Swinburne Research Bank http://researchbank.swinburne.edu.au Clarke, SR & Norman, JM (2012). Optimal challenges in tennis. Accepted for publication in Journal of the Operational Research Society. Published version will be available after publication from: http://www.palgrave-journals.com/jors/. Copyright 2011 Operational Research Society Ltd. All rights reserved. This is the author s version of the work. It is posted here with the permission of the publisher for your personal use. No further distribution is permitted. If your library has a subscription to this journal, you may also be able to access the published version via the library catalogue. Accessed from Swinburne Research Bank: http://hdl.handle.net/1959.3/211853
Optimal challenges in tennis Stephen R Clarke 1 and John M Norman 2 1 Faculty of Life and Social Sciences, Swinburne University, Hawthorn, Victoria, AUSTRALIA: sclarke@swin.edu.au 2 Sheffield University Management School, Sheffield, UK. Abstract. The use of technology in sport to assist umpires has been gradually introduced into several sports. This has now been extended to allow players to call upon technology to arbitrate when they disagree with the umpire s decision. Both tennis and cricket now allow players to challenge a doubtful decision, which is reversed if the evidence shows it to be incorrect. However the number of challenges is limited, and players must balance any possible immediate gain with the loss of a future right to challenge. With similar challenge rules expected to be introduced in other sports, this situation has been a motivation to consider challenges more widely. We use Dynamic Programming to investigate the optimal challenge strategy and obtain some general rules. In a traditional set of tennis, players should be more aggressive in challenging in the latter stages of the games and sets, and when their opponent is ahead. Optimal challenge strategy can increase a player s chance of winning an otherwise even 5 set match to 59%. Keywords: Sports, Dynamic Programming, Optimisation, Challenge, Tennis Introduction Technology is being used increasingly in sport to assist, and in some cases replace, judges and referees. The photo finish has been used to help the steward decide close horse races since the camera was invented. In some sports, such as swimming, we have seen the judges replaced entirely by electronic timing. In tennis we have had the electronic service line machine for many years, and now also a device for picking up net cords. More recently, some forms of Rugby have used a video referee to decide on the legality of tries. Cricket followed suit, allowing umpires to refer decisions on run outs, stumpings and catches to a third umpire with access to video footage. In many other sports the media use video replays or other technology (such as the hot spot in cricket) to provide evidence for or against the umpire s decision. In 2008 the Australian Tennis Open saw an interesting development with the introduction of Hawkeye. This system, relying on several video cameras and some mathematical modelling, was originally used in cricket, where it claims to show where the ball actually went or would have gone had it not hit the batsman. In tennis, it displays a schema of the court lines along with a mark where the ball is believed to have bounced, along with a decision on whether it was in or out. (Interestingly, the path of the ball is never shown with any error bounds: the public and players appear to accept that it is exact and infallible). The interesting
development in tennis was that the players, not the umpires, under certain conditions were allowed to challenge the umpire s decision by referring to Hawkeye. If Hawkeye sided with the appealing player, the umpire s decision was reversed. The International Cricket Committee has now introduced a similar rule into the playing conditions of some cricket series, and it seems inevitable that allowing players to challenge umpires decisions will play an increasing part in many sports. But some poor decision making by players shows they do not always make good use of their right to challenge. The number of challenges is limited, and players must balance any possible immediate gain with the loss of a future right to challenge. This provides the motivation to investigate the optimal strategy in the use of challenges in a wider context. Norman (1995) gives several examples of the use of Dynamic programming to find optimal strategies in sport, and Norman (1985) uses it to find an optimal serving strategy in tennis. We show here how it can be used here to find the optimal challenge strategy under simplified