Symmetric Sequential Analysis: The Efficiencies of Sports Scoring Systems (With Particular Reference to Those of Tennis)

Abstract

SUMMARY Bernoulli‐type models are adopted for the sequence of contested points between two players (or teams). In bipoints, there are two types of point – “A serving” and “B serving” – necessary for a realistic model of top (class) men's tennis. Unipoints is the special case of bipoints in which the probabilities that A (and hence B) wins each type of point are the same–a reasonable model for top men's squash rackets. Sports scoring systems based upon such play (“uniformats” and “biformats”) may be regarded as sequential statistical tests concerning the identity of the “better player”. Provided they satisfy certain fairness criteria, these tests are symmetric, with equal probabilities of type I and II error, and a unique efficiency p; then formally 1 – p represents the average fraction of play being virtually wasted through the use of a suboptimal scoring system. Largely because in top men's tennis the proportions of service points won by both players are so high, the efficiency of traditional tennis scoring in such play is unduly low. Whilst the introduction of tiebreaker games has further reduced efficiency, it is shown how a simple entropy‐increasing modification of the traditional scoring system serves to increase efficiency.

Optimal challenges in tennis

“…In addition, Clarke (1988), Clarke and Norman (1999) and Preston and Thomas (2000) are three examples of many that have applied the technique to cricket. Papers such as Schutz (1970), George (1973), Miles (1984), Croucher (1998), Riddle (1988), Klaassen and Magnus (2003) and Barnett et al (2006) demonstrate a long and successful use of constant probability Markov chains to model tennis. Barnett et al (2004) use probability models to investigate optimal allocation of resources, namely points in a game and games in a set where players should use extra energy.…”

Probability of Winning at Tennis I. Theory and Data

“…A recent statistical analysis of 4 years of Wimbledon data [9] shows that although points in tennis are not iid, for most purposes this is not a bad assumption as the divergence from iid is small. Other aspects of tennis that have been analyzed using probabilistic models include optimal serving strategies [10], the efficiency of various scoring systems [11], and the question of which is the most important point [12]. Statistical methods have also been used to study the effects of new balls [13], service dominance [14], and the probabilities of winning the final set of a match [15].…”

The Effect of New Balls in Tennis: Four Years at Wimbledon

“…Klaassen and Magnus (1998) analyse whether the assumption of ®xed probabilities of winning a point on service is realistic. Another series of papers deals with the tennis scoring system and its effect on the probability of winning a match; see Maisel (1966), Miles (1984), Riddle (1988Riddle ( , 1989 and the comments by Jackson (1989). Finally, the service and the ®rst±second service strategy have been analysed by George (1973) and Gillman (1985).…”