Horseshoe Pitchers’ Hot Hands

Gary Smith

Department of Economics

Pomona College

Claremont, California 91711

gsmith@pomona.edu

909.607.3135

Fax 909.621.8576

 * I am very grateful to Walter Ray Williams, Jr., for sharing his data with me and for his generous and patient answers to my many questions about competitive horseshoes.

Abstract

Gilovich, Vallone, and Tversky’s analysis of basketball data indicate that a player’s chances of making a shot are not affected by the results of earlier shots. However, their basketball data do not control for several confounding influences. An analysis of horseshoe pitching, which does not have these defects, indicates that players do have modest hot and cold spells.

key words: hot hands, horseshoes, self-efficacy

Horseshoe Pitchers’ Hot Hands

Gilovich, Vallone, and Tversky (1985) present evidence against the popular belief that basketball players get “hot hands.” Their survey of 100 basketball fans showed an overwhelming belief that a player has a better chance of making a basket after having made shots than after missing shots. Five of seven Philadelphia-76ers basketball players also believed this.

However, their analysis of the performance of individual Philadelphia 76ers during the 1980–1981 season found that the probability of making a shot was usually somewhat lower after having made shots than after missing shots, though the observed differences were not statistically persuasive. Similarly, an analysis of the number of runs of hits and misses by individual players across all games and within individual games generally found slightly more runs than would be expected if shots were independent (the opposite of the hot-hands phenomena), though again the results were seldom statistically persuasive.

One weakness of their analysis is that it ignores the time interval between shots. A player’s two successive shots might be taken several minutes apart, before and after the halftime intermission, or even in different games. The results of such widely separated shots is not what fans mean by a hot hand. Another problem is that a player’s shot selection may be affected by his recent hits or misses. A player who has made several shots in a row may be tempted to try more difficult shots, thereby reducing his success probability. A player who has been missing shots may pass up attempts he would normally take and shoot easier shots. Also, the score or other game considerations may affect team strategies. For example, the opposing team may guard hot players more closely and cold players more loosely. A team that is behind may take more low-percentage shots by shooting more quickly and attempting more 3-point shots. A person may play less aggressively if he is in foul trouble and more aggressively if his defender is in foul trouble.

The average starter takes 10-to-20 shots a game, 5-to-10 in each half. If the hot hand is a relatively modest phenomenon, a statistical test based on 5-to-10 shots will have very little power and may be hidden in data sets that combine shots of varying difficulty separated by long periods of time. When data are aggregated from different games, the confounding influences include home-court advantage, the number of days between games, and the amount of travel to get to a game.

Gilovich, Vallone, and Tversky also examined free-throw accuracy for nine Boston Celtics players during the 1980–81 and 1981–82 seasons. When shooting pairs of free throws, four players were more likely to make the second shot after making the first shot and five were more likely to make the second shot after missing the first shot; none were statistically significant. This is arguably their most persuasive evidence; however, the aggregation of infrequent shots taken in different games doesn’t really encompass the usual notion of hot streaks.

Finally, they conducted a controlled experiment in which 26 Cornell players took 100 shots moving along arcs drawn from the basket. Overall, the players were slightly more likely to make a basket after having made one-to-three shots, but the differences were statistically persuasive for only one player.

Gilden and Wilson (1995) present some experimental evidence of nonconstant success probabilities in golf putting and dart throwing. However, their results might be criticized for their artificiality, as their experiments involved volunteers making 300 repetitions and being paid $5 plus (in three of the four studies) five cents for each hit. With so many trials and such small stakes, it is conceivable that there are substantial fluctuations in the attentiveness of poorly motivated volunteers. If success rates are higher when they are focused on their assignment and lower when they are bored, hits and misses will cluster in the data and exhibit the statistical patterns associated with streakiness; for example, higher hit rates after hits than after misses and fewer runs of hits and misses than expected with a constant hit rate.

