Adding a road or bridge can slow traffic! (Braess's paradox)

Miscellaneous and Personal Stuff I Must Share
1

I wonder how many Dopers are familiar with Braess’s Paradox. The paradox is that adding a new road can slow traffic. This doesn’t just arise in contrived situations, but actually applies frequently in the real world. In 1990, traffic sped up the day New York’s 42nd street was closed. The concept is applied by real-world traffic engineers, e.g. “A few years ago, traffic planners made the decision to tear down a 6-lane highway in Seoul, South Korea and replace it with a five-mile long park. Many transportation professionals were shocked, but surprised when the traffic congestion actually improved.” Here’s another closely related paradox.

It might be fun to identify some other mathematical counterexamples with real-world interest, but please do so in a different thread. I intend to follow up here, by using Braess’s Paradox to segue into a larger, more important point.

The Wikipedia article does a good job of explaining the “paradox” but I’ll spend the rest of this post summarizing their example.

Drivers need to commute from Startburg to Endtown and have two choices.

Option A) They can take the road to Ableton, then the $45 ferry from Ableton to Endtown. A complication is that time is money and the time cost of traveling the road to Ableton varies with traffic: zero if the road is otherwise empty, $40 if all the vehicles traveling from Startburg to Endtown have taken this road, and proportional to the traffic for any amount in between. As we will see, half the traffic will use this road, so its time cost is $20, and the total cost of travel for drivers who use Option A is 20+45 or $65.

Option B) This option is very similar in effect to Option B. Take the $45 ferry from Startburg to Bakersville, then the road from Bakersville to Endtown, which has just the same cost schedule as the road from Startburg to Ableton.

Options A and B are essentially the same so half the traffic will pick (A) at random, for a total cost of $65, the other half will pick (B) for a $65 cost. If the use of (A) rises, its cost will rise while B’s cost declines; smart consumers will then cause equilibrium to be restored.

Now an entrepreneur builds a new bridge from Ableton to Bakersville and charges only $2 as toll. The first driver to use this bridge is delighted. He drives to Ableton, losing only $20 in wait time, since half the traffic is still taking the old ferry from Startburg to Bakersville, pays $2 toll on the new bridge, then takes the Bakersville-Endtown road for another $20. Total cost is $42, significantly better than the $65 everyone else pays.

But the other drivers soon figure out the new route and they all take it. Since 100% of the traffic is now using both the Startburg-Ableton road and the Bakersville-Endtown road, each of those roads now costs $40 in wait time. With the $2 toll on the new bridge, the total cost to travel from Startburg to Endtown has gone up to $82, compared with the old $65 cost!

Now, at this point, what decisions do the market participants make? A driver can take the old ferry to Bakersville, but then he still has to travel on the congested Bakersville-Endtown road and ends up spending $85 instead of $82. Similarly, if he takes the old ferry from Ableton he ends up spending $85. Drivers can do no better than $82 or $85, even though they made the trip for $65 before the bridge was built!

And what about the entrepreneur who’s operating the bridge? He can increase the toll to $4 and the drivers will still all be better off using his bridge. Drivers would all be better off if none of them took the bridge, but, regardless of other drivers’ actions, each driver individually is better off paying $4 to take the bridge.

2

Interesting, and I’ve seen it at work.

Let’s say that A is outside the city, B is in the city. You can drive from A to the city as soon as possible, or you can use the beltway to go around the city to the point where the beltway is closest to B, then drive in. My experience in both Pamplona and Bilbao is that so much of the traffic has transferred to the beltways that cutting across town is usually faster. I’m not so sure about Madrid, which has so much traffic it’s congested at all but the wee-est hours.

That paradox seems like a good reason to improve existing roads rather than building alternates, inasmuch as it’s possible.

3

Apart from the issue that improving existing roads may be more expensive that building new ones, it doesn’t change the paradox. Suppose, in septimus’ case, that there was an existing road between Ableton and Bakersville, but it was so long and winding that it cost $60 to drive on it, even with no traffic. That means that it was never an option for the drive between Startburg and Endtown, but the new bridge is effectively an upgrade to that existing road, and still creates the paradox.

4

I’m talking about things like straightening roads, turning a 1-lane-each-way with no median (max speed 80km/h) into a 3-lanes-each-way with a median road (max speed 120km/h), etc. It improves traffic for the improved stretch more than building another route would.

“A good reason for doing something” does not mean there are no reasons for the opposite. “Improving the old road will cost a Peru” is a reason for not improving the old road where applicable, but it does not deny the existance of reasons for.

5

NM

6

When they built the Triborough Bridge, it was supposed to alleviate congestion at the Brooklyn & Manhattan bridges. Instead, within 4 years it was locked just as badly, if not worse, than the other two (while congestion on the Bklyn & Manhattan remained as bad as ever). Making driving easier simply encourages more people to drive.

