Exponential lower bounds for finding Brouwer fixed points

The Brouwer fixed point theorem has become a major tool for modeling economic systems during the 20th century. It was intractable to use the theorem in a computational manner until 1965 when Scarf provided the first practical algorithm for finding a fixed point of a Brouwer map. Scarf\´s work left open the question of worstcase complexity, although he hypothesized that his algorithm had "typical" behavior of polynomial time in the number of variables of the problem. Here we show that any algorithm for fixed points based on function evaluation (which includes all general purpose fixed-point algorithrna) must in the worst case take a number of steps which is exponential both in the number of digits of accuracy and in the number of variables.