The hot-hands question is whether highly skilled and motivated athletes sometimes encounter hot and cold spells that cannot be easily explained as chance fluctuations about a constant success rate. Serious athletes competing for meaningful stakes are likely to remain focused on their task and to provide meaningful data for testing whether hot hands are real or an illusion.

Horseshoe Pitching Data

One sport that has very few confounding influences is horseshoe pitching. The rules described here have been used at recent world championships and represent the most commonly followed norms. A horseshoe court has two stakes, placed 40 feet apart, with each stake centered in a 6-foot-square pitcher’s box that contains pitching platforms on each side of a 3-foot-wide pit. The players can pitch their shoes from the front of either pitching platform, reducing the distance to the opposite stake to 37 feet. Extended platforms are used in women’s competition to reduce the pitching distance to 27 feet.

In each inning, a player pitches two shoes and then the other player pitches two shoes, with the score tallied after all four shoes have been pitched. A shoe that encircles the stake is a ringer worth 3 points, a nonringer that is within 6 inches of the stake is a “shoe in count” worth 1 point. In conventional cancellation scoring, only one contestant can score in each inning. Ringers thrown by both players cancel each other (“dead ringers”), as do shoes in count that are equidistant from the stake. For instance, if both players throw double ringers, these are all dead ringers and no points are scored. If one player throws double ringers and the opponent throws one ringer and a shoe in count, the live ringer scores 3 points.

A shoe is flipped like a coin toss to determine who pitches first in the first inning. Thereafter, the player who scores pitches first in the next inning. If neither player scores, the player who pitched second pitches first in the next inning. The first player to score 40 or more points is the winner. At the World Championships, 16 players qualify for the men’s and women’s championship matches and each player pitches against each of the 15 other players. The final standings are determined by the players’ overall won-loss records.

In top competition, players typically throw 60–80% ringers and games last 20-30 innings. One of the greatest games of all time occurred at the 1965 World Championships when Ray Martin threw 89.7 percent ringers and lost a 2 1/2 hour, 97-inning marathon to Glen "Red" Henton, who pitched 90.2 percent ringers.

Walter Ray Williams, Jr, six times world champion, generously provided detailed data from the 2000 and 2001 World Championships for men and women (Williams, 2002). These score sheets record each player’s live and dead ringers, shoes in count, and misses in each inning, but do not distinguish between a player’s first and second pitch.

Analysis

In comparison to basketball data, horseshoes have many valuable properties for testing streakiness in performance. There are no confounding influences from team play, defenses, or strategy based on the score. World-class pitchers always try to throw ringers and every pitch is from the same distance and separated by only brief intervals of time.

Conditional Probabilities

The key issue is whether the number of ringers a player pitches in an inning is independent of the number of ringers in the previous inning. Each game was analyzed separately in that I did not consider whether the first inning of a game is influenced by the last inning of the previous game. Because double misses are unusual for world-class pitchers, each player’s inning was characterized as a double ringer or not a double ringer. The top male and female players throw doubles roughly half the time, making a nice analogy to coin flips.

Table 1 shows the overall frequencies with which players pitched doubles after doubles or nondoubles in the preceding one or two innings. These overall frequencies might be affected by the fact that the best pitchers are more likely to be pitching after throwing doubles. This effect should be small since these are all championship-caliber pitchers; nonetheless, Table 1 also shows simple unweighted averages of the individual player frequencies.

Men and women were both somewhat more likely to throw a double after a double than after a non-double, and also more likely to throw a double after two doubles in the preceding two innings than after two non-doubles. (These hot-hand patterns imply analogous cold hands; if a hit is more likely after a hit than after a miss, then a miss is more likely after a miss than after a hit.)

Using Fisher’s exact test, of those pitchers who were more likely to throw a double after a double, 8 had one-sided p values less than 0.025; of those pitchers who were more likely to throw a double after two doubles, 8 had one-sided p values less than 0.025. (None of the pitchers who were less likely to throw a double after a double or after two doubles had p values below 0.025). If each of the 64 pitchers had a 0.025 probability of a p value below 0.025, the binomial distribution shows that the probability that as many as 8 would have p values below 0.025 is 0.0002.

