“Who's number 1?” is a question asked and analyzed by sports announcers before, during, and sometimes even after a season. “We're number one!” is the cheer athletes chant in the pandemonium of success! In this chapter, we learn to apply linear algebra to sports ranking. We compute mathematical “number 1s” by ranking teams. We'll see how the rankings can assist in predicting future play. We'll apply the methods to two tournaments: March Madness and FIFA World Cup. You can learn more about these in [5, 1, 2].

Soon we'll learn how to compute teams' ratings to determine who is number 1 and also who is number 2, 3, 5, 8 and so forth. Here lies the important distinction between ratings and rankings. A rating is a value assigned to each team from which we can then form a ranking, an ordered list of teams based on a rating method. Solving the linear systems that we will derive produces the ratings. Sorting the ratings in descending order creates a ranked list, from first to last place.

When someone wants to rank teams, a natural rating to start with is winning percentage. This is done by computing each team's ratio of number of games won to the total number of games played. This includes information only about each game's end result and can lead to misleading information. For instance, if Team A only plays one game and they win, then their winning percentage rating would be 1. If Team B plays 20 games and goes undefeated, their winning percentage rating would also be 1. Intuitively, Team B is more impressive since they went undefeated for 20 games, while Team A only won once. But, both teams have the same winning percentage, so this rating system would produce a tie when ranking these two teams.