|
All points are not equal |
||
All players realize that tennis's unique
scoring system makes some situations
more
important than others. Careless or
lax play
is of course never acceptable. But
there
are times when things become extremely
critical,
when the player must bear down to the
fullest,
raising his game and using every reserve
of energy and tactic to give himself
the
best chance of winning the immediate
exchange.
Conversely, there are times where less
is
at stake, when longer-term considerations
may and probably should prevail. At
such
times a player might choose to mix
in varied
tactics, for example, thereby reducing
his
predictability in later situations.
He might
seek a prolonged point to weaken the
opponent's
stamina, or perhaps try to end the
point
quickly to preserve his own strength.
Often
a player may hold off using a favorable
ploy,
thereby preserving and enhancing its
surprise
value for a more-critical juncture.Making such choices intelligently calls for an understanding of how points differ in importance. Armed with a PC home computer and spreadsheet software, we can readily calculate the purely numerical importance of all situations. Note, however, that such calculations cannot be a perfect guide to playing the score, as it neglects the unpredictable mental aspects that can arise in various situations. 1. The basic situations in tennisThe importance of each point played within a given game is measured by how its outcome affects the game-winning probabilities of the players. In the examples that follow, we assume that the two players are of equal ability and that each when serving has a 60% chance of winning a given point.To illustrate the calculations, let us examine the first point of a game. At the point's outset, server's probability of winning the game is .73573, or 73.6%. If server wins the first point, making the score 15-love, his game-winning probability becomes .84207. He has thus improved his game-winning probability by .10634. But if he loses the point, reaching love-15, his game-winning probability falls to .57622, having declined by .15951. Averaging the two values (.10634 and .15951) weighted in proportion to server's 60% point-winning frequency, we determine that the average effect of the first point on the game's outcome is to alter the probabilities by .12761. Thus, the first point, at .12761, is of moderate importance. Not surprisingly, the calculations confirm that the most important situation occurs at score 30-40, or "ad-out," when server's game-winning probability is .41538. If server wins, the score becomes deuce and server's game-winning probability improves to .69231. If server loses, his probability becomes zero. The weighted mean change in the outcome probabilities is .33231, representing the point's importance. The 30-40 is thus roughly 2.6 times more important than the game's first point and, from the table below, is 14 times more important than a point played at 40-love. Listed here are the fifteen possible situations in a standard (non-tiebreak) game, along with the calculated importance of each. 30-40 or ad-out, .33231 30-30 or deuce, .22154 15-30, .21268 15-40, .19938 love-30, .17546 love-15, .16589 15-15, .15951 40-30 or ad-in, .14769 first point, .12761 30-15, .12406 love-40, .11963 15-love, .10208 30-love, .06360 40-15, .05908 40-love, .02363 It can be seen that more-important points generally occur when server is behind. Perhaps surprising is the importance of the 15-30 point--the third-most-important situation, closely behind the deuce situation in importance and ahead of the 15-40. The finality of winning at 40-30 would seem to make that situation very important, and perhaps it is in a psychological way. But the calculations show that the 15-30 is considerably more important than the 40-30 in its effect on the ultimate outcome probabilities. Indeed, it can be suggested that the 15-30 is the most widely undervalued point in tennis. The least important situations, and thus the ones most suitable for alternative, longer-term strategies, occur when server leads by two or three points, at 40-15, 30-love, or 40-love. Note that when server leads by only one point, as at 15-love and 30-15, the cost if points are lost becomes considerably heavier. Things are very different when it is receiver who leads by two points, at love-30 or 15-40. These are among the high-importance situations, and it would seem that the player should use optimum tactics to best assure winning these points. The numerical importance of any point is the same for both server and receiver. But because receiver is the underdog for winning any point, there are slight differences in how each may choose to play the score. At 30-40, for example, server should bear down ruthlessly to exploit his 60% inherent advantage. Receiver, meanwhile, might be more inclined to employ a surprise variant, one that may offer a one-time improvement in his 40% point-winning odds. Different calculations are required when no-ad scoring is used. In no-ad play, whenever the score reaches deuce, the game is decided by the next, seventh point. The calculations confirm the seventh point's