Braess's Paradox

What’s going on

Every morning 4,000 drivers travel from Start to End. There are two ways to go, one through town A and one through town B. Each way has two roads:

• a narrow road that clogs up: it takes as many minutes as the number of cars on it divided by 100. With 2,000 cars it takes 20 minutes; with 4,000 cars, 40;
• a wide road that never clogs: it always takes 45 minutes.

The way through A starts with the narrow road; the way through B ends with one. The two routes are mirror images.

Drivers are selfish, in the plainest sense: each one wants the fastest trip for themselves. If one route is quicker, some drivers move to it, which makes it slower. They settle when both routes take the same time: 2,000 cars each way and 20 + 45 = 65 minutes for everybody. That is the opening picture in the scene above.

A free shortcut

Now the town builds a superb new road from A to B, so short that it takes no time at all. Press Open the shortcut.

A driver going the zigzag way, Start → A → B → End, uses both narrow roads and neither wide one. Right after opening, the narrow roads carry about 2,000 cars each, so the zigzag takes about 20 + 0 + 20 = 40 minutes. The first drivers to try it save 25 minutes.

Of course, others follow. A driver who switches from the route through A keeps the first narrow road and adds a car to the second one, which slows down the zigzag and the route through B. Watch the days go by: the narrow roads fill up and every route gets slower, but the zigzag stays the quickest, so drivers keep moving onto it.

Where does it stop? When all 4,000 drivers take the zigzag. Each narrow road then has 4,000 cars and takes 40 minutes, so the trip takes 40 + 0 + 40 = 80 minutes. Could anyone do better by going back to an old route? No: that would be 40 minutes on a narrow road plus 45 on the wide one, 85 minutes.

So the new road made every trip 15 minutes longer, and nobody can fix it on their own. A situation where no one gains by changing only their own choice is called a Nash equilibrium. Here the equilibrium is worse for everyone than the one before. Close the shortcut, and everybody is back to 65 minutes.

What went wrong?

Nobody was foolish. Each driver made the best choice for themselves. But a zigzag driver uses two narrow roads and slows down everybody on both of them, and that delay to others never shows up in their own trip time. Selfish choices add up to a result that nobody wanted.

Try the slider. A shortcut that takes 10 minutes still hurts: drivers settle at 80 minutes. Only when it takes 25 minutes or more does nobody bother with it, and the trip is back to 65.

Try it yourself

1. Light traffic

Suppose only 2,000 drivers make the trip. How long does it take without the shortcut, and with a free shortcut?

Answer

Without it: 1,000 cars on each route, 10 + 45 = 55 minutes. With it: everyone zigzags, 20 + 0 + 20 = 40 minutes, and an old route would take 20 + 45 = 65. Here the shortcut really helps. It starts to hurt above 3,000 drivers, where both answers are 60 minutes.

2. A traffic planner

A planner could tell each driver which route to take. Can she use the free shortcut to make the average trip shorter than 65 minutes?

Answer

Yes, a little. Send 500 drivers on the zigzag and 1,750 each way on the old routes. Each narrow road then carries 2,250 cars. Zigzag drivers take 22.5 + 22.5 = 45 minutes, everyone else 22.5 + 45 = 67.5, and the average is about 64.7 minutes. But 3,500 drivers would be unhappy, and each of them could switch to save time. Without rules or a toll, the plan falls apart.

3. Just one driver

At the 80-minute equilibrium, one driver decides to take the route through A without the shortcut. What happens to their time, and to everyone else’s?

Answer

Their trip becomes 40 + 45 = 85 minutes, 5 minutes worse. Everyone else gains almost nothing: one fewer car on the second narrow road saves the others just 1/100 of a minute. That is why nobody moves on their own.

Go deeper

• The German mathematician Dietrich Braess described this effect in 1968.
• Tim Roughgarden and Éva Tardos proved that when travel times grow in a straight line with traffic, as they do here, selfish routing is never worse than 4/3 of the best possible average. Here it is 80 against about 64.7.
• The same paradox appears outside traffic: Joel Cohen and Paul Horowitz showed in 1991 that cutting a string in a network of springs and strings can make a hanging weight rise.