 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 hasNow 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, andThe 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.