A stochastic analysis of scoring systems

Abstract

Many scoring systems can be seen as statistical tests of hypotheses. In tennis singles, for example, the scoring system used can be seen as a test involving 2 binomial probabilities pa and pb, where pa (pb) is the probability player A (player B) wins a point initiated by player A (player B). Tennis singles is thus a “bipoints” game. The tennis scoring system is an inefficient test relative to the sequential probability ratio test (SPRT) based on pairs of these points. When pa + Pb > 1 (the tennis context), an SPRT based on the “play-the-loser” (PL) rule is superefficient. Chapter 2 shows that, when pa + pb > 1, there is in fact a spectrum of super-efficient tests (with even durations) based on “partia1-PLV (PPL) rules. The most efficient tests within this spectrum, when pa + pb > 1, are the SPRT based on the (full) PL rule. Chapter 3 extends this spectrum of tests to produce the total spectrum of tests (including those with odd durations). Points within the tennis scoring system have different “importances” whereas points within any member of the above (efficient) spectrum of PPL systems are equally “important” when pa = pb. Intuitively, the differing importances of the points within the tennis scoring system contribute to the inefficiency of that system. Chapter 4 establishes a relationship between the efficiency of a bipoints scoring system and the importances of the points within it; a relationship which is used in Chapter 5 to show that the SPRT based on the PL rule has an optimal efficiency property when pa + Pb > 1. Chapters 6 and 7 address the question as to whether the super-efficiency of the PL rule carries over to the case of tennis doubles in which there axe essentially 4 binomial probabilities pai, pa2 , Pbi and pb2 . Some asymptotic results axe achieved although, generally speaking, they are of little practical relevance. The particular scoring system used in tennis is analysed in detail in Chapter 8 and the methodology used is seen to be useful for analysing any “nested” scoring system (e.g. tennis is 3-nested: points - games - sets). It was the study of this specific scoring system and its inherent inefficiency which lead to the theory of Chapters 2 to 7. A new tennis scoring system is proposed in Chapter 8. Chapter 9 contains a brief discussion of some of the characteristics the designer of a scoring system needs to consider and some results are given. The study of the importances of points is extended in Chapter 10 and in Chapter 11 team play with associated countback rules is investigated. The general conclusion is that “upwardnested” countback systems (e.g. points - games - sets, in tennis) axe preferable to “downward-nested” systems (sets - games - points). In Chapter 12 it is shown that the classical scoring system used in multiple choice examinations can be considerably improved by modifying tha t scoring system and instructing the examinees to cross any boxes known to be incorrect when the correct box for that question is unknown.