach for solving a Multi-Goal Path Finding (MGPF) problem in high dimensional spaces. The approach alternates between sampling points from selected regions in the search space and de-emphasizing regions that may not lead to good solutions for MGPF. Our approach provides an asymptotic, 2-approximation guarantee for MGPF. We also present extensive numerical results to illustrate the advantages of our proposed approach over uniform sampling in terms of the quality of the solutions found and computation speed.