2010/108. A Stochastic Analysis of Some Two-Person Sports II. Yves DOMINICY Christophe LEY Yvik SWAN

Transcription

1 2010/108 Stochastic nalysis of Some Two-Person Sports II Yves DOMINICY Christophe LEY Yvik SWN

2 Stochastic nalysis of Some Two-Person Sports II Yves Dominicy, Christophe Ley and Yvik Swan E.C..R.E.S. and Département de Mathématique Université Lire de Bruxelles, Brussels, Belgium stract We estalish a general formula for the game-winning proailities in tale tennis, which extends previous results y Schulman and Hamdan We use this formula to derive the proaility distriution and hence the expectation and variance of the numer of rallies necessary to achieve any given score. Our results also encompass, with minor modifications, similar results for the game of tennis. We use these findings to investigate theoretically, and via Monte Carlo simulations the implications of recent rule change imposed y the International Tale Tennis Federation ITTF. Finally, we riefly indicate how our results can lead to more efficient estimation techniques of individual players ailities. 1 Introduction. In this paper, we analyze the following situation. Two players or teams, referred to as and B, play a sequence of rallies after each of which either or B is declared winner. Every rally is initiated y a server the other player is then called the receiver and a point is scored y the winner after each rally. The players continue competing until a Research supported y an IP P6/07 contract, from the IP program Belgian Scientific Policy: Economic policy and finance in the gloal economy. Research supported y a Mandat d spirant of the Fonds National de la Recherche Scientifique, Communauté française de Belgique. Research supported y Mandat de Chargé de Recherches of the Fonds National de la Recherche Scientifique, Communauté française de Belgique. 1

3 match-winner is declared, with a match eing composed of several sets according to the m, n, G-scoring system, which we define as follows. The m, n, G-scoring system: i set consists of a sequence of independent rallies; the winner of a set is the first player to score n points or, in case of a tie at n 1, the first player to create a difference of two points after the tie. ii match consists of a sequence of independent sets; the winner of a match is the first player to win G sets. iii The server in the first rally of the first set is determined y flipping a coin; if G 2, the right to serve in the first rally of each susequent set alternates etween the two opponents. iv Within a set, the right to serve changes etween the opponents after each sequence of m consecutive rallies y a server until either a set-winner is declared or a tie is reached at n 1, n 1. v fter a tie at n 1, n 1, the right to serve alternates after each rally. To the est of our knowledge, tale tennis is the only sport currently using the aovedescried m, n, G-scoring system, with G eing, according to the rules of the International Tale Tennis Federation ITTF, any odd numer of sets, usually 3, 5 or 7. Tennis can nevertheless e shown after a little thought and minor ut straightforward adaptations to also fit within the aove scoring system with m = 1 and n = 6. This, however, is not the purpose of the present article, and the applications of our results to tennis is still the suject of ongoing research. See the concluding section of this paper for more information. Until 2001, tale tennis was played according to the 5, 21, G-scoring system, with G = 3. In order to reduce the length of the game and to make its duration more predictale, the ITTF decided to switch to the 2, 11, G-scoring system, with G either 3 or 4. Besides the aims announced efore, the idea ehind the system change was to make the game more exciting y enaling potentially more crucial points, and hence to make it more appealing to television viewers. s in most of the mathematical literature on this kind of prolem, we will restrict our attention to two well-known models. The first is the so-called server model, in which 2

