Braess's paradox

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Braess’s paradox is the surprising idea that adding new roads to a traffic network can make overall travel slower. It happens because each driver chooses the route that looks fastest for themselves, not the one that minimizes total congestion for everyone. When many drivers act this way, the best personal choice might not add up to the best overall result.

Key idea in plain terms
- Each driver picks a route trying to minimize their own travel time.
- The quickest individual choice can cause more people to use the same shortcut, slowing everyone down.
- The result at equilibrium (no one can do better by changing routes) can be worse for all than if the network were kept smaller.

A simple illustration
Imagine 4,000 drivers traveling from Start to End with two available paths:
- Start → A → End: Start to A takes T/100 minutes, and A to End takes 45 minutes.
- Start → B → End: Start to B takes 45 minutes, and B to End takes T/100 minutes.
If the two routes are the only options, traffic splits evenly so each path takes 65 minutes.

Now add a new road directly connecting A and B, with almost no travel time. A single driver using Start–A–B–End might complete in about 40 minutes, which seems great. But as more drivers try the new path, it becomes crowded and the time on that route rises. When the numbers balance out (about 2,500 on the new path and 1,500 on the old), everyone ends up slower than before. In this example, the new route ends up making all drivers take about 80 minutes. If instead the new route hadn’t existed, everyone would have been faster by about 15 minutes. The paradox shows that removing or not adding certain connections can sometimes improve overall traffic.

What makes this work in theory
- The situation can be described as a game where each player (driver) chooses a route.
- A Nash equilibrium is reached when no driver can improve their own time by switching routes, given what everyone else is doing.
- This equilibrium need not be the fastest possible overall (the socially optimal flow).
- If the road costs depend linearly on how many cars use them, the total travel time at equilibrium can be noticeably worse than the social optimum—sometimes by a factor of up to 4/3.

Broader implications and examples
- Real-world traffic: Cities have observed that closing a major road or removing a shortcut can reduce overall congestion, while opening or expanding another link can sometimes increase congestion.
- Seoul’s Cheonggye Expressway removal helped traffic after the project.
- In Manhattan, a temporary 42nd Street closure for Earth Day reduced congestion locally.
- New York’s 2009 closures of Broadway segments and other experiments showed improvements in some cases.
- A 2012 study noted that removing main roads doesn’t necessarily make conditions worse and can transfer traffic rather than simply increasing it.
- Beyond cars: Braess’s paradox has been seen in other networks:
- Power grids with decentralized generation can experience slower overall flow when a new link is added.
- In mesoscopic electronic systems, adding a path can reduce conductance.
- Mechanical analogies with springs show similar counter-intuitive behavior when connections are added or removed.
- In ecology and biology, removing a perturbed part of a network can sometimes prevent cascading failures.
- In blockchain payment networks (layer-2), adding a new channel can raise routing costs rather than lowering them, and closing channels can reduce fees.

Takeaways
- The paradox teaches a fundamental lesson: selfish routing does not guarantee the best global outcome.
- Network designers should be aware that adding capacity can backfire and that sometimes pruning paths or channels improves overall performance.
- Understanding the balance between individual incentives and system-wide efficiency is crucial for planning transportation, power, communication, and other complex networks.