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Gambler’s fallacy makes baseball and life unfair

Suppose you’re watching a baseball game, and your favorite player, a terrific hitter with a .320 average, has struck out three times in a row. If you’re like most people, you might think, “He’s due!” — and conclude that on his fourth at-bat, he’s likely to get a hit.

Now suppose that you are working in a college admissions office. Your job is to evaluate 200 applicants, about 50 of whom will be admitted. You’ve just accepted three in a row, and now you might be inclined to think that the next two are unlikely to deserve admission. You might even evaluate their applications with that skeptical thought in mind.

A lot of people are prone to this “gambler’s fallacy” — the mistaken belief that a small sequence of events will look a lot like a bigger one. Flip a coin 1,000 times, and there’s a very high probability it will come up heads half the time. But flip a coin five times, and you'll find some surprises. Heads-tails-heads-tails-heads is no more likely than heads-heads-heads-heads-tails, for example. When we’re dealing with small numbers, our intuition leads us the wrong way.

Does the gambler’s fallacy lead to major mistakes in the real world? Until recently, we haven’t had a good answer to that question. But economists Daniel Chen, Tobias Moskowitz and Kelly Shue have now found evidence that the fallacy leads asylum judges, loan officers and baseball umpires to make major mistakes. The result can be serious unfairness.

Asylum judges have a lot of power. If foreign nationals can prove that they have “a well-founded fear of persecution” in their own countries, they are allowed to stay in the U.S. — potentially avoiding torture, death or imprisonment. Here’s the question: After granting asylum in one case, are asylum judges less likely to grant it in the next one?

Indeed they are. The difference is not huge, but it is real — 1.5 percentage points less than if the previous decision had been a denial. Things get worse if a judge has granted two requests; then, the likelihood of a denial is 2.1 percentage points lower than after two denials.

These might not sound like big numbers, but they add up; over long periods of time, hundreds of asylum applicants can be affected. The alarming implication is that a lot of people are being denied asylum in the U.S. not on the merits of their requests, but because of what happened in previous cases.

Interestingly, the effect is limited to judges who have less than 10 years of experience. Those who have been deciding asylum cases a long time don’t “autocorrect” in light of their previous judgments.

To test the decisions made by loan officers, Chen and his coauthors recruited a large number of officers from banks (who had an average of 10 years’ experience) and asked them to screen actual applications.

The gamblers’ fallacy played a large role here. Under various versions of the experiment, loan officers were 5 to 8 percentage points less likely to approve a loan if they had approved the previous application.

To investigate baseball umpires’ decisions, the researchers used an extraordinary data set consisting of 1.5 million called pitches from 2008 through 2012 in major league baseball, including information about the location and trajectory of each pitch as it crossed home plate. Because they could tell whether calls were actually right, the researchers could see whether the umpires made mistakes as a result of the gambler’s fallacy.

Here again, the effect was small but real: Umpires were 0.9 percent less likely to call a pitch a strike if they had called the most recent called pitch a strike. If the two most recently called pitches were strikes, umpires were 1.3 percentage points less likely to call a pitch a strike. Over the course of any team’s season, hundreds of calls will go the wrong way. And when the researchers restricted their analysis to consecutive calls (excluding pitches where the hitter swings), they found that umpires were even more likely to be affected by their previous decision.

When the researchers looked at all their data, they found that the gambler’s fallacy leads asylum judges, loan officers and baseball umpires to reverse as many as 5 percent of all their decisions. That’s a large number, suggesting that intuition leads to a lot of unfairness. It also raises the question: What other important decisions are being skewed by the gambler’s fallacy?

Cass R. Sunstein, a Bloomberg View columnist, is director of the Harvard Law School’s program on behavioral economics and public policy.

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