Applied Mathematics Colloquium: Dr. Carolyn Yarnall

Title: Approval Voting in Circular Societies

Abstract: In the system of approval voting, individuals vote for all candidates they find acceptable. This method of voting is of interest in that it satisfies many "fairness" criteria and it is the way most individuals handle scheduling problems. We can model voting societies geometrically using lines, circles, planes, etc. to represent the options voters are choosing from and subsets of these objects to model their preferences. For instance, if scheduling a meeting at a particular time, we might use the real line to represent the options available and intervals on the line for the times that individuals are available to meet. We are interested in two items: (1) the "agreement number" or the maximum number of simultaneously intersecting sets and (2) the "piercing number" or minimum number of points needed to "pierce" all sets. In the context of scheduling, the former will tell us how many people could meet at the "winning time" and the latter demonstrates how many meetings would need to be held so everyone could attend at least one. In this talk, we will investigate these ideas in societies modeled by arcs on a circle circles, give bounds on piercing and agreement numbers, and present probabilistic results about the average piercing number for such societies.