A mathematical analysis of fairness in shootouts

IF 1.9 3区工程技术 Q3 MANAGEMENT

IMA Journal of Management Mathematics Pub Date : 2021-08-01 DOI:10.1093/imaman/dpaa023

Roel Lambers;Frits C R Spieksma

引用次数: 7

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.

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对枪战公平性的数学分析

点球大战是确定两队比赛获胜者的一种流行机制。比赛分为几轮，每支队伍依次获得一次得分的机会。经验表明，在一轮中先射还是后射对得分概率有影响。这就提出了一个公平问题:是否有可能指定一个顺序，使相同的球队有相同的获胜机会?我们表明，对于突然死亡，没有重复的序列是公平的。此外，我们还证明了所谓的Prohuet-Thue-Morse序列是不公平的。然而，只要存在一个序列，就有一个算法输出一个公平的序列。我们还分析了流行的两局两胜的点球大战，并表明在这种情况下不存在公平的顺序。此外，我们还发现了best-of-$k$点球大战中不公平程度的显式表达式;这使得体育管理者可以评估点球时间长短对不公平程度的影响。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

IMA Journal of Management Mathematics OPERATIONS RESEARCH & MANAGEMENT SCIENCE-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

4.70

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The mission of this quarterly journal is to publish mathematical research of the highest quality, impact and relevance that can be directly utilised or have demonstrable potential to be employed by managers in profit, not-for-profit, third party and governmental/public organisations to improve their practices. Thus the research must be quantitative and of the highest quality if it is to be published in the journal. Furthermore, the outcome of the research must be ultimately useful for managers. The journal also publishes novel meta-analyses of the literature, reviews of the "state-of-the art" in a manner that provides new insight, and genuine applications of mathematics to real-world problems in the form of case studies. The journal welcomes papers dealing with topics in Operational Research and Management Science, Operations Management, Decision Sciences, Transportation Science, Marketing Science, Analytics, and Financial and Risk Modelling.