You are one of 4,000 drivers heading across town. A new bridge opens that lets every driver cut across a river instantly. Everybody saves time, right? In 1968 the German mathematician Dietrich Braess showed that the opposite can happen: a shortcut can make every single trip slower, and no one is behaving stupidly.
Drivers go from Start to End. There are two ways. The top route has a narrow road that gets slower with traffic (x/100 minutes, where x is the number of cars on it), then a wide motorway that always takes 45 minutes. The bottom route is the same thing the other way round: a 45 minute motorway first, then a narrow road. With 4,000 drivers splitting evenly, 2,000 per route, each trip takes 2,000/100 + 45 = 65 minutes. Nobody can do better by switching, so the roads are at balance.
Now add a free link between the two middle points. A driver can take the top narrow road, hop across, and use the bottom narrow road, skipping both 45 minute stretches. Try it below.
Start with the shortcut open and the 2,000 / 2,000 split. A zig-zag driver pays only the two narrow-road delays, about 40 minutes, against 65 for everyone else. So drivers move to the zig-zag, one after another. Each switch is rational: at every moment the zig-zag is the quicker option for the person switching.
But each switcher also adds traffic to both narrow roads, making the trip slower for everyone using them. Press the button and watch: the zig-zag stays slightly faster than the outer routes all the way up to 4,000 drivers, so nobody stops switching. At the end, every driver takes the narrow road twice: 4,000/100 + 4,000/100 = 80 minutes. Before the bridge it was 65. Everyone is 15 minutes worse off, and no individual can improve by going back to an outer route, which would now cost 85 minutes.
This is a Nash equilibrium: a state where nobody gains by changing their own choice. The trouble is that it is not the best state for the group. Economists measure the gap with the "price of anarchy". For networks like this one, where delays rise in a straight line with traffic, a theorem says that adding a road can never make equilibrium travel time worse by more than a factor of 4/3. Our example is 80 against 65, about 1.23, so it is inside that limit.
The cause is an externality. When you choose a road, you feel your own delay but not the extra delay you cause for others. Close the shortcut and the drivers are forced back into a split that, in this toy network, is better for all of them.
There are reports that it does, though real networks are messier than a four-point diagram. According to the usual accounts, in Stuttgart in 1969 traffic did not improve after new road investment until a newly built section was closed again. In New York in 1990, closing 42nd Street for Earth Day reportedly reduced congestion in the area. And in Seoul, traffic is reported to have moved faster after the Cheonggye Expressway was removed as part of the Cheonggyecheon river restoration. These stories are suggestive rather than proof, since many things change at once in a city. The solid result is the mathematics: whenever selfish route choice meets delays that depend on traffic, the paradox is possible.
The same structure shows up outside roads. Braess-style effects have been described in electrical grids, in the flow of data on networks, and in mechanical spring systems, anywhere that independent choices share a network with congestion.
More capacity is not always better capacity. When people choose routes for themselves, adding an option changes what everyone else faces. Sometimes the best improvement is to take something away.