William S. Zwicker is the William D. Williams Professor of Mathematics at Union College, and was trained at Harvard University (undergraduate) and MIT (Ph.D.). He has held visiting positions at York University (Toronto), the Autonoma University of Barcelona, Alicante University, the London School of Economics, Balliol College Oxford, Université de Caen, and Université Paris Dauphine. He received the 2009 Stillman Prize for teaching, and currently serves on the editorial boards of Mathematical Social Sciences and of Studies in Economic Design. His principal research areas include set theory and logic, cooperative game theory, social choice theory, and fair division. He authored the Voting chapter in the Handbook of Computational Social Choice (Cambridge, 2016) and is co-author, with Alan D. Taylor, of the research monograph Simple Games (Princeton, 1999). Bill is particularly enthusiastic about questions of a geometric or combinatorial nature that arise from the social sciences and are of independent mathematical interest.
Aggregation fuses disparate information from multiple sources into one collective view or decision. When we vote, individual ballots are aggregated into an election outcome; in cluster analysis, similarities and differences among objects are aggregated into a grouping of objects; and in judgment aggregation, discordant views on which statements are true and which false are combined into a logically consistent collective judgment. A variety of aggregation methods exist, and the choice matters. For example, when we vote for president with ballots that rank several candidates in order of preference, different voting methods applied to the same ballots may declare different winners. Who, then, is the “right” winner?
We lack any general theory of aggregation explaining why methods disagree, or helping us choose the right method for a given context, but there is now a promising direction to explore. The Median Procedure yields a wide variety of methods as special cases and is described via a certain measure of distance; the election winner is the candidate “closest” to all ballots cast, for example. For the space in which distance is measured, some directions turn out to be perpendicular to others, and encode different, independent types of information. When different information types are actively aggregated or suppressed, we get different methods. The key research questions are now: How far can this type of analysis be pushed? Can classifying types of information tell us which method is best?