Braess's paradox is one of the counter-intuitive consequences of Nash's equilibrium.
When a system is moved from its state of equilibrium, the new state of equilibrium may be worse than the previous state, even if the intention was to improve the cost function for everyone.
Take the case where 4000 cars have to go every day from START to END. There are two routes available, one that goes through A and one that goes through B.
The START-A leg takes t=a/100 minutes, where a is the number of cars going through A, and A-END takes a constant time of t=45 minutes.
Similarly, START-B takes t=45 minutes, and B-END takes t=b/100 (b being the number of cars going through B).
Since the total duration of the two routes is the same, in a state of equilibrium half the cars would go through A, and half the cars would go through B. If we have a total of 4000 cars, both the START-A-END and START-B-END routes will therefore take 2000/100 + 45 = 65 minutes.
Suppose that we decide to break this equilibrium, and make commuting faster by building a new fast route between A and B. Suppose that it's a super-short but super-wide highway and the time required to cross it is practically zero. How will the new equilibrium state look like?
All the drivers may now pick the START-A leg, because it takes 4000/100=40 minutes, compared to START-B which is guaranteed to take 45 minutes. Then they would commute from A to B (which takes approximately zero), and then B-END, which takes 4000/100=40 minutes, compared to A-END which takes always 45 minutes.
So, in this new state of equilibrium where all the cars choose the new optimal route, compared to the previous 50/50 split, the total time will be 40+40=80 minutes. 23% worse than the initial state, even though the initial idea was to speed up commuting times for everyone.
https://resources.mpi-inf.mpg.de/departments/d1/teaching/ws12/ct/Braess-paradox.pdf