4 it is assumed that rally-outcomes are mutually independent and are, conditionally upon the server, identically distriuted. Denoting y p a resp., p the proaility that player resp., player B wins a rally he/she initiates, the game is then entirely governed y the ivariate parameter p a, p 0, 1 0, 1. For a discussion on the validity and fallacies of this model, see e.g. Paindaveine and Swan The second model under investigation is the simpler no-server model, in which it is assumed that every rally is won y each player with proailities that do not depend on the server; the latter model is thus a particular case of the former, with parameters p a = 1 p = p 0, 1. While an important literature has een devoted to sports such as tennis see, e.g., Hsi and Burich 1971, George 1973, Carter and Crews 1974, Riddle 1988 or Klaassen and Magnus 2003 or adminton see, e.g., Phillips 1978, Simmons 1989 or the recent contriutions y Percy 2009 and Paindaveine and Swan 2010, the corresponding literature on taletennis is, to say the least, scant. Indeed, to the est of our knowledge, only one paper Schulman and Hamdan 1977 addresses this prolem, and proposes a solution only for m, n, G = 5, 21, 2. This lack of interest perhaps originates in the apparent simplicity of the stochastics underlying the game of tale tennis. Indeed, whereas in sports such as adminton or tennis the numer of consecutive serves y any player is random, the right to serve in tale tennis changes according to a deterministic rule, and thus rather simple aleit delicate cominatoric arguments allow for otaining all the corresponding proaility distriutions see Section 2. Moreover the formulae we otain are see ppendix B rather lengthy and cumersome. This dismissive view of the prolem is, in our opinion, wrong. Indeed, as already riefly mentioned aove, our formulae can e applied with only minor adaptations to the game of tennis. Moreover, when viewed in comination with the recent findings for the game of adminton reported in Paindaveine and Swan 2010, one sees that, together, oth articles provide a complete and unified proailistic description of a whole class of games, encompassing sports such as tennis, volleyall, racquetall, tale tennis, adminton, etc. Finally these findings open new angles of approach to the interesting prolem of estimating the parameters p a, p in an efficient way and of developing statistical testing procedures for 3

5 these. This latter aspect of our research is, of course, outside the scope of this short paper and will e the suject of a later pulication see Dominicy et al The outline of the paper is as follows. In Section 2, we derive the distriution of the score in the two general server models descried aove, and exploit this result to otain the proailities of winning a set and a match. The corresponding moments of the duration are given in Section 3. On the asis of our stochastic models, a detailed comparison etween what we will from now on refer to as the old and the new scoring system in tale tennis is provided in Section 4. Section 5 descries the methodology ehind the Monte Carlo simulations that are provided in every illustration given in the paper. Some final comments and an outlook on future research are stated in Section 6, while an appendix collects the more technical proof. 2 Game-winning proailities in a single set. In this section we descrie the scoring process within a single set y having recourse to cominatorial arguments. Before stating our results, we need to introduce some notations. We will throughout denote scores y couples of integers, where the first entry resp., the second entry stands for the numer of points scored y player resp., y player B. We will reserve the use of the notation n, k resp., k, n to indicate the final score in a set won y resp., y B without a tie. General intermediate scores will e denoted α, β. We call -set resp., B-set a set in which player resp., player B is the first server. Note the symmetric roles played y and B, which will often allow us to state our results in terms of -sets only. For C 1, C 2 {, B}, we denote y p C 2 C 1 the proaility that C 2 wins a C 1 -set. Now, similarly to Schulman and Hamdan 1977, it can e shown that, for m dividing n 1, p = p B resp., pb = pb B, i.e. the proaility for each player to win a single set is not affected y the choice of the first server in the set. Consequently, the proaility of player resp., player B winning a match can e otained in a straightforward manner y simple conditioning. Since the values of m chosen y international federations always satisfy the aforementioned divisiility constraint, we can and will restrict most of our attention 4

