Traffic Equilibrium in the Real World
http://www.nytimes.com/1990/12/25/health/what-if-they-closed-42d-street-and-nobody-noticed.html
This article from The New York Times investigates the closing of 42nd Street in New York City on Earth Day, 1990. At the time, this road was extremely congested, and most people unsurprisingly thought that its closing would create a doomsday scenario on alternate routes. However, the opposite effect occurred, and the traffic flow actually improved once the road was closed. This situation exhibits Braess’ Paradox, which states that adding a new street to an already-crowded network only inhibits the movement of traffic. The rationale for this theorem is removing a major road would reduce congestion on the roads that allow access to it, while adding traffic to jammed roads would not make much difference. Mathematicians say that adding a new street exacerbates the problem at least half the time. Queuing theory is an analytic method of examining traffic networks that describes traffic jams in terms of the vehicles lining up on the streets. Mathematicians Dr. Joel Cohen and Dr. Frank Kelly found a simple network for which an added street increases travel time: a V-shaped divide causing people to choose left or right. The left path consists of a garbage truck forcing cars into one lane followed by a traffic light, while the right path has a traffic light and then construction that creates a single-lane road as well. The experts claim that if an engineer tries to improve the situation by adding a street from the garbage truck to the construction site, flow slows by 50% due to the inherent selfishness of drivers.
The article’s discussion is a real-world example of a traffic equilibrium problem. In a traffic network of people traveling from A to B, the addition or subtraction of a path can have a major effect on the flow efficiency. Let us take an example network with 4000 travelers trying to get from A to B. There are currently two routes: A to C to B and A to D to B, where AC and DB take 45 minutes while CB and AD take p/100 minutes where p is the number of people on that edge. At equilibrium each route should take the same amount of time, so half the total travelers, or 2000 travelers, would take each one. This would result in everyone taking 45 + 2000/100 = 65 minutes. Now say a road from D to C is built and it takes no additional time. The new traffic equilibrium is all travelers take path ADCB because taking a p/100 edge is always better than taking a 45-minute edge, and nobody has an incentive to deviate from this choice. The new total travel time is 4000/100 + 4000/100 = 80 minutes. The extra road has worsened the traffic situation. The selfishness of drivers is what creates this phenomenon because each one takes what road is best for them. As such, any potential shortcuts simply become overused, and drivers direct themselves to paths that become congested. Essentially, the p/100 road may be best for any individual, but each person who chooses it hurts every other driver that chooses the same path. This idea is represented in Braess’ Paradox.