24 - May the Best Team Win: Determining the Winner of a Cross Country Race

Summary

Abstract

Finding the winner of an athletic contest or sporting event should be a simple process: the competitor with the most runs, goals, points, or the quickest time should be the winner. In practice, however, choosing a team winner can be a challenge, especially when many teams are involved in a tournament. In this article, we explore some of the specific dilemmas associated with finding the winning team in a cross country running race.

Standard scoring of a cross country race is straightforward. A team typically consists of seven runners, and a team's score is the sum of the placings of its first five runners. A team's sixth and seventh runners do not score points towards their team's total, but their place can serve to increase the team score of their opponents. Teams are ranked by the order of their scores from lowest to highest. While teams often face off head-to-head in dual meets, invitationals of up to 30 teams are also common.

Though simple to implement, this race scoring system can yield surprising and somewhat counterintuitive outcomes. Chief among these are failures of binary independence: the relative ranking of two teams in an invitational can depend upon the presence and performance of the other teams in the race. For example, one team (the Acorns) might finish ahead of a second team (the Buckeyes) when a third team (the Chestnuts) has a good day.