Cognition 139 (2015) 168–170
Contents lists available at ScienceDirect
Cognition journal homepage: www.elsevier.com/locate/COGNIT
Discussion
A fallacious ‘‘Gambler’s Fallacy’’? Commentary on Xu and Harvey (2014) Heath A. Demaree a,⇑, Joseph S. Weaver b, James Juergensen a a b
Department of Psychological Sciences, Case Western Reserve University, Cleveland, OH, USA Department of Psychology, DePauw University, Greencastle, IN, USA
a r t i c l e
i n f o
Article history: Received 13 June 2014 Revised 25 August 2014 Accepted 26 August 2014 Available online 18 September 2014 Keywords: Risk-taking Gambling Cognition Statistics Bias
a b s t r a c t In their recent article in Cognition, Xu and Harvey (2014) suggested that people who placed wagers on an online gambling site demonstrated very different wagering preferences depending on whether they were on winning or losing streaks. Specifically, they reported that people on winning streaks were more likely to win their subsequent wagers because they chose increasingly ‘‘safer,’’ higher-probability bets as the win streak continued. People on losing streaks were more likely to lose their subsequent wagers because they chose ‘‘riskier,’’ lower-probability wagers as the losing streak progressed. The authors suggested that individuals on winning and losing streaks both fell prey to the Gambler’s Fallacy. Specifically, individuals on winning streaks combatted their expectancy to lose soon by choosing higher-probability wagers (with lower payoffs). Conversely, people on losing streaks expected to win soon and thus preferred lower-probability wagers with higher payoffs. Though their paper is fascinating and contains a remarkable data set, we note that the statistical methods employed by Xu and Harvey are prone to a serious selection bias, such that participants on winning or losing streaks may have already been choosing safer and riskier wagers, respectively, prior to the beginning of their streaks. We suggest easy, intuitive analyses to determine whether the effects reported in Xu and Harvey (2014) are real. Ó 2014 Elsevier B.V. All rights reserved.
Xu and Harvey (2014) presented fascinating real-world, multi-trial gambling data. They found that people on winning streaks were more likely to win subsequent wagers because they gradually opted for higher-probability (‘‘safer’’) bets (i.e., they had a ‘‘hot hand’’). Conversely, people on losing streaks were more likely to lose their subsequent wagers because they gradually preferred lowerprobability (‘‘riskier’’) wagers with higher payouts. Xu and Harvey (2014) stated that individuals on both winning and losing streaks expected their luck to reverse (i.e., they
⇑ Corresponding author. Address: Department of Psychological Sciences, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7123, USA. Tel.: +1 216 368 6468; fax: +1 216 368 4891. E-mail address: [email protected] (H.A. Demaree). http://dx.doi.org/10.1016/j.cognition.2014.08.016 0010-0277/Ó 2014 Elsevier B.V. All rights reserved.
became prone to the ‘‘Gambler’s Fallacy’’). As a result, individuals on winning streaks preferred higher-probability wagers to help offset their expected bad luck whereas individuals on losing streaks expected to win soon and thus increasingly preferred ‘‘long shots’’ with high payouts. The paper is fascinating and, as such, it quickly received enormous attention: it has been written about on websites including The New Yorker, The Economist, Business Insider, The Wall Street Journal, and many, many others. As exciting as the data are, we believe that the statistical analyses used introduced a serious selection bias leading to a misunderstanding of the data. In their article, Xu and Harvey (2014) analyzed the probabilities of the wagers placed by individuals who just won 1, 2, 3, 4, 5, and 6 times in a row (i.e., bettors with a ‘‘hot hand’’), comparing them to the probabilities of the
169
H.A. Demaree et al. / Cognition 139 (2015) 168–170
Fig. 1. Hypothetical Trial 1 and Trial 2 data from repeated winners (left panel, a) and repeated losers (right panel, b). When evaluating the influence of repeated winning on subsequent wagers (Panel a), we imagine a normal distribution of Trial 1 data with some individuals preferring lower- and higher-probability wagers (represented by thinner and thicker vertical lines, respectively). The identification of 1-trial winners would filter out those who prefer ‘‘long shots,’’ leading to a negatively skewed population with higher P(W) preferences. The continued selection of 2-trial, 3-trial. . . 6-trial winners (not shown) would lead to populations with greater and greater negative skew and higher and higher P(W) averages. The mirror opposite would be expected when identifying individuals on losing streaks (Panel b; please note that thicker vertical lines in Panel b represent lower probabilities of winning).
