Luck, time, and price

Praise, blame, and luck

As a young psychologist in Israel in the early to mid 1960s, Daniel Kahneman was asked to teach flight instructors in the Air Force how to train cadets. He argued that praise works better than punishment. A senior instructor interrupted. Whenever he praised a cadet for a clean maneuver, he said, the cadet did worse on the next attempt; whenever he screamed at a cadet for a bad one, the cadet did better. So, he concluded, praise spoils people and shouting helps.

Kahneman later called it the most satisfying eureka moment of his career, because he suddenly saw what was happening. The instructor was right about the facts and wrong about the cause. A cadet who has just flown unusually well was probably both good and lucky that day; the luck will not repeat, so the next flight is likely to be closer to average. A cadet who has just flown unusually badly was probably unlucky, and the next flight will tend to be better. This is regression to the mean. Kahneman put it in a sentence: instructors are punished for praising and rewarded for scolding, and neither has anything to do with the praise or the scolding. He then ran a small demonstration: the instructors tossed coins at a target behind their backs, and the best first throws mostly got worse on the second.

The word “regression” comes from Francis Galton, who in 1886 measured the heights of adult children and their parents and found that children of unusually tall parents are on average tall, but not as tall as their parents: their deviation from the average was about two-thirds of the parents’. Wherever a result mixes skill and luck, the extremes drift back. Try the simulation below.

Regression to the mean explains a great deal that looks like cause and effect: why a player with an exceptional season often has a worse one next year, why a patient who comes in at their worst gets better with almost any treatment, and why a school’s best-ever year is followed by an ordinary one.

Eight daughters in a row

A couple has had eight daughters. Is a boy now “due”?

We expect small samples to look like the population they come from, and we are surprised by streaks. In a 1972 paper, Kahneman and Tversky posed the “hospital problem.” A town has two hospitals. In the larger, about 45 babies are born each day; in the smaller, about 15. About half of all babies are boys, but on any single day the exact share varies. Over a year, which hospital records more days on which more than 60 percent of the babies born are boys? Of 50 Stanford students, 28 said “about the same,” 12 said the larger hospital, and 10 the smaller. The right answer is the smaller one. A small sample strays further from the average: with 15 births, more than 60 percent boys (ten or more) happens on about 15 percent of days; with 45 births, on about 7 percent.

The opposite mistake is also common. In a famous series of studies, Thomas Gilovich, Robert Vallone, and Tversky (1985) examined the shooting records of the Philadelphia 76ers and the free throws of the Boston Celtics and found no evidence that a player who has just scored is more likely to score again. Yet 91 percent of the fans they surveyed believed in the “hot hand.” For thirty years this was the textbook example of seeing a pattern in randomness.

The gambler’s fallacy, the belief that after a run the other outcome is “due,” appears in real work. Daniel Chen, Tobias Moskowitz, and Kelly Shue found in 2016 that after granting an asylum case, immigration judges were slightly less likely to grant the next (0.5 percentage points less, and more after two grants in a row); that loan officers in a field experiment were 8 points less likely to approve a loan right after approving one; and that baseball umpires were 1.5 points less likely to call a strike on a pitch after calling the previous one a strike. People deciding independent cases treated them as if the run had to even out. (You will see the story of the Monte Carlo casino, where black came up 26 times in a row and gamblers lost millions betting on red. The run is widely repeated, but the earliest source for the 1913 date is a popular book of 1959, and mid-century writers dated it to 1931, so the story is a legend as told.)

Why plans run late

In 1979 Kahneman and Tversky named a related error, the planning fallacy: we underestimate how long a project will take, even when we know that similar projects ran late. In 1994 the psychologist Roger Buehler and colleagues asked 37 honors students at the University of Waterloo how long their theses would take. The average best guess was 33.9 days; the “worst case” guess was 48.6 days. The actual average was 55.5 days. Only 30 percent finished by their best guess, and fewer than half by their worst-case guess.