and more realistic rules for tennis, and thus formulate some general and more specific rules. A simple model to determine challenge policy In a game of tennis, on some courts with the required technology, at the end of a point a player may challenge a line call. For example, if his opponent s ball had been called in when in fact it was out, then if his challenge is successful, he may win the point (or if the miscall occurred on his opponent s first serve, the opponent may be required to serve again). Again, if his own ball had been called out when in fact it was in, he may earn a replay of the point, or even be awarded the point if it is deemed that his opponent would have been unable to return the ball. A player may make up to three unsuccessful challenges in a set, up to four if a tie break is reached. The scoring system in tennis makes the game difficult to model and we first consider here a simpler game in which two players compete to be the first to gain n points. It could be thought of as tennis with different scoring. For example with n = 21 it is not very different from the game of table tennis played under the old rule of first to 21, and with n = 7 or 10 similar to a tiebreaker or super tiebreaker game in tennis. Suppose A (our man) is playing B. With probability p A wins the point outright. Of the points A loses, with probability p c a challenge opportunity may occur, and if A makes a challenge the probability of success is s c. The point has just been played and one of three states occurs: W: A is about to be given the point outright; L: B is about to be given the point and A thinks this is right; C: A call has been made such that if it stands B will win the point, but A thinks there is a good chance he would be awarded it if he challenges the call. The probabilities of the three states W, L and C occurring are respectively p, (1-p)*(1- p c )and (1-p)*p c Before the umpire says anything, we take the score to be i-j. A has m challenges left. We take the state of the system to be (i, j, m, θ) where θ can take the values W, L and C. Define f(i, j, m, θ) as the maximum probability of A winning the game, with the score i-j and A having m challenges left, with θ the state of play. Then 2
f(i, j, m, W) = p*f(i+1, j, m, W) + (1-p)*(1- p c )*f(i+1, j, m, L) + (1-p)*p c *f(i+1, j, m, C) (1) f(i, j, m, L) = p*f(i, j+1, m, W) + (1-p)*(1- p c )*f(i, j+1, m, L) + (1-p)*p c *f(i, j+1, m, C) (2) f(i, j, m, C) = max don t challenge: p*f(i, j+1, m, W) + (1-p)*(1- p c )*f(i, j+1, m, L) + (1-p)*p c *f(i, j+1, m, C) challenge: s c *[ p*f(i+1, j, m, W) + (1-p)*(1- p c )*f(i+1, j, m, L) + (1-p)*p c *f(i+1, j, m, C)] + (1- s c )*[ p*f(i, j+1, m-1, W) + (1-p)*(1- p c )*f(i, j+1, m-1, L) + (1-p)*p c *f(i, j+1, m-1, C)] (3) Since having an extra challenge can never decrease A s chance of winning (i.e. f(i, j, m, θ) >= f(i, j, m-1, θ)), it is easily shown that the challenge test quantity is monotone increasing in s c. Suppose the two test quantities are equal when s c = π. Then for s c > π it is better to challenge and for s c < π it is better not to challenge. Thus the form of the optimal policy is Challenge if and only if the probability of success is greater than some probability π. Such a strategy would apply in a single tiebreaker game of tennis. A computable model for tennis One (big) disadvantage of this formulation is the number of variables in the state description. For computational simplicity, we may reduce the number of variables by one by taking the time at which a decision is made to be when player A is about to serve (or receive a serve). We suppose that he then asks himself whether if an opportunity to challenge occurs, he will take it. While the occurrences of challenge opportunities with different positive probabilities of success will follow a continuous probability distribution, we replace this with a finite set of challenge types. The state of the system is (i, j, m) where i - j is the score: i and j are the points each player (A and B, respectively) has earned so far in the game. m is the number of challenges left. We consider A as the player who decides whether or not to challenge, and who has three challenges available at the start of the game. We suppose that challenge possibilities are of two types, occurring with probabilities p 1 and p 2. If player A makes a challenge, his probability of success is s 1 and s 2 respectively (s 1 >s 2 ). Our player A wins a point with probability p (taking into account any successful challenges by his opponent). More correctly, the umpire initially awards him the point with probability p. But a proportion p 1 + p 2 of all rallies (and hence of rallies that A loses) finish with a questionable line call that the losing player may challenge. If A challenges successfully he wins the point and the state of the system becomes (i+1, j, m). If the challenge fails, he loses the point and one right to challenge, and the state becomes (i, j+1, m-1). Just before a point is played, player A may consider three possibilities: A: With probability p, A wins the point and with probability (1-p)*(1- p 1 - p 2 ) A loses the point and there is no possibility of a successful challenge. 3
