There are situations when even full information leads to failure. Tragedy of the commons (overgrazing) is the most known example.
Another less known one is as follows: assume two routes from A to B, where one is shorter (say, 1 hour when not congested) but prone to congestion and the other one longer - say 2 hours regardless of traffic.
The selfish decision is to take the shorter route as long as its congestion hasn't slowed it down to the level of the longer route. So the "anarchy" equilibrium leads to everybody spending 2 hours.
A benevolent dictator (or enlightened citizens able to come up with a social contract) could of course come with a better solution - every day choose a limited number(#) of people who can choose the shorter road without congesting it.
(#) The optimal number obviously depends on the exact characteristic of the dependency between the number of people choosing the road and the latency generated. This just a model after all.