6 on the outcome of a single -set. For C {, B}, we denote y E α,β,c the event associated with a sequence of rallies in an -set that gives rise to α resp., β points scored y player resp., y player B and sees player C score the last point. The latter condition entails that we necessarily assume α > 0 resp., β > 0 when C = resp., when C = B. We denote the corresponding proaility y p α,β,c = P[E α,β,c ]. 2.1 With these notations, an -set is won y resp., y B on the score n, k resp., k, n with proaility p n,k, resp., p k,n,b, for k n 2. Oviously these quantities do not suffice to compute the proaility p C that C wins a set initiated y, since we still need to account for what happens in case of a tie at n 1, an event which occurs with proaility p n 1,n 1, + p n 1,n 1,B. For this we introduce the notation p tie, resp., p tie,b to represent the proaility that resp., B scores 2 more consecutive points than his opponent after the tie; this quantity will have to e computed differently than those defined aove, since the rules governing the game after a tie are not the same as those which were applicale efore this event. With these notations we get p := n 2 k=0 p n,k, + p n 1,n 1, + p n 1,n 1,B p tie,, 2.2 and p B = n 2 k=0 p k,n,b + p n 1,n 1, + p n 1,n 1,B p tie,b. 2.3 The corresponding quantities for B-sets are defined y switching the roles of the two players in these definitions. Determining 2.1 and computing p tie,c is therefore the key to our understanding of the prolem, and this derivation is the suject of the rest of the current section. In order to give the reader a feeling for what kind of mechanics are at play, we will first state our results in the very simple setting of the no-server model; we will then use the intuition provided y these findings in order to solve the general prolem. 5

7 2.1 Game-winning proailities in the no-server model. Within the framework of the no-server model, every rally is won y player resp., y player B with proaility p resp., 1 p, irrespective of the server. The process therefore reduces to a succession of independent Bernoulli trials with success-proaility p. The score distriution is thus a straightforward consequence of standard arguments. Theorem 2.1 Fix α, β N, and let p α,β,c := P[E α,β,c ] with C {, B}. Then p α,β, = α + β 1 p α 1 p β α 1 and p α,β,b = α + β 1 p α 1 p β. α Now note that the numer of points played after a tie and until victory of either or B clearly follows a geometric distriution with parameter 2p1 p. Hence the proaility of resp., of B winning the set after a tie at n 1 is given y p tie, = p 2 /[1 2p1 p] resp., y p tie,b = 1 p 2 /[1 2p1 p]. Hence, from Theorem 2.1 and equations 2.2 and 2.3, we get n 2 p := p n k=0 n + k 1 2n 3 1 p k + 2 n 1 n 1 p n+1 1 p n 1 1 2p1 p, and n 2 p B = 1 p n k=0 n + k 1 2n 3 p k + 2 k n 1 p n 1 1 p n+1 1 2p1 p. 2.2 Game-winning proailities in the server model. The case of the server model is trickier. The scoring process of resp., of B is a succession of m independent Bernoulli trials with success-proaility p a resp., 1 p a, followed y m independent trials with success-proaility 1 p resp., p, followed again y m Bernoulli trials identical to the first and so on, until the end of the set or until the tie is reached. We aim at deriving the proaility of the event E α,β,c. From what we have just written, 6

8 we know that this event is equivalent to a succession of independent aleit not identically distriuted inomial experiments with parameter m, p a or m, 1 p. Hence all related distriutions will e relatively straightforward to derive as soon as we are ale to compute, for any given score, the numer K, say, of serve changes and the numer R, say, of serves y either or B during the last service sequence. This is performed in the following lemma which we state without proof. Lemma 2.1 Let 0 α, β n 1. Define the integers K and R as the unique solutions of α+β = Km+R with R m 1. Then the event E α,β,c can e decomposed into k 1 := K/2 service sequences of and k 2 := K/2 service sequences y B, ended y R serves of resp., of B when K is even resp., when K is odd. Let us now consider an -set won y C C {, B} on the score α, β. From the nature of the game it is clear that, given the score, the only random quantity left is the numer of points or, equivalently, B has otained on his/her own serve. is natural to define the events E α,β,c j E α,β,c Hence it for which the scoring sequence occurs with scoring exactly j points on his/her serve. Denoting y p α,β,c j the corresponding proaility, these quantities can then e derived y exploring the essentially inomial nature of the experiment. These results are the oject of the following Lemma. Lemma 2.2 Let α, β N, and define K, R as in Lemma 2.1. lso, for C {, B}, define δ C as the indicator of the equality etween and C. Then, i if R > 0 and K is even or R = 0 and K is odd, p α,β,c K 2 j = m + R 1 j δ C p j a1 p a K 2 m+r j K 2 m 1 p α j p K 2 m α+j, α j where j {maxδ C, α K 2 m,..., minα, K 2 m + R 1 + δc }; ii if R > 0 and K is odd or R = 0 and K is even, p α,β,c K j = 2 m p j j a1 p a K 2 m j K 2 m + R 1 α j δ C 1 p α j p K 2 m+r α+j, where j {max0, α K 2 m R + 1 δc,..., minα δc, K 2 m}. 7

