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Apparently the price of anarchy post was fairly popular, and I was recently invited to give a guest post on the traffic blog http://ecostreets.net.  So I did.  Please go check it out: http://studentdev.jour.unr.edu/ecostreets/?p=653.  The topic is “three reasons we don’t understand traffic”.

As a teaser, I’ll give you a picture of what I call a “traffic soliton”:

UPDATE:  It seems that ecostreets.net no longer exists.  Below is the original post, as it once appeared on that blog.

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Three reasons we don’t understand traffic

Being stuck in traffic is frustrating in a way that few other things are.  Something about the helplessness of it makes us extremely irritable, and we become inclined to think that the people around us are idiots.  The guy who just cut you off, the lady talking on her cell phone with her blinker left on, the construction workers standing around and looking at a hole in the ground instead of actually fixing the road.  Everyone seems incompetent and unintelligent when you’re stuck in traffic.

Not uncommonly, this line of thinking extends to the people that decided how and where to build the roads.  Their (lack of) planning can seem like pure idiocy, and the solution for fixing the terrible gridlock can seem painfully obvious.

I understand this type of thinking all too well.  My very first job was in the heart of Washington DC, which meant that I spent about an hour and a half stuck in traffic every day.  And during that time I thought about traffic, how terrible it was, and how it should be a solved problem by now.  It seemed to me that after decades of intense mathematical research and super-powered computer simulations we should understand traffic.  After all, we know where people live, where they work, and when they work.  Why can’t we “solve” for the most efficient pattern of roads and intersections?  Why do the world’s most modern cities still suffer from terrible gridlock?

During the eight years since then, I have thought off and on about traffic from a scientific perspective.  And the more I think/read/hear about it, the more difficult it seems to understand.  In this post, I’ll give a few reasons for our continued inability to “figure out” traffic patterns.

#1:  Traffic is a network problem

In its essence, the problem of how to prevent traffic jams is the problem of how to go from point A to point B.  Between points A and B are a series of roads: links between the two destinations.  It’s easy to draw out the possibilities for travel between the two places.  Why can’t we just put some kind of mathematical efficiency formula to each path, and figure out the optimal travel time and route?

It is possible, in theory.  The problem is that the number of possible travel routes between points increases exponentially with the number of roads and intersections.  The problem also gets exponentially harder with increasing numbers of drivers and destinations.  It doesn’t take much to arrive at a problem so complicated that you have no hope of facing it head-on.

Take Las Vegas as an example.  “Solving traffic” for Las Vegas would require you to find the ideal transit route for something like 1 million drivers with 20,000 separate destinations.  Sure, there is an ideal solution.  But the world’s most powerful computer working continuously for the lifetime of the universe wouldn’t be able to compute it.  The problem is just too large.

And size isn’t the only hard part about network problems.  Even simple, completely solvable problems exhibit strange behavior.  They are subject, for example, to Braess’s Paradox: adding a new road can make traffic worse and tearing up a road can make it better.  But since real-world traffic networks are way too hard to describe quantitatively, we never know when we will be hit by Braess’s Paradox.  If you’re the mayor of a town planning a new construction project, you face the possibility that your initiative to add more streets or widen the highways might actually make traffic worse.  And there’s no way to predict it with any kind of formula; our science just isn’t good enough yet.

#2: Traffic is a chaotic problem

Imagine, for a second, that we actually had an infinitely powerful computer.  Then we could feed it our city maps and demographic information, and it would immediately spit out the answer: where all our roads should be, how many lanes they should have, etc.  Wouldn’t that be great?  Traffic = solved.  Then every driver could know that they were reaching their destination in the shortest possible time, by the best possible route.

But there’s a problem.  Traffic patterns, to a large extent, are the domain of chaos theory.  A traffic jam is a chaotic entity; it responds dramatically to small changes in external conditions.  And science is not good at addressing chaotic problems.  Scientists much prefer “linear” problems, where the change you get is directly proportional to the change you make.  These sorts of problems are easy to think about and easy to describe mathematically; they make sense to us.

For example, if you were a scientist on Mars who had never seen a traffic jam, you might think that closing one lane of a four-lane highway would slow traffic down by 25%.  Makes sense, right? Close 25% of the road, traffic becomes 25% slower.  But traffic does not make sense.  Anyone who has been on a highway during construction season knows that closing one lane of a highway can make an enormous difference for traffic, not just 25%.  It can turn a 20 minute commute into a 60 minute commute.  This is a consequence of the chaotic nature of traffic flows.  A small input (closing 1/4 of the road) gets greatly magnified (making your commute 3x longer).  What’s more, the “magnification” in a chaotic problem is impossible to predict beyond a very short time scale.