B: With probability (1-p)*p 1, player A loses the rally but A thinks there is a good chance (probability s 1 ) a challenge would be successful and he will gain the point. If it is unsuccessful; he will lose the point and lose one right to challenge. C: With probability (1-p)*p 2, player A loses the rally but A thinks there is a medium chance (probability s 2 < s 1 ) a challenge would be successful and he will gain the point. If it is unsuccessful; he will lose the point and lose one right to challenge. The decision problem faced by A before the point is played is thus to choose one of three alternatives: 0: Not to challenge, even if a possibility occurs 1: To challenge if and only if possibility B occurs 2: To challenge if and only if either possibility B or possibility C occurs (since s 1 >s 2 it is never optimal to challenge only if possibility C occurs ) Define f(i, j, m) as the maximum probability of A winning the set, with the score i j and A having m challenges left. Then the functional equation is f(i, j, m) = p* f(i+1, j, m) + (1-p)*max 0: f(i, j+1, m) 1: (1- p 1 ) * f(i, j+1, m) + p 1 *[ s 1 * f(i+1, j, m) +(1- s 1 )*f(i, j+1, m-1)] 2: (1- p 1 - p 2 ) * f(i, j+1, m) + p 1 *[ s 1 * f(i+1, j, m) +(1- s 1 )*f(i, j+1, m-1)] + p 2 *[ s 2 *f(i+1, j, m) +(1-s 2 )*f(i, j+1, m-1)] for i, j < n, m = 1, 2, 3. (4) f(i, j, 0) = p*f(i+1, j, 0) + (1-p)*f(i, j+1, 0) for i, j < n (5) f(n, j, m) = 1 and f(i, n, m) = 0, for all i, j < n, m = 0, 1, 2, 3. (6) The form of the functional equations for each decision is more clearly seen in the table below, where the body of the table contains the coefficients of the functional equation for each decision. The pattern shows how the formulation is easily extended to increase the number of challenge types to better approximate what in reality would be a continuous function. Alternatively the model can be simplified to one challenge type by simply equating the superfluous p 2 to zero. -------------------------------------------------------------- Table 1: Coefficients of objective equation ---------------------Table 1 about here ------------ 4
Model calibration Table 2 summarises data on the success rate of challenges during the 2009 Wimbledon championship, obtained from http://www.wimbledon.org/en_gb/scores/challenge/index.html accessed 11 April 2003. -------------------------------------------------------------- Table 2 Statistics on Challenges during Wimbledon 2009. ------------------Table 2 about here---------------------- The total challenges in Table 1 relate to matches played with Hawkeye, a subset of 47 out of 127. Assuming the presence of Hawkeye has no effect on the number of sets in a match, we can calculate the number of challenges per set. We would really like to have average challenges per set, so we looked at the men s singles results. 474 sets were played in 125 matches played to completion, giving the average sets per match as 473/125 = 3.8. The average number of challenges per set is thus 6.7/3.8 = 1.8. We first consider the above model with challenges limited to two types, with probabilities of success 0.4 and 0.2. (They can t be close to 1, as the line judges rarely make bad mistakes, and they can t be close to 0 for then a challenge would not be worthwhile). How often do challenge opportunities occur? More guesswork is needed. Suppose, on average, that one type 1 opportunity and two type 2 opportunities occur per set. We might suppose that players takes up all type 1 opportunities and half of type 2, giving an average of two challenges per set and an average success rate of (0.4 + 0.2)/2 = 0.3 or 30%. It s easy to juggle with these figures. If only a quarter of type 2 opportunities are taken up, then there will be an average of 1.5 challenges per set and an average success rate of (0.4 + 0.1)/1.5 = 0.33 or 33%. The point of all this is to suggest a set of credible values. Let s take the ones in the preceding paragraph and suppose that, on average, a player has 0.5 type 1 opportunity and 1.0 type 2 opportunity per set. These values seem reasonable. A player is allowed up to three unsuccessful challenges