**ABSTRACT **

**Key words**: Efficiency of tennis scoring systems, play-the-loser, play-the-winner, volleyball, team tennis.

**INTRODUCTION **

**Unipoints**

Many sports consist of playing a sequence of points each of which is won by either player A or player B. If there is only one type of point in the match, we have unipoints in which p (q) is the probability player A wins (loses) each point (p + q = 1). Player A is the better player if p > 0.5. Miles, 1984 considered fair scoring systems possessing sensible regularity conditions and only one type of point, which he called uniformats.

For testing the hypothesis H0: Player A better than player B, versus

the alternative hypothesis H1: Player B better than player A,

he applied the result of Wald and Wolfowitz (1948) to conclude that there is a unique class of optimally efficient uniformats, given by the Sequential Probability Ratio Test, and that it is {Wn} (n = 1,2,3,…) where Wn is the uniformat in which the winner is the first player to achieve a lead of n points over his opponent. The key characteristics of Wn are given by

and the efficiency of a general uniformat with key characteristics P and ? is given by

**Bipoints**

If there are two types of points (a-points when player A serves, and b-points when player B serves), we have bipoints. The probability player A wins a point on service is pa, and the probability player B wins a point on service is pb. Player A is the better player if pa > pb. For the bipoints situation, Miles, 1984 considered fair scoring systems with sensible regularity conditions, which he called biformats.

Noting that Wald (1947) had recommended the use of paired trials for comparing two binomial probabilities (H0: pa > pb Vs H1: pb > pa), Miles (1984) set up {Wn(point-pairs)}, where n = 1, 2, 3, …, as the standard biformat family of scoring systems (with unit efficiency) against which the efficiency of any tennis singles scoring system could be measured. An example of this is given by Pollard and Noble (2004). Here, point-pairs represent the playing of pairs of points consisting of an a-point and a b-point.

Miles (1984) showed that the efficiency of a bipoints scoring system with mean ? and probability P that the better player wins, is given by

*where Q = 1 – P, qa = 1 - pa and qb = 1 - pb. He also set up what he called the reversal biformat W1(WnALa, WnALb), where the superscript represents the first type of point played, and AL represents alternating point types played (e.g. ababab…), and (WnALa, WnALb), operating like a point-pair, results in a win by player A, a draw, or a loss by player A. Pollard, 1992 showed that W1(WnALa, WnALb) was stochastically equivalent to the scoring system Wn(point-pairs). Note that in both of these scoring systems, P and ? do not depend on the type of the first point played.*

Using the play-the-loser mechanism (PL) in which a win by player A (B) is followed by a b- (a-) point, Miles (1984) showed that the family of reversal biformats {W1(WnPLa,WnPLb)} (n = 2, 3, 4, …) was slightly more efficient than {Wn(point-pairs)} (n = 1, 2, 3, …) when pa + pb > 1, which is the tennis context. Pollard (1992) showed that this PL family of scoring systems was not only more efficient than the AL or point-pairs family of systems, but that it (and its stochastic equivalent, for example, one based on PL generalized point-pairs) was the most efficient possible. (Note that when using PL generalized point-pairs, a win to A (AA) is followed by a (b, b) point-pair, a draw (AB) is followed by an (a, b) point-pair, and a loss to A (BB) is followed by an (a, a) point-pair.) He showed correspondingly that the family of play-the-winner (PW) reversal biformats {W1(WnPWa,WnPWb)} (n = 2, 3, 4, …) is optimally efficient when pa + pb < 1 (which is the volleyball situation).

If Pk (Qk) denotes the probability that player A wins (loses) in k points, then a scoring system is said to have the constant probability ratio property (c.p.r.) if Pk/Qk is constant for all k for which Qk > 0. Pollard (1992) showed that the above families of AL, PL and PW scoring systems possessed the c.p.r. property.

