When forming a collective decision, how should we aggregate diverse individual views? Our goal will be to make this question more precise while retaining a high level of generality. We’ll encode individual views as binary relations from some specified input class I (equivalence relations, weak orders, digraphs with no cycles, . . .) and represent a collective decision as a binary relation from a possible different output class O. We will describe a small family of general quasi-linear aggregation procedures that yield a wealth of known aggregation rules – including, but not limited to, a number of well-known voting rules – when specialized to particular choices of I and O. Three orthogonal decompositions then explain many of the similarities and differences among these procedures and their specializations. A useful metaphor is that of a block sitting on an inclined plane; the effect of gravitational force FG can be understood via a vector decomposition into orthogonal vectors that have distinct effects on the block. In the case of aggregation procedures, such distinct effects correspond to the different types of information being aggregated.

Professor William S Zwicker is the William D Williams Professor of Mathematics at Union College, in New York. His original training was in logic and set theory, and more recent scholarship has been in the mathematical social sciences: social choice theory, cooperative game theory, and fair division.

The lectures are funded by a generous benefaction from Professor Oliver Smithies, which enables Balliol to bring distinguished visitors to the University of Oxford.

www.balliol.ox.ac.uk/events/2018/february/26/oliver-smithies-lecture