First thing we do, let's close all the roads...

If you were in the German city of Stuttgart in 1969 – and that’s a big if – you got to witness first hand a truly bizarre paradox. In an effort to alleviate downtown traffic, the city built a new road. Surprisingly, travel time got worse. After extensive investigation and hand-wringing, they closed the new road. Travel times got better.

It’s not a one-off, either. In 2003, Seoul demolished the elevated Cheonggye Expressway - a six-lane highway carrying 160,000 cars a day - to restore a buried stream underneath. By every principle of common sense, this should have been a traffic apocalypse. It wasn't. Travel times through the area actually improved for drivers who used to rely on the highway. The effect has been seen in NYC as well - shutting down part of 42nd street eased traffic congestion.

This is Braess's Paradox, named for the mathematician who studied it, Dietrich Paradox.

Let's dive in.

The Setup

Four nodes: Start, A, B, and End. Everybody wants to get from Start to End. There are two obvious ways to do that:

• Top route: Start → A → End
• Bottom route: Start → B → End

Now we give each route one "fast" leg and one "slow" leg. The slow legs are simple: they just take 45 minutes, no matter what. Think Broadway at rush hour, or the FDR on any given weekday - totally saturated, you're moving at crawl speed regardless of whether you're car #1 or car #10,000.

The fast legs are different. They're uncongested right up until they aren't. Specifically, if t cars per hour are using the fast leg, it takes t/100 minutes. So 1000 cars per hour means a 10-minute leg. 4000 cars per hour means a 40-minute leg.

Here's the asymmetry that matters: we put the fast leg at the start of the top route, and at the end of the bottom route.

• Top route: fast leg (t/100) → 45 min constant
• Bottom route: 45 min constant → fast leg (t/100)

Now suppose 4000 cars per hour need to make this trip. In equilibrium, they split 50/50 between the two routes (by symmetry - no reason to prefer one over the other). So:

• 2000 cars on each route
• Fast leg: 2000/100 = 20 min
• Slow leg: 45 min
• Total: 65 minutes

Everyone's moderately annoyed. As God intended.

Enter the Shortcut

Somebody in the Department of Transportation has a bright idea. "What if," they say, "we connect A and B directly? With, like, a really fast road. We'll make it so fast it's essentially free - zero minutes."

(This is admittedly a bit of a cartoon assumption, but it makes the math pristine and doesn't change the qualitative result. If it bothers you, imagine a very fast road with a 1 or 2 minute travel time and redo the arithmetic. The paradox is robust to this.)

Now you're a selfish driver staring at your options. You could take the old top route: 20 + 45 = 65 minutes. You could take the old bottom route: 45 + 20 = 65 minutes. Or you could take the brand new Start → A → B → End route: two fast legs stitched together, no 45-minute penalty in sight. How could that not be better?

So you take it. And here's the thing - so does everyone else. Every single driver in the network has the same thought at the same time, and they all pile onto the same route.

Now all 4000 cars are on Start→A. And all 4000 are on B→End.

• Start → A: 4000/100 = 40 min
• A → B: 0 min (the shortcut)
• B → End: 4000/100 = 40 min
• Total: 80 minutes

Adding the shortcut made everyone's commute 15 minutes longer.

I built an interactive simulator for this one - a little four-node graph where you can drag a slider to change traffic volume and toggle the shortcut on and off. Cars flow continuously, edges fatten and redden as they get congested, and a live "paradox indicator" tells you whether the shortcut is currently helping or hurting. Go mess with it.

There's no going back...

The natural question is: okay, so it's worse - but why don't some drivers just switch back to the old routes? If we all pretended the new road didn't exist, wouldn't everybody be faster?

Well yes... and no. We can compare this to the classical Prisoner's Dilemma. For any individual car, taking the shortcut is a dominant strategy (it optimizes travel time, regardless of what other cars do) so it's akin to telling people to 'pretend they have to cooperate' in the Prisoner's Dilemma. If everyone DID pretend the new road didn't exist, traffic would improve. But one by one, drivers would start using the shortcut again, because for them it is faster. Let's take a look.

Suppose one driver - call her Alice - decides to be a hero and go back to the old route. She takes the pure top route: Start → A → End.

• Her Start → A leg: she's on a road with 4000 cars (everyone else still uses it, plus her). That's 40 minutes.
• Her A → End leg: 45 minutes, always.
• Alice's total: 85 minutes.

Alice just made her own commute longer by ignoring the new road. So she won't. Neither will any other individual driver, by exactly the same argument. The 80-minute configuration is a Nash equilibrium - nobody can improve by unilaterally switching. Everybody is optimally screwed.

The only fix is to remove the option entirely, i.e. shut down the new road.

Worth noting: if the total volume is low enough, the shortcut actually does help everybody. If you solve for the crossover, you'll find that below about 3000 cars per hour the shortcut is a genuine improvement, at 3000 it's a wash, and above 3000 it starts hurting - more so the higher traffic gets. At 4500 cars per hour the shortcut costs you about 25 minutes per trip.

The math works. The optics don't.

So if removing roads can make everyone faster, why don't we do it more? Well for the most part, people tend to get irritated when you shut down perfectly good roads, and adopt a "shout first, examine network graphs later" approach. Imagine Mayor Mamdani coming out and saying "In an effort to reduce traffic, we're shutting down part of 42nd Street." It would certainly be a bold play, and if traffic didn't immediately improve - and it might not, Braess paradoxes are actually pretty tough to identify from the outside - well, he could end up being nearly as unpopular as Eric Adams.

Can star athletes hurt the team?

This is very related to a phenomenon sometimes seen in sports, where a star player leaves a team or is injured and the team actually does better. This is known as the Ewing Theory, named for basketball star Patrick Theory. In a way I think it's a little more intuitive. When Rob Gronkowski is on the field, you have to throw it to him. Everyone knows that - including the defense. When he's off the field, it's much harder to predict who the ball is going to. This is of course much less consistent, and plenty of analysts think it's flawed. But hey, it's just a theory.

What's next?

A few threads I want to pull on in future posts:

1. The general Price of Anarchy story. Roughgarden's 4/3 bound is one of those beautifully tight results that deserves its own full treatment. It's also a great entry point into thinking about when markets go wrong and by how much.
2. Designing networks to be paradox-free. There's a whole line of work on road networks where Braess-style effects are provably impossible. Turns out it's related to matroid theory, which is always fun.

Also, to be clear: I am not proposing we close any actual roads in New York City. But I am proposing we build a new one in Stuttgart, Germany as a fun prank.