Picture a town where 4,000 people drive from the same start to the same finish every morning. Two routes lead there, each with one slow bridge and one jam-prone road. Now the council builds a brand-new link between them. Nothing is taken away. It is pure extra road. And every single commute gets longer.
That is Braess's paradox, and it is not a trick. It falls straight out of arithmetic and a simple assumption: each driver picks whatever route is quickest for them.
Call the start S and the finish E. The upper route goes S to A, then A to E. The lower route goes S to B, then B to E.
The road from S to A is a narrow street that gets slower with traffic: it takes T ÷ 100 minutes, where T is the number of cars using it. Ten cars, 0.1 minutes. 4,000 cars, 40 minutes. The road from A to E is a wide motorway that always takes 45 minutes, jam or no jam. The lower route is the mirror image: a fixed 45 minutes from S to B, then a jam-prone road from B to E.
If everyone chose the upper route, the narrow street would take 40 minutes, for a total of 85. Same for the lower. So drivers spread out. When 2,000 take each route, the jam-prone road takes 2,000 ÷ 100 = 20 minutes, and every trip is 20 + 45 = 65 minutes. Nobody can do better by switching, because the other route is exactly as slow. That balance has a name: a Nash equilibrium.
Now build a new road from A to B. Press the button below to open it, and drag the slider to set how long it takes. At 0 minutes it is a teleporter.
With a free shortcut, the route S to A to B to E costs one narrow-street trip plus one jam-prone trip, with nothing in between. The first driver to try it finds it far quicker than 65 minutes, so they take it. So does the next, and the next. Soon the narrow streets carry everyone.
When all 4,000 are on the shortcut route, the road from S to A has 4,000 cars (40 minutes) and so does the road from B to E (another 40), so each trip takes 80 minutes. Could anyone do better by leaving? The old upper route would now be 40 + 45 = 85 minutes. The old lower route, 45 + 40 = 85. Both are worse. Everyone is stuck at 80, each making the best choice available to them.
Notice what went wrong. Each driver counts their own delay but not the delay they add to the others. The motorways, which never slow down, are abandoned, and the roads that do slow down fill up. The new link tempts everyone into using the part of the network that jams.
Drag the shortcut time up. Make it 5 minutes and the commute is 85. It gets longer than the free version. Past 5 minutes, some drivers decide it isn't worth it and return to the old routes. Between 5 and 25 minutes the trip shortens by exactly one minute for each extra minute of shortcut, so a slower shortcut makes the commute quicker. Only at 25 minutes does the shortcut stop attracting anyone, and the town is back at 65. For any time shorter than that, the new road is a net loss for every single driver.
The German mathematician Dietrich Braess described the effect in 1968, and the economist Arthur Pigou had noticed a similar idea in 1920. The numbers above are a textbook model: real networks have many roads, drivers who don't know the whole map, and changing demand, so the effect has to be argued case by case. There are reports that traffic in Stuttgart improved in 1969 when a newly built road section was closed again, and that traffic around Seoul flowed better after a downtown expressway was removed for the Cheonggyecheon stream restoration. Treat those as suggestive stories rather than controlled experiments.
What is solid is the limit of the damage. For roads whose delay grows in proportion to traffic, as the narrow streets here do, mathematicians have shown that selfish routing is never worse than 4/3 of the best possible outcome. Our town sits at 80 against a coordinated optimum of 65, a ratio of about 1.23, comfortably inside that bound.
When everyone chases their own best option, adding choices can leave everyone worse off. The same shape shows up in networks of all kinds, from computer data to power grids. The fix is rarely a clever road. It is changing what each driver sees as the price of their choice, which is why economists like tolls that make people pay for the jam they cause.