This Elegant Math Problem Could Help You Make the Best Choice in House-Hunting and Even Love

This Elegant Math Problem Helps You Find the Best Choice for Hiring, House-Hunting and Even Love

By Jack Murtagh

Imagine cruising down the highway when you notice your fuel tank running low. Your GPS indicates 10 gas stations lie ahead on your route. Naturally, you want the cheapest option. You pass the first handful and observe their prices before approaching one with a seemingly good deal. Do you stop, not knowing how sweet the bargains could get up the road? Or do you continue exploring and risk regret for rejecting the bird in hand? You won’t double back, so you face a now-or-never choice. What strategy maximizes your chances of picking the cheapest station?

Researchers have studied this so-called best-choice problem and its many variants extensively, attracted by its real-world appeal and surprisingly elegant solution. Empirical studies suggest that humans tend to fall short of the optimal strategy, so learning the secret might just make you a better decision-maker—everywhere from the gas pump to your dating profile.

The scenario goes by several names: “the secretary problem,” where instead of ranking gas stations or the like by prices, you rank job applicants by their qualifications; and “the marriage problem,” where you rank suitors by eligibility, for two. All incarnations share the same underlying mathematical structure, in which a known number of rankable opportunities present themselves one at a time. You must commit yourself to accept or reject each of them on the spot with no take-backs (if you decline all of them, you’ll be stuck with the last choice). The opportunities can arrive in any order, so you have no reason to suspect that better candidates are more likely to reside at the front or back of the line.

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Let’s test your intuition. If the highway were lined with 1,000 gas stations (or your office with 1,000 applicants, or dating profile with 1,000 matches), and you had to evaluate each sequentially and choose when to stop, what are the chances that you would pick the absolute best option? If you chose at random, you would only find the best 0.1 percent of the time. Even if you tried a strategy cleverer than random guessing, you could get unlucky if the best option happened to show up quite early when you lacked the comparative information to detect it, or quite late at which point you might have already settled for fear of dwindling opportunities.

Amazingly, the optimal strategy results in you selecting your number one pick almost 37 percent of the time. Its success rate also doesn’t depend on the number of candidates. Even with a billion options and a refusal to settle for second best, you could find your needle-in-a-haystack over a third of the time. The winning strategy is simple: Reject the first approximately 37 percent no matter what. Then choose the first option that is better than all the others you’ve encountered so far (if you never find such an option, then you’ll take the final one).

Adding to the fun, mathematicians’ favorite little constant, e = 2.7183… rears its head in the solution. Also known as Euler’s number, e holds fame for cropping up all across the mathematical landscape in seemingly unrelated settings. Including, it seems, the best-choice problem. Under the hood, those references to 37 percent in the optimal strategy and corresponding probability of success are actually 1/e or about 0.368. The magic number comes from the tension between wanting to see enough samples to inform you about the distribution of options, but not wanting to wait too long lest the best pass you by. The proof argues that 1/e balances these forces.

The first known reference to the best-choice problem in writing actually appeared in Martin Gardner’s beloved “Mathematical Games” column here at Scientific American. The problem spread by word of mouth in the mathematical community in the 1950s, and Gardner posed it as a little puzzle in the February 1960 issue under the name “Googol,” following up with a solution the next month. Today the problem generates thousands of hits on Google Scholar as mathematicians continue to study its many variants: What if you’re allowed to pick more than one option, and you win if any of your choices are the best? What if an adversary chose the ordering of the options to trick you? What if you don’t require the absolute best choice and would feel satisfied with second or third? Researchers study these and countless other when-to-stop scenarios in a branch of math called “optimal stopping theory.”

Looking for a house—or a spouse? Math curriculum designer David Wees applied the best-choice strategy to his personal life. While apartment hunting, Wees recognized that to compete in a seller’s market, he would have to commit to an apartment on the spot at the viewing before another buyer snatched it. With his pace of viewings and six-month deadline, he extrapolated that he had time to visit 26 units. And 37 percent of 26 rounds up to 10, so Wees rejected the first 10 places and signed the first subsequent apartment that he preferred to all the previous ones. Without inspecting the remaining batch, he couldn’t know if he had in fact secured the best, but he could at least rest easy knowing he maximized his chances.

When he was in his 20s, Michael Trick, now dean of Carnegie Mellon University in Qatar, applied similar reasoning to his love life. He figured that people begin dating at 18 and assumed that he would no longer date after 40, and that he’d have a consistent rate of meeting potential partners. Taking 37 percent of this timespan would put him at age 26, at which point he vowed to propose to the first woman he met whom he liked more than all his previous dates. He met Ms. Right, knelt down on one knee, and promptly got rejected. The best-choice problem doesn’t cover cases where opportunities may turn you down. Perhaps it’s best we leave math out of romance.

Empirical research finds that people tend to stop their search too early when faced with best-choice scenarios. So learning the 37 percent rule could improve your decision-making, but be sure to double-check that your situation meets all of the conditions of the problem: a known number of rankable options presented one at a time in any order, and you want the best, and you can’t double back. Nearly every conceivable variant of the problem has been analyzed, and tweaking the conditions can change the optimal strategy in ways large and small. For example, Wees and Trick didn’t really know their total number of potential candidates so they substituted in reasonable estimates instead. If decisions don’t need to be made on the spot, then this nullifies the need for a strategy entirely: simply evaluate every candidate and pick your favorite. If you relax the requirement of picking the absolute best option and instead just want a broadly good outcome, then a similar strategy still works, but a different threshold, typically sooner than 37 percent, becomes the optimal (see discussions here and here). Whatever dilemma you face, there’s probably a best-choice strategy that will help you quit while you’re ahead.