Mathic Bus Research Puts Techer in Driver’s Seat

By Barbara Ellis

It’s a decision almost everyone has faced. You’re waiting for a bus that never seems to come. Should you walk to your destination, if it’s just a mile or two away? Or should you tough it out at the bus stop?

In January, a paper posted on an open-access preprint site offered a mathematical solution to this conundrum, implicit in the title, “Walk versus Wait: The Lazy Mathematician Wins.” The article soon had everyone walking its way. First up was a piece in Britain’s New Scientist that caught the attention of most of the British press. Soon the news spread to other regions of the world where buses play a large part in moving people around and are, presumably, often late, including Australia, India, Southeast Asia, South Africa, the United Arab Emirates, and Estonia. “Laziness Pays Off: It’s Official,” trumpeted one headline, while others proclaimed “The Waiting Game Pays Off at Bus Stop, Mathematicians Find,” or “Why It’s a Mistake Not to Wait Lazily for the Bus.” The story even attracted attention in Germany and Switzerland, two countries known for their very reliable bus service.

Somehow lost in the initial coverage was the fact that the “university mathematicians” who had done this research were undergraduates. One of the authors was Caltech physics major Justin Chen ’09, who acquired a doctorate in some of the reports. The other two, Scott Kominers, a math, music, and ethnology major, and Robert Sinnott, majoring in statistics, are juniors at Harvard. This misunderstanding caused some dissent among the rave reviews. One German blog, the Too Much Cookies Network, called the article “a very shallow analysis of the problem,” writing, “With this paper mathematicians strive further to prove once more that they are, in fact, very disconnected from the real world.”

The paper was never intended to be taken that seriously, says Chen, but he’s gotten a kick out of all the publicity. (It was Chen who was interviewed for the BBC from the recording studios of NPR station KPCC at Pasadena City College.) In fact, he and his coauthors have plans to write a book in which they mathematically tackle other problems of everyday life, such as how to work out which line to stand in at a supermarket, and how to make your way through traffic congestion.

The students’ inquiry started at a bus stop in Boston last fall, says Chen. “My friend Scott was trying to get from MIT to Harvard. He was waiting for the longest time for a bus and so he ended up having to walk the whole way. As he walked, constantly looking over his shoulder in case a bus came, he wondered what the best strategy is, whether to walk or to wait.”

Chen and Kominers are friends from high school in Bethesda, Maryland. When they both went home for winter break, Kominers told Chen about his bus stop dilemma and suggested that they try to work out the optimal strategy mathematically. “So we thought about it and sat down one afternoon and wrote a paper,” Chen says. To work out their equations, they “went online to the Massachusetts Bay Transport Authority (MBTA) website and looked up the distance between bus stops and how often the buses arrived.” They didn’t do any practical research in Bethesda, where “a screen at the Metro stations displays the arrival time of the next bus. Our equation doesn’t matter then. Most of it depends on not knowing when the next bus is coming,” Chen says.

When Kominers got back to Harvard he had his friend Robert Sinnott check the math, and on January 1 the trio submitted the paper to arXiv.org, Cornell University’s open access e-print site (where you can see it online here), stating clearly that it was a recreational mathematics note, a caveat that seems to have been trampled in the stampede of press coverage. The paper attributes the bus-stop adventure to Chen, but otherwise it’s “pretty much the exact same problem that Scott had,” he says. Early readers included Todd Kaplan ’89, now an economics professor at the University of Exeter in England, who was one of two academics to contact the students with some corrections, which are in the current online version. The paper has now been submitted to the undergraduate research journal Math Horizons, with publication slated for later this year.

The gist of the study is a short series of increasingly complex equations that the authors say demonstrate that it’s always best to wait patiently at the stop. If you do decide to walk, perhaps because you’re cold, uncomfortable, or just plain bored at the bus stop, you should give up on the bus completely and plan to walk the whole way, without looking back in case there’s a bus coming. “Of course, it all depends on the timing of the bus and the distance,” Chen says. “There’s no point in waiting if the bus only comes once an hour and the destination is a mile away, as it’s always going to be quicker to walk.” There’s a distance constraint on the equation too. “It’s for walkable distances, perhaps five miles max,” he says.

Although Chen, who chose Caltech over Harvard “because of the weather,” now has an international reputation as a public-transportation expert, he always uses his car. He did once try to get from Pasadena to Long Beach on the weekend using mass transit, but “it involved two buses and three different rail lines, and took two hours and fifteen minutes, compared to less than an hour by car.” And although the final lines of the paper have Chen shouting “’Eureka!’ . . . upon realizing that the laziest possible waiting strategy would prevail,” he’s far from indolent. As well as spending the weekends either surfing at Bolsa Chica State Beach or (weather permitting) snowboarding at Big Bear and Snow Summit, he wheels around campus on his longboard.

When Boston’s Daily Free Press reported on Chen’s work, they also contacted an MBTA spokesperson for comments. She insisted that the city’s bus and trolley systems are generally on time. So how was it that Kominers walked all the way from MIT to Harvard, a distance of two miles, without seeing a single bus? Chen laughs: “It’s really funny. When Scott got back to campus, he found out the buses weren’t actually running that day. If he hadn’t walked, he would never have gotten there.”