obvious importance, which at .48000, far exceeds all others. The next three situations in importance under no-ad scoring are the 30-40, the 30-30, and the 15-30. (For data, see tabulation at end of this essay.) Things change somewhat if we increase the server's dominance in the calculations. Service breaks become rarer, so that all break points, indeed all points where server trails, are increased in importance. Points where server is ahead or where the score is even become less important. In calculations where server's edge for winning each point is 70% (ad scoring), the ad-out point's importance is increased to .35483, and the 15-40 at .24838 becomes the second-most-important situation. (See tabulation at the end.) 2. Situations in the tiebreak gameThe calculations for tiebreak-game situations produce few surprises. The most important situations occur late in the game when the score is close. (We return to the assumption where each player has a 60% chance of winning a served point.)point score 6-5, 5-6, 5-5, or higher, .24000 point score 5-4, 4-5, or 4-4, .18240 point score 4-3, 3-4, or 3-3, .15245 server leads 6-4, .14400 point score 3-2, 2-3, or 2-2, .13356 server leads 5-3, .13248 ... first point, .11030 ... server trails 0-6, .00553 3. Game-score situationsHere, we calculate the importance of each game-score situation in terms of its importance to the outcome of the set.The most important game is the tiebreak game, which whenever played always decides the set. Its average effect on set-winning probabilities is always .50000 provided that the players are of equal ability, regardless of dominance of serve. Listed below is the calculated importance of each game-situation in the context of a given set. Generally, the most important games occur late in the set when the score is close. (We return to our assumption that each player has a point-winning probability of 60% when serving.) game-score 6-6 (tiebreak game), .50000 each player has 4 games or more, .22643 4-3, 3-4, and 3-3, .18240 5-3, .16659 3-2, 2-3, and 2-2, .15550 4-2, .14584 2-1, 1-2, and 1-1, .13739 3-1, .13089 1-0, 0-1, and 0-0, .12429 2-0, .11959 0-2, .08782 1-3, .08698 2-4, .08059 3-0, 0-3, .07002 4-1, 1-4, ,06237 3-5, .05984 4-0, .04815 5-2, 2-5, .04403 5-1, .03239 0-4, .02278 1-5, .01163 5-0, 0-5, .00856 Note that the legend elevating the importance of a set's fifth game has no numerical validity. When the score is 2-2, for example, the next--the fifth--game's importance is the same as the sixth's, and is lower than in the 3-3, the 4-4, or the 5-5 situations. 4. Point-and-game-score situationsWe can use the above values to calculate the importance of any point-and-game situation in a given set.In the case of the tiebreak game, the game-winning probabilities and the set-winning probabilities of any point score are the same. In all other games, the game-winning importance of any point must be adjusted (by multiplication) for the set-winning importance of the given game in progress. Here are the most important points in a set: tiebreak game, each player has at least 5 points, .50000 tiebreak game, point score 5-4, 4-5, or 4-4, .38000 each player has at least 4 games, point score 30-40 or ad-out, .34615 tiebreak game, point score 4-3, 3-4, or 3-3, .31760 tiebreak game, point score server leads 6-4, .30000 game score 4-3, 3-4, or 3-3, point score 30-40 or ad-out, .27885 tiebreak game, point score 3-2, 2-3, or 2-2, .27824 tiebreak game, server leads 6-4, .27600 game score 5-3, point score 30-40 or ad-out, .25467 game score 3-2, 2-3, or 2-2, point score 30-40 or ad-out, .23772 each player has 4 games or more, point score 30-30 or deuce, .23077 each player has 4 games or more, point score 15-30, .22154 each player has 4 games or more, point score 15-40, .20769 ... first point of set, .01586 ... game score 5-0, point score 40-love, .00020 We note that the most important situation that can occur--a point occurring during an extended tiebreak game--is 31.5 times more important than the first point of a set. Comparing the absolute extremes, the most important and the least important possible situations in a given set differ in importance by an astonishing factor of 2,500. 5. The weight of each setUnder our assumption that the players are of equal ability, calculations to tell the importance of each set are straightforward. Numerically the first and second sets are exactly equal in importance. Any situation in the third set is twice as important as the same situation occuring in the earlier sets. (This can be glimpsed logically, as a match is won either by winning the first two sets or by winning the third set.)Thus, the most important point in tennis--any point during a third-set extended tiebreaker--is 5,000 times more important than the least important possible situation (at 40-love when server leads by 5-0 in the first or second set). The most important point is 63 times more important than the first point of the match. In a best-of-five-set match, the fourth set is half as important numerically as the fifth. The importance of the third set is also half that of the fifth if the first two sets were split, but is