Braess' paradox

Braess' paradox or Braess's paradox is a proposed explanation for a seeming improvement to a road network being able to impede traffic through it. It was discovered in 1968 by mathematician Dietrich Braess, who noticed that adding a road to a congested road traffic network could increase overall journey time, and it has been used to explain instances of improved traffic flow when existing major roads are closed.

The paradox may have analogues in electrical power grids and biological systems. It has been suggested that in theory, the improvement of a malfunctioning network could be accomplished by removing certain parts of it.

Dietrich Braess, a mathematician at Ruhr University, Germany, noticed the flow in a road network could be impeded by adding a new road, when he was working on traffic modelling. His idea was that if each driver is making the optimal self-interested decision as to which route is quickest, a shortcut could be chosen too often for drivers to have the shortest travel times possible. More formally, the idea behind Braess' discovery is that the Nash equilibrium may not equate with the best overall flow through a network.

The paradox is stated as follows:

"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."

Adding extra capacity to a network when the moving entities selfishly choose their route can in some cases reduce overall performance. That is because the Nash equilibrium of such a system is not necessarily optimal. The network change induces a new game structure which leads to a (multiplayer) prisoner's dilemma. In a Nash equilibrium, drivers have no incentive to change their routes. While the system is not in a Nash equilibrium, individual drivers are able to improve their respective travel times by changing the routes they take. In the case of Braess' paradox, drivers will continue to switch until they reach Nash equilibrium despite the reduction in overall performance.

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