Imagine a simple road network in a major city. People try to get from point $A$ to point $B$ and they have two choices: either travel through $X$ or travel through $Y$.
Either way, the total travel time is 1 hour. Going through $X$, the first leg is shorter:
\begin{align}
AX &{} = 0.2~\text{hours},\\
XB &{} = 0.8~\text{hours}.
\end{align}
Going through $Y$, it's the other way around:
\begin{align}
AY &{} = 0.8~\text{hours},\\
YB &{} = 0.2~\text{hours}.
\end{align}
So city council, wisely, decides to fix the problem. Surely, a high-speed link between $X$ and $Y$ will make things much more efficient?
They design the bypass so that people can travel from $X$ to $Y$ in $XY = 0.3$ hours. So they calculate that now, $AX + XY + YB = 0.2 + 0.3 + 0.2 = 0.7$ hours. That's a very significant improvement! Surely, the city will be better off.
After spending a few billion dollars, the express route is completed. It is opened to the public with great fanfare. The public is enthusiastic, and everyone who previously traveled congested city streets now takes the route, $AX - XY - YB$. Unfortunately, this doubles the traffic on $AX$ and $YB$. These city streets were not designed to handle so much extra traffic, so everything slows to a crawl. Travel time on these short stretches of road now doubles. So the total travel time from $A$ to $B$, using the new bypass, is
$$2AX + XY + 2YB = 0.4 + 0.3 + 0.4 = 1.1~\text{hours}.$$
Oops. This is no good. People notice. Some brave souls decide that maybe the bypass is not for them. Perhaps they should return to the old route. Unfortunately, whether they drive through $X$ or $Y$ makes no difference: they will have to take one of the congested stretches of road, be it $AX$ or $YB$. So either way, their travel time is
$$2AX + XB = AY + 2YB =1.2~\text{hours},$$
which is even worse than taking the bypass.
The debate rages. There are repercussions, accusations in city council. The mayor resigns. The central government cuts funding to further highway developments, seeing this fiasco as a political embarrassment. Eventually, a new city government is elected that, listening to the radical advice of traffic engineers (denounced as rabid environmentalists in certain corners), decides to shut down and once and for all demolish the $XY$ bypass. The bypass is closed to traffic that evening.
And lo and behold... the next day, travel time from $A$ to $B$ is back to the old normal, the usual 1 hour.
What happened? How can removing a bypass improve traffic?
The explanation lies in behavior and the nature of optimization. When it comes to traffic decisions, drivers optimize "selfishly": That is, they try to minimize their own travel time, which of course is the only sensible thing to do from an individual perspective.
However, what is optimal for the individual is not necessarily optimal for the community. And if this sounds like a sneaky way to introduce collectivist propaganda, it isn't: it's hard math. Even in the presence of the bypass, people could choose other routes. But why would you, as an individual, choose a route that leaves you at a disadvantage if others do not cooperate? And while there is a sweet spot -- when just the right number of cars take the bypass vs. the old routes -- that sweet spot represents an unstable equilibrium of the system.
This is known as Braess's paradox, first documented by Dietrich Braess at the Institut für numerische und instrumentelle Mathematik in Münster, Germany, in 1968.