Another way to look at the data is that 25 of the 32 male pitchers and 26 of the 32 females were more likely to throw a double after a double than after a non-double. If each pitcher had a 0.5 probability of being more successful after a double, the binomial distribution shows that there is only a 0.0000009 probability that as many as 51 of 64 players would be more successful after a double. In addition, 23 of the 32 men and 25 of 31 women were more likely to throw a double after two doubles than after two non-doubles (one woman was equally likely in 2000). If each pitcher had a 0.5 probability of being more successful after two doubles, there is only a 0.00002 probability that as many as 48 of 63 players would be more successful after two doubles.

Runs

Independence can also be tested by the length of the longest run and by the number of runs in a game. Bateman (1948) shows that, in the case of first-order dependence, a test based on the number of runs in a sequence is more powerful than a test based on the length of the longest run.

After categorizing each player’s throws in each inning as either a double or nondouble, the exact probability that the number of runs would be as small as actually observed (indicating the presence of hot and cold streaks) can be calculated (Stevens, 1939). However, in addition to the low power with small sample sizes, the possible p values are not continuous. Consider, for example, a game with 10 double and 10 non-double innings. If double and non-double innings are independent, the probability of 6 or fewer runs is 0.0185 and the probability of 7 or fewer runs is 0.0513. The probability of a p value less than 0.025 is 0.0185, not 0.025.

Another complication is that the number of doubles and non-doubles may be such that the number of runs cannot possibly be statistically persuasive. For example, if a player throws all doubles, this seems evidence of a hot streak, but the statistical fact that such data will always yield exactly one run means that a runs test, which looks for the clumping of doubles and non-doubles, cannot provide statistically persuasive evidence against the null hypothesis that doubles and non-doubles are independent.

There were no perfect games at these World Championships, but there were several close enough to make a runs test useless. In one game in 2000, Alan Francis pitched 13 doubles in 14 innings. Under the null hypothesis, there is a 2/14 = 0.143 probability of 2 runs and a 12/14 = 0.857 probability of 3 runs. There is no chance of rejecting the null hypothesis at the 5 percent level.

One way to avoid these problems is to calculate the expected value of the number of runs in each game under the null hypothesis of independence and then see whether the actual number of runs was higher or lower than this expected value. I then tabulated the total number of games in which the actual number of runs was above and below the expected value and used the binomial distribution to test the null hypothesis that the actual number of runs is equally likely to be above or below its expected value.

Table 2 summarizes the results. Too few runs (evidence of streakiness in performance) were consistently more likely than too many runs. Overall, there is only a 0.000009 probability of such an imbalance between the number of games with fewer runs than expected and those with more runs than expected.

Ringer Percentages by Game

Another kind of evidence of streakiness would be if a pitcher’s performance fluctuated from game to game more than would be expected if horseshoe pitching were a Bernoulli process with constant success probability. Table 3 shows an example. Mary Ann Peninger threw 74.4% ringers in her 15 games at the 2000 championships. If her chances of throwing a ringer were the same in each game, the expected value of the number of ringers in each game would be .744 multiplied by the number of pitches in that game. In practice, her actual game ringer percentages varied from .574 to .827. The chi-square statistic gauges whether the variations between the actual and expected values shown in Table 3 are improbably large. For these particular data, the p value is .139, not low enough to reject the null hypothesis at the 5 percent level.

Similar calculations for all pitchers found that 5 of 32 men and 8 of 32 women had p values less than 0.05. If the null hypothesis were true, so that each pitcher has a 0.05 probability of a p value less than 0.05, the probability that as many as 13 of 64 pitchers (20.3%) would have p values less than 0.05 is 0.00001.