9 Note that the aove condensed formulae are easier to read if spelt out explicitly in each of the 8 possile cases. We therefore provide a complete version of Lemma 2.2 in the ppendix, accompanied y a formal proof. The distriution of the score in an -set and the proailities that each player wins the set easily follow from Lemma 2.2 y summing over all possile values of j > 0. When the parameters are chosen so as to satisfy the constraint p a = 1 p = p, the corresponding results concur with those otained in Section 2.1. In all other cases, these results do not allow for an agreeale form so we dispense their explicit expression. Theorem 2.2 Fix α, β N, and let p α,β,c C {, B}. Then p α,β, = j=0 p α,β,c j are defined in Lemma 2.2. := P[E α,β,c p α,β, j and p α,β,b = ], where E α,β,c j=0 := j Eα,β,C j with p α,β,b j, where the proailities rguments similar to those given in Section 2.1 see also Schulman and Hamdan 1977 show that p tie, := p a 1 p /[1 p a p 1 p a 1 p ] resp., p tie,b := p 1 p a /[1 p a p 1 p a 1 p ]. Hence, Theorem 2.2 and equations 2.2 and 2.3 readily allow to otain the proailities p and pb. 3 Moments of the numer of rallies. The distriution of the score estalished in the previous section allows us now to investigate the numer of rallies, D say, needed to end a single set. Let S denote the random variale recording the first server of a set and, for C {, B} and E an event, denote y P C [E] := P[E S = C]. The main oject of interest in this section is the function d P [D = d] for d N. Oviously, ecause a point is scored on every rally, the numer of rallies needed to reach a score α, β is equal to α + β. P [D = d] = p n,d n, Hence P [D = d] = 0 for all d n 1 and + p d n,n,b for all n d 2n 1. Finally, for d 2n 1, such length can only e achieved through the occurrence of a tie so that P [D = d] = 0 if d = 2n 1 and otherwise P [D = d] = p n 1,n 1, + p n 1,n 1,B p d,n tie where the latter 8

10 quantity gives the proaility that there are exactly d 2n 1 rallies after the tie. Note that p d,n tie = 0 if d 2n 1 is odd. We again start y illustrating the situation in the setting of the no-server model, in which the different quantities are easily derived from standard cominatorial arguments. We then turn our attention to the general model. 3.1 Moments in the no-server model. Theorem 2.1 oviously provides us with nearly all the proailities we need to derive the distriution of D; it therefore only remains to compute p d,n tie, which is easily achieved in view of the essentially geometric nature of the process. This yields p d,n tie = 2p1 pl 1 p p 2, with l = d 2n 1. This yields the distriution of D, and hence all corresponding moments. 3.2 Moments in the server model. gain the distriution of D is a trivial consequence of the scores, as soon as the prolem of the ties is solved. Now note that under the hypothesis that m divides n 1, the first server at the start of the tie situation is the first server of the set. Hence, in order to have l = d 2n 1 rallies after the tie, it suffices that and B win alternatively during the first l 2 rallies, and that one of the two players scores 2 successive points thereafter. This yields p d,n tie = p a1 p + 1 p a p 1 p a 1 p + p a p l/2 1, and the distriution of D readily follows. 9

