Fixed Points and Equilibria
In 1909, L. E. J. Brouwer proved a fixed point theorem that is illustrated by this scenario: At dawn, the surface of an oval swimming pool is perfectly still. Then a breeze begins to blow. The wind is strong enough to create waves, but not breakers. At dusk, the wind dies down, and the surface becomes still again. Each point on the surface of the pool may have shifted continuously during the day, but each point that began on the surface remains there throughout the day. Brouwer’s theorem guarantees that at least one point on the surface ends up where it began.