A mathematical analysis of fairness in shootouts. (12th November 2020)

• Record Type: Journal Article
• Authors: Lambers, Roel; Spieksma, Frits C R
• Is Part Of: IMA journal of management mathematics. Volume 32:Number 4(2021)
• Page Start: 411
• Page End: 424
• Publication Date: 2020-11-12
• Subjects: sports -- fairness -- shootout -- prouhet-thue-morse sequence; Management -- Mathematical models -- Periodicals; Management science -- Mathematical models -- Periodicals; Business mathematics -- Periodicals; 650.01513
• DOI: 10.1093/imaman/dpaa023
• Languages: English
• ISSNs: 1471-678X

Abstract

Abstract: A shootout is a popular mechanism to identify a winner of a match between two teams. It consists of rounds in which each team gets, sequentially, an opportunity to score a point. It has been shown empirically that shooting first or shooting second in a round has an impact on the scoring probability. This raises a fairness question: is it possible to specify a sequence such that identical teams have equal chance of winning? We show that, for a sudden death, no repetitive sequence can be fair. In addition, we show that the so-called Prohuet–Thue–Morse sequence is not fair. There is, however, an algorithm that outputs a fair sequence whenever one exists. We also analyze the popular best-of-$k$ shootouts and show that no fair sequence exists in this situation. In addition, we find explicit expressions for the degree of unfairness in a best-of-$k$ shootout; this allows sports administrators to asses the effect of the length of the shootout on the degree of unfairness.