Route Assignment Models

Definitions of Network Equilibrium

The first reference to minimizing individual travel times was stated in 1841, but it was not until the late 1910s, when the subject of route choice over a two-route road network was examined. In 1924, the economist Frank Knight developed the argument that forms the basis of traffic equilibrium: he presented two highways connecting two terminals, one broad enough to accommodate a large number of vehicles but poorly graded and surfaced, and the second one with much better quality but with limited capacity. He stated that the trucks operating between the two terminals will tend to distribute themselves between both roads in such proportions that the cost per unit of transportation will be the same for all trucks on both routes. He concluded that the congestion created in the road with the limited capacity will reach the level at which both roads will be equally profitable.

In 1951, John Nash published the expected flow of traffic in a network, using the Nash equilibrium concept he developed within the game theory framework. He proved the existence and uniqueness of the equilibrium solutions.

In 1952, John Glen Wardrop stated two formal principles of the concept of equilibrium:

• First principle. The journey times on all the routes actually used are equal, and less than those that would be experienced by a single vehicle on any unused route.
• Second principle. At equilibrium, the average journey time is at minimum.

Various alternative definitions have been presented for the first principle. Let r and s be two different used routes in OD pair (p, q), let h be the path-flow vector that contains the flow on each path for all the OD pairs, let Drs be a path-shifting flow vector of the same dimension as h, which is zero in every position except those corresponding to the routes r and s, where the values of the vector are minus 1 and 1, respectively.

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