A mathematical analysis of fairness in shootouts

Roel Lambers (Corresponding author), Frits C.R. Spieksma

Research output: Contribution to journalArticleAcademicpeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)411-424
Number of pages14
JournalIMA Journal of Management Mathematics
Issue number4
Publication statusPublished - 1 Oct 2021

Bibliographical note

Funding Information:
The NWO Gravitation Project NETWORKS (grant number 024.002.003 to F.C.R. Spieksma, partial funding).

Publisher Copyright:
© 2020 The Author(s) 2019.


  • Fairness
  • Prouhet-thue-morse sequence
  • Shootout
  • Sports


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