If John Nash submitted his paper on Nash equilibria to an AI conference, the reviewers would say “Where’s the formal proof? Where are the experiments?” and reject it. And then the growth of game theory would be stunted.
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I believe his paper had a rigorous proof (I wouldn’t call it “formal”, but few proofs are ever formalized).
If you call this a rigorous proof: "Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani’s theorem that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point."
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Why is this not rigorous? The previous paragraph proves closeness and convexity. Then it's a direct application of Kakutani's theorem.
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Yes-rigorous (albeit not “formal”) by a mathematician’s standards.
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