wagers made by those without such winning streaks. They found that as the winning streak increased in size, so did the probability of winning the subsequent bet. Why? Xu and Harvey (2014) argued that it was because winning streaks gradually caused Gamblers to make ‘‘safer’’ and ‘‘safer,’’ higher probability, wagers. But didn’t the statistical analyses employed by Xu and Harvey (2014) gradually select out those bettors who preferred to make ‘‘safer,’’ higher-probability wagers (with lower payouts)? For example, Xu and Harvey (2014) first divided Bet 1 data, identifying those who won from those who lost. It is highly
intuitive that those who won their first wager were more likely to have placed higher-probability bets than those who lost their first wager. Thus, the analysis of Bet 2 data would be biased because it compared those who preferred ‘‘surer bets’’ (higher probability wagers with smaller payouts) at Bet 1 with those who preferred ‘‘long-shots’’ (bets with a lower probability of winning higher payoffs) at Bet 1. Once again, Bet 2 data were divided into winners and losers. Who were the individuals most likely to win 2 bets in a row? Those who preferred higher-probability wagers, of course! Repeated 6 times, we believe that Xu and Harvey (2014) gradually (but inadvertently) filtered out those individuals with the strongest preference for highprobability wagers and compared them to those without such a preference (please see Fig. 1 for a graphic depiction of this selection bias). Thus, in fact, Xu and Harvey’s (2014) analysis did not show that winning caused people to alter their betting strategy towards ‘‘safer,’’ higher-probability wagers. Rather, their analysis simply identified those individuals who preferred such wagers. We also suspect that a symmetrical function occurred when Xu and Harvey (2014) analyzed the data of Gamblers on losing streaks. When analyzing the wagers of individuals who just lost 1, 2, 3, 4, 5, and 6 straight bets, the researchers (inadvertently) gradually identified those individuals who preferred ‘‘long shots’’ compared to ‘‘sure things’’ (please see Fig. 1). Accordingly, it is no surprise that a person’s probability of winning a wager declined with the number of consecutive losses they just experienced (i.e., these individuals preferred long-shots). However, rather than the consecutive losses being a causal factor in increasing risk-taking propensities, the results found may be due to individual differences in gambling preferences. Indeed, people with different personalities consistently prefer different types of wagers. For example, Demaree, DeDonno, Burns, and Everhart (2008) found that individuals with higher levels of Behavioral Inhibition System (BIS) strength
Table 1 The data required to perform the suggested repeated-measures ANOVAs. Hot hand
Time After After After After After After
0 1 2 3 4 5 6
win wins wins wins wins wins
1 Straight win N = 178,947
2 Straight wins N = 88,036
3 Straight wins N = 50,300
4 Straight wins N = 33,871
5 Straight wins N = 24,390
6 Straight wins N = 18,190
A1 B1
A2 B2 C2
A3 B3 C3 D3
A4 B4 C4 D4 E4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6 G6
1 Straight loss N = 192,359
2 Straight losses N = 101,595
3 Straight losses N = 60,739
4 Straight losses N = 41,595*
5 Straight losses N = 11,231*
6 Straight losses N = 2808*
A1 B1
A2 B2 C2
A3 B3 C3 D3
A4 B4 C4 D4 E4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6 G6
Gambler’s Fallacy
Time After After After After After After *
0 1 2 3 4 5 6
loss losses losses losses losses losses
N not provided by Xu and Harvey, but calculated as approximates based on the data provided.
