This elegant math problem could help you make the best decisions in your home search or relationship

This elegant math problem can help you find the best choices in employment, house hunting, and even love

Jack Murtagh

Imagine you’re driving down the highway and realize your fuel tank is low. Your GPS shows that there are 10 gas stations ahead of your route. Naturally, you choose the cheapest one. After passing the first few and checking the prices, you approach one that offers a seemingly good deal. Do you stop, not knowing how much better the deal could be up ahead? Or do you keep exploring, risking regretting turning down the one you have? With no intention of turning back, you’re faced with a now-or-never choice. What’s the strategy that will maximize your chances of picking the cheapest station?

Researchers have thoroughly studied this so-called best-choice problem and its many variants, drawn by its real-world appeal and surprisingly elegant solutions. Empirical studies suggest that humans tend to adopt suboptimal strategies, so learning its secrets could make us better decision-makers in everything from gas stations to dating profiles.

This scenario has several names. The “Secretary Problem” ranks job applicants by their qualifications, rather than ranking things like gas stations by price. The “Marriage Problem” ranks suitors by their qualifications. All forms share the same mathematical structure, Known Number of Can be ranked Opportunity Arise One by oneYou must commit to accepting or rejecting each offer then and there, irrevocably (rejecting them all leaves you with only the last option). Opportunities can come your way in any order, so there’s no reason to suspect that a better candidate is more likely to be at the front or back of the queue.

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Let’s test your intuition. If there were 1,000 gas stations lined up on the highway (or an office with 1,000 applicants, or a dating profile with 1,000 matches) and you had to evaluate each one in turn and choose when to stop, what are the chances that you would pick the absolute best option? If you chose randomly, you would only have a 0.1 percent chance of finding the best option. Even if you tried a smarter strategy than random guessing, you could still be out of luck if the best option appeared too early to be found due to lack of comparative information, or too late, when you have already compromised for fear of losing opportunities.

Surprisingly, the optimal strategy is Number One You have a chance of making a choice nearly 37 percent of the time, and your success rate doesn’t depend on the number of candidates. Even with a billion choices, if you don’t choose the suboptimal option, you’ll end up searching for a needle in a haystack more than one-third of the time. The winning strategy is simple: reject the first roughly 37 percent at all costs. Then choose the first choice that’s better than all the others you’ve encountered so far (or, if you can’t find one, choose the last one).

Even more fun, mathematicians’ favorite little constant is e = 2.7183… appears in the solution, also known as Euler’s number. e It is notorious for turning up frequently in mathematical settings that are seemingly unrelated, such as best choice problems. Under the hood, any mention of an optimal strategy and its corresponding probability of success of 37% is actually a 1/e or about 0.368. This magic number comes from the desire to see enough samples to get an idea of ​​the distribution of choices, but not wanting to wait too long for fear of missing the best ones.e Balance these forces.

The first publication of best-choice problems was by Martin Gardner in his “Mathematical Games” column. Scientific AmericanThe problem went viral in math circles in the 1950s, when Gardner posed it as a little puzzle he called “Googol” in a February 1960 issue and published the solution the following month. Today, the problem generates thousands of hits on Google Scholar, and mathematicians continue to study variations of the problem. What if you could choose multiple options, and any one of them would be the best and you won? What if your opponent chose the order of the options to trick you? What if you didn’t need the absolute best choice, and would be happy with second or third? Researchers study these and countless other stopping scenarios in a branch of mathematics called “optimal stopping theory.”

Are you looking for a home or a spouse? Math curriculum designer David Wees applied the Best Choice strategy to his personal life. While apartment hunting, he realized that to compete in a seller’s market, he had to make a decision on an apartment on the spot during the viewing before other buyers snatched it up. Given the pace of viewings and a six-month deadline, he estimated he had time to visit 26 properties. And since 37 percent of the 26 were close to a 10, Wees rejected the first 10 and made a deal on the first next apartment he liked better than all the previous ones. Without checking out the rest of the properties, he wouldn’t know if he’d really secured the best deal, but he had the peace of mind of knowing he’d at least maximized his chances.

Michael Trick, now president of Carnegie Mellon University in Qatar, applied a similar idea to his own love life when he was in his 20s. He assumed that people start dating at 18, stop dating after 40, and meet a certain percentage of potential partners. If he took 37% of this period, he would be 26, at which point he vowed to propose to the first woman he liked better than any date he had ever had. He met the woman of his dreams, got down on his knees, and was quickly rejected. The best choice problem doesn’t cover the cases where chance might turn you down. Maybe love is better off without math.

Empirical studies have shown that people tend to quit their search too early when faced with a best-choice scenario. So while mastering the 37 percent rule may improve your decision-making, double-check that your situation meets all of the conditions of the problem: that the number of rankable options is known, they are presented one at a time in any order, you want the best, and there is no going back. Nearly every possible variation of the problem has been analyzed, and tweaking the conditions can change the optimal strategy in ways both big and small. For example, Wees and Trick didn’t actually know the total number of potential candidates, so they substituted a reasonable estimate instead. If you don’t have to make an on-the-fly decision, the need for a strategy disappears entirely: just evaluate all the candidates and pick your favorite. If you relax the requirement of choosing the absolute best option and instead want a generally good outcome, a similar strategy will still work, but a different threshold earlier than 37 percent will usually be optimal (see discussion here and here). Whatever dilemma you face, there’s probably a best-choice strategy that can help you quit before you’re even close to winning.