only one-fourth if either player leads by two sets. The first and second sets are each three-eighths (37.5%) as important as a fifth set. 6. Playing to the scoreThere can be danger in playing to the score. In tennis, a player who is losing--no matter how one-sided the match to date--will be the victor if he can steadily win more than half the remaining points. This can most readily occur where server's advantage is small, on slow courts for example. Axiomatically, therefore, it is more important to stay ahead than to get ahead. Lax play when well ahead, even though the numerical importance of points may be low until the score tightens, can lead to disaster. For the same reasons, the player who trails by a large amount must not lose heart.Our calculations cannot, of course, bear on whether any particular tactic--very aggressive play, for example--is the optimum tactic in a pressure situation. Most players recognize that unexpected aggressiveness can have surprise value, or can put extra demands on the opponent at such a time. But this is a matter for the player's judgment. It could be, for example, that a retreat to defensive play, emphasizing the absolute avoidance of errors, is the soundest tactic under pressure. Even as the intelligent player plays to the score, he must avoid becoming too predictable in doing so. There is need for variety in critical situations as well as in overall play. If a player invariably attacks opponent's weaker side in big points, his opponent may recognize the tactic and counter in some way. Finally, we must give a final notice to the mental aspects. All players understand the importance of confidence and momentum. The loss of a few relatively unimportant points can lead to a shift in confidence in playing the ensuing more-critical points. The loss of a game after an early lead can be mentally devastating. Tennis's unusual scoring system has great beauty. It can produce crucial situations at frequent intervals during a match and offers a losing player persistent opportunity to reverse the trend of events. Our calculations have yielded a few interesting results. I hope they have stirred the reader's thoughts on playing to the score, and that they will enhance his or her enjoyment of the game. Footnote: For those loyal readers who have read the above data carefully, two seeming anomalies in the calculations merit comment. (1) Why is it that, according to the calculations, the 5-3 game has greater importance than the 3-5? To understand this, consider the following. At 5-3, If server wins the ensuing game, he has won the set. If he loses the game, he has lost his service-break advantage, and his opponent needs only to hold serve to equalize the set. The swing is patently significant. But at 3-5, if server wins the next game he must still break serve to equalize the set, so he remains a strong underdog. Thus the probabilities in the set's outcome change more radically in the 5-3 game than at 3-5. (2) Why is it that the games played after 4-all are not exactly one-half as important as the tiebreak game? The answer is that they would be if server had no point-winning advantage. Consider the extreme case where server's advantage in any game is 100%. All games prior to the tiebreak would inevitably be won by server and, being absolutely predictable, these outcomes would have no effect on the set's ultimate outcome probabilities. If server's advantage is just 99%, however, there would be a small possibility that receiver could win a given game and thus affect the set's probable outcome. As server's advantage is further reduced, the mean effect of games on outcome probabilities is increased until reaching the case where server had no advantage and the penultimate games would have exactly one-half the importance as the final tiebreaker. Addenda for the real fanatic: Importance of situations where server has 70% edge on each point (compare with 60% tabulated early in essay) 30-40 or ad-out, .35483 15-40, .24838 love-30, .19586 15-30, .18096 love-40, .17387 30-30 or deuce, .15207 love-15, .12071 15-15, .09580 first point, .06706 40-30 or ad-in, .06517 30-15, .05931 15-love, .04407 30-love, .02190 40-15, .01955 40-love, .00587 Importance of situations in no-ad scoring (server has 60% edge on each point) deuce, .48000 30-40, .31680 30-30, .23040 15-30, .20736 15-40, .17280 40-30, .17280 love-30, .16589 15-15, .16589 love-15, .15898 30-15, .13824 first point, .13271 15-love, .11059 love-40, .10368 40-15, .07680 30-love, .07373 40-love, .03072 The notion that "success breeds success and failure breeds failure" is a widely held belief that people apply to a variety of situations, from business to sports. Initial success generates confidence, which increases the probability of success in subsequent trials. Similarly, an initial failure may be so discouraging that subsequent failures become practically inevitable, leading to a massive defeat. That sort of effect is sometimes called psychological momentum. Indeed, sports commentators and fans often can't resist accounting for even the most mundane strings of wins and losses as the result of some sort of psychological