Kahneman fell into it himself. He was once part of a team writing a textbook on judgment and decision-making for Israel’s Ministry of Education. Asked to estimate how long it would take, each member wrote down a number, and the estimates clustered around two years, from one and a half to two and a half. Then Kahneman asked the team’s curriculum expert how long such projects had taken in other places. The expert said that about 40 percent never finished, and that the rest took seven to ten years. The textbook was finished eight years later, after Kahneman had left, and the ministry never used it.

The same happens on a grand scale. The Sydney Opera House was estimated at about £3.5 million (A$7 million) when construction began in 1959; its first stage was finished two years late, and it opened on 20 October 1973 at many times that cost. The economist Bent Flyvbjerg, who studies megaprojects, found with colleagues in 2002 that costs were underestimated in nearly nine of ten of 258 transport projects worldwide: rail by an average of 44.7 percent, bridges and tunnels 33.8 percent, roads 20.4 percent. A 2016 study of 95 road and rail projects in China built between 1984 and 2008 found that 75 percent went over budget, by an average of 30.6 percent, a record no better and no worse than in rich democracies. Flyvbjerg argues that the cause is not only optimism but also the incentive to make a project look cheap in order to get it approved.

The remedy Kahneman and Flyvbjerg recommend is the outside view, or reference-class forecasting: instead of asking how long this project will take given its particular plan, ask how long projects like it have taken, and begin from that number. An “inside view” is made of the story of the project. The outside view is made of the record.

Optimism is not confined to planners. In a 1988 survey of entrepreneurs, 81 percent rated their own chances of success at seven in ten or better, and a third said their chance of failing was zero. Official statistics tell a different story: about 80 percent of new private-sector establishments in the United States are still open after a year and about half after five. In 1981 the Swedish psychologist Ola Svenson found that 93 percent of 41 American students rated themselves above the median driver in skill, a classic result, though a statistical argument by Jean-Pierre Benoît and Juan Dubra shows that some “better than average” results can be consistent with accurate self-knowledge.

The price in your head

We do not carry a bill of rates in our heads. We carry accounts, comparisons, and feelings. Two problems from Tversky and Kahneman (1981) show how. First: you are on your way to a play, and you discover that you have lost a $10 bill. Do you still buy a $10 ticket? Second: you have bought the ticket, and discover on arrival that you have lost it, and it cannot be replaced. Do you buy another? In the first case 88 percent of respondents said yes; in the second, only 46 percent. The loss of $10 is the same, but a lost ticket makes the evening cost $20, while a lost bill belongs to another mental account.

In the same paper, 68 percent of respondents would drive 20 minutes to save $5 on a $15 calculator, but only 29 percent to save $5 on a $125 one. A 2018 replication with more than 7,000 people found the same direction at less than half the size, 49 percent against 32 percent. Judging a saving by its proportion rather than by what it buys is why a “50 percent off” sign on a cheap item beats a $20 saving on an expensive one.

The same accounting traps us after we have paid. In a 1985 field experiment by Hal Arkes and Catherine Blumer, buyers of season tickets to a university theater were randomly given discounts. Those who paid full price attended more of the plays in the first half of the season, 4.1 against 3.3, though the money was gone either way. That is the sunk-cost fallacy: past spending that cannot be recovered still pulls at the next decision. (The study was small, and later studies, such as one by Daniel Friedman and colleagues in 2007, find the effect in the laboratory smaller.) A related puzzle is the gym. Stefano DellaVigna and Ulrike Malmendier found in 2006 that members on monthly contracts paid more than $70 a month and visited about four times, so that each visit cost over $17, when a ten-visit pass cost $10 a visit. They estimated that members lost about $600 over the length of a membership by not choosing the pass: a monthly fee bought good intentions.

An unusually expensive example is the American Airlines “AAirpass.” In 1981 the airline began selling unlimited first-class travel for life at $250,000, with a companion pass for $150,000 more; the price went to $600,000 with a companion in 1990 and to over a million dollars in 1993, and sales stopped in 1994. A buyer called Steven Rothstein, who bought in 1987, flew more than 30 million miles; American later estimated that he and another holder, Jacques Vroom, each cost it more than $1 million a year, and revoked both passes in 2008, accusing them of breaking the pass’s terms. For the buyer, every flight after the purchase cost nothing, so he flew constantly; for the airline, a price set for the average traveler met an extreme one. (The popular story that one holder flew to London sixteen times in 25 days belongs to a different holder, Mike Joyce, in 2012.)

