How did Berezovsky decide it was time to leave Russia? Is there a way, perhaps, to think mathematically about the advice to “quit while you’re ahead”? Berezovsky in particular might have considered this very question himself, since the topic he had worked on all those years ago as a mathematician was none other than optimal stopping; he authored the first (and, so far, the only) book entirely devoted to the secretary problem.
The problem of quitting while you’re ahead has been analyzed under several different guises, but perhaps the most appropriate to Berezovsky’s case—with apologies to Russian oligarchs—is known as the “burglar problem.” In this problem, a burglar has the opportunity to carry out a sequence of robberies. Each robbery provides some reward, and there’s a chance of getting away with it each time. But if the burglar is caught, he gets arrested and loses all his accumulated gains. What algorithm should he follow to maximize his expected take?
The fact that this problem has a solution is bad news for heist movie screenplays: when the team is trying to lure the old burglar out of retirement for one last job, the canny thief need only crunch the numbers. Moreover, the results are pretty intuitive: the number of robberies you should carry out is roughly equal to the chance you get away, divided by the chance you get caught. If you’re a skilled burglar and have a 90% chance of pulling off each robbery (and a 10% chance of losing it all), then retire after 90/10 = 9 robberies. A ham-fisted amateur with a 50/50 chance of success? The first time you have nothing to lose, but don’t push your luck more than once.
Despite his expertise in optimal stopping, Berezovsky’s story ends sadly. He died in March 2013, found by a bodyguard in the locked bathroom of his house in Berkshire with a ligature around his neck. The official conclusion of a postmortem examination was that he had committed suicide, hanging himself after losing much of his wealth through a series of high-profile legal cases involving his enemies in Russia. Perhaps he should have stopped sooner—amassing just a few tens of millions of dollars, say, and not getting into politics. But, alas, that was not his style. One of his mathematician friends, Leonid Boguslavsky, told a story about Berezovsky from when they were both young researchers: on a water-skiing trip to a lake near Moscow, the boat they had planned to use broke down. Here’s how David Hoffman tells it in his book The Oligarchs:
While their friends went to the beach and lit a bonfire, Boguslavsky and Berezovsky headed to the dock to try to repair the motor.… Three hours later, they had taken apart and reassembled the motor. It was still dead. They had missed most of the party, yet Berezovsky insisted they had to keep trying. “We tried this and that,” Boguslavsky recalled. Berezovsky would not give up.
Surprisingly, not giving up—ever—also makes an appearance in the optimal stopping literature. It might not seem like it from the wide range of problems we have discussed, but there are sequential decision-making problems for which there is no optimal stopping rule. A simple example is the game of “triple or nothing.” Imagine you have $1.00, and can play the following game as many times as you want: bet all your money, and have a 50% chance of receiving triple the amount and a 50% chance of losing your entire stake. How many times should you play? Despite its simplicity, there is no optimal stopping rule for this problem, since each time you play, your average gains are a little higher. Starting with $1.00, you will get $3.00 half the time and $0.00 half the time, so on average you expect to end the first round with $1.50 in your pocket. Then, if you were lucky in the first round, the two possibilities from the $3.00 you’ve just won are $9.00 and $0.00—for an average return of $4.50 from the second bet. The math shows that you should always keep playing. But if you follow this strategy, you will eventually lose everything. Some problems are better avoided than solved.
Always Be Stopping
I expect to pass through this world but once. Any good therefore that I can do, or any kindness that I can show to any fellow creature, let me do it now. Let me not defer or neglect it, for I shall not pass this way again.
—STEPHEN GRELLET
Spend the afternoon. You can’t take it with you.
—ANNIE DILLARD
We’ve looked at specific cases of people confronting stopping problems in their lives, and it’s clear that most of us encounter these kinds of problems, in one form or another, daily. Whether it involves secretaries, fiancé(e)s, or apartments, life is full of optimal stopping. So the irresistible question is whether—by evolution or education or intuition—we actually do follow the best strategies.
At first glance, the answer is no. About a dozen studies have produced the same result: people tend to stop early, leaving better applicants unseen. To get a better sense for these findings, we talked to UC Riverside’s Amnon Rapoport, who has been running optimal stopping experiments in the laboratory for more than forty years.
The study that most closely follows the classical secretary problem was run in the 1990s by Rapoport and his collaborator Darryl Seale. In this study people went through numerous repetitions of the secretary problem, with either 40 or 80 applicants each time. The overall rate at which people found the best possible applicant was pretty good: about 31%, not far from the optimal 37%. Most people acted in a way that was consistent with the Look-Then-Leap Rule, but they leapt sooner than they should have more than four-fifths of the time.
Rapoport told us that he keeps this in mind when solving optimal stopping problems in his own life. In searching for an apartment, for instance, he fights his own urge to commit quickly. “Despite the fact that by nature I am very impatient and I want to take the first apartment, I try to control myself!”
But that impatience suggests another consideration that isn’t taken into account in the classical secretary problem: the role of time. After all, the whole time you’re searching for a secretary, you don’t have a secretary. What’s more, you’re spending the day conducting interviews instead of getting your own work done.
This type of cost offers a potential explanation for why people stop early when solving a secretary problem in the lab. Seale and Rapoport showed that if the cost of seeing each applicant is imagined to be, for instance, 1% of the value of finding the best secretary, then the optimal strategy would perfectly align with where people actually switched from looking to leaping in their experiment.
The mystery is that in Seale and Rapoport’s study, there wasn’t a cost for search. So why might people in the laboratory be acting like there was one?
Because for people there’s always a time cost. It doesn’t come from the design of the experiment. It comes from people’s lives.
The “endogenous” time costs of searching, which aren’t usually captured by optimal stopping models, might thus provide an explanation for why human decision-making routinely diverges from the prescriptions of those models. As optimal stopping researcher Neil Bearden puts it, “After searching for a while, we humans just tend to get bored. It’s not irrational to get bored, but it’s hard to model that rigorously.”
But this doesn’t make optimal stopping problems less important; it actually makes them more important, because the flow of time turns all decision-making into optimal stopping.
“The theory of optimal stopping is concerned with the problem of choosing a time to take a given action,” opens the definitive textbook on optimal stopping, and it’s hard to think of a more concise description of the human condition. We decide the right time to buy stocks and the right time to sell them, sure; but also the right time to open the bottle of wine we’ve been keeping around for a special occasion, the right moment to interrupt someone, the right moment to kiss them.
Viewed this way, the secretary problem’s most fundamental yet most unbelievable assumption—its strict seriality, its inexorable one-way march—is revealed to be the nature of time itself. As such, the explicit premise of the optimal stopping problem is the implicit premise of what it is to be alive. It’s this that forces us to decide based on possibilities we’ve not yet seen, this that forces us to embrace