## “Nash and Walras Equilibrium via Brouwer,” J. Geanakoplos (2003)

A number of famous proofs in economics involve the following argument: use Berge’s Maximum Theorem to show a correspondence is uhc, then invoke Kakutani’s FPT to show a fixed point exists. For instance, the existence of Nash equilibria and the existence of Walrasian equilibrium can be proved in this manner. Most people (myself included) find Kakutani much more difficult to understand than Brouwer’s FPT. Geanakoplos shows in this unpublished paper than many of these proofs can actually be done with Brouwer’s FPT by writing the maximization problem as one whose argmax is a function rather than a correspondence, and using an analogue of the Maximum Principle called the Satisficing Principle to guarantee continuity rather than just upper hemicontinuity.

As an example, consider a concave game with finite strategy sets and N players. Let u=(u1,u2….uN) concave, and let qi(s)=argmax(pi)[u(i)(s(i),s(-i))-|pi-s(i)|^2], where p is a potential mixed strategy, and s is the hypothesized Nash equilibrium, with agent i choosing. Note that the first term is concave and the second is negative quadratic, hence the whole term is strictly concave. Since u is continuous and the mixed extension is compact, the argmax is unique, therefore qi is a continuous function, as is q=(q1,q2,q3…qN), so by Brouwer a fixed point s exists. To show s is Nash, assume that u(i)(p,s(-i))-u(i)(s)>E>0 for some other strategy p. Then by concavity, u(i)[ep+(1-e)s(i),s(-i)]-u(i)(s)>eE for some e>0. This contradicts that s was an argmax of qi, since |ep+(1-e)s(i)-s(i)|^2=e|p-s(i)|^2<eE for e small enough. So s is Nash.

This proof is quite a bit simpler than the standard proof for concave games, and the proof of Walrasian existence is simplified in a similar manner.