Proving existence of a Nash equilibrium without fixed-point theorems

Question

Is there a way to prove the existence of a Nash equilibrium in finite games without using Kakutani or Brouwer's fixed-point theorems?

Answer 1

There are ways to do so for specific types of games. Some examples:

• For zero-sum games, you can prove existence using Farka's Lemma.
• For supermodular games, you can prove existence using Topkis Theorem and the Knaster-Tarski Fixed-Point Theorem.
• For potential games, you can prove existence using Weierstrass Maximum Theorem.
• You can show that all compact-continuous games have rationalizable strategies without a fixed-point argument. Also, if a game has a unique rationalizable strategy-profile, it must be a Nash equilibrium.