Free is special

Free is a price like no other. In 2007 Kristina Shampanier, Nina Mazar, and Dan Ariely offered shoppers a choice between a Lindt truffle at 15 cents and a Hershey’s Kiss at 1 cent: 73 percent took the truffle. When each price was lowered by a cent, to 14 cents and free, 69 percent switched to the Kiss. The relative difference had not changed. Independent work supports the idea that “free” has a pull of its own, such as the extra draw of a free hotel breakfast and of free app tiers, but we could find no direct repeat of the chocolate experiment itself, and one of its authors has had other work questioned, as described below.

Prices speak in digits as well as in amounts. A price that ends in 9 seems lower only when the leftmost digit changes: $2.99 feels much cheaper than $3.00, but $2.89 feels no cheaper than $2.90 (Thomas and Morwitz, 2005). In Chinese-speaking markets the digits carry meaning too: 八 (bā) sounds like 发 (fā), “to get rich,” and 四 (sì) sounds like 死 (sǐ), “death.” A study of Chinese-language advertising found 8 over-used and 4 avoided in price endings (Simmons and Schindler, 2003). A study of about 117,000 house sales in Greater Vancouver found that addresses ending in 8 sold for 2.5 percent more and those ending in 4 for 2.2 percent less (Fortin, Hill, and Huang, 2014). American buildings often have no thirteenth floor, and many Chinese buildings have no fourth.

When money spoils a favor

Our last lesson in price is that some things should not have one. In 2000 Uri Gneezy and Aldo Rustichini studied ten day-care centers in Haifa, Israel. Six began to charge parents a small fine, about $3, for picking up children more than ten minutes late. Late pickups roughly doubled, and did not drop when the fine was removed. The fine, they argued, turned an obligation to the teachers into a price that could be paid. In Sweden, Erik Mellström and Magnus Johannesson found in 2008 that paying people about $7 to donate blood cut the number of women who signed up by almost half, and that letting them give the payment to charity cancelled the drop. A larger American study found that incentives raised donations, though a good part of the gain came at the expense of nearby blood drives.

This is the difference between social norms, in which things are given and kindness is repaid by kindness, and market norms, in which things are exchanged for money and the accounts are settled on the spot. Mixing the two can offend both. In a 2004 experiment, Dan Ariely and James Heyman paid people to drag circles into a box on a computer screen; people paid 50 cents did far worse than people paid $5, and people asked to do it as a favor did as well as or better than either. (The journal that published the study issued an expression of concern in 2021 because the data could not be checked, so we count this as an illustration, not as evidence.)

What can be done?

If our minds mislead us this reliably, can we do anything about it? Kahneman was candid in his book’s conclusion: apart from the effects of age, his own intuitions were as prone to overconfidence and the planning fallacy as ever. What he had learned was to recognize situations, minefields, in which a mistake is likely, and to slow down there. Organizations do this better than individuals, with checklists, reference-class forecasting, and the pre-mortem, in which a team imagines that a plan has failed and asks why.

There is some evidence that training helps. In a 2015 study by Carey Morewedge and colleagues, a short video game that taught people about bias reduced their bias scores by at least 32 percent right away and at least 24 percent two months later; a video lecture did about as well at two months (19 percent). A later field study of students found that trained students were 19 percent less likely to pick the inferior, confirmation-biased answer on a business case. The effect is modest, but it lasts. Kahneman, Olivier Sibony, and Cass Sunstein turned to a related problem in their 2021 book Noise: the unwanted variation in judgments that should be the same.

Kahneman died on 27 March 2024, aged 90. His legacy is a set of experiments that have held up, a set of stories that outran their evidence, and a method for telling which is which: ask who else has repeated it. Thinking slowly about our own thinking is the habit all of this adds up to.