The Mathematics of Congestion
The Paradox
In 1968, mathematician Dietrich Braess proved something shocking: adding extra capacity
to a network can make it less efficient for everyone.
When all drivers optimize selfishly, the equilibrium gets worse!
Consider our road network: 4,000 drivers travel from A to D.
Each road has a travel time that depends on traffic:
Road A→B
T/100 minutes
(T = traffic)
Road B→D
45 minutes
(constant)
Road A→C
45 minutes
(constant)
Road C→D
T/100 minutes
(T = traffic)
New Road B→C
0 minutes
(instant shortcut!)
Without the New Road:
Drivers split evenly: 2,000 take the upper route (A→B→D), 2,000 take the lower route (A→C→D).
Upper route: 2000/100 + 45 = 65 minutes
Lower route: 45 + 2000/100 = 65 minutes
This is a Nash equilibrium: no driver can improve by switching routes.
With the New Road:
Now there's a shortcut from B to C that takes 0 minutes. Every rational driver reasons:
"The A→B road is fast, and I can instantly jump to C, then take C→D."
Result: Everyone takes A→B→C→D!
Time: 4000/100 + 0 + 4000/100 = 80 minutes!
The new equilibrium is 15 minutes worse for everyone!
Why Does This Happen?
The new road creates a dominant strategy: regardless of what
others do, taking A→B→C→D is always better for an individual driver.
But when everyone follows this logic, the congestion-sensitive roads A→B and C→D
both get overloaded. The "shortcut" becomes a trap!
Real-World Examples
🏙️ Seoul, South Korea (2003)
When the Cheonggyecheon highway was demolished and replaced with a park,
traffic flow in the city actually improved! Removing the road
forced drivers to use diverse routes, reducing overall congestion.
🇩🇪 Stuttgart, Germany (1969)
A newly opened street had to be closed because it made traffic worse.
This was one of the first real-world confirmations of Braess's Paradox.
🗽 New York City (1990)
During Earth Day 1990, 42nd Street was closed to traffic. Instead of
gridlock, traffic throughout Midtown improved dramatically.
🌐 Internet Routing
The same paradox applies to computer networks! Adding bandwidth can
sometimes increase latency when packets are selfishly routed.
The Solution?
Congestion pricing (like London's or Singapore's) can fix Braess's Paradox
by making drivers internalize the cost they impose on others. Alternatively, closing strategic
roads or implementing coordinated routing can restore efficiency.
The lesson: individual optimization ≠ collective optimization.