A mathematical analysis of fairness in shootouts

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.