Braess's Paradox

Braess's paradox, credited to the German mathematician Dietrich Braess (de), states that adding extra capacity to a network when the moving entities selfishly choose their route, can in some cases reduce overall performance. This is because the Nash equilibrium of such a system is not necessarily optimal.

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."

The reason for this is that in a Nash equilibrium, drivers have no incentive to change their routes. If the system is not in a Nash equilibrium, selfish drivers must be able to improve their respective travel times by changing the routes they take. In the case of Braess's paradox, drivers will continue to switch until they reach Nash equilibrium, despite the reduction in overall performance.

If the latency functions are linear then adding an edge can never make total travel time at equilibrium worse by a factor of more than 4/3.