in a set. This is presumably thought to be a reasonable maximum. If a player makes one challenge in a set, on average, then if the number of challenges follows a Poisson distribution, the probability that he makes three or fewer challenges is 0.98, a 2% chance that he cannot make as many challenges as he would like. As any number of points between n and 2n-1 points could be played we take an average of 1.5n points. Thus to ensure approximately 2 challenges per set we take p 1 = 1/(1.5n) and p 2 = 2/(1.5n). s 1 = 0.4 and s 2 = 0.2 as suggested earlier. In an even set, half of these or 1 per set will occur for each player. Results Barnett & Clarke (2002) show how an Excel spreadsheet can be used to calculate a player s chance of winning and the expected number of points in a tennis match using a model effectively the same as above but 5
with no challenge opportunity (i.e. decision 0 is the only option). The extra decisions are easily included by incorporating extra sheets. An Excel spreadsheet has been written to solve equations (4)-(6) for n up to 30, and allowing all other parameters to be altered. In all games, with more than one challenge left, in almost every situation, the optimal decision is Option 2 to take up every challenge opportunity. When the game is virtually over with the better player very close to winning and leading by many points, there is no computable difference in the strategies. For example, when A has a 60% chance of winning any point and leads 28 2 in the first to 30 game, his probability of winning is very close to one and it matters little which decision he makes. p = 0.6 might be relevant for a seeded player playing a non-seeded player, p = 0.4 for his opponent. These might be reasonable values for many of the matches played on the show courts for which the challenge system is used. The decision table for a game to 30 with one challenge left for values of p of 0.4 and 0.6 is shown in table 3. With only one challenge left, the choice depends mainly on the opponent s score, and is reasonably insensitive to changes in p. Only very mismatched players would have p values outside this range. With these values the respective players would have less than 6% and more than 94% chance of winning, yet the optimal strategy differs in only a small range when the opponent has won about 5 or 6 points. As the player becomes weaker, his relative reward for a successful challenge becomes greater, and he should challenge more often. Note however that for a weaker player with p = 0.4, the score moves in a random walk, with the player expecting to score less, perhaps considerably less, than his opponent, so that this player would likely hit the Challenge at every opportunity area very early in the game, around the score 3-5 or even earlier. On the other hand, a stronger player with p = 0.6 will expect to score more, perhaps considerably more, than his opponent, so he would not change his decision rule until later, perhaps around 12-8. For two even players (p = 0.5) playing up to 30 a near-optimal policy for either player would be Option 2 if the other player s score is 8 or more and Option 1 otherwise. For an even game up to 20 the critical opponent score is 5, and for first to 7 it is 2. In games first to n the critical score for B is about n/4. If this decision rule were applied to a set in tennis, then the recommended rule would be: take every opportunity to challenge, but with only one challenge remaining, if your opponent has won two games or fewer, challenge only if you have a (relatively) high chance of success -------------------------------------------------------------- Table 3. Optimal strategy in first to 30 game, with one challenge left ------------------Table 3 about here---------------------- A more realistic model of tennis. To make the model more realistic we need to introduce the nested scoring system used in tennis. A set of tennis consists of games nested within sets. A player s score consists of a set score (number of games won so far in the set), and a game score (number of points won in a game). When a player wins a game, his set score advances by one, and both players game scores revert to zero for the next game. In practice there is a third 6
