The Braess Paradox

Interactive Traffic Simulator

There are two routes from START to END. The segments START→B and A→END take a fixed 45 minutes. The segments START→A and B→END are congestible, taking R/K minutes, where R is the total number of drivers and K is the bridge capacity.

Why Does This Happen?

The paradox arises from the conflict between individual self-interest and collective good. In game theory terms, drivers settle into a Nash Equilibrium, where no single driver can improve their time by changing routes, given what everyone else is doing. This isn't always the best outcome for the group, known as the System Optimum.

• Before Freeway: Traffic splits evenly. The travel time on both routes is equal, creating a stable, efficient equilibrium. This state is both a Nash Equilibrium and the System Optimum.
• After Freeway: The new A→B path seems like a great deal. Drivers on the START→A→END route see a chance to bypass the slow second half. They switch to START→A→B→END.
• The Trap: As everyone makes this "rational" choice, they flood the first leg (START→A) and the last leg (B→END), increasing the travel time on those links for everyone. The new equilibrium is stable (no one wants to switch back to an even slower route), but the travel time for every single driver has increased.

System Optimum Demonstration

So is adding the freeway just a waste? No, you can still find a way to make it work by limiting the number of drivers who use the freeway (something akin to congestion-pricing). Adjust the same parameters (R, K, c) and choose how many drivers r use the freeway route START→A→B→END. The remaining drivers split evenly across the two original routes. Compare total travel times and see the optimal freeway usage.