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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 $( N , B )$ be a finite directed [[Graph|graph]], with node set $N$ and arc, branch or link set $B$ (cf. also [[Graph, oriented|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 $W$ of O/D pairs, and for each $w \in W$, supposes a flow demand $d _ { w } &gt; 0$ to be given. Let $R _ { w }$ be a set of routes joining $w$. For each $w \in W$, and $r \in R _ { w }$, one considers $F _ { r } \geq 0$ such that $\sum _ { r \in R _ { W } } F _ { r } = d _ { W }$, giving a route flow vector $F = ( F _ { r } ) _ { r \in R _ { W } , w \in W }$. This route flow induces a link flow $f = ( f _ { b } ) _ { b \in B }$, by $f _ { b } = \sum _ { r \ni b } F _ { r }$ for each $b$, where one identifies a route with the set of its links. For each link $a$, one supposes a link cost <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130210/b13021018.png"/>, where $g _ { ab }$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130210/b13021020.png"/> are given. For $r \in R _ { w }$, $w \in W$, one defines a route cost by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130210/b13021023.png"/>.

A route flow $H$ is a user equilibrium if it satisfies the condition that for all $r , s \in R _ { w }$, $w \in W$, if $C _ { r } &lt; C _ { s }$ then $H _ { S } = 0$. In other words, there is, for each $w$, a common route cost $\gamma _ { w }$ for all routes $r \in R _ { w }$ with non-zero $H _ { r }$. A user equilibrium flow exists always [[#References|[a5]]], and if the matrix $g = g _ { a b }$ is such that $g + g ^ { T }$ is positive-definite, then the equilibrium link flows, and hence the route costs $\gamma _ { w }$, are unique. In [[#References|[a3]]], Braess's paradox is said to occur if adding a new route $r$ to some $R _ { w }$ results in $\gamma _ { w }$ being increased. See also [[#References|[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, $t$, such as $\gamma _ { w }$. On the network given by any one link $b$, $t$ depends on the flow $f$ and a parameter $k_b$. For example, $t = 1 / ( k _ { b } - f )$ if the link is given by an M/M/1 queue (cf. also [[Queue|Queue]]). Note that $t$ decreases if $k_b$ increases, for $f$ 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, $t$ increases when some $k_b$ increases. Adding links may be thought of as changing a parameter $k_b$ 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 [[#References|[a4]]] is a particular type of Braess's paradox. If one distinguishes these paradoxes mathematically, it is by requiring the link costs $t$ in the Braess paradox to be increasing functions of the link flow $f$, while in the Downs–Thomson paradox there is a link with $t$ a decreasing function of $f$.

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

[[#References|[a2]]] contains an ample list of references.

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