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H.A. Demaree et al. / Cognition 139 (2015) 168–170
and higher levels of Sensation Seeking (SS) prefer lowerprobability, higher payout, wagers. When identifying people who experienced long winning and losing streaks, Xu and Harvey (2014) may have simply identified people with different personalities (e.g., high BIS, low SS versus low BIS, high SS) and different wagering preferences (e.g., ‘‘sure things’’ versus ‘‘long shots’’). To help determine whether a selection bias was present in their research, we used the information provided by Xu and Harvey (2014) to determine how closely the comparison groups’ probabilities of winning aligned with the winning percentage of the entire population. In all analyses, the probability of winning for the comparison group was significantly different from the entire population (all ps < .001), even when the comparison group contained 99.2% of all bets made. For analyses investigating the ‘‘hot hand’’ (which we believe identified those who preferred ‘‘safer,’’ high-probability wagers) and ‘‘Gambler’s Fallacy’’ (identifying those who preferred ‘‘riskier,’’ low-probability wagers), the winning percentage of the comparison group was always significantly lower and higher, respectively, compared to the entire population. These results strongly suggest that a selection bias was present and likely had a strong influence on Xu and Harvey’s (2014) results. The analysis of repeated (multi-trial) gambling data with multiple participants is very difficult. We believe that Xu and Harvey’s (2014) data would be better analyzed by evaluating how all individuals who achieved a consecutive
winning/losing streak of N trials wagered on trials N 1, N 2. . . N N. These data should be analyzed using repeated-measures analyses of variance (ANOVAs) and the dependent measure should be the statistical odds of winning each wager. The data would reflect Table 1. To support Xu and Harvey’s (2014) claims regarding the ‘‘Hot Hand’’ following wins (top panel), one would expect A > B > C > D > E > F > G (with each comparison being statistically significant using Post-hoc tests). To support their ‘‘Gambler’s Fallacy’’ explanation following losses (bottom panel), one would expect A < B < C < D < E < F < G. Xu and Harvey (2014) claimed that people select wagers with significantly lower and lower odds as they go from winning 1 in a row to winning 6 in a row (from about 8–1 odds to about 1–1 odds). Likewise, as losing streaks continued from 1 to 6 trials, they concluded that gamblers gradually chose ‘‘riskier’’ wagers (from odds of about 8–1 to about 17–1). We contend that more appropriate statistical analyses would reveal, at a minimum, a drastically attenuated influence of winning- and losing-streaks on risk-taking. Moreover, it is very plausible that the null hypothesis would be accepted. References Demaree, H. A., DeDonno, M. A., Burns, K. J., & Everhart, D. E. (2008). You bet: how personality differences affect risk-taking preferences. Personality and Individual Differences, 44, 1484–1494. Xu, J., & Harvey, N. (2014). Carry on winning: the gambler’s fallacy creates hot hand effects in online gambling. Cognition, 131, 173–180.
Contents lists available at ScienceDirect
Cognition journal homepage: www.elsevier.com/locate/COGNIT
Discussion
A fallacious ‘‘Gambler’s Fallacy’’? Commentary on Xu and Harvey (2014) Heath A. Demaree a,⇑, Joseph S. Weaver b, James Juergensen a a b
Department of Psychological Sciences, Case Western Reserve University, Cleveland, OH, USA Department of Psychology, DePauw University, Greencastle, IN, USA
a r t i c l e
i n f o
Article history: Received 13 June 2014 Revised 25 August 2014 Accepted 26 August 2014 Available online 18 September 2014 Keywords: Risk-taking Gambling Cognition Statistics Bias
a b s t r a c t In their recent article in Cognition, Xu and Harvey (2014) suggested that people who placed wagers on an online gambling site demonstrated very different wagering preferences depending on whether they were on winning or losing streaks. Specifically, they reported that people on winning streaks were more likely to win their subsequent wagers because they chose increasingly ‘‘safer,’’ higher-probability bets as the win streak continued. People on losing streaks were more likely to lose their subsequent wagers because they chose ‘‘riskier,’’ lower-probability wagers as the losing streak progressed. The authors suggested that individuals on winning and losing streaks both fell prey to the Gambler’s Fallacy. Specifically, individuals on winning streaks combatted their expectancy to lose soon by choosing higher-probability wagers (with lower payoffs). Conversely, people on losing streaks expected to win soon and thus preferred lower-probability wagers with higher payoffs. Though their paper is fascinating and contains a remarkable data set, we note that the statistical methods employed by Xu and Harvey are prone to a serious selection bias, such that participants on winning or losing streaks may have already been choosing safer and riskier wagers, respectively, prior to the beginning of their streaks. We suggest easy, intuitive analyses to determine whether the effects reported in Xu and Harvey (2014) are real. Ó 2014 Elsevier B.V. All rights reserved.
