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I have a more general question on the importance of fixed-point theorems. In mathematics youre being introduced to so many fixed-point theorems but i still could not figure out why they are so important. Why would be a simply looking statement as $f(x)=x$ be so important. I would appreciate any answer. Thanks in advance. On wikipedia it says nothing about the importance, contextualisation of theorems in mathematics is sometimes not given.

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One important reason is that the existence of solutions to systems of equations are equivalent to fixed-points of appropriate functions. Suppose you want to show $f(x)=0$ for some $x$. This is equivalent to $f(x)+x=x$, which means that the function $F$ defined by $F(x)=f(x)+x$ has a fixed-point.

If you want to discuss properties of solutions to equations that you might not be able to solve explicitly, it is useful to know that such solutions exist in the first place.

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I second lisyarus' link. Nash's thesis on the existence of Nash equilibria in normal form games relied on the Brouwer Fixed Point Theorem. It was famously dismissed by von Neumann (who did work in both Game Theory and Functional Analysis) as being "just a fixed point theorem." More modern proofs rely on Kakutani's Fixed Point Theorem instead.

Computing Brouwer Fixed Points is also an interesting question. This problem is complete for the complexity class PPAD, as are a number of associated problems in theoretical economics (https://en.wikipedia.org/wiki/List_of_PPAD-complete_problems). Note that Nash's theorem guarantees the existence of Nash equilibria. So deciding whether a normal form game admits a Nash equilibrium is in NP, where every instance has an answer of YES. Thus, computing Nash equilibria is very unlikely to be NP-complete. However, it is not easy to find Nash equilibria. Thus, PPAD-completeness is a weaker notion of intractability than NP-completeness.

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