level of nesting, match score in sets, but for our purposes this can be ignored as the number of challenges left always resets at the start of a set. Thus the ordered pair (i,j) used to designate the game score in the above model is actually an ordered pair of ordered pairs ((i 1,i 2 ), (j 1,j 2 )), being ((Player A set score in games, player A game score in points), (Player B set score in games, player B game score in points)). This complicates the transition equations (4) - (6). Normally, a point won by player A transforms ((i 1, i 2 ), (j 1,j 2 )) to ((i 1, i 2 +1), (j 1,j 2 )), but at game point for player A it transforms ((i 1, i 2 ), (j 1,j 2 )) to ((i 1 +1,0), (j 1,0)), with similar transitions when player B wins a point. There are various rules for what constitutes a game. For example, in the mixed doubles at the Australian Open 2010, a game is first to 4 points. (For simplicity in exposition, we refer to 0, 1, 2 and 3 instead of the conventional love, 15, 30 and 40). Thus the alternative transition rule applies whenever i 2 or j 2 = 3. More usually, advantage games are played, where a player has to be ahead by two points. Thus at 4 points to 3 up (advantage A), player A wins the game if he wins the next point, but if he loses the score reverts to 3 all (deuce). In our notation ((i 1,4), (j 1,3)) becomes ((i 1 +1,0), (j 1,0)) if A wins the point, but becomes ((i 1,3), (j 1,3)) if he loses it. There are also alternative rules for what constitutes a set. An advantage set is won by the first player to reach 6 games, provided they are at least 2 games ahead. Thus 6 games to 5 in a set is analogous to 4-3(advantage A) in a game. It is now becoming more common at 6 games all to play a single first to 7 points tiebreaker game to decide the winner of the set. While all these refinements could be incorporated into equations 4-6, they would be very complicated and do little to increase our understanding. However they are relatively straight forward to implement on an Excel spreadsheet. The recursive nature of deuce and advantage sets can be handled within Excel by enabling the iterative calculations option. It is also possible to introduce other refinements into the model. In tennis the player serving alternates each game. As the server in men s tennis can win up to 70% of his points, his chance of winning a point changes significantly each game. This can be allowed for by replaced the parameter p with p A and p B depending on whether i 1 + j 1 is odd or even. To better approximate a continuous function, we also allow for a third challenge type, with half the chance of success and twice the frequency of occurrence as the previous type 2 challenge. Although this increases the average number of challenges per set, this is not unreasonable. The Australian Open website http://www.australianopen.com/en_au/index.html accessed 11 April 2003 shows that although the chance of success was still around 30%, the number of challenges per match increased by over 20% when compared with the previous Wimbledon. Perhaps players are becoming more used to the challenge system. There are now four choices: 0: to never challenge; 1: to challenge only if an opportunity with the highest chance of success occurs; 2: to challenge if a high or medium chance opportunity occurs; and 3: to challenge at every opportunity. A separate sheet on the spreadsheet can be used to show the average number of points in such a set is between 50 and 60 for a range of values of p A and p B. Hence we use p 1 = 1/55, p 2 = 2/55, p 3 = 4/55 and s 1 = 0.4, s 2 =0.2, s 3 = 0.1. 7
We illustrate here the results obtained for an advantage six game set, each game being first to 4 points, which allows for different chances of winning a point depending on whether player A is serving or receiving. This model illustrates all the salient points from a more complicated model allowing for advantage games. In this first example we look at two equal players each with a winning service percentage of 60%. Thus p A = 0.6, p B = 0.4, and A serves for the first game of the set. The spreadsheet confirms that with no challenges left, player A has a 50% chance of winning the set. However the spreadsheet shows that with one challenge left he has a 53% chance of winning the set, which increases to 54% and 55% with 2 and 3 challenges left. A 55% chance of winning the set results in a 59% chance of winning a 5 set match. This demonstrates the value of having challenges in reserve and using the challenge system optimally. With 3 challenges left player A should always take up a maximum chance challenge, and take up any challenge at 2 or 3 all in a game. Later in the set (after the fourth game) virtually always challenge except on a few occasions when well up or down in the game. Earlier in the set (first 3 games) he should take up a medium challenge, except when leading 40-0 on serve, or down 0-40 on his opponents serve, he should only challenge if there is a high chance of success. With 2 challenges left, he generally always challenges later in the set (after the sixth