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The words "Braess's paradox" refer to a surprising decrease in performance for some network, which is a result of it being improved locally. The occurrence of this decrease has been studied in mathematical models of equilibrium flow in road/rail traffic, computer networks, telephone networks, water supply systems, electrical circuits, spring systems, and so on. This has been done under various assumptions on routing schemes, such as being state-dependent or fixed.

Below, the occurrence of Braess's paradox in a classical model of user equilibrium traffic flow is given.

Let be a finite directed graph, with node set and arc, branch or link set (cf. also Graph, oriented). A path in which all links are similarly directed is called a route, with the initial and final nodes forming an origin/destination pair (or O/D pair). One considers a set of O/D pairs, and for each , supposes a flow demand to be given. Let be a set of routes joining . For each , and , one considers such that , giving a route flow vector . This route flow induces a link flow , by for each , where one identifies a route with the set of its links. For each link , one supposes a link cost , where and are given. For , , one defines a route cost by .

A route flow is a user equilibrium if it satisfies the condition that for all , , if then . In other words, there is, for each , a common route cost for all routes with non-zero . A user equilibrium flow exists always [a5], and if the matrix is such that is positive-definite, then the equilibrium link flows, and hence the route costs , are unique. In [a3], Braess's paradox is said to occur if adding a new route to some results in being increased. See also [a3] for necessary and sufficient conditions for this to happen under the assumption that there is a strictly positive equilibrium flow in all routes.

There is no fixed definition of Braess's paradox in all systems, but there is a common theme. One assumes some measure of performance, , such as . On the network given by any one link , depends on the flow and a parameter . For example, if the link is given by an M/M/1 queue (cf. also Queue). Note that decreases if increases, for fixed. Suppose some flow demand is fixed and link flows are given by some requirement about equilibria, or by a given dynamical process not in equilibrium. One says that Braess's paradox occurs if, for the network as a whole, increases when some increases. Adding links may be thought of as changing a parameter from zero or infinity. A different language would be used to describe certain other types of networks, such as electrical circuits.

Under this description, the Downs–Thomson paradox [a4] is a particular type of Braess's paradox. If one distinguishes these paradoxes mathematically, it is by requiring the link costs in the Braess paradox to be increasing functions of the link flow , while in the Downs–Thomson paradox there is a link with a decreasing function of .

Independent discoveries of Braess's paradox can be attributed to D. Braess [a1], A. Downs [a4], J.M. Thomson [a6] and C.A. Zukowski and J.L. Wyatt [a7].

[a2] contains an ample list of references.

References

 [a1] D. Braess, "Über ein Paradoxon aus der Verkehrsplannung" Unternehmensforschung , 12 (1968) pp. 258–268 [a2] D. Braess, http://homepage.ruhr-uni-bochum.de/Dietrich.Braess (2000) [a3] S. Dafermos, A. Nagurney, "On some traffic theory equilibrium paradoxes" Transportation Res. B , 18 (1984) pp. 101–110 [a4] A. Downs, "The law of peak-hour expressway congestion" Traffic Quart. , 16 (1962) pp. 393–409 [a5] M.J. Smith, "The existence, uniqueness, and stability of traffic equilibria" Transportation Res. B , 13 (1979) pp. 295–304 [a6] J.M. Thomson, "Great cities and their traffic" , Gollancz, London (1977) [a7] C.A. Zukowski, J.L. Wyatt, "Sensitivity of nonlinear one-port resistor networks" IEEE Trans. Circuits Syst. , CAS-31 (1984) pp. 1048–1051
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