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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, with node set $N$ and arc, branch or link set $B$ (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 $W$ of O/D pairs, and for each $w \in W$, supposes a flow demand $d _ { w } > 0$ 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 , where $g _ { ab }$ and are given. For $r \in R _ { w }$, $w \in W$, one defines a route cost by .

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 $C _ { r } < C _ { s }$ 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 [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 [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 [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). 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 [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 [a1], A. Downs [a4], J.M. Thomson [a6] and C.A. Zukowski and J.L. Wyatt [a7].

[a2] contains an ample list of references.

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