game), generally near the end of a game and always at 3 all or when down 2-3 on his own serve. This pattern of increasing frequency of challenging later in the set, later in the game, and when in trouble on serve is shown more clearly in Figure 1, which shows the optimal strategy with one challenge left. Clearly the player should be much more conservative with the optimal choice generally decreasing by one compared with the strategy with 2 challenges left. Early in the set a player should not challenge at all if up 40-0 on service or down 0-40 on his opponents, and in the early parts of early games should only make challenges with a maximum chance of success. The tendency to challenge increases as you get further into the game, further into the set, and when behind. Thus most of the challenge at every opportunity states occur in the upper half (ie above the leading diagonal) of the small grids in the upper half of the larger grid. Of the 100 states in which the player should always challenge, only 12 occur in the first 3 points of the game, only 17 in the first 5 games of the set, and only 20 when the player is ahead. It seems players should use risky challenges to avoid disaster rather than press home an advantage. This is most graphically illustrated with the contrast between the last column and last row. When one game away from losing, always challenge is the most common strategy, whereas it rarely is when one game away from winning. Presumably this is because when ahead there is a higher probability of the player getting a good challenge chance later in the set if the set is extended due to the opponent making a comeback. Clearly a player has to balance the importance of the point, along with the chance of another challenge opportunity arising later in the set. Points become more important later in the game and in the set, but the chance of another challenge opportunity arising decreases as the set progresses. There are some challenges in which the successful challenger wins only a replay of the point. If an out call on a ball which the opponent could have played is successfully challenged, the point is replayed. While we have not included this possibility in any of the models, since the rewards for a successful challenge are not as good, clearly the player should be less aggressive in challenging such calls. 8
---------------------------------------------------------------- Figure 1 Optimal strategy with one challenge left for player serving first game, both players winning 60% of their service points. ------------------Figure 1 about here---------------------- Conclusion Analysing simplified rules can be helpful in generating simple rules towards optimal challenge strategy. Results suggest that in a simple first to game, the optimal strategy will also be fairly simple always challenge when you have a good chance of success, and take any challenge once you get deep enough into the game that it looks as if you might not use all your challenges. However the decision rules are not as straightforward with the nested scoring system used in tennis. In the first to n game analysed here, there is no sense in saving a challenge until a more important stage of the game. Once a challenge opportunity with the maximum chance of success arises, there is nothing to be gained by saving that challenge to later in the game, as if the challenge is successful that point stays on your score until the game is over. Keeping it in a first to 20 game in case you have a similar challenge opportunity at 19 all is futile, since if you used it successfully earlier you would already be 20-18. The only consideration is then how long into the match does it become unlikely that another maximum chance challenge opportunity will arise at that stage you may take a lesser chance challenge. This is not true in the nested scoring system used in many racquet sports, in particular tennis. A set consists of first to six games, and once a game is won there is no advantage in winning it to love as opposed to winning it to 15. A challenge opportunity arising at 40-love might not be taken up, as the game will probably be won anyway. Morris (1977) defines the importance of points in tennis. This preliminary analysis of the actual scoring system used in tennis suggests the optimal strategy depends on the importance of the point the more important the point in winning the set, the more likely a player should challenge. Since importance increases in later points of close games, and in later games of close sets, this implies that players should save their challenges until needed deeper into close games and sets. However this must be balanced against the possibility that another challenge opportunity may not arise. A player well ahead will have more chances of challenge opportunities should his opponent make a comeback, and so might be sensible to save his challenge. But a player well behind may not get another opportunity, so should be more aggressive with his challenges. This analysis shows that optimal use of the 3 challenges available can increase a player s chance of winning a set to 55% in an otherwise even contest. This increases their chance of winning a best of 3 set match to 58%, and a best of 5 set match to 59%. Clearly players should give careful consideration to their challenge strategy. 9
