Centennial Honors College
Western Illinois University
Undergraduate Research Day 2012
Podium Presentation
Rank Injustice? How the Scoring Method for Cross-Country Running
Competitions Violates Major Social Choice Principles: A Comment
Adam Winn
Faculty Mentors: Shane Sanders and Justin Ehrlich
Economics and Computer Science
Given the rising cultural and economic significance of sport, Hammond (2007) notes the
importance of understanding the social choice properties of sporting outcomes. He
shows that sum of positions team scoring (rank-ordering) in cross-country running
violates two major social choice principles: transitivity and “independence from
irrelevant teams.” Using manual computation, he concludes that ranking cycles occur in
5.17 percent of outcome sequences featuring three teams (m=3) and three individuals
per team (k=3). We show that Hammond inadvertently but crucially conditions the
analysis. For all outcome sequences featuring three teams (m=3) and three individuals
per team (k=3), 30 of 1680 (1.79 percent) feature a ranking cycle. Hammond does not
calculate this likelihood over the much larger (realistic) set where m=3 and k=5. We use
computational methods to generate each possible outcome sequence for the realistic
set (n=756,756) and conclude that the likelihood of a cycle drops to 1.37 percent
(10,392 of 756,756 sequences) for this set. Coaches interviewed by Hammond believe
ranking cycles to be rare in cross-country competition. The corrected results more
closely support this general (practitioner) view. Rather than somewhat common but
non-salient, ranking cycles in sum of positions scoring may be salient but rare.