A traffic tangle

by Burkard Polster and Marty Ross

The Age, 10 August 2009

Once upon a time, in the faraway kingdom of Ausland, there were the two mighty cities of Melbaville and Sydtown. The cities were connected by two highways, each consisting of a red section and a green section. A red section always took two hours to traverse, but time on the green sections varied. When traffic was heavy a green section would also take two hours, but in light traffic an hour sufficed.

Overall, neither highway offered an advantage, and motorists chose between the two routes more or less equally. This resulted in traffic being sufficiently sparse that the green sections were quickly traversed, and the total travel time between cities was three hours. Everyone was destined to drive happily ever after.

But then King Kevin the Klever had an idea. Halfway along, the highways passed through the rustic townships of Canburg and Queanbee. Although very close, no road connected the two towns. King Kevin decreed that a superfast road be built, after which travelling from Canburg to Queanbee took a matter of seconds.

King Kevin was pleased. For about a day. Then Kevin’s loyal subjects began petitioning him, complaining that it now took four hours to travel between Sydtown and Melbaville!

What had happened? King Kevin summoned the royal mathematicians. They explained that the new super-road gave travellers more options. Quite reasonably, most people chose to begin along a green section, switch along the super-road and then traverse the second green section. But, because so many chose this route, traffic was clogged, and the trip now took four hours.

King Kevin was still puzzled, but satisfied. He gave big research grants to the mathematicians to design Ausland’s roads. And everyone lived happily ever after, particularly the mathematicians.

That ends the fairy tale, but what about in real life? Obviously, actual road networks are not so simple. Nonetheless, it can genuinely happen that adding a new road slows things down for everybody. And, conversely, closing down roads can result in a speeding up of traffic.

The possibility of such counterintuitive consequences was first predicted in 1969 by the mathematician Dietrich Braess. Subsequently, real-life examples of “Braess’s Paradox” were observed. For example, road closures in Seoul, Stuttgart and New York resulted in a speeding-up of traffic. In other cities roads have been identified whose closure is predicted would have the same effect.

We do not know whether Braeess’s Paradox affects cities in Australia. However, there is a similar traffic phenomenon well known to all Melburnians. A fourth lane is being added to the M1 freeway. Presumably, the hope is that this added lane will speed up traffic. And, it probably will, at least for a while.

But, of course, faster travel time will mean that longer commutes become feasible, which means that more people will buy houses in distant locations, and then add to the traffic. Finally, way too soon, we will have back the same nightmarish traffic. How do we know? Well, lanes have been added to the M1 before…

Is Melbourne’s frustrating M1 in the nature of Braess’s Paradox? Or is it just an instance of a Really Dumb Idea? We’ll let the reader decide.

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