11 Figure 1: Distriution of the scores: 5,21,1-scoring system Both sufigures refer to an -set played under the 5, 21, 1-scoring system. Sufigure a: for p a, p =.7,.5,.6,.5,.5,.5 and.4,.5, proailities p n,k, that player wins the set on the score n, k. Sufigure : the corresponding values for victories of B. Estimated proailities ased on 100 replications are also reported thinner lines in plots. Figure 2: Distriution of the scores: 2,11,1-scoring system Both sufigures refer to an -set played under the 2, 11, 1-scoring system. Sufigure a: for p a, p =.7,.5,.6,.5,.5,.5 and.4,.5, proailities p n,k, that player wins the set on the score n, k. Sufigure : the corresponding values for victories of B. Estimated proailities ased on 100 replications are also reported thinner lines in plots. 4 Comparison of the two scoring systems in tale tennis. In this section, we provide an in-depth comparison of the 5, 21, G-scoring system to the new 2, 11, G-scoring system in terms of score distriutions, match-winning proailities and durations. Figures 1 and 2 present, for n = 21 and n = 11 respectively, the score distriution associated with p a, p =.7,.5,.6,.5,.5,.5 and.4,.5. The left suplots contain the 10

12 proaility that player wins an -set in which the opponent scores a total of k points for instance, in the new scoring system, k = 0,..., 9 corresponds to a victory of on the score 11, 0,...,11, 9; in case of a tie, for which k 10, player wins on scores of the form k + 2, k and the right suplots give the proaility for B winning an -game where this time player scores k points, p still eing fixed at.5 and p a varying from.4 to.7. It appears that these distriutions are extremely sensitive to p a, p ; oviously, this extends to the corresponding match-winning proailities. In the 5, 21, G-scoring system, p varies etween.91 and.25 and p B etween.09 and.75, when, for fixed p =.5, p a ranges from.7 to.4. For the other scoring system, p takes values etween.84 and.31, while pb ranges from.15 to.68. Note that as is getting stronger with respect to B, his winning proaility dramatically increases see Figure 1 and Figure 2. Thus, tight matches will happen very rarely. Note that, whatever the value of p a, white-washes seldom occur. Finally note that the score distriutions are not monotone in p a. The dependence of the set-winning proailities on p a, p is of primary importance. In what follows, we will investigate this dependence visually y comparing oth scoring systems in the no-server model p = p a = 1 p. The results are illustrated in Figure 3. Figure 3 a shows the proaility that player wins a set as a function of his strength p. The lue curve gives the proaility for the new scoring system and the red curve for the old scoring system. The plot apparently supports the claim that, for any fixed value of p, the choice of the scoring system arely influences set-winning proailities. In other words, whatever the rules considered, the proaility of winning appears to e almost the same in oth scoring systems. This claim, however, is dampened y the plot reported in Figure 3, where the ratio etween the 5, 21, 1-scoring system and the 2, 11, 1-scoring system is given as a function of p. This graph reveals that although the proaility that player wins an -set is essentially the same for oth scoring systems if he/she is the est player p,old /p,new 1 for p >.7, with evident notations, for p [.5,.7] the proaility that wins the -set in the old scoring system is slightly higher than in the new one, and when p <.5, player is more likely to win the set in the new system than in the old one. Finally, the ratio approaches 0 as p tends to 0 for p <.2, say. This last remark indicates that the 11

13 Figure 3: Comparison s a function of p = p a = 1 p that is, the no-server model, Sufigure a contains the proailities p 21,k, in red that player wins an -set on the score 21,k, along with the proailities p 11,k, in lue that player wins an -set on the score 11,k. Sufigure represents the ratio etween the old scoring system and the new one. Sufigures c and d represent the expectation and the standard deviation of the numer of rallies needed to complete the corresponding set, unconditional on the winner. Estimated proailities, expectations and standard errors ased on 100 replications at each value of p = 0,.0025,...,.9975 are also reported thinner lines. old scoring system leaves severely less winning chances to when he is weak than the new one, i.e. the new scoring system appears more alanced. The duration plots simply confirm what common-sense already tells us. Indeed, we see in Figure 3 c that the expectation of D is, uniformly in p 0, 1, smaller for the new scoring system than for the old one; this is of course a direct consequence of the dramatic reduction in the numer of points necessary to achieve victory of one of the two players. Since the numer of sets necessary to win a match has not een significantly increased y the passage from the old to the new scoring system, this property carries through at the match-level. 12

