Soon after Maes’s work made its debut, online stores quickly understood the value of having a recommendation system, and today most Web sites selling entertainment products have one. Most of them use some variant of collaborative filtering — like Amazon’s “Customers Who Bought This Item Also Bought” function. Some setups ask you to actively rate products, as Netflix does. But others also rely on passive information. They keep track of your everyday behavior, looking for clues to your preferences. (For example, many music-recommendation engines — like the Genius feature on Apple’s iTunes, Microsoft’s Mixview music recommender or the Audioscrobbler program at Last.fm — can register every time you listen to a song on your computer or MP3 player.) And a few rare services actually pay people to evaluate products; the Pandora music-streaming service has 50 employees who listen to songs and tag them with descriptors — “upbeat,” “minor key,” “prominent vocal harmonies.”

Netflix came late to the party. The company opened for business in 1997, but for the first three years it offered no recommendations. This wasn’t such a big problem when Netflix stocked only 1,000 titles or so, because customers could sift through those pretty quickly. But Netflix grew, and today, it stocks more than 100,000 movies. “I think that once you get beyond 1,000 choices, a recommendation system becomes critical,” Hastings, the Netflix C.E.O., told me. “People have limited cognitive time they want to spend on picking a movie.”

Cinematch was introduced in 2000, but the first version worked poorly — “a mix of insightful and boneheaded recommendations,” according to Hastings. His programmers slowly began improving the algorithms. They could tell how much better they were getting by trying to replicate how a customer rated movies in the past. They took the customer’s ratings from, say, 2001, and used them to predict their ratings for 2002. Because Netflix actually had those later ratings, it could discern what a “perfect” prediction would look like. Soon, Cinematch reached the point where it could tease out some fairly nuanced — and surprising — connections. For example, it found that people who enjoy “The Patriot” also tend to like “Pearl Harbor,” which you’d expect, since they’re both history-war-action movies; but it also discovered that they like the heartstring-tugging drama “Pay It Forward” and the sci-fi movie “I, Robot.”

Cinematch has, in fact, become a video-store roboclerk: its suggestions now drive a surprising 60 percent of Netflix’s rentals. It also often steers a customer’s attention away from big-grossing hits toward smaller, independent movies. Traditional video stores depend on hits; just-out-of-the-theaters blockbusters account for 80 percent of what they rent. At Netflix, by contrast, 70 percent of what it sends out is from the backlist — older movies or small, independent ones. A good recommendation system, in other words, does not merely help people find new stuff. As Netflix has discovered, it also spurs them to consume more stuff.

For Netflix, this is doubly important. Customers pay a flat monthly rate, generally $16.99 (although cheaper plans are available), to check out as many movies as they want. The problem with this business model is that new members often have a couple of dozen movies in mind that they want to see, but after that they’re not sure what to check out next, and their requests slow. And a customer paying $17 a month for only one movie every month or two is at risk of canceling his subscription; the plan makes financial sense, from a user’s point of view, only if you rent a lot of movies. (My wife and I once quit Netflix for precisely this reason.) Every time Hastings increases the quality of Cinematch even slightly, it keeps his customers active.

But by 2006, Cinematch’s improving performance had plateaued. Netflix’s programmers couldn’t go any further on their own. They suspected that there was a big breakthrough out there; the science of recommendation systems was booming, and computer scientists were publishing hundreds of papers each year on the subject. At a staff meeting in the summer of 2006, Hastings suggested a radical idea: Why not have a public contest? Netflix’s recommendation system was powered by the wisdom of crowds; now it would tap the wisdom of crowds to get better too.

AS HASTINGS HOPED, the contest has galvanized nerds around the world. The Top 10 list for the Netflix Prize currently includes a group of programmers in Austria (who are at No. 2), a trained psychologist and Web consultant in Britain who uses his teenage daughter to perform his calculus (No. 9), a lone Ph.D. candidate in Boston who calls himself My Brain and His Chain (a reference to a Ben Folds song; he’s at No. 6) and Pragmatic Theory — two French-Canadian guys in Montreal (No. 3). Nearly every team is working on the prize in its spare time. In October, when I dropped by the house of Martin Chabbert, a 32-year-old member of the Pragmatic Theory duo, it was only 8:30 at night, but we had to whisper: his four children, including a 2-month-old baby, had just gone to bed upstairs. In his small dining room, a laptop sat open next to children’s books like “Les Robots: Au Service de L’homme” and a “Star Wars” picture book in French.