The following visualization demonstrates both independence and intransitivity violations that can occur in rank-sum scoring, which is used at the NCAA cross country tournament. Each bar can be clicked to remove or add the team to the tournament. Notice when the independence option is displayed (default) that with team Z, team Y wins and without team Z, team X win. When the intransitivity label is pressed, team X beats team Y, team Y beats team Z, and team Z beats team X, causing a cycle when the teams are considered in pairs. This was featured in Justin Ehrlich’s talk at the 2016 Midwest Sports Analytics Meeting in Pella, Iowa:
• J. Boudreau, J.A. Ehrlich, S. Sanders, and A. Winn “Social Choice Violations in Rank Sum Scoring: A Formalization of Conditions and Corrective Probability Computations,” Mathematical Social Sciences, vol. 71, 2014, pp. 20-29; DOI 10.1016/j.mathsocsci.2014.03.004
• J. A. Ehrlich, S. Sanders, and J. Boudreau “Rank Sum Aggregation: Origin, Applications, and Social Choice Characteristics” presented at the 2015 Illinois Economic Association conference, Chicago, IL, 2015.
• J. Ehrlich, “Computational/Visual Analysis of How the Scoring Method for Cross-Country Running Competitions Violates Major Social Choice Principles,” presented at the Midwest Sports Analytics Meeting, Central College, Pella, IA, 19-Nov-2016.