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Surprised by the Gambler’s and Hot Hand Fallacies? A Truth in the Law of Small Numbers

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  • Joshua B. Miller
  • Adam Sanjurjo

Abstract

We prove that a subtle but substantial bias exists in a standard measure of the conditional dependence of present outcomes on streaks of past outcomes in sequential data. The magnitude of this novel form of selection bias generally decreases as the sequence gets longer, but increases in streak length, and remains substantial for a range of sequence lengths often used in empirical work. The bias has important implications for the literature that investigates incorrect beliefs in sequential decision making - most notably the Hot Hand Fallacy and the Gambler's Fallacy. Upon correcting for the bias, the conclusions of prominent studies in the hot hand fallacy literature are reversed. The bias also provides a novel structural explanation for how belief in the law of small numbers can persist in the face of experience. JEL Classification Numbers: C12; C14; C18;C19; C91; D03; G02. Keywords: Law of Small Numbers; Alternation Bias; Negative Recency Bias; Gambler's Fallacy; Hot Hand Fallacy; Hot Hand Effect; Sequential Decision Making; Sequential Data; Selection Bias; Finite Sample Bias; Small Sample Bias.

Suggested Citation

  • Joshua B. Miller & Adam Sanjurjo, 2015. "Surprised by the Gambler’s and Hot Hand Fallacies? A Truth in the Law of Small Numbers," Working Papers 552, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    • Handle: RePEc:igi:igierp:552
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    Cited by:

    1. David M. Ritzwoller & Joseph P. Romano, 2019. "Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences," Papers 1908.01406, arXiv.org, revised Apr 2021.
    2. Brett Green & Jeffrey Zwiebel, 2018. "The Hot-Hand Fallacy: Cognitive Mistakes or Equilibrium Adjustments? Evidence from Major League Baseball," Management Science, INFORMS, vol. 64(11), pages 5315-5348, November.
    3. Kovic, Marko & Kristiansen, Silje, 2016. "The gambler's fallacy fallacy (fallacy)," SocArXiv xdsxg, Center for Open Science.
    4. Robert M. Lantis & Erik T. Nesson, 2019. "Hot Shots: An Analysis of the ‘Hot Hand’ in NBA Field Goal and Free Throw Shooting," NBER Working Papers 26510, National Bureau of Economic Research, Inc.
    5. Cotton, Christopher S. & McIntyre, Frank & Nordstrom, Ardyn & Price, Joseph, 2019. "Correcting for bias in hot hand analysis: An application to youth golf," Journal of Economic Psychology, Elsevier, vol. 75(PB).
    6. Florian Peters & Simas Kucinskas, 2018. "Measuring Biases in Expectation Formation," Tinbergen Institute Discussion Papers 18-058/IV, Tinbergen Institute.
    7. Ambroise Descamps & Changxia Ke & Lionel Page, 2022. "How success breeds success," Quantitative Economics, Econometric Society, vol. 13(1), pages 355-385, January.
    8. Colella, Fabrizio & Dalton, Patricio & Giusti, G., 2018. "You'll Never Walk Alone : The Effect of Moral Support on Performance," Other publications TiSEM 1dac53ca-9483-48f5-84b0-0, Tilburg University, School of Economics and Management.
    9. Miller, Joshua Benjamin & Sanjurjo, Adam, 2018. "How Experience Confirms the Gambler's Fallacy when Sample Size is Neglected," OSF Preprints m5xsk, Center for Open Science.
    10. Robert Lantis & Erik Nesson, 2021. "Hot Shots: An Analysis of the “Hot Hand†in NBA Field Goal and Free Throw Shooting," Journal of Sports Economics, , vol. 22(6), pages 639-677, August.
    11. Christopher Cotton & Frank McIntyre & Joseph P. Price, 2016. "Correcting For Bias In Hot Hand Analysis: Analyzing Performance Streaks In Youth Golf," Working Paper 1366, Economics Department, Queen's University.
    12. Daniel J. Benjamin & Don A. Moore & Matthew Rabin, 2017. "Biased Beliefs About Random Samples: Evidence from Two Integrated Experiments," NBER Working Papers 23927, National Bureau of Economic Research, Inc.
    13. Kononovicius, A., 2019. "Illusion of persistence in NBA 1995–2018 regular season data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 520(C), pages 250-256.
    14. Legge, Stefan & Schmid, Lukas, 2016. "Media attention and betting markets," European Economic Review, Elsevier, vol. 87(C), pages 304-333.
    15. Daniel F. Stone & Jeremy Arkes, 2016. "Reference Points, Prospect Theory, and Momentum on the PGA Tour," Journal of Sports Economics, , vol. 17(5), pages 453-482, June.
    16. Oyarzun, Carlos & Sanjurjo, Adam & Nguyen, Hien, 2017. "Response functions," European Economic Review, Elsevier, vol. 98(C), pages 1-31.
    17. Yosef Rinott & Maya Bar-Hillel, 2015. "Comments on a “Hot Hand” Paper by Miller and Sanjurjo (2015)," Discussion Paper Series dp688, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

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    More about this item

    Keywords

    law of small numbers; alternation bias; negative recency bias; gambler's fallacy; hot hand fallacy; hot hand effect; sequential decision making; sequential data; selection bias; finite sample bias; small sample bias.;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other
    • C91 - Mathematical and Quantitative Methods - - Design of Experiments - - - Laboratory, Individual Behavior
    • D03 - Microeconomics - - General - - - Behavioral Microeconomics: Underlying Principles
    • G02 - Financial Economics - - General - - - Behavioral Finance: Underlying Principles

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