factor. One arena where it's possible to test the apparent relationship between psychological momentum and sequences of successes or failures is best-of-five tennis matches. The question is whether the result of the first set changes the probability of success or failure in subsequent sets. If psychological momentum is truly a factor, the probability of winning a set would depend on the results of previous sets instead of remaining essentially constant for a given match. To investigate this effect, statisticians David Jackson and Krzysztof Mosurski of Trinity College in Dublin, Ireland, have taken a close look at the results of professional tennis matches. They describe their findings in a recent issue of Chance. Earlier work by Jackson had already demonstrated that a "success breeds success" model provides a much better fit to data from the 1987 Wimbledon and the U.S. Open tennis tournaments than an independent-sets model. "These data exhibit far more heavy defeats than can be accommodated by the independence model," Jackson and Mosurski say. However, there is a possible alternative explanation for the apparent overabundance of heavy defeats (when one player wins the match three sets to zero) -- random variation in player ability from day to day. "A random-effects model for player ability provides a good explanation of a common occurrence in sport in which a player inflicts a heavy defeat on his opponent on one day but himself suffers a heavy defeat from the same opponent on the next day," the statisticians note. To distinguish between psychological momentum and random day-to-day fluctuations in ability as alternative explanations of a preponderance of overwhelming defeats, Jackson and Mosurski analyzed two years of data from men's singles matches at the Wimbledon and U.S. Open tournaments. Their models included such factors as the official ranking of the players. The results were clear. "Whatever the contribution of random variation in a player's ability from day to day may be, our analysis suggests that psychological momentum is certainly a major factor in the outcome of matches at the Wimbledon and U.S. Open tennis tournaments," Jackson and Mosurski conclude. Evidence of the fundamental dependency of the outcome of one set on that of another within a match can also be seen in the distribution of wins and losses. For example, when a player wins by a score of 3 to 1, there are three different sequences that may occur: LWWW, WLWW, and WWLW. For a 3-to-2 result, there are six possible sequences. If the sets were independent, each of these outcomes would be equally likely. In both cases, "there is evidence that the set or sets lost by the winner in these matches occurred earlier rather than later in the match, which would not be so for independent sets," the statisticians say. For example, in 158 3-to-1 matches, there are 60 results in which the loss occurred in the first set (LWWW) and 41 in which the loss occurred in the third set (WWLW). The analysis, however, does not mean that we should reject the common-sense idea that player ability can vary from day to day. It's just that such variability doesn't account sufficiently well for the overwhelming defeats often inflicted at major tennis tournaments. It's also possible to examine the head-to-head records of particular pairs of players to see if psychological momentum plays a role. Jackson and Mosurski looked at the epic battles between Ivan Lendl and Jimmy Connors from 1982 to 1985 and between John McEnroe and Bjorn Borg from 1978 to 1981. Interestingly, in the Borg-McEnroe series, there is no evidence that the probability of winning a set in any of their matches was influenced by the score or that the probability of winning a set varied from match to match. The Lendl-Connors data suggest that either psychological momentum or day-to-day variation in ability was a factor in their series of matches. In other sports, however, situations arise where psychological momentum doesn't appear to play a role -- contrary to the perceptions of players, coaches, commentators, and fans. Amos Tversky and Thomas Gilovich showed that the chances of a basketball player hitting a shot are as good after a miss as after a basket. In baseball, analyses of hitting streaks also failed to detect any significant effect of the player's recent history of successes and failures on the probability of making a hit. There's something about tennis that really brings the mind into play. References: Albright, S.C. 1993. A statistical analysis of hitting streaks in baseball. Journal of the American Statistical Association 88:1175-1188. Jackson, David, and Krzysztof Mosurski. 1997. Heavy defeats in tennis: Psychological momentum or random effect? Chance 10(No. 2):27-34. Peterson, Ivars. 1997. The Jungles of Randomness: A Mathematical Safari. New York: Wiley. Tversky, Amos, and Thomas Gilovich. 1989. The cold facts about the 'hot hand' in basketball. Chance 2(No. 1):16-21. Wardrop, Robert L. 1995. Simpson's paradox and the hot hand in basketball. American Statistician 49(February):24-28. |
|||
| Следующая статья | Предыдущая статья | Все статьи Обсудить на форуме |