Xu and Harvey (2014) presented fascinating real-world, multi-trial gambling data. They found that people on winning streaks were more likely to win subsequent wagers because they gradually opted for higher-probability (‘‘safer’’) bets (i.e., they had a ‘‘hot hand’’). Conversely, people on losing streaks were more likely to lose their subsequent wagers because they gradually preferred lowerprobability (‘‘riskier’’) wagers with higher payouts. Xu and Harvey (2014) stated that individuals on both winning and losing streaks expected their luck to reverse (i.e., they
⇑ Corresponding author. Address: Department of Psychological Sciences, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7123, USA. Tel.: +1 216 368 6468; fax: +1 216 368 4891. E-mail address: [email protected] (H.A. Demaree). http://dx.doi.org/10.1016/j.cognition.2014.08.016 0010-0277/Ó 2014 Elsevier B.V. All rights reserved.
became prone to the ‘‘Gambler’s Fallacy’’). As a result, individuals on winning streaks preferred higher-probability wagers to help offset their expected bad luck whereas individuals on losing streaks expected to win soon and thus increasingly preferred ‘‘long shots’’ with high payouts. The paper is fascinating and, as such, it quickly received enormous attention: it has been written about on websites including The New Yorker, The Economist, Business Insider, The Wall Street Journal, and many, many others. As exciting as the data are, we believe that the statistical analyses used introduced a serious selection bias leading to a misunderstanding of the data. In their article, Xu and Harvey (2014) analyzed the probabilities of the wagers placed by individuals who just won 1, 2, 3, 4, 5, and 6 times in a row (i.e., bettors with a ‘‘hot hand’’), comparing them to the probabilities of the
169
H.A. Demaree et al. / Cognition 139 (2015) 168–170
Fig. 1. Hypothetical Trial 1 and Trial 2 data from repeated winners (left panel, a) and repeated losers (right panel, b). When evaluating the influence of repeated winning on subsequent wagers (Panel a), we imagine a normal distribution of Trial 1 data with some individuals preferring lower- and higher-probability wagers (represented by thinner and thicker vertical lines, respectively). The identification of 1-trial winners would filter out those who prefer ‘‘long shots,’’ leading to a negatively skewed population with higher P(W) preferences. The continued selection of 2-trial, 3-trial. . . 6-trial winners (not shown) would lead to populations with greater and greater negative skew and higher and higher P(W) averages. The mirror opposite would be expected when identifying individuals on losing streaks (Panel b; please note that thicker vertical lines in Panel b represent lower probabilities of winning).
wagers made by those without such winning streaks. They found that as the winning streak increased in size, so did the probability of winning the subsequent bet. Why? Xu and Harvey (2014) argued that it was because winning streaks gradually caused Gamblers to make ‘‘safer’’ and ‘‘safer,’’ higher probability, wagers. But didn’t the statistical analyses employed by Xu and Harvey (2014) gradually select out those bettors who preferred to make ‘‘safer,’’ higher-probability wagers (with lower payouts)? For example, Xu and Harvey (2014) first divided Bet 1 data, identifying those who won from those who lost. It is highly
intuitive that those who won their first wager were more likely to have placed higher-probability bets than those who lost their first wager. Thus, the analysis of Bet 2 data would be biased because it compared those who preferred ‘‘surer bets’’ (higher probability wagers with smaller payouts) at Bet 1 with those who preferred ‘‘long-shots’’ (bets with a lower probability of winning higher payoffs) at Bet 1. Once again, Bet 2 data were divided into winners and losers. Who were the individuals most likely to win 2 bets in a row? Those who preferred higher-probability wagers, of course! Repeated 6 times, we believe that Xu and Harvey (2014) gradually (but inadvertently) filtered out those individuals with the strongest preference for highprobability wagers and compared them to those without such a preference (please see Fig. 1 for a graphic depiction of this selection bias). Thus, in fact, Xu and Harvey’s (2014) analysis did not show that winning caused people to alter their betting strategy towards ‘‘safer,’’ higher-probability wagers. Rather, their analysis simply identified those individuals who preferred such wagers. We also suspect that a symmetrical function occurred when Xu and Harvey (2014) analyzed the data of Gamblers on losing streaks. When analyzing the wagers of individuals who just lost 1, 2, 3, 4, 5, and 6 straight bets, the researchers (inadvertently) gradually identified those individuals who preferred ‘‘long shots’’ compared to ‘‘sure things’’ (please see Fig. 1). Accordingly, it is no surprise that a person’s probability of winning a wager declined with the number of consecutive losses they just experienced (i.e., these individuals preferred long-shots). However, rather than the consecutive losses being a causal factor in increasing risk-taking propensities, the results found may be due to individual differences in gambling preferences. Indeed, people with different personalities consistently prefer different types of wagers. For example, Demaree, DeDonno, Burns, and Everhart (2008) found that individuals with higher levels of Behavioral Inhibition System (BIS) strength
