The following definition and characterization appear in Characterizing Braess's Paradox for Traffic Networks, J. N. Hagstrom and R. A. Abrams, Proceedings of IEEE 2001 Conference on Intelligent Transportation Systems 837-842.
Given an equilibrium distribution of flows in a traffic network, a generalized Braess Paradox occurs if there exists some other distribution of flows for which some travelers have improved travel costs and no travelers have worse travel costs than in equilibrium.
In Über ein Paradox der Verkerhsplannung, Dietrich Braess, Unternehmenstorchung 12:258-268 (1968), the author provided an example of a traffic network in which travel costs increased when a link was added to the network. Braess used the common assumption that traffic distributes itself according to Wardrop's user-equilibrium principle. An alternate way of viewing this example is to consider the network with the additional link present and note that deleting the link (or equivalently, blocking it by imposing a large toll on the link) improves travel costs. A generalization of this is to define a Braess paradox to occur any time there exists an alternate distribution of traffic flow which makes some travelers better off and no travelers worse off than in equilibrium. This generalization encompasses all examples of a Braess paradox occurring in the literature.
If we consider vehicular traffic in light of a continuum of players in a game, a Wardrop user-equilibrium corresponds to a Nash equilibrium in the field of game theory. In game theory, a vector of player strategies which satisfies the property that any change made by a single player causes some player to have a smaller payoff is said to be strongly Pareto optimal. The definition of a generalized Braess Paradox above then corresponds to the property that a Nash equilibrium need not be strongly Pareto optimal.
We establish that a generalized Braess Paradox occurs if and only if the equilibrium distribution of flows is not optimal for the Braess Optimization Problem (pdf file, Adobe Reader required).
Revised, 3 December, 2001, by Jane Hagstrom.