First and Second Pitchers

While horseshoe data are in many ways ideal, one possible confounding influence is that there may be a physical or psychological difference between pitching first and second. For example, shoes may be more likely to bounce off the stake if they land on another shoe than if they land in a bare pit. The high ringer percentages in championship tournaments indicate that this doesn’t happen frequently. Still, it might give a slight advantage to the player pitching first.

Since the player who scores in any inning pitches first in the next inning, it is conceivable that the hot and cold streaks documented above simply reflect the advantages of pitching first. On the other hand, the pitching order rotates when neither player scores, which happened in 29% of the innings.

Conditional Probabilities

One way to control for pitching order is to separate the conditional probabilities for those pitching first from the conditional probabilities for those pitching second. Unfortunately, the reduced sample sizes make it more difficult to reject the null hypothesis if the differences in conditional probabilities are relatively small. On average, individual players pitched first in roughly 160 innings after throwing a double in the preceding inning and in roughly 50 innings after throwing a nondouble in the preceding inning, with the numbers reversed for players pitchers second. Using these sample sizes, Figure 1 shows, for various values of the differences in ringer probabilities p₁ - p₂, the probability of a sufficiently large observed difference in ringer frequencies to reject the null hypothesis p₁ = p₂. This power function shows that for modest differences in ringer probability, the sample sizes are too small to reject the null hypothesis consistently. For example, if the probability of a double after a double is p₁ = 0.55 and the probability of a double after a nondouble is p₂ = 0.45, there is only 0.234 probability that the observed performance differences will be sufficiently large to reject the null hypothesis. If p₁ - p₂ = 0.05, the probability of rejecting the null hypothesis is only 0.090.

Nonetheless, there is statistically persuasive evidence here of nonconstant success probabilities. Table 4 shows that the chances of throwing a double are somewhat lower when pitching second but that, for both first and second pitchers, the chances of throwing a double are generally higher after throwing a double in the preceding one or two innings. A comparison with Table 1 indicates that controlling for pitching order makes the differences in observed conditional probabilities somewhat smaller, but does not eliminate them.

If, when either pitching first or pitching second, the probability of throwing a double did not depend on previous results, each player should be equally likely to be more successful after previous doubles than after previous nondoubles. I tabulated the number of players who were more successful after doubles and the number who were more successful after nondoubles and used the binomial distribution to test this null hypothesis. In 81 of 128 cases (63%), players were more likely to throw a double after a double than after a non-double (p = .0017), and in 85 of 127 cases (67%) they were more likely to throw a double after two doubles than after two non-doubles (p = 0.00009).

Correlations

If individual ringer probabilities vary from game to game, a player who is hot should have elevated success probabilities whether he is throwing first or second; a player who is cold should have deflated success probabilities. If so, looking across games we might find a positive correlation between a player’s success rate when throwing first and when throwing second. To examine this issue, correlation coefficients were calculated for each player using data for every game in which the player had at least 10 innings of first pitches and 10 innings of second pitches.

With 64 players, we expect to find some positive and negative correlations. The question is whether the number and magnitudes are greater than expected by chance. Of the 64 players, 42 (66%) had positive correlation coefficients. If positive and negative correlation coefficients were equally likely, the binomial distribution tells us that probability of so many positive correlations is 0.0084. Eight of 64 players had positive correlation coefficients with p values less than 0.025. If the null hypothesis is true, the probability of so many statistically significant correlation coefficients is 0.0002.

Ringer Percentages by Game

We can also use a chi-square statistic to test the null hypothesis that a player’s pitching-first and pitching-second ringer percentages do not vary from game to game. Table 5 shows the data used for Mary Ann Peninger in 2000. The calculations compare the actual and expected values as in Table 3, but the details are not shown here. The expected values are calculated by assuming that she has a .761 probability of throwing a ringer when pitching first and a .732 probability of throwing a ringer when pitching second. The p value is 0.0013, demonstrating that it is extremely unlikely that the observed game-to-game variations in her accuracy are simply random fluctuations about constant success probabilities. Overall, 9 of 64 players have p values less than 0.05 and 5 of these have p values less than 0.01. If success probabilities do not vary by game, the probabilities of so many low p values are 0.0044 and 0.0005, respectively.

