Braess paradox: Maximum penalty in a minimal critical network

Abstract

A ‘simplest anti-symmetric’ two-path network is described which exhibits the well-known Braess paradox: the user travel costs being higher after the paths are joined by a transversal link (bridge). This network, herein named a ‘Minimal Critical Network’, clearly demonstrates the essence of the paradox with the minimum number of independent parameters, a minimum of mathematical complexity and a maximum Braess penalty. Although the Braess paradox has been studied extensively in the past, this ‘simplest’ network has been overlooked. The critical ranges of flow, and of user travel cost, all agree with the theorems of Frank, thus extending the validity of those theorems to a wider range of networks. Only one result, showing the effects of bridge congestion, contrasts with Frank's conclusion. Examples are given of techniques, some old and some new, which modify or eliminate this paradoxical behavior. A discussion of the good effects (non-paradoxical) of a bridge (especially a two-way bridge) is also included for the first time. Our Minimal Critical Network and graphical solution technique give a clear understanding of the paradox for this network. They are also especially useful for analysis of sensitivity to such extensions as, e.g. changes in parameters, elastic demand, general non-linear (even non-continuous) cost functions, two-way bridges, tolls and other methods to control the paradox, and diverse populations of users. We show that the paradox occurs in a simpler network than previously noted, and with a larger Braess penalty than previously noted.