Discovery, May 1992
Road to Ruin
If highways are jammed, common sense says that an additional road will make things better. Mathematics says otherwise.
BY THOMAS BASS
JOEL COHEN is leaning over the balcony of his office at Rockefeller University; staring at six lanes of bumper-to-bumper traffic creeping along New York City's FDR Drive. "It's appalling!" Cohen yells over the road noise. What a waste of everybody's time! The people headed for the George Washington Bridge, on the other side of the city, will take an hour to get there. If you asked them, everybody out there would probably think it was a good idea to build a new crosstown expressway offering a shortcut to the bridge. 'Don't do it!' I'd tell the traffic commissioner. 'You could run into Braess's paradox, in which case your shortcut is going to be a long cut.' I haven't looked at the data, but I suspect these drivers would fall into a paradoxical situation. Everybody, including the cars on the old road and the new one, would travel slower than before." • When we build a new highway, we expect to put the pedal to the metal and get where we are going faster than before. So by what logic does Cohen suspect that an extra six lanes of macadam across Fun City will make the trip slower? • The paradox he cites—discovered in 1968 by German operations researcher Dietrich Braess—initially described how increased capacity in congested electronic networks slows down communication. It is, to say the least, counterintuitive. It implies the world is not necessarily the way it seems. But in the hands of Joel Cohen, who takes puckish delight in turning received ideas on their head and who might be described as a connoisseur of the counterintuitive, examples of Braess's paradox have started popping up all over the place: in telephone exchanges, air routes, military supply lines. Everywhere he looks, Cohen discovers new examples of networks where added capacity can make things worse than before.
|IN JOEL COHEN'S mechanical pun on the traffic paradox, length equals time. A) With an extra string connecting the springs in the middle, the weight hangs low. B) But when that string is cut, instead of dropping lower, the weight rises.|
Officially, Cohen's position at Rockefeller is listed as professor of populations. "I made it up myself," he says of his job title at this venerable research institution on New York's East Side. "A population is any ensemble of living creatures or other things whose characteristics you're interested in studying." These ensembles range from species interactions in nature to cars streaming along the FDR Drive.
Cohen's main area of expertise lies in mathematical biology; in the study of food webs—"a tally of who eats whom." In the late 1970s, after spending ten years analyzing 14 vast food webs, Cohen discovered that contrary to what ecologists believed, prey species did not significantly outnumber predator species, but rather existed with them in nearly equal numbers. "The bottom of the food chain was being underreported," says Cohen.
The particular ratio Cohen found, though, was not as important as the fact that one exists. "It implies an orderliness to nature that no one had ever suspected," he says. Termed a methodological tour de force, Cohen work on populations garnered him a MacArthur genius award in 1981.
He got sidetracked from food webs to traffic jams because, he explains, "I'm looking for paradoxes in ecology; but I'm willing to borrow them from any field in which I can find them." The detour presented itself a few years ago when he attended a lecture by mathematician Richard Steinberg of AT&T Bell Labs entitled "Paradox in Traffic Flow."
"That was the first I'd heard about Braess's paradox," Cohen says. "I just couldn't believe that if you added capacity to a crowded network you could actually slow things down. It was mind-blowing. It reminded me of Ceausescu's Romania, where every time the government said, 'We are going to make some improvements,' it meant there was going to be less bread or higher prices."
Steinberg was talking about slowdowns in electronic data, but Cohen began thinking about the general mathematical principles behind the data jams. He realized that Braess's paradox also applies to networks of roads, hydraulics, and mechanical links. Any system consisting of several pathways that transmit cars, data bits, or energy is a candidate. The key is that all the paths of a network vulnerable to the paradox are not equal. Some slow down the data—or cars or energy—more than others because they squeeze the traffic flow into a narrow channel; a pothole blocking a highway lane, for example, causes just such a bottleneck. When one car travels the network by itself, bottlenecks don't make a difference. But when more than one car is on the road, things can get dicey.