So even with an infinitely-powerful computer and the “perfect” travel plan, we’re not guaranteed smooth sailling.  Our utopian scheme might be destroyed by a flat tire on Central Avenue, or a rain storm, or an old man going 30 mph on the highway.  The consequences of these small perturbations on the city-wide traffic network could be immense, and they would be impossible to predict.

To finish making the point, I’ll close this section with a personal story.  On one particular hot summer morning, my normal 45-minute commute into Washington DC slowed to a crawl.  I was literally in first gear for nearly an hour, inching in to DC.  I assumed that there was some big accident on the highway, but when I finally got to the front of the jam I found the source: a police car was parked on the side of the road.  That was it.  A cop had parked on the shoulder of the highway, no doubt to remind people not to speed (ironically).  Earlier that morning, as drivers had rounded a corner of I-275 and seen the parked police car, they stomped on their brakes to avoid a speed ticket.  Their sudden slow-down caused a wave of people stomping on their brakes to avoid rear-ending the person in front of them.  And that wave of “oh shit” brake-stomping spread backward through the line of cars, getting more and more pronounced, until finally it become a full-blown traffic jam.  It’s like that Ben Franklin proverb about the horseshoe nail; I couldn’t believe that one cop’s decision to stop on the side of the road could lead to a 10,000 man-hour traffic jam.  That’s why traffic patterns are the domain of chaos theory; no mathematics could have predicted that.

#3: Traffic is a wave problem

The last example leads me to an interesting way of describing traffic: as a wave phenomenon.  Each person responds to the conditions directly ahead of them, but with a certain time delay.  As a result, high density “jammed” regions travels backward down the road, only eventually affecting cars down the line.  If you’ve ever been stuck in a traffic jam for a long time, and then reached the front only to find that the original cause of the jam has long been cleared away, you know what I mean.  Traffic jams have a finite propagation speed, just like waves do.

Here’s a visualization of a long line of cars inching their way down the highway (I’ll do the standard physics thing here and assume that a car is a sphere):

Traffic jams are full of these kinds of waves, which in physics are called “solitons” (solitary traveling waves).  The reason you’ve probably never heard of a soliton (whereas you are likely to have heard of other kinds of waves, like sines and cosines) is that they are fairly complicated mathematically.  And when two such waves meet together, who knows how they interact?  A description of the merging of two “traffic solitons” may require scientific theories that we don’t have yet.

In the end, this post probably hasn’t taught you anything new.  My only point was that traffic both is fascinating and frustrating.  For a scientific-minded person, it displays a surprising number of fascinating phenomena.  Unfortunately for the driver, most of those phenomena are infuriating.  But maybe next the time you’re stuck on the highway, you’ll give scientists and politicians a little more credit for not having it figured out yet.

June 7, 2009 1:44 am

So, here’s a question…
Is it better to take the “short cut” through smaller streets with stop signs or to take the long way around through major streets with faster speed limits? (or is this also a problem with too many variables to actually compute?) 🙂

June 7, 2009 10:38 am

There’s probably no good general answer to that question. But I would say that if lots of other people know about the “short cut” then it’s unlikely to be faster.

July 6, 2009 6:05 pm

Though a little late, the blog “The Infrastructurist” has a post today on real world examples of Braess’ Paradox and induced demand. That is, examples of when removing a major road has reduced traffic congestion.

July 6, 2009 10:11 pm

In general, we should be careful pointing to real-world examples as “proof” of the paradox, because it’s so easy to make one of those “after which, therefore because of” logical fallacies. But it’s certainly food for thought.

3. August 17, 2009 8:12 pm

I know this is an old post, but…

Although there’s certainly a component of chaos in traffic (!), some of its properties are perfectly predictable by mathematics, such as the way that a minor event can precipitate a full-scale traffic jam in a previously (mostly) free-moving road. That happens a lot here in Los Angeles, where the freeways are generally close to saturation. This can be hidden to the careless eye because even at 75 to 80 percent capacity, traffic is slowed only slightly. But within that last 20 to 25 percent occurs most of the potential slowdown, and you can use that up by reducing the capacity of the road by that amount (by lane closure, for instance, or some other event that reduces the rate at which cars can pass by on the road). Voila, instant traffic jam.

4. January 6, 2011 1:22 pm

defies the laws of physics, when you pinch a hose (eliminating a lane) the water comes out faster! so when a lane is closed, people need to speed up proportionally, increasing the speaces between cars to allow the merge.