Braess's Paradox

The Road That Made Everything Worse

In 1990, New York City closed 42nd Street for Earth Day. Traffic engineers braced for gridlock. Instead, traffic improved. The cars didn't just vanish—they redistributed, and the network breathed easier without one of its busiest arteries. More road had meant more congestion. Less road meant less.

This sounds like the kind of thing a crank would say at a city council meeting, the sort of claim that makes traffic engineers roll their eyes. Except the traffic engineers were the ones who couldn't explain it. And 42nd Street wasn't even the first time.

In Stuttgart, Germany, in 1969, a new road was built through the city center. Traffic got worse. When the road was later closed for construction, traffic got better.1 In Seoul, South Korea, in 2003, the city demolished an entire elevated highway—the Cheonggyecheon Expressway—and replaced it with a stream and a park. Experts predicted catastrophe. Instead, traffic speeds in the surrounding area actually increased.2

These aren't flukes. They're manifestations of a mathematical phenomenon discovered in 1968 by a German mathematician named Dietrich Braess, who proved something that sounds impossible: in a network of selfish agents, adding capacity can make everyone worse off.3

The Diamond Network

Here's the simplest version. Imagine a city with 4,000 drivers who all need to get from point S (Start) to point E (End). There are two routes through intermediate nodes A and B:

The key is that some edges have fixed costs (like a long highway—takes 45 minutes no matter what) and some have variable costs that depend on how many people use them (like a city street—more cars means more congestion). If x drivers use a variable-cost edge, it takes x/100 minutes.

With 4,000 drivers splitting between two symmetric routes, equilibrium is easy: 2,000 go via A and 2,000 go via B. Each driver's travel time is:

Great. Everyone gets to work in 65 minutes. Now some ambitious urban planner builds a magical shortcut—a new road connecting A to B that takes essentially zero time. A gift to commuters! What could go wrong?

Everything Goes Wrong

Think about it from a single driver's perspective. You're on route S→A→E, taking 65 minutes. But now there's a free road from A to B. From A, you could zip over to B, then take the variable-cost road B→E. If enough people are still on the old routes, that B→E road might be fast.

In fact, every driver reasons the same way. The shortcut creates an irresistible temptation: take S→A (variable, fast if few people), then the free shortcut A→B, then B→E (variable, fast if few people). The problem is that everyone finds it irresistible.

When all 4,000 drivers pile onto route S→A→B→E:

Wait—could someone switch back to, say, S→A→E? That route now costs 4000/100 + 45 = 85 minutes (since all 4,000 are on S→A). Or S→B→E costs 45 + 4000/100 = 85 minutes. Either alternative is worse than staying on the shortcut route. So 80 minutes is the new Nash equilibrium. Nobody can individually improve by switching.4

Try it yourself:

The Selfish Herd

What makes Braess's paradox so unsettling is that nobody is behaving irrationally. Each driver is making the best possible choice given what everyone else is doing. That's what Nash equilibrium means—it's the stable state where no individual can benefit by unilaterally changing strategy. The problem is that the Nash equilibrium of the system with the shortcut is worse for everyone than the Nash equilibrium without it.

This is a Prisoner's Dilemma with four thousand players and a highway system. Each driver, acting alone, makes the perfectly reasonable choice to take the shortcut. But when everyone makes that perfectly reasonable choice, they all end up sitting in worse traffic than before. The invisible hand, in this case, slaps everyone in the face.

Economists have a name for the gap between what selfish individuals achieve and what a benevolent dictator could arrange: the Price of Anarchy.5 In our diamond network without the shortcut, selfish routing happens to be optimal—65 minutes is the best anyone could do. But add the shortcut, and selfish routing gives 80 minutes while the optimal assignment (just ignore the shortcut!) would still be 65. The Price of Anarchy is 80/65 ≈ 1.23—selfishness costs us 23% more than cooperation would.

The mechanism is worth understanding deeply, because it shows up everywhere. The shortcut creates a coordination problem. If everyone could agree to ignore it, they'd all benefit. But no individual has an incentive to ignore it. The new road is like a candy bowl on a diet—each piece seems harmless, but collective consumption is ruinous.

