James' World 2

Click the link below the picture

.

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.

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.”

.

Jonathan Kitchen/Getty Images

.

.

Click the link below for the article:

.

__________________________________________