Braess’s paradox is the discovery that adding a road to a network can make every driver’s trip slower. It happens because each driver picks the route that is quickest for them, and those individually sensible choices can add up to a worse result for everyone. In a simple model with 4,000 drivers, opening a shortcut that takes zero minutes raises everyone’s journey from 65 minutes to 80, about 23% longer, and no single driver can do better by switching back. It is a model, not a forecast for every road project.
The road network before the shortcut
Draw four points: a start S, two towns A and B, and a destination T. There are two routes from S to T, one through A and one through B, and 4,000 drivers want to make the trip.
Each route has one fixed road and one crowded road. The fixed roads, A to T and S to B, always take 45 minutes. The crowded roads, S to A and B to T, take x/100 minutes when x drivers use them, so the more cars, the slower they get.
The drivers settle into an even split, 2,000 on each route. Each crowded road then takes 2,000/100 = 20 minutes, and every driver’s trip is:
Nobody gains by moving: a driver who switches only makes the other crowded road a little slower for themselves.
Add a free shortcut
Now open a one-way road from A to B that takes zero minutes. A driver can go S to A, cut across to B, and finish B to T, using both crowded roads and neither 45-minute road.
The lure is plain. Even with all 4,000 drivers on it, a crowded road takes at most 4,000/100 = 40 minutes, which beats 45. The first driver to try the shortcut takes 2,000/100 + 2,001/100 = 40.01 minutes, a saving of almost 25 minutes. So more drivers try it, and the crowded roads fill up.
They fill up all the way. With every driver on the new route, each crowded road takes 40 minutes:
Could anyone escape? A driver who leaves alone for an old route still uses one crowded road, at about 40 minutes, plus a 45-minute road: about 85 minutes. Nobody gains by leaving. Everyone is stuck at 80 minutes, 15 more than before, despite having an extra option. The video lets you pause and guess before the counters run.
Why Braess’s paradox happens
A state like this is called a Nash equilibrium: nobody can improve their own result by changing only their own choice. Both the 65-minute split and the 80-minute jam are equilibria for their networks, and the paradox shows that an equilibrium need not be the best outcome for the group. Each driver’s choice slows everyone else on the crowded roads, a cost no single driver has a reason to count.
There is a limit to the damage. When travel times grow in a straight line with traffic, as they do here, adding a road can never make total travel time at equilibrium worse by more than a factor of 4/3. This example, 80 against 65, stays under that bound.
Who found it, and where it shows up
The German mathematician Dietrich Braess noticed the effect while working on traffic modeling, and published it in 1968 in a paper titled “Über ein Paradoxon aus der Verkehrsplanung”, on a paradox of traffic planning. An English translation appeared in 2005.
The paradox has been used to explain cases where traffic improved after roads closed. In Seoul, traffic in the area sped up when the Cheonggye Expressway was removed to restore a creek. In Stuttgart, investment in the road network in 1969 did not help until a newly built section was closed again. In 1990, the temporary closing of 42nd Street in Manhattan for Earth Day reduced congestion in the area. None of this means every new road backfires; a network can’t be judged by counting its roads.
For another puzzle where the obvious answer is wrong, see why switching doors wins twice as often.
One free road, 4,000 sensible drivers, 15 lost minutes each.