To predict just how dicey, or whether a network will fall into the paradox, Cohen plugs numbers signifying the amount of congestion on different legs of the network into a five-page mathematical equation. "I’ll spare you the details," he says. "But the gist of it is that in a vulnerable network—and there are enough of these to be worrisome—you might stand a fifty-fifty chance of falling into Braess's paradox."
What follows is an example of why more is not necessarily merrier. It also proves that what is good for the individual is not necessarily good for the group.
|1. THE TIME IT TAKES a car to travel each road here is related to the number of other cars on the road, written in the equations next to the road as f. One car will take (10 x 1) + (1 + 50) = 61 minutes, no matter which route it takes.||2. ADDING AN EXTRA expressway going from the west side to the east side across the park shortens the trip for one car. That car can now make the journey from downtown to uptown in (10 x 1) + (1 + 10) + (10 x 1) = 31 minutes.|
|3. WITH SIX CARS going from downtown to uptown, and no CrossPark Expressway, the commute is slower. The best drivers can do is (10 x 3) + (3 + 50) = 83 minutes. No one can switch routes without making the trip even longer.||4. BUT ADDING THAT expressway just makes matters worse. With six cars taking each of the three available routes, the former shortcut takes (10 x 4) + (2 + 10) + (10 x 4) = 92 minutes—longer than it did before the expressway existed.|
Diagram 1 represents Joe Commuter, a mild-mannered reporter for a major metropolitan newspaper, on his way to work in the city of Metropolis. Every day Joe has to drive from his home downtown to his office uptown. He can take one of two possible routes (downtown→west side→uptown or downtown→east side→uptown), either of which gets him to work in the same amount of time.
His driving time is written as a function of the number of cars on the road, f. For example, three cars traveling a stretch of road marked 10f will each take (10 x 3) = 30 minutes to get to the end. It's a two-lane road with a traffic light and with a pothole blocking one lane, so cars squeezing over into the remaining lane will slow everything down. If Joe has the road to himself, he can roll along with no one cutting him off and travel the road in 10x1=10 minutes.→
On the other hand, three cars driving a stretch marked f + 50 will each take 53 minutes. This is a clear six-lane highway with no obstructions and thus fewer chances for cars to interfere with one another. If Joe is on this road alone, he'll make it to the end in 1+50=51 minutes. Traffic doesn't make that much difference here; it just takes about 50 minutes to drive the road.
Let's make Joe Commuter King for a Day and start him driving through Metropolis all by himself. No matter which of the two alternate routes he rakes, Joe's total driving time for getting to work in the morning is (10 x 1) + (1 + 50) = 61 minutes. Since he makes the same commute home after work, Joe is spending more than two hours a day in his car getting hot under the collar.
It would be nice if Joe could zip straight uptown from his home; unfortunately, there are no roads going that way. But what if we paved over a strip of the park in the center of the city and gave him an alternate route to work? Look what happens in diagram 2 when Joe zips along the new CrossPark Expressway. His route now goes downtown→west side→east side→uptown, and his total elapsed driving time is a new land speed record of (10 x 1) + (1 + 10) + (10 x 1) = 31 minutes! While not faster than a speeding bullet, this is still half the time it took him to get to work the old way. Joe is a happy man.
Now, though, let's inject a little reality into Joe's life. He actually drives to the office when a lot of other working stiffs are clogging the road. In addition to our diagram of the Lone Commuter, we need to see what happens when the rest of society gets involved. "You have to remember," says Cohen, stating what might be called his first law of traffic, "that most people are traveling when it's most crowded." And the following example can be read as a parable about the difference between self-interest and the collective good.
Cohen's second law is "Traffic is a noncooperative game." This means that if someone driving from downtown to uptown can switch routes and beat out his neighbor by five minutes, he will. But his fellow drivers will soon catch on, and some of them will also switch routes, until the five-minute edge disappears into a state of equilibrium. This is the natural state of traffic flow, and it means that all commuters traveling a common network of roads from the same origin to the same destination will generally get where they are going at the same time.