“Building bigger roads is like giving a fat man bigger pants.” It does not alleviate the actual problem; it encourages the problem to develop further by making a bad situation suddenly more comfortable.

You have to make it undesirable for everyone to drive to the same place by the same route, one to a car. You have to either give people a new place to go, such as by developing commercial centers that support a counter-commute, or develop mass transit to get them off the road, encourage carpooling by developing HOV lanes, or engage in congestion pricing - where you are severely tolled, not for going from A to B, but for entering the congested region by car at all.

7

Braess’s paradox is a great example of how a simple game can have a pretty serious price of anarchy.

8

This paradox pops up in computer networking as well. Packets are sent through the fastest route for them from server A to server B, so it has the same result as drivers picking their routes even though there’s no ‘irrational’ human mind involved. It’s one of the stranger things I’ve run across.

9

All of Northern Virginia is an example of this paradox. They keep building roads and traffic keeps getting worse.

10

Braess’s paradox only works if the new route spoils part of each of the old routes. If they had built a road that doesn’t intersect the old roads, this wouldn’t have been a problem. Because a driver is unable to take an old route without hitting new-route traffic, his old system is spoiled. This is the same thing that happens when you mix tuna salad and ice cream. You might’ve been happy with either, but since you now can’t get one without the other, you’re worse off getting “both” instead of “either”.

11

This doesn’t continue into perpetuity, though. It’s not like infinite cars will keep appearing when you build more roads. Some severely cramped places (like NYC) simply don’t have the room to support enough roads for people to drive on comfortably. It’s not a failing of the people who want to drive, though–it’s a failure of geography. If the way to remedy the failure is tolling or higher gas taxes, so be it. But there’s no need to assign “blame” to people who decide to drive because it’s recently become more convenient to do so.

12

That’s exactly what’s going on, though: everybody does what’s best for them personally, and ends up losing out on more than if they followed some coordinated strategy.

13

I do note some possibly unjustified assumptions in the OP’s scenario, that might change the final answer: First, the price of the ferries is fixed, and does not depend on amount of traffic. For ferries, this might actually be true, but it’s likely to break down if we replaced them with more roads. And second, the OP’s scenario assumes that a negligible number of people want to travel from Abelton to Bakerville, or any other start-end pair other than Startville to Endville. Clearly, the bridge is a significant benefit to anyone traveling Abelton-Bakerville.

14

:confused: :confused:

We crawl before we try to walk. We deal with simplified examples. I think you know that. Just a moment ago I noticed you implied in another thread that information transmission is proportional to bandwidth; I’m sure you’re quite aware that this is quite wrong (you need to multiply by log(1+SNR) or such), but I saw no need to nitpick.

And did you miss the part of the post that pointed out that Yes, Braess’s Paradox does apply to real-world traffic analysis, and, yes, real traffic engineers really do shut down roads to really improve traffic in the real world?

Hope this helps.

15

Actually, this is exactly what happens if the roads point to someplace everyone wants to be. See also, Washington, D.C.

It’s not blame, it’s just understanding you can’t solve the problem this particular way. If traffic is not congested it won’t become so by widening the road (then again a widening would not be needed if its not congested) – but if traffic is congested you can never solve it by widening the road. Ever.

Let me put it another way. All things being equal people will flock to locations with the best, most convenient, emptiest roads that go to a place they want to be. Widen a road, and Population increases around the road, until it reaches an identical level of density to that which prompted the original widening.

16

I’ll bump to complete this thought.

My intent was to use Braess’s Paradox as a counterexample to the oft-repeated misconception that “free markets always lead to optimal allocations.” Although largely unrelated to issues like Braess’s Paradox, I’d intended to post a link to a fun article in Atlantic Monthly.

But, alas, I didn’t start the second thread. I couldn’t even decide what forum would be appropriate:
IMHO: “Skies are not always blue. Markets are not always optimal.”
Great Debates: “What types of government interventions in the economy are appropriate?”
BBQ Pit: “Why is #$@@!% such a frigging dolt?”

17

And this is why, in congested traffic, it is often faster to drive in the right lane than the left. Everyone thinks the left lane is faster and merges into it. Despite the added burden from added cars coming in, the right lane winds up being faster.

18

I believe that adding a temporary lane can slow down total throughput on a highway. By this I mean a 2-lane road that has 3 lanes for a certain space–say a mile or so. I have seen this done in a lot of places, and my belief is that this slows traffic considerably after it reaches a certain density–the time cost of merging the 3 lanes down to 2 more than elmininates any time savings of the third lane.

I don’t know if this is directly related to Braess’s Paradox or not.

19

On a related note, there’s talk of removing connections from the internet to improve its performance:

I don’t know off the top of my head how closely the BA model agrees with the internet as it is, but this could be a big deal.