**Quadpoints**

In tennis doubles, we have four probabilities, pa1, pa2, pb1 and pb2 where pa1 is player A1’s probability of winning a point on service, pa2 is player A2’s probability of winning a point on service, pb1 is player B1’s probability of winning a point on service, and pb2 is player B2’s probability of winning a point on service. Thus, it is natural to call this situation quadpoints. Using these four probabilities, Pollard, 2005 showed that a first to six games rule for a set of tennis could be unfair in some doubles situations, as could the present ‘first to seven’ and ‘first to ten’ tiebreak games rules. Also, Pollard (1990) carried out research on the asymptotic efficiencies of some quadpoints Wn systems (for large n). He noted that some complex Wn systems could be decomposed into smaller independent components called modules, which could in turn be analyzed to produce approximate asymptotic values for P, ? and ? for the whole system. In a natural extension of point-pairs, he set up a basic module consisting of the 4 points {a1, b1, a2, b2}, which we call ‘point-quads’. He also considered a second module, called PLteams, using PL and generalized point-pairs in the following way. The first section of the PLteams module starts with an (a1, b1) point-pair, and the PL mechanism operates in the following way. A point-pair win to pair A, (AA), is followed by a (b1, b2) point-pair whilst a point-pair win to pair B, (BB), is followed by an (a1, a2) point-pair. The first section of the module finishes as soon as a draw, (AB), occurs, and the second section of the module begins with a point-pair (a2, b2), and proceeds in the same manner as above until a draw occurs, thus completing the whole module. He showed that, provided the modulus of (pa1 - pa2) equals the modulus of (pb1 - pb2), PLteams (PWteams) modules used in conjunction with Wn systems (n large) are asymptotically more efficient than the {Wn(point-quads)} (n large) system when ((pa1+pa2)/2 + (pb1+pb2)/2) is greater (less) than 1. Note that the situation when ((pa1+pa2)/2 + (pb1+pb2)/2) is greater than 1 is referred to as the ‘tennis context’ since it corresponds to the typical advantage of serving (rather than receiving) in tennis.

In this paper some quadpoints results are established for non-asymptotic cases.

**METHODS **

It was noted above that in doubles we have four service probabilities pa1, pa2, pb1 and pb2. In order to establish the efficiency of a doubles scoring system we need to set up an appropriate structure so that P and ? do not depend on the order of the four points being played. This suggests setting up the family of scoring systems {Wn(W1(W2AL(a1, b1), W2AL(a2, b2)), W1(W2AL(a1, b2), W2AL(a2, b1)))} (n = 1, 2, 3, …) as the standard scoring system against which the efficiency of any doubles scoring system can be measured. Here, for example, AL(a1, b1) represents the playing of alternating a1 and b1 points. Note that AL(a1, b1) is stochastically equivalent to (a1, b1) point-pairs when used with Wn (n even).

An expression for the efficiency of a doubles scoring system is now derived. Firstly we note that the above scoring system has four components, each with an AL structure. The component listed first, W2AL(a1, b1) has characteristics P1, ?1 and Q1 given by

where P1 is the probability pair A wins this first component, and ?1 is the mean duration of (or the expected number of points played in) this first component.

It is noted that W2AL(a1, b1) has the c.p.r. property, and that ?1 is equal to the mean number of points conditional on pair A winning, and is also equal to the mean number of points conditional on pair A losing (Pollard, 1992). This fact is used in the following analysis.

Corresponding expressions for P2, Q2 and ?2 for the second component can be written down.

The first half of the above scoring system, W1(W2AL(a1, b1), W2AL(a2, b2)), is now analysed. This first half of the scoring system amounts to playing W2AL(a1, b1) and W2AL(a2, b2) until one pair wins both components. If these two components are won by different pairs, the process is repeated until one pair wins both. It can be shown that W1(W2AL(a1, b1), W2AL(a2, b2)) has the c.p.r. property. The probability pair A wins this half of the above scoring system is given by

and the expected duration of this first half is given by

where R1,2 is the probability that the pairs each win one component of this half, and is given by

The second half of the above scoring system can be analyzed similarly giving corresponding results. The subscipts 3 and 4 are used for this second half. Combining the results for the two halves of the scoring system, we have

where P1,2,3,4 is the probability player A wins under this scoring system (when n = 1), and µ1,2,3,4 is the expected number of points played.

