scenario also being 61%.
For Kepler, the difference between reality and the classical secretary problem brought with it a happy ending. In fact, the twist on the classical problem worked out well for Trick, too. After the rejection, he completed his degree and took a job in Germany. There, he “walked into a bar, fell in love with a beautiful woman, moved in together three weeks later, [and] invited her to live in the United States ‘for a while.’” She agreed—and six years later, they were wed.
Knowing a Good Thing When You See It: Full Information
The first set of variants we considered—rejection and recall—altered the classical secretary problem’s assumptions that timely proposals are always accepted, and tardy proposals, never. For these variants, the best approach remained the same as in the original: look noncommittally for a time, then be ready to leap.
But there’s an even more fundamental assumption of the secretary problem that we might call into question. Namely, in the secretary problem we know nothing about the applicants other than how they compare to one another. We don’t have an objective or preexisting sense of what makes for a good or a bad applicant; moreover, when we compare two of them, we know which of the two is better, but not by how much. It’s this fact that gives rise to the unavoidable “look” phase, in which we risk passing up a superb early applicant while we calibrate our expectations and standards. Mathematicians refer to this genre of optimal stopping problems as “no-information games.”
This setup is arguably a far cry from most searches for an apartment, a partner, or even a secretary. Imagine instead that we had some kind of objective criterion—if every secretary, for instance, had taken a typing exam scored by percentile, in the fashion of the SAT or GRE or LSAT. That is, every applicant’s score will tell us where they fall among all the typists who took the test: a 51st-percentile typist is just above average, a 75th-percentile typist is better than three test takers out of four, and so on.
Suppose that our applicant pool is representative of the population at large and isn’t skewed or self-selected in any way. Furthermore, suppose we decide that typing speed is the only thing that matters about our applicants. Then we have what mathematicians call “full information,” and everything changes. “No buildup of experience is needed to set a standard,” as the seminal 1966 paper on the problem put it, “and a profitable choice can sometimes be made immediately.” In other words, if a 95th-percentile applicant happens to be the first one we evaluate, we know it instantly and can confidently hire her on the spot—that is, of course, assuming we don’t think there’s a 96th-percentile applicant in the pool.
And there’s the rub. If our goal is, again, to get the single best person for the job, we still need to weigh the likelihood that there’s a stronger applicant out there. However, the fact that we have full information gives us everything we need to calculate those odds directly. The chance that our next applicant is in the 96th percentile or higher will always be 1 in 20, for instance. Thus the decision of whether to stop comes down entirely to how many applicants we have left to see. Full information means that we don’t need to look before we leap. We can instead use the Threshold Rule, where we immediately accept an applicant if she is above a certain percentile. We don’t need to look at an initial group of candidates to set this threshold—but we do, however, need to be keenly aware of how much looking remains available.
The math shows that when there are a lot of applicants left in the pool, you should pass up even a very good applicant in the hopes of finding someone still better than that—but as your options dwindle, you should be prepared to hire anyone who’s simply better than average. It’s a familiar, if not exactly inspiring, message: in the face of slim pickings, lower your standards. It also makes clear the converse: with more fish in the sea, raise them. In both cases, crucially, the math tells you exactly by how much.
The easiest way to understand the numbers for this scenario is to start at the end and think backward. If you’re down to the last applicant, of course, you are necessarily forced to choose her. But when looking at the next-to-last applicant, the question becomes: is she above the 50th percentile? If yes, then hire her; if not, it’s worth rolling the dice on the last applicant instead, since her odds of being above the 50th percentile are 50/50 by definition. Likewise, you should choose the third-to-last applicant if she’s above the 69th percentile, the fourth-to-last applicant if she’s above the 78th, and so on, being more choosy the more applicants are left. No matter what, never hire someone who’s below average unless you’re totally out of options. (And since you’re still interested only in finding the very best person in the applicant pool, never hire someone who isn’t the best you’ve seen so far.)
The chance of ending up with the single best applicant in this full-information version of the secretary problem comes to 58%—still far from a guarantee, but considerably better than the 37% success rate offered by the 37% Rule in the no-information game. If you have all the facts, you can succeed more often than not, even as the applicant pool grows arbitrarily large.
Optimal stopping thresholds in the full-information secretary problem.
The full-information game thus offers an unexpected and somewhat bizarre takeaway. Gold digging is more likely to succeed than a quest for love. If you’re evaluating your partners based on any kind of objective criterion—say, their income percentile—then you’ve got a lot more information at your disposal than if you’re after a nebulous emotional response (“love”) that might require both experience and comparison to calibrate.
Of course, there’s no reason that net worth—or, for that matter, typing speed—needs to be the thing that you’re measuring. Any yardstick that provides full information on where an applicant stands relative to the population at large will change the solution from the Look-Then-Leap Rule to the Threshold Rule and will dramatically boost your chances of finding the single best applicant in the group.
There are many more variants of the secretary problem that modify its other assumptions, perhaps bringing it more in line with the real-world challenges of finding love (or a secretary). But the lessons to be learned from optimal stopping aren’t limited to dating or hiring. In fact, trying to make the best choice when options only present themselves one by one is also the basic structure of selling a house, parking a car, and quitting when you’re ahead. And they’re all, to some degree or other, solved problems.
When to Sell
If we alter two more aspects of the classical secretary problem, we find ourselves catapulted from the realm of dating to the realm of real estate. Earlier we talked about the process of renting an apartment as an optimal stopping problem, but owning a home has no shortage of optimal stopping either.
Imagine selling a house, for instance. After consulting with several real estate agents, you put your place on the market; a new coat of paint, some landscaping, and then it’s just a matter of waiting for the offers to come in. As each offer arrives, you typically have to decide whether to accept it or turn it down. But turning down an offer comes at a cost—another week (or month) of mortgage payments while you wait for the next offer, which isn’t guaranteed to be any better.
Selling a house is similar to the full-information game. We know the objective dollar value of the offers, telling us not only which ones are better than which, but also by how much. What’s more, we have information about the broader state of the market, which enables us to at least roughly predict the range of offers to expect. (This gives us the same “percentile” information about each offer that we had with the typing exam above.) The difference here, however, is that our goal isn’t actually to secure the single best offer—it’s to make the most money through the process overall. Given that waiting has a cost measured in dollars, a good offer today beats
