LogicalMANIPULATIVE

Gambler's Fallacy

What it is

Treating independent random events as if they were self-correcting — a number that has not come up is “due”, a streak must end — and selling, betting or arguing on the strength of that expectation.

How it works

A fair coin that has landed heads five times has exactly a 50 percent chance of heads on the sixth throw; the coin has no memory and no obligation to even out. People feel otherwise because they expect short sequences to look like the long-run proportions — Tversky and Kahneman's belief in the law of small numbers — so a run of one outcome feels like a debt the other outcome must repay. On a reported night in 1913 the roulette wheel at Monte Carlo came up black twenty-six times in a row while players bet ever larger sums on red, certain each spin that the streak had to end; the twenty-seventh spin's odds were what they had been on the first. The bias is not confined to casinos. Chen, Moskowitz and Shue found that asylum judges, loan officers and baseball umpires were all measurably less likely to decide a case one way if they had just decided the previous case that way. The persuasion move is to sell the correction: lottery “overdue numbers”, hot-and-cold boards at roulette tables, tipsters and market pundits announcing that a run is due to break, campaign talk that a district is due to flip. The related hot-hand question in sport is more nuanced than once thought — Miller and Sanjurjo showed the classic 1985 analysis was itself biased — but that concerns skill, not dice.

Real-world examples

  • •The 1913 Monte Carlo run of twenty-six blacks is the textbook case: the casino reportedly gained millions of francs from players betting on red with growing conviction as the streak lengthened.
  • •Chen, Moskowitz and Shue (2016) analyzed US asylum decisions, loan approvals and umpires' ball-strike calls and found in each a tendency to alternate after a run of similar decisions, consistent with a belief that outcomes should even out.
  • •Casinos install displays showing the last twenty roulette results, and lottery operators publish “hot” and “cold” numbers; both feed a belief that past draws carry information, which they do not, and both increase play.
  • •Croson and Sundali (2005) used casino video records to show that roulette bettors did bet against streaks — the gambler's fallacy in real money — while also betting more after their own wins, the hot-hand belief.
  • •Commentary that a party is “due” to win because the other has held office for several terms, or that a stock is due to rise because it has fallen for weeks, imports the same intuition into events that are not dice throws either — where the error is compounded by the fact that the events are not independent in any simple way.

Ethical guidelines

  • ●Do not market, forecast or advise on the basis that independent events are owed a correction; if you display past results of a random process, say plainly that they do not predict the next one.
  • ●When an event is not independent of its predecessors, say what the actual dependence is instead of borrowing the language of streaks and dues.
  • ●Decision-makers who process cases in sequence should be told about the alternation tendency and use structured criteria that do not depend on the previous case.

How to defend against it

  • ►Ask whether the next outcome physically depends on the previous ones. For coins, wheels, lottery balls and most market ticks the answer is no, and any “due” claim is empty.
  • ►Rewrite the streak as a single question: what is the chance of this outcome on one trial? That is the chance now, whatever came before.
  • ►Notice when you are being shown history for a random process — recent results, hot numbers — and ask what the display is for, since it cannot be for prediction.
  • ►If you make sequential decisions, keep a log and check whether your approvals cluster or alternate; if they alternate, you are being influenced by the previous case and can correct for it.

References

  1. Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76(2), 105-110 · link
    The expectation that short random sequences should resemble long-run proportions, the root of the gambler's fallacy.
  2. Chen, D. L., Moskowitz, T. J., & Shue, K. (2016). Decision making under the gambler's fallacy: Evidence from asylum judges, loan officers, and baseball umpires. Quarterly Journal of Economics, 131(3), 1181-1242
    Field evidence of alternation after runs of similar decisions among professional decision-makers.
  3. Croson, R., & Sundali, J. (2005). The gambler's fallacy and the hot hand: Empirical data from casinos. Journal of Risk and Uncertainty, 30(3), 195-209
    Casino video data showing bets against streaks in roulette and increased betting after personal wins.
  4. Miller, J. B., & Sanjurjo, A. (2018). Surprised by the hot hand fallacy? A truth in the law of small numbers. Econometrica, 86(6), 2019-2047
    The finding that the classic 1985 test for the hot hand was biased, so the sport question is more open than the dice question.
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