Thus, for the complete scoring system (general n), which also possesses the c.p.r. property, the probability pair A wins, Pn, and the mean number of points played, ?n , satisfy the equations

The efficiency of a doubles scoring system with mean ? and probability pair A wins equal to P, is now considered. The efficiency is equal to ?n/? where ?n is the mean of the above standard scoring system with the same P value (i.e. P = Pn). Noting

and the above expression for ?n , the efficiency ? of a doubles scoring system with probability P and mean ? is given by

Expressing ?1,2,3,4 and (P1,2,3,4 – Q1,2,3,4) as functions of pa1, pa2, pb1 and pb2, we have, after some algebra,

where

This is the general expression for the efficiency of a doubles scoring system. Note that when pa1 = pa2 = pa and pb1 = pb2 = pb, this expression is equal to the expression given in the introduction for the efficiency of tennis singles scoring systems.

Thus, it can be seen that when comparing two doubles scoring systems, we simply need to compare their respective values for the expression ((P – Q)*ln(P/Q))/?, known as the relative efficiency, as the rest of the expression for ? above is simply a function of the parameters for the players. Thus, interestingly, this result for unipoints and bipoints carries over to the quadpoints situation.

**The efficiency of {Wn(a1, a2, b1, b2)}**

The efficiency of the Wn systems using sets of the four points at a time, called point-quads, is considered. (Note that this system involves playing sets of 4 points, and then making a decision as to whether pair A has won or lost, or the match continues). This system is a natural extension of point-pairs to the quadpoints case, and we might expect it to have unit efficiency, as in the point-pairs case. As in the paper by Pollard, 1990, this set of four points is called a module. The independent modules effectively become the steps of a general one-dimensional walk in discrete time. Using the approach and notation of Cox and Miller, 1965, the steps in the random walk, Zi , are mutually independent random variables on the integers …, -2, -1, 0, 1, 2,… and the moment generating function (m.g.f.) of Zi is defined by

If P(Q) represent the probability of absorption in states [a, infinity) ((-infinity, -b]), and E(N) is the expected number of steps to absorption, then, neglecting the excess over the barriers,

when E(Zi) ? 0 and

when E(Zi) ? 0,

where ?0 is the non-zero solution of the equation f*(?) = 1. For Wn we set a and b equal to n. Also, the random variable S ( = Zi) is used to represent the increase in pair A’s score (i.e. the better pair’s score) during the play of one module, and D is used to represent the number of points in the play of one module. It can be seen from above that the efficiency of system 1 relative to system 2 is given by the expression

Using the above expressions for the ratio P/Q and E(N), and representing Wn(point-quads) as system 1, and the above standard system as system 2, we have

where ?1 and ?2 are the non-zero solutions of their respective equations.

It turns out that the efficiency of Wn(point-quads) (n = 1, 2, 3, …) is slightly less than unity, even when n is large. For example, when pa1 = 0.9, pa2 = 0.8, pb1 = 0.7, and pb2 = 0.6, it can be shown that E(D1) = 4, E(S1) = 0.8, and exp(?1) = 1.7650, and E(D2) = 39.7541, E(S2) = 7.8502, and exp(?2) = 1.7908 (note that exp(4?2) = (pa1pa2qb1qb2)/(pb1pb2qa1qa2)), and it follows that, when n = 30 say, ?1/ ?2 = 0.9876, which is slightly less than 1. A few differences between systems 1 and 2 (or their modules) are noted here. A module of system 2 has the same expected number of a- and b- points in total, but the expected number of a1 points is not equal to the expected number of a2 points, and the expected number of b1 points is not equal to the expected number of b2 points (For example, these are 10.3159, 9.5611, 10.8192 and 9.0579 respectively in the above example). Also system 1 can have excesses of 1, 2 or 3 over the boundaries, whereas system 2 cannot have any excesses.

