Applying Majority Judgment over a Polyhedral Candidate Space

Most of the existing voting methods deal with a moderate (or at least finite) number of candidates. In practice, there are important voting applications where candidate space is huge or even of infinite size. We describe new methods in voting by extending the Majority Judgment voting and ranking method to handle a candidate space of infinite size. Specifically, the candidate space is modeled as a polyhedral set. Two approaches are developed. The first approach relies on multiple rounds of grading and iterative candidate generation. The candidate generation employs a novel mixed-integer programming model. The second approach employs a robust optimization framework and only takes as input each voters most preferred candidate. This results in an output vector which is the candidate that has the best worst-case guarantee in terms of majority grade. We demonstrate the effectiveness of our approaches through two case studies involving voting over polyhedral candidate space.