14 Interestingly, the same conclusion happens to e true for the standard deviation of D the old scoring system varies etween 3.26 and 4.29, the new scoring system etween 2.42 and 3.22, as illustrated in Figure 3 d. The system change has thus indeed rendered the length of the matches more predictale. The twin-peak shape of oth standard error curves is most certainly due to the equality in rally-winning proailities which lead to tighter final scores; this property had already een noted in Paindaveine and Swan Finally note that oth curves of Figures 3 c and d are symmetric aout p =.5, as expected. 5 Simulations. We performed several Monte Carlo simulations, one for each figure considered in the previous section. To descrie the general procedure, we focus on the Monte Carlo experiment associated with the new scoring system in Figure 3 a results for the other figures are otained similarly. For each of the 399 values of p considered in Figure 3, the corresponding values of p p, e p and v p, where e p stands for ED and v p for VarD, were estimated on the asis of J = 100 independent replications of an -set played in the no-server model p = p a = 1 p. Of course, for each fixed value of p, the set-winning proaility p C p is simply estimated y the proportion of sets won y C in the J corresponding -sets: ˆp C := J C J := 1 J J I;j, C j=1 where I;j C, j = 1,..., J, equals one resp., zero if player C has won resp., lost the jth set. The corresponding estimates for e p and v p are given y ê p := 1 J J d ;j, j=1 and ˆv p := 1 J J d ;j ê p 2, j=1 where d ;j, j = 1,..., J, clearly represents the total numer of rallies in the jth set. These estimates are plotted as thin lue lines in Figure 3. Note that we have restricted the 13

15 numer of replications to a lowly 100, since for larger values of J the empirical curves ecome indistinguishale from the theoretical, in all cases except Figure 3. In the latter, where the ratio p,old /p,new is reported, 100 simulations are not enough to contain any victories of when p < 0.4. Hence this ratio is undefined for nearly all values of p in this interval. 6 Final comments. This paper provides a complete proailistic description for games played under the m, n, G- scoring system. It complements the previous contriution y Schulman and Hamdan 1977 and, although we do not state it explicitly, encompasses previous corresponding results on tennis see e.g. Hsi and Burich The formulae provided, although cumersome, can easily e implemented for numerical purposes, and yield some striking illustrations. Our findings also allow for constructing elegant estimation procedures of the respective strength of the players. s will e shown in future work, maximum likelihood estimators for p a and p take on a short and appealing form when final scores and durations are taken into account. Based on Bradley-Terry paired comparison methods, these estimators will allow for constructing interesting new ranking methods within round-roin tournaments. Finally, since our results of Lemma 2.2 are valid for general intermediate scores α, β, an extension to the case where the initial score differs from 0, 0 is readily implemented. This naturally paves the way to a more dynamic analysis of a match, à la Klaassen and Magnus 2003, in which the winning proailities and duration are sequentially estimated throughout the course of a given match. ppendix..1 Complete formulae. Lemma.1 Fix m N 0 and α, β N, and define K, R N via the Euclidian division α + β = Km + R. Then we have 14

16 1. if R = 0 and K is even, then p α,β, x = K 2 m x p x a1 p a K 2 m x K 2 m 1 α x 1 1 p α x p K 2 m α+x, where x {max0, α K 2 m,..., minα 1, K 2 m}, and p α,β,b x = K 2 m x p x a1 p a K 2 m x K 2 m 1 α x 1 p α x p K 2 m α+x, where x {max0, α K 2 m + 1,..., minα, K 2 m}. 2. if R = 0 and K is odd, then p α,β, K x = 2 m 1 p x x 1 a1 p a K 2 m x K 2 m α x 1 p α x p K 2 m α+x where x {max1, α K 2 m,..., minα, K 2 m}, and p α,β,b K x = 2 m 1 p x x a1 p a K 2 m x K 2 m 1 p α x p K 2 m α+x, α x where x {max0, α K 2 m,..., minα, K 2 m 1}. 3. if R > 0 and K is even, then p α,β, x = K 2 m + R 1 x 1 p x a1 p a K 2 m+r x K 2 m α x 1 p α x p K 2 m α+x, where x {max1, α K 2 m,..., minα, K 2 m + R}, and p α,β,b x = K 2 m + R 1 x p x a1 p a K 2 m+r x K 2 m α x 1 p α x p K 2 m α+x, where x {max0, α K 2 m,..., minα, K 2 m + R 1}. 15