References Barnett TJ and Clarke SR (2002). Using Microsoft Excel to model a tennis match. In: Cohen G and T. Langtry (eds.) The Sixth Australian Conference on Mathematics and Computers in Sport. University of Technology Sydney: Sydney, Australia, pp 63-68. Morris C (1977). The most important points in tennis. In: Ladany SP and Machol RE (eds.) Optimal Strategies in Sports. North Holland: Amsterdam, pp 131-140. Norman JM (1985). Dynamic Programming in tennis - when to use a fast serve. J Opl Res Soc, 36: 75-77. Norman JM (1995). Dynamic programming in sport: A survey of applications. IMA J Math Appl Business Ind 6(December): 171 176. 10
Figure 1 Optimal strategy with one challenge left for player serving first game, both players winning 60% of their service points. Opponent Score 00 0 1 2 3 01 0 1 2 3 02 0 1 2 3 03 0 1 2 3 04 0 1 2 3 05 0 1 2 3 0 1 1 1 1 0 1 1 1 0 0 1 1 2 1 0 2 1 1 0 0 1 2 2 2 0 2 3 3 3 1 1 1 2 1 1 1 1 1 1 1 1 1 2 2 1 2 2 2 1 1 1 1 2 3 1 2 2 3 3 2 1 1 2 2 2 1 2 2 2 2 0 1 2 3 2 1 2 2 2 2 0 1 2 3 2 1 2 3 3 3 0 1 1 3 3 1 1 2 3 3 0 0 1 3 3 1 1 2 3 3 0 0 1 3 3 0 1 2 3 10 0 1 2 3 11 0 1 2 3 12 0 1 2 3 13 0 1 2 3 14 0 1 2 3 15 0 1 2 3 0 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 1 2 2 2 0 2 2 2 1 0 1 2 3 3 1 1 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 1 2 2 1 2 2 2 2 1 1 1 3 3 2 1 2 2 1 2 1 1 2 2 2 1 2 2 2 2 0 1 2 3 2 1 2 3 3 2 0 1 2 3 P 3 1 1 2 3 3 0 1 2 3 3 1 1 2 3 3 0 0 2 3 3 1 1 2 3 3 0 0 1 3 l a 20 0 1 2 3 21 0 1 2 3 22 0 1 2 3 23 0 1 2 3 24 0 1 2 3 25 0 1 2 3 y 0 1 1 1 1 0 1 1 1 0 0 1 2 2 1 0 2 1 1 0 0 1 2 2 2 0 2 3 3 3 e 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 1 2 2 2 1 1 1 2 2 3 1 2 3 3 3 r 2 1 1 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 1 1 2 3 2 2 2 3 3 3 0 1 2 3 3 1 2 2 3 3 0 1 2 3 3 1 2 2 3 3 0 1 2 3 3 1 1 3 3 S c 30 0 1 2 3 31 0 1 2 3 32 0 1 2 3 33 0 1 2 3 34 0 1 2 3 35 0 1 2 3 o 0 1 1 0 0 0 1 2 1 1 0 1 1 1 0 0 2 2 2 1 0 2 2 2 1 0 1 2 3 3 r 1 1 1 1 0 1 1 2 2 1 1 2 2 1 1 1 1 2 2 2 1 2 2 2 2 1 1 2 3 3 e 2 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2 1 2 2 3 2 2 2 3 3 2 1 1 3 3 3 1 2 2 2 3 0 1 2 3 3 1 2 2 3 3 0 1 2 3 3 1 2 3 3 3 0 1 2 3 40 0 1 2 3 41 0 1 2 3 42 0 1 2 3 43 0 1 2 3 44 0 1 2 3 45 0 1 2 3 0 1 1 1 1 0 1 1 0 0 0 2 2 2 1 0 2 1 1 0 0 2 2 2 2 0 3 3 3 3 1 2 2 2 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 1 1 2 2 3 3 1 3 3 3 3 2 1 2 2 2 2 2 2 2 1 2 1 2 2 2 2 2 2 2 2 2 1 2 3 3 2 2 3 3 3 3 1 2 2 2 3 2 2 2 3 3 1 2 2 3 3 2 2 3 3 3 0 1 2 3 3 2 2 3 3 50 0 1 2 3 51 0 1 2 3 52 0 1 2 3 53 0 1 2 3 54 0 1 2 3 55 0 1 2 3 0 1 1 0 0 0 2 2 1 1 0 1 1 1 0 0 2 2 2 1 0 2 2 1 0 0 2 2 2 2 1 1 1 1 0 1 2 2 2 1 1 2 1 1 1 1 2 2 2 2 1 2 2 2 1 1 2 2 3 3 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 3 3 2 2 1 2 3 3 3 2 2 2 2 3 2 2 2 2 3 2 2 2 3 3 2 3 3 3 3 3 3 3 3 3 0 1 2 3 11
Table 1: Coefficients of objective equation Decis Functional equation term ion f(i+1, j, m) f(i, j+1, m) f(i, j+1, m-1) 0 p (1-p) 0 1 p+(1-p)*p 1 *s 1 (1-p)*(1-p 1 ) (1-p)*p 1 *(1-s 1 ) 2 p+(1-p)*[p 1 *s 1 +p 2 *s 2 ] (1-p)*(1-p 1 -p 2 ) (1-p)*[p 1 *(1-s 1 )+p 2 *(1-s 2 )] 12
Table 2 Statistics on Challenges during Wimbledon 2009. Men Women Total challenges 314 130 Successful challenges 93 38 Unsuccessful challenges 221 92 Percentage successful 29.6 29.2 Average challenges per match 6.7 3.8 13
Table 3 Optimal strategy in first to 30 game, with one challenge left B S c o r e 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >=15 0 1 p=0.4 2 3 p=0.6 4 5 6 A 7 S 8 c 9 OPTION 1 OPTION 2 o 10 CHALLENGE CHALLENGE r 11 ONLY IF HIGH AT EVERY e 12 PROBABILITY OPPORTUNITY 13 OF SUCCESS 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 14
Captions for figures and tables Table 1: Coefficients of objective equation Table 2 Statistics on Challenges during Wimbledon 2009. Table 3 Optimal strategy in first to 30 game, with one challenge left Figure 1 Optimal strategy with one challenge left for player serving first game, both players winning 60% of their service points. 15