Education

Abstract

The Education section of this issue focuses on fair games. The article, “Making the Rules of Sports Fairer,” by Steven J. Brams and Mehmet S. Ismail, defines a fair game as one in which equally skilled competitors have the same probability of winning. The authors discuss mainly rules in soccer and tiebreaker rules in tennis, with brief comments on rules in three racquetball sports, football, and volleyball. The starting point of the discussion is the tie in a game by the end of regular time, in which case a coin toss determines further actions. For example, in football, the coin toss determines which team decides whether to kick or to receive in the overtime period. The authors stipulate that most of the time the winner of the coin toss elects to receive, and this gives it a substantial advantage of winning the game. In soccer, which is a low-scoring game, a knockout tournament is very frequent. In such a tournament, after a tie in regular time, a five-round penalty-kick shootout takes place. Several datasets of shootouts from 1970 to 2013 indicate that the team who had the right to kick first won the entire game more frequently than the team who went second. A survey of coaches and players indicates an overwhelming preference to go first in the shootout. The issue of fairness in a game arises when an element of chance is introduced, which has a potential to provide a substantial advantage. It has been a matter of discussion in the context of soccer or other games with various proposals such as bidding, handicapping, and others. In this paper, it is assumed that a series of contests takes place where the chance (e.g., coin toss) has given an advantage to a player or a team. The contest can result in a win, a loss, or be unresolved. Three rules are analyzed in detail for such a sequence of contests. The first one is the currently existing rule in soccer, called the Standard Rule: the coin toss determines the advantage for all contests. The other two rules are the Catch-Up Rule, which is advocated by the authors, and the Behind-First, Alternating Order Rule, which was proposed previously. The Catch-Up Rule states that the first contest should be conducted according to the coin toss, while in every subsequent contest, the player or team who lost becomes advantaged. If one of the contests is unresolved, then the advantage for the subsequent contest is given to the team that was disadvantaged in the unresolved contest. The Behind First, Alternating Order Rule takes into account the current score: the player or team that is behind gets the advantage in the next round; if the score is tied, the advantage alternates. The authors illustrate in detail how the rules function and how they differ. Analysis of the probability to win for equally strong competitors under the different rules is provided. The main assumption to evaluate the probability of winning is that the advantage gives a higher probability to win a single contest, and the result of each contest is independent of the results of the previous ones. The authors show that the Standard Rule increases substantially the probability of winning over a longer sequence of contests, while the other two rules give substantially smaller power to the lucky coin toss. Additional discussion includes the evaluation of the probability of winning the entire game in the case of two differently skilled teams. The conclusions are similar. The authors specifically consider the penalty shootout in soccer and a tiebreaker in tennis. In the latter case, the contest is either won or lost; it cannot be unresolved. In the context of tiebreaker rules in tennis, the authors analyze only the case of differently skilled players and conclude that the current Standard Rules are fairer than the Catch-Up Rule. In the last section of the paper, the authors comment on the applicability of those rules to various sports, when one of the teams is advantaged.