Claire Chang

Thesis Advisor: Dr. Jamie Haddock (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Michael Orrison (Harvey Mudd College - Department of Mathematics) 

Contact: cschang@hmc.edu

The Sensitivity of a Laplacian Family of Ranking Methods

Have you ever wondered if scoring a few more points in that last game would have made your favorite football team have a higher standing? Personally, I'm curious about how there would have been different sport climbing medalists in the Tokyo Olympics if the rankings for the three disciplines were added, rather than multiplied. How are rankings determined in sports and other applications, and how can we ensure that they are reasonable?  By analyzing the sensitivity and stability of various ranking methods, we will be able to better characterize their fairness and susceptibility to manipulation.

Ranking from pairwise comparisons is a particularly rich subset of ranking problems, which involves combining potentially incomplete or contradictory orderings of pairs of alternatives into a single complete ordering of all alternatives. This group of problems has applications as diverse as choosing political candidates, ranking web pages, filtering job applicants, and comparing sports teams. In this work, we focus on the sensitivity of a family of ranking methods for pairwise comparisons which encompasses the well-known Massey, Colley, and Markov methods. We will accomplish two objectives. First, we will consider a network diffusion interpretation for this family. Second, we will analyze the sensitivity of this family by studying the “maximal upset” where the weight of an arc between the highest and lowest ranked alternatives is modified. Through these analyses, we will build intuition to answer the question “what are the characteristics of robust ranking methods?” to ensure fair rankings in a variety of applications.