# The Math That Shows How Fewer Roads Can Lead to Less Traffic Congestion

New roads don't fix traffic. In fact, they often make it worse. This has not only been observed in many real-life cases, but, as the mathematician Josefina Alvarez pointed out recently in Plus magazine, it's also a mathematical reality. In 1968, maths thinker Dietrich Braess even proved this: "an extension of a road network by an additional road can cause a redistribution of the flow in such a way that the travel time increases." It's called the Braess paradox, and has implications well beyond roads.

First off, what we're talking about isn't really a paradox. It's just counterintuitive. Here is the not-really-a-paradox stated more completely:

For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times.

Rush hour can be viewed as a non-cooperative game. Every driver is trying to minimize their time on the road and over many repetitions of rush hour across the same road network they achieve a kind of equilibrium, which is known as Nash equilibrium (for the idea's genesis, John Nash). The idea is that a driver will settle on a route once they're convinced that switching routes won't result in a shorter drive-time. Thus, the travel time is the same for all drivers on every route.

This whole arrangement is a collective decision brought to life by selfish acts. There may in fact be a way of decreasing the total amount of travel time for all drivers on every route together, giving a collective optimal traffic scheme, but selfish decision-makers won't have it if the optimal traffic scheme involves other drivers getting to their destinations faster. Hence, we have Nash equilibrium, in which the sum total of all drive times may be greater than it could be for unselfish actors.

To see the math, here's an example. Imagine a point A and a point B, with two different routes between them. (To bring this to life, pretend that point A is downtown and point B is some suburb.) Each route has a long section in which the travel time will be constant no matter what and a smaller section in which the travel time is determined as a function of total traffic. So, each route has a portion that it will take a guaranteed 45 minutes of driving and then it has a section where the travel time is given by the number of cars x divided by 100. It looks like this:

The example is pretty general and widely used, but I'm pulling it here from Networks, Crowds, and Markets: Reasoning about a Highly Connected World, which is a textbook by economist David Easley and computer scientist Jon Kleinberg, both of Cornell University. Here's how it looks with numbers:

Now, suppose that 4000 cars want to get from A to B as part of the morning commute. There are two possible routes that each car can choose: the upper route through C, or the lower route through D. For example, if each car takes the upper route (through C), then the total travel time for everyone is 85 minutes, since 4000/100 + 45 = 85. The same is true if everyone takes the lower route. On the other hand, if the cars divide up evenly between the two routes, so that each carries 2000 cars, then the total travel time for people on both routes is 2000/100 + 45 = 65.