Table 1 The data required to perform the suggested repeated-measures ANOVAs. Hot hand
Time After After After After After After
0 1 2 3 4 5 6
win wins wins wins wins wins
1 Straight win N = 178,947
2 Straight wins N = 88,036
3 Straight wins N = 50,300
4 Straight wins N = 33,871
5 Straight wins N = 24,390
6 Straight wins N = 18,190
A1 B1
A2 B2 C2
A3 B3 C3 D3
A4 B4 C4 D4 E4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6 G6
1 Straight loss N = 192,359
2 Straight losses N = 101,595
3 Straight losses N = 60,739
4 Straight losses N = 41,595*
5 Straight losses N = 11,231*
6 Straight losses N = 2808*
A1 B1
A2 B2 C2
A3 B3 C3 D3
A4 B4 C4 D4 E4
A5 B5 C5 D5 E5 F5
A6 B6 C6 D6 E6 F6 G6
Gambler’s Fallacy
Time After After After After After After *
0 1 2 3 4 5 6
loss losses losses losses losses losses
N not provided by Xu and Harvey, but calculated as approximates based on the data provided.
170
H.A. Demaree et al. / Cognition 139 (2015) 168–170
and higher levels of Sensation Seeking (SS) prefer lowerprobability, higher payout, wagers. When identifying people who experienced long winning and losing streaks, Xu and Harvey (2014) may have simply identified people with different personalities (e.g., high BIS, low SS versus low BIS, high SS) and different wagering preferences (e.g., ‘‘sure things’’ versus ‘‘long shots’’). To help determine whether a selection bias was present in their research, we used the information provided by Xu and Harvey (2014) to determine how closely the comparison groups’ probabilities of winning aligned with the winning percentage of the entire population. In all analyses, the probability of winning for the comparison group was significantly different from the entire population (all ps < .001), even when the comparison group contained 99.2% of all bets made. For analyses investigating the ‘‘hot hand’’ (which we believe identified those who preferred ‘‘safer,’’ high-probability wagers) and ‘‘Gambler’s Fallacy’’ (identifying those who preferred ‘‘riskier,’’ low-probability wagers), the winning percentage of the comparison group was always significantly lower and higher, respectively, compared to the entire population. These results strongly suggest that a selection bias was present and likely had a strong influence on Xu and Harvey’s (2014) results. The analysis of repeated (multi-trial) gambling data with multiple participants is very difficult. We believe that Xu and Harvey’s (2014) data would be better analyzed by evaluating how all individuals who achieved a consecutive
winning/losing streak of N trials wagered on trials N 1, N 2. . . N N. These data should be analyzed using repeated-measures analyses of variance (ANOVAs) and the dependent measure should be the statistical odds of winning each wager. The data would reflect Table 1. To support Xu and Harvey’s (2014) claims regarding the ‘‘Hot Hand’’ following wins (top panel), one would expect A > B > C > D > E > F > G (with each comparison being statistically significant using Post-hoc tests). To support their ‘‘Gambler’s Fallacy’’ explanation following losses (bottom panel), one would expect A < B < C < D < E < F < G. Xu and Harvey (2014) claimed that people select wagers with significantly lower and lower odds as they go from winning 1 in a row to winning 6 in a row (from about 8–1 odds to about 1–1 odds). Likewise, as losing streaks continued from 1 to 6 trials, they concluded that gamblers gradually chose ‘‘riskier’’ wagers (from odds of about 8–1 to about 17–1). We contend that more appropriate statistical analyses would reveal, at a minimum, a drastically attenuated influence of winning- and losing-streaks on risk-taking. Moreover, it is very plausible that the null hypothesis would be accepted. References Demaree, H. A., DeDonno, M. A., Burns, K. J., & Everhart, D. E. (2008). You bet: how personality differences affect risk-taking preferences. Personality and Individual Differences, 44, 1484–1494. Xu, J., & Harvey, N. (2014). Carry on winning: the gambler’s fallacy creates hot hand effects in online gambling. Cognition, 131, 173–180.
