Fixed Point Theorems

Many existence problems in economics – for example existence of competitive equilibrium in general equilibrium theory, existence of Nash in equilibrium in game theory – can be formulated as fixed point problems. Because of this, theorems giving sufficient conditions for existence of fixed points have played an important role in economics. My treatment is schematic, focusing on only a few representative theorems and omitting many proofs. An excellent introduction to fixed point theory is Border (1985). McLennan (2008) is a more recent concise survey that provides a treatment more sophisticated than the one here. The remainder of these notes is divided into three sections. Section 2 focuses on fixed point theorems where the goal is to find restrictions on the set X strong enough to guarantee that every continuous function on X has a fixed point. The prototype of theorems in this class is the Brouwer Fixed Point Theorem, which states that a fixed point exists provided X is a compact and convex subset of RN . In contrast, the Contraction Mapping Theorem (Section 3) imposes a strong continuity condition on f but only very weak conditions on X. Finally, the Tarski Fixed Point Theorem (Section 4) requires that f be weakly increasing, but not necessarily continuous, and that X be, loosely, a generalized rectangle (possibly with holes).