Fixed-point theorems

Fixed-point theorems are one of the major tools economists use for proving existence, etc. One of the oldest fixed-point theorems - Brouwer's - was developed in 1910 and already by 1928, John von Neumann was using it to prove the existence of a "minimax" solution to two-agent games (which translates itself mathematically into the existence of a saddlepoint). von Neumann (1937) used a generalization of Brouwer's theorem to prove existence again for a saddlepoint - this time for a balanced growth equilibrium for his expanding economy. This generalization was later simplified by Kakutani (1941). Working on the theory of games, John Nash (1950) was among the first to use Kakutani's Fixed Point Theorem. Gerard Debreu (1952), generalizing upon Nash, came across this.