Diagram 3 represents Joe Commuter driving bumper to bumper with five other cars. Metropolis has yet to be improved, and there's a choice of only two routes for getting from downtown to uptown. The network is at equilibrium, with three cars traveling one route and three cars the other, so for each route f=3. Everyone is creeping along at the same rate, and no one can switch routes without penalizing himself by making the trip longer.
Joe's rush-hour trip now takes (10x3)+(3+50)=83 minutes. Joe's boss, a crusty city editor, doesn't like him to be late, and Joe is not a happy man.
But hold on to your gearshifts, because we are about to enter the strange terrain of a mathematical paradox. Diagram 4 represents Joe's rush-hour commute via the new CrossPark Expressway. Remember that when Joe drove it alone, this was the road that cut his travel time in half. But now he is bumper to bumper with his neighbors, choosing one of three alternate routes uptown (downtown→west side→uptown, downtown→east side→uptown, downtown→west side→east side→uptown).
The diagram again represents traffic flows at equilibrium. Joe has more choices in a network that he knows should zip him to work before his boss can say "Great Caesar's ghost!" Yet look what happens when other people come into the picture.
Any of the three possible routes will give us the same result, but let's follow Joe's trajectory along the supposed shortcut, the route downtown→west side→east side→uptown. His travel time is now (10 x 4) + (2 +10) + (10 x 4) = 92 minutes! This is more than 10 percent slower than his old commute.
"The remarkable, and utterly counterintuitive, feature of this new equilibrium is that everyone is worse off!" says Cohen. "The cost per car of getting to work has risen from 83 to 92 minutes. In this and similar examples, Adam Smith's invisible hand leads everyone astray."
Cohen is referring, of course, to the eighteenth-century Scottish political philosopher and his belief in laissez-faire economics, whereby individuals pursuing their own self-interest are expected to bring about the common good. "Here is an example of everyone being worse off if they pursued their own self-interest," Cohen says. A highway system good for a solitary driver crossing Manhattan at three in the morning makes a hash of things for the rest of us trying to get to work by nine.
"Now that communism has collapsed in Eastern Europe," says Cohen, "we're giving everyone the message that they should submit to Adam Smith's invisible hand. But Braess's paradox proves that self-interest does not always work for the public good. There are better ways to organize society than the ideology of the Wild West and the Lone Ranger.
"It's a paradox. You have a set of expectations for one circumstance, and these expectations are violated in another circumstance. We assume that adding choices for drivers will shorten the time it takes them to drive from point A to point D. Everybody thinks that adding roads will make things faster. But it doesn't. And once we know that, the next thing to realize is that all of us would be better off if we closed down the new road and went back to driving the old route.
"I know of at least one real-life example. In Stuttgart they built a new road through the area around the Schlossplatz. But traffic through the city moved even slower than before, so they closed down the road. Nobody has looked at it mathematically, but I wouldn't be surprised if this was a case of Braess's paradox at work."
Cohen has even found the paradox at work in a Rube Goldberg-like contraption of strings, springs, and weights that he put together in his garage. At the top is a weight, and hanging from it is spring number one. A short string leads from the bottom of this spring to the top of a second, similar spring, which hangs down in a mirror image of the first assembly, with a weight anchored to its bottom. Two long strings complete the device. One string leads from the top weight to the top of spring number two. The second long string goes from the bottom of spring number one to the weight at the bottom of spring number two.
"We've made a mechanical pun," says Cohen. "Here length equals time." The weights, like the cars on the road, are the load on the network. The springs, which stretch when a greater weight is put on them, are like the 10f roads, which take longer to drive when more cars are on them. The strings, which don't stretch, are like the f + 50 and f + 10 roads, which aren't affected very much by the amount of traffic. When the contraption is hung vertically, the short string in the middle is pulled taut while the long strings hang loosely. Then Cohen cuts the short siring. You expect the bottom weight to drop even lower, since it is now suspended only by the longer strings. But presto! It rises to a higher point.
"Once you start looking for them," says Cohen with a grin, "there are a lot of examples where following your intuition can lead you astray."