First Algorithms for Approximating Fixed Points on the Unit Simplex

Abstract

In this chapter we consider the problem of computing fixed points of a continuous function f from the unit simplex S ⁿ into itself. As we see in Chapter 1, Brouwer theorem guarantees the existence of a fixed point on S ⁿ but does not offer any insight about the location of such fixed points. Nevertheless it is clear to us that in order to approximate a fixed point, one possible way is to design a systematic procedure for finding completely labelled simplices. Such procedure did not exist until 1967. In his pioneering work Scarf [1967b] developed the first such procedure. Starting with a specific primitive set, this procedure generates a path of adjacent primitive sets and terminates within a finite number of steps with a completely labelled primitive set which yields a good approximation of a fixed point. (Completely labelled primitive sets are analogue of completely labelled simplices.) To prove the convergence of the procedure, Scarf utilized an argument of Lemke and Howson [1964] and Lemke [1965] which is a purely combinatorial argument and does not rely on any monotonicity property. Later Kuhn [1968,1969] proposed two alternative procedures by using simplices and triangulations instead of primitive sets. These very first fixed point methods are the central topic of this chapter.