Sunday, January 25, 2009

Braess's Paradox and the Nash Equilibrium

One interesting fact about driving in Phoenix is that depending on traffic conditions, including daily traffic patterns, there is no particular advantage to driving on one of our lovely freeways. I finally read enough to learn some of the underlying reasons why this occurs. One of the major factors is a principle called Braess's paradox. The formulation of this paradox is credited to the mathematician Dietrich Braess. Essentially, "[f]or 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."

The reason why this occurs is due to the fact that the drivers have no incentive to change their routes, that is the system is in Nash equilibrium and that the equilibrium is not necessarily optimal. "If the system is not in a Nash equilibrium, selfish drivers must be able to improve their travels time by changing the routes they take. In the case of Braess's paradox, drivers will continue to switch until the Nash equilibrium despite the fact that overall performance is reduced."

What I had observed in driving across Phoenix is that in many cases taking a variety of surface streets can equal or reduce the driving time involving freeways. Very simply, the freeways don't always go where you are going and driving to and from the freeway may add more time to a specific trip than simply taking a more direct surface street. Another contravening factor is the vast distances involved in travel in the Salt River Valley. Even traveling at the legal speed limit on freeways it can take more than an hour to reach the edge of the city.

Yet another limitation on the full application of the paradox is the availability of side streets that are seldom used by commuters. If you find a line of cars you can turn down the first available side street and avoid the congestion, if you know the streets.

The real reason this paradox is interesting is that it tends to show that adding freeways and widening streets may increase congestion and travel time, rather than reduce it. Each driver in turn takes this slightly quicker route, at a small saving for himself, but causing substantial extra delay to a number of other drivers. By the time everyone has decided whether to take the short cut, the total congestion gained by the system far outweighs the savings in distance.

We are presently being told by our new President, that he is going to improve roads and pour huge sums of money into infrastructure. The solution in Phoenix, so far, has been to add more freeways, i.e. increase the complexity of the network. Maybe it is time to think of the alternative? Reduce the number of cars instead of just adding another route to the network.

1 comment:

  1. I thought that light rail was supposed to reduce the number of cars? Not likely. Government can't come up with the solutions because the government thought process usually has no relation to reality.

    On another note, Phoenix traffic is bad, but not yet critical. When I lived in California, it took 45 minutes to drive six miles to work each way. And that was with taking all the back streets, through yards and alleys, and down the median. Staying on the main streets was insanity.