= Solution
<Braess paradox> is the possibility that adding a route reduces the attainable equilibrium performance, despite enlarging the feasible set for a planner.
Take unit demand from $O$ to $D$. Initially the routes are $O\to U\to D$ and $O\to V\to D$. The links $O\to U$ and $V\to D$ have delay equal to their own flow; $U\to D$ and $O\to V$ have constant delay one. If the upper route has flow $h$, its delay is $h+1$, while the lower delay is $2-h$. The <Wardrop equilibrium> therefore splits traffic equally and has
$$
\boxed{\text{original equilibrium delay}=3/2}.
$$
Now add a directed zero-delay link $U\to V$. Let upper, lower and middle route flows be $a,b,c$ with $a+b+c=1$. Their delays are respectively $2-b$, $2-a$ and $2-a-b$. If $a>0$, its route would be strictly more expensive than the middle route; likewise $b>0$ is impossible at equilibrium. Hence $a=b=0$, $c=1$. All three routes then have delay two, so this is an equilibrium, and
$$
\boxed{\text{new equilibrium delay}=2>3/2}.
$$
The cheaper-looking cross-link tempts everyone onto both flow-dependent links. No individual can improve after congestion has built up. A planner could retain the old split and ignore the new link, so the feasible optimum cannot worsen. The paradox concerns selfish equilibrium, not the physical disappearance of the earlier allocation.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-37-braess-network.png]
{title=Braess paradox: equilibrium delay rises from three halves to two after adding a free directed cross-link}
{height=360}
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