17 4. if R > 0 and K is odd, then p α,β, K x = 2 m p x x a1 p a K 2 m x K 2 m + R 1 α x 1 1 p α x p K 2 m+r α+x, where x {max0, α K 2 m R,..., minα 1, K 2 m}, and p α,β,b K x = 2 m p x x a1 p a K 2 m x K 2 m + R 1 1 p α x p K 2 m+r α+x, α x where x {max0, α K 2 m R + 1,..., minα, K 2 m}..2 Proof of the complete formulae. Since all the formulae follow a very similar pattern, we only provide formal proofs for two of the eight expressions, namely for p α,β, when R > 0 and K is odd. x when R = 0 and K is even and for p α,β,b x Suppose α and β are such that α + β = Km with K even, and player scores the last point. In this setting, the last point is served y player B, hence only the outcome of K 2 m 1 serves of player B are random, while all the K 2 m serves of player are random. Thanks to the independence of the rallies, we may consider the serves of player and player B separately. Moreover, as explained in Section 2, each sequence of serves corresponds to a inomial distriution, with parameters K 2 m, p a for player serving and parameters K 2 m 1, 1 p for player B serving, where we adopt each time the point of view of player. Since player scores x points on his own serve, he necessarily wins α x rallies initiated y player B. The last point eing considered apart, we are left with α x 1 successes for the inomial distriution with parameters K 2 m 1, 1 p. Comining all these facts yields the announced formula, and it remains to estalish the domain of the possile values of x. s player B serves exactly K 2 m times, it is clear that, whenever α > K 2 m, necessarily player has to score at least α K 2 m points on his own serve. Otherwise, he/she may very well score only on player B s serve. These two oservations readily yield the lower ound 16

18 for x. Regarding the upper ound, as player scores at least one point not served y himself, he cannot score more than α 1 points on his/her own serve. This directly yields the upper ound, as player is limited to K 2 m serves. Suppose now that α + β = Km + R with K odd and R > 0, and player B scores the last point. In this case, player B also serves the last point; would R e zero, then the oddness of K would give the last serve to player. With this said, player serves exactly K 2 m points, while player B initiates K 2 m + R exchanges. The formula for pα,β,b x and the corresponding lower and upper ound for x follow along the same lines as for p α,β, x, which concludes the proof. References W.H. Carter & S.L. Crews n analysis of the game of tennis. The merican Statistician, 28, Y. Dominicy, C. Ley, D. Paindaveine & Y. Swan Estimation and testing for rallywinning proailities in two-person sports. Manuscript in preparation. S.L. George Optimal strategy in tennis: a simple proailistic model. Journal of the Royal Statistical Society Series C, 22, B.P. Hsi & D.M. Burich Games of two players. Journal of the Royal Statistical Society Series C, 20, F.J.G.M. Klaassen &, J.R. Magnus Forecasting the winner of a tennis match. European Journal of Operations Research, 148, D. Paindaveine & Y. Swan stochastic analysis of some two-person sports. Sumitted. D.F. Percy mathematical analysis of adminton scoring systems. Journal of the Operational Research Society, 60, M.J. Phillips Sums of random variales having the modified geometric distriution 17

19 with application to two-person games. dvances in pplied Proaility, 10, G.H. Riddle Proaility Models for Tennis Scoring Systems. Journal of the Royal Statistical Society Series C, 37, R.S. Schulman & M.. Hamdan proailistic model for tale tennis. The Canadian Journal of Statistics, 5, J. Simmons proailistic model of squash: strategies and applications. Journal of the Royal Statistical Society Series C, 38,