Discussion

Gilovich, Vallone, and Tversky argue that basketball performances are misperceived by fans and players to contain remarkable hot and cold spells. In addition to contradicting fan perceptions, their results seemingly contradict substantial evidence (for example, Bandura, 1977; Taylor, 1979) from a wide variety of sports indicating that athletic performance is enhanced by a person’s self-efficacy, the personal assessment of one’s ability to perform a specific task. In an arm-wrestling study (Nelson & Furst, 1972), the weaker subjects won ten of twelve matches when both contestants incorrectly believed the weaker person to be stronger than his opponent; when the contestants correctly identified the weaker contestant, the stronger subjects won all twelve matches. Other studies have found that positive self-talk improves the performance of basketball players (Kendall, Hrycaiko, Martin, & Kendall, 1990), skiers (Rushall, Hall, Roux, Sasseville, & Rushall, 1988), and swimmers (Rushall & Shewchuk, 1989).

It is plausible that the positive reinforcement provided by an athlete’s success increases self-efficacy and thereby enhances performance. But perhaps this enhancement is relatively small for professional basketball players in relation to such confounding factors as shot selection, lengthy spells between shots, and strategic adjustments. Championship horseshoe data are cleaner in that every pitch is from the same distance and made at regular, brief intervals with intense concentration and little or no strategy. Variations in player performances within games and between games at the 2000 and 2001 World Championships indicate that success probabilities are not completely independent of previous outcomes.

References

Bateman, G. (1948). On the Power Function of the Longest Run as a Test for Randomness in a Sequence of Alternatives. Biometrika, 35, 97—112.

Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change. Psychological Review, 84, 191—215.

Gilden, D. L. & Wilson, S. G. (1995). Streaks in skilled performance. Psychonomic Bulletin & Review, 2, 260–265.

Gilovich, T., Vallone, R., & Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17, 295—314.

Kendall, G., Hrycaiko, D., Martin, G. L., & Kendall, T. (1990). The effects of an imagery rehearsal, relaxation, and self-talk package on basketball game performance. Journal of Sport and Exercise Psychology, 12, 157—166.

Nelson, L. R., & Furst, M. L. (1972). An objective study of the effects of expectation on competitive performance. Journal of Psychology, 81, 69—72.

Rushall, B. S., Hall, M., Roux, L., Sasseville, J., & Rushall, A. C. (1988). Effects of three types of thought content instructions on skiing performance. The Sport Psychologist, 2, 283—297.

Rushall, B. S., & Shewchuk, M. L. (1989). Effects of thought content instructions on swimming performance. The Journal of Sports Medicine and Physical Fitness, 29, 326—334.

Stevens, W. L. (1939). "Distribution of Groups in a Sequence of Alternatives," Annals of Eugenics, 9, 10—17.

Taylor, D. E. M. (1979). Human endurance: Mind or muscle? British Journal of Sports Medicine, 12, 179—184.

Williams, W. R., Jr. (2002). [score sheets from 2000 and 2001 world championships]. Unpublished raw data.

Table 1 Frequency of Doubles Following One or Two Doubles or Nondoubles, with Unweighted Averages in Parentheses

Table 2 Number of Games with Fewer or More Runs than Expected with Independence

Table 3 Mary Ann Peninger’s Ringers by Game in 2000, with Expected Values in Parentheses

Table 4 Frequency of Doubles Following One or Two Doubles or Nondoubles, with Unweighted Averages in Parentheses

Table 5 Mary Ann Peninger’s Ringers by Game in 2000, When Pitching First and When Pitching Second

 Figure 1 The probability of a sufficiently large difference in success frequencies to reject the null hypothesis that the success probabilities p₁ and p₂ are equal.