Chapter summary

In this chapter we present some basic results from different areas of mathematics required for various proofs in the book. In Section 23.1 we state and prove several fixed point theorems. The main and best known is Brouwer's Fixed Point Theorem, which states that every continuous function from a compact and convex subset of a Euclidean space to itself has a fixed point. This theorem is used in Chapter 5 to prove the existence of a Nash equilibrium in mixed strategies. Using Brouwer's Fixed Point Theorem we prove Kakutani's Fixed Point Theorem, which states that every upper semi-continuous convex-valued correspondence from a compact and convex subset of a Euclidean space to itself has a fixed point. This result provides a shorter proof for the existence of a Nash equilibrium in mixed strategies in strategic-form games. We then prove the KKM theorem, which is used to prove the nonemptiness of the bargaining set (Theorem 19.19, page 790). The main tool for proving both Brouwer's Fixed Point Theorem and the KKM Theorem is Sperner's Lemma, which is stated and proved first.

In Section 23.2 we prove the Separating Hyperplane Theorem, which states that for every convex set in a Euclidean space and a point not in the set there is a hyperplane separating the set and the point. This theorem is used in Chapter 14 to prove that every B-set is an approachable set.