**The efficiency of some PL and PW quadpoints systems**

In this section we consider whether the PL and PW service exchange mechanisms can be used to find super-efficient scoring systems for quadpoints, as has been noted was possible in the case of bipoints. We firstly consider {Wn(W1(W2mPL(a1, b1), W2mPL(a2, b2)), W1(W2mPL(a1, b2), W2mPL(a2, b1)))} (n = 1, 2, 3, …; m = 1, 2, 3, …). Note that W2mPL(a1, b1), for example, is the generalized PL point-pair structure described earlier that makes use of the W2m stopping rule.

For the first of the four components, W2mPL(a1, b1), we have, using a similar notation to above,

Similar expressions for the other three components can be written down. Firstly, the case when n = 1 is considered. Using similar methods to above, expressions for P1,2,3,4 and ?1,2,3,4 can be determined for this PL scoring system, and it can be shown by substituting these expressions into the formula for ? above that the efficiency of this PL system is less than unity for all m = 2, 3, 4, …. For example, Table 1 has values for the efficiency of this PL system when n = 1 and m = 1, 2, 3, 4 and 5, when pa1 = 0.9, pa2 = 0.8, pb1 = 0.7 and pb2 = 0.6. This table also gives values for the efficiency of the corresponding alternating and play-the-winner scoring systems {Wn(W1(W2mAL(a1, b1), W2mAL(a2, b2)), W1(W2mAL(a1, b2), W2mAL(a2, b1)))} (n = 1, 2, 3, …; m = 1, 2, 3, …) and {Wn(W1(W2mPW(a1, b1), W2mPW(a2, b2)), W1(W2mPW(a1, b2), W2mPW(a2, b1)))} (n = 1, 2, 3, …; m = 1, 2, 3, …), for the cases when n = 1 and m = 1, 2, 3, 4 and 5.

It was observed that the efficiency of all of these systems (when n = 1) is less than unity as soon as m is greater than unity. Noting that these scoring systems with general n amount to playing the corresponding system with n = 1 n or more times, it is clear that the efficiency of these systems when n is greater than 1 (and m is not unity) must also be less than unity. This follows since efficiency under nesting is essentially multiplicative (Miles, 1984). Thus, it follows that it is not possible to use these PL and PW mechanisms in the above manner to increase efficiency above 1.

**The efficiency of some other PL and PW quadpoints systems**

We now consider an alternative application of the PL mechanism, and consider the PLteams module approach mentioned in the introduction. For example, for this module as defined, when pa1 = 0.9, pa2 = 0.8, pb1 = 0.7 and pb2 = 0.6, it was shown numerically that ? was approximately 0.8642 and E(D)/E(S) was 7.5 for each ‘half’ of the module. (It was verified numerically that E(D)/E(S) for this module is in general equal to (pa1 + pa2 + pb1 + pb2)/(pa1 + pa2 – pb1 – pb2). The value for ?, however, needed to be calculated numerically.) Thus, for these parameter values, E(D2)/(?2E(S2)) is equal to 8.6909 (from above), and E(D1)/(?1E(S1)) is equal to 8.6785, so the ratio is 1.0014. When n = 30, exp(exp(-n?2)-exp(-n?1)) = 1.0000, so the efficiency of this W30(PLteams) relative to the standard system is equal to 1.0014 when pa1 = 0.9, pa2 = 0.8, pb1 = 0.7 and pb2 = 0.6. Thus, in the tennis context, for quadpoints just as for bipoints, it is possible to use the PL mechanism to achieve efficiencies greater than 1. It is noted that the PLteams module as defined in the introduction should be extended so that it includes two additional components beginning with (a1, b2) and (a2, b1). Pollard, 1990 noted that the values of ?0 and of E(D)/E(S) were unaffected by reversing the roles of b1 and b2; namely the case in which the first section of the module started with an (a1, b2) point-pair and the second section with an (a2, b1) point-pair. Thus, as in the bipoints case, any increases in efficiency above 1 are miniscule. Correspondingly, when the underlying p-values are less than 0.5 as in volleyball, the PW mechanism can be used to achieve efficiencies marginally greater than 1.