Tim Roughgarden, a computer scientist at Columbia, proved something remarkable in 2002: for networks with linear cost functions (like our x/100), the Price of Anarchy never exceeds 4/3.6 That is, selfish routing can never be more than 33% worse than optimal. Our example, with its 23% penalty, is already close to the theoretical worst case. But for nonlinear cost functions—the kind that describe real traffic, where congestion grows faster than linearly—the Price of Anarchy can be much worse.

Beyond Roads

Braess's paradox isn't really about traffic. It's about any network where selfish agents choose paths through shared resources.

The internet. In 2006, researchers showed that adding bandwidth to a computer network could slow down data transmission, because routing protocols (like drivers) selfishly choose the fastest-looking path.7 Your Netflix stream might actually benefit from a cable being cut somewhere in Ohio.

Power grids. Adding a transmission line to an electrical grid can cause blackouts, because power flows redistribute according to physics (Kirchhoff's laws) in ways analogous to selfish routing. This is known as the Braess paradox in electrical engineering, and it has caused real problems.8

Mechanical springs. In 2012, physicists at the Max Planck Institute built a physical network of springs and strings that exhibited Braess's paradox: adding a connection between two points caused the system to sag more under a load, not less. The paradox isn't metaphorical—it's literally mechanical.9

Basketball. Removing a strong player from a team can sometimes improve the team's performance—because the remaining players redistribute their effort more evenly, like drivers finding less-congested routes. Brian Skinner showed this using Price of Anarchy theory applied to shot selection.10

When Does Adding Help vs. Hurt?

The natural question: is every new road bad? Obviously not. Braess's paradox requires specific conditions—a mix of fixed and variable costs, a network topology that creates a tempting shortcut that overloads shared edges. In many networks, adding capacity genuinely helps.

The question is: when? And this turns out to depend on the relationship between the parameters. If the fixed costs are high relative to the variable costs, the shortcut won't tempt enough drivers to cause problems. If the shortcut itself has a significant cost (a toll, or just being slow), it dampens the paradox. And if there are enough drivers to overwhelm the variable-cost edges regardless, the shortcut can actually relieve pressure.

Explore it yourself:

What Do We Do About It?

Braess's paradox is not just a mathematical curiosity. It's a genuine policy problem. And there are really only three solutions:

1. Remove the road. This is what Seoul did. If you can identify a Braess edge—one whose existence makes the equilibrium worse—you can improve everyone's life by closing it. Of course, identifying which road is the problem requires knowing the full network structure, costs, and demand. This is not easy. In a city like Los Angeles, with thousands of road segments and millions of drivers, finding the Braess edges is a computational challenge that researchers are still working on.

2. Charge a toll. Congestion pricing forces drivers to internalize the cost they impose on others. If the shortcut costs enough, some drivers will avoid it, and the equilibrium can shift back to the efficient outcome. London's congestion charge, Singapore's ERP system, and Stockholm's toll ring are all real-world implementations of this idea—nudging the Nash equilibrium closer to the social optimum. The math is elegant: the optimal toll on each edge equals the marginal externality—the extra delay one driver imposes on everyone else by using that edge. This is the Pigouvian tax, transplanted from economics into graph theory.

3. Centralize routing. If a central planner controls all drivers (as Waze and Google Maps increasingly do), they can route traffic optimally. But this requires trust, compliance, and giving up individual agency—something humans are famously reluctant to do. And there's an irony here: if Waze routes everyone optimally and you know the routes are optimal, you have no incentive to deviate, which means selfish behavior and central planning converge. But if the algorithm makes even small errors, selfish deviators can unravel the whole scheme.

There's a fourth approach that Braess himself might have appreciated: understand the math before you build. Run the model. Simulate the equilibrium. Check whether your shiny new road is a Braess edge before you pour the concrete. This is what modern transportation engineers increasingly do, using software that computes Wardrop equilibria—the continuous version of Nash equilibrium for traffic networks—to predict the effects of network changes.

Ellenberg's great theme in How Not to Be Wrong is that mathematics forces us to confront uncomfortable truths about how the world works. Braess's paradox is one of the most uncomfortable: the world is not always improved by giving people more options. Sometimes the generous, obvious, well-intentioned intervention—a new road, a new link, a new shortcut—makes everything worse. And the reason it makes everything worse is precisely because each individual is acting rationally.

The math doesn't tell you to stop building roads. It tells you to think about the system, not just the parts. It tells you that networks have emergent behaviors that resist common sense. And it tells you that the gap between individual rationality and collective welfare is not a bug in human nature—it's a theorem.