Ivars Peterson's MathTrek

December 1, 1997

# Getting Slammed in Tennis

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.