The above module, PLteams, made use of point-pairs. However, a corresponding module making use of points rather than point-pairs can be formed. This is done by using the PL mechanism between teams but alternating the points within each team. It was verified that the value of E(D)/E(S) was again given by (pa1 + pa2 + pb1 + pb2)/(pa1 + pa2 – pb1 – pb2), and that exp(?0) was the square root (since points rather than point-pairs are involved) of its value for the PLteams module. Thus, these two modules are equivalent. The advantage of this formulation, however, is that it can be extended to multi-points (a1, a2, a3, …; b1, b2, b3, …) by ‘rotating’ or ‘cycling’ through the relevant points rather than alternating between the pair of them. This result is relevant to the situation where there are two teams of two doubles pairs (i.e. a1, a2, a3, a4; b1, b2, b3, b4), even though ‘rotating’ or ‘cycling’ would be impractical as it would involve (amongst other things) players going on and off the court continuously. Nevertheless, a relative measure of efficiency in this team situation could be evaluated. It can be seen that this paragraph is related to the next section.

**An extension of the quadpoints systems**

Suppose team A has two doubles pairs represented by [(a1, a2), (a3, a4)] and team B has two doubles pairs represented by [(b1, b2), (b3, b4)]. Then, denoting the above quadpoints standard scoring system {W1(W1(W2AL(a1, b1), W2AL(a2, b2)), W1(W2AL(a1, b2), W2AL(a2, b1)))} by SS(a1, a2; b1,b2), it follows that Wn(W1(SS(a1, a2; b1,b2), SS(a3, a4; b3,b4)), W1(SS(a1, a2; b3,b4), SS(a3, a4; b1,b2))) (n = 1, 2, 3,…) has the c.p.r. property, and is the corresponding standard family of scoring systems for two teams with two doubles pairs per team. Thus, it can be used as the standard family of scoring systems against which to assess the efficiency of team doubles with two pairs per team. In the same way, this process can be extended to teams of 4 doubles pairs, 8 doubles pairs, etcetera.

**RESULTS **

A doubles scoring system based on simply playing groups of the four point-types at a time and using the Wn stopping rule has been shown to have slightly smaller efficiency than the framework scoring systems.

Two scoring systems using the PL rule have been considered. In the tennis context, one has efficiency less than the framework family of scoring systems, and the other has efficiency greater than that family.

It has been demonstrated how the methods of this paper can be further developed to find the efficiency of scoring systems used in a contest between two teams of 2, 4, 8, …doubles pairs.

**CONCLUSION **

An expression for the relative efficiency of two doubles scoring systems has been identified, and it has been shown to be identical to that for tennis singles and other sports such as squash which typically need just one parameter when modelling.

The methods of this paper can be applied to a match between two teams of 2, 4, 8, etc. doubles pairs. That is, it is possible to establish a yardstick, and to measure the efficiency of such team contests.

**KEY POINTS **

The expression for the relative efficiency of two tennis doubles scoring systems is a simple one, and is the same as that for two singles scoring systems.

The methodology of this paper can be used and extended so that the efficiency of a scoring system for a match between two teams of doubles pairs can be measured.

**AUTHORS BIOGRAPHY **

Geoff POLLARD

Employment: President, Tennis Australia, Melbourne, Australia and PhD student, Swinburne University, Melbourne, Australia.

Degree: MSc in mathematical statistics.

Research interests: Mathematical statistics applied to sport, demographic studies.

E-mail: gpollard@tennis.com.au

Graham POLLARD

Employment: Emeritus Professor (Statistics), University of Canberra, Australia.

Degree: PhD.

Research interests: Scoring systems in sport and in assessment. Optimal learning.

E-mail: graham@foulsham.com.au