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After having studied carefully Simpson's book SOSOA (Subsystems of second order arithmetic) I've naturally arrived at the question about the connection of Game theory with Reverse mathematics. Is there such a thing? Results such as this is of interest for me: any finite normal form game has a Nash Equilibrium iff $textsfWKL_0$ holds over $textsfRCA_0$. It is just an example, I do not claim it is true.

There is an extensive body of research on games (specifically determinacy principles) and reverse mathematics. Just to mention a few results:

• WKL$_0$ is equivalent to clopen determinacy for games on $0,1$ (= "Finite-length, finite-option games have winning strategies").



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  ATR$_0$ is equivalent to both clopen determinacy on $omega$ and to open determinacy on $omega$ - this is due to Steel.

  
  
  
  • Incidentally, clopen and open determinacy for games on $mathbbR$ are not equivalent in higher reverse mathematics; this was initially proved by me via forcing, then shortly after given a much better proof by Hachtman via fine structure theory, and apparently there's another proof by Sato (although it hasn't appeared yet) via proof theory.
  
  

(From now on, games are on $omega$.)

• $Pi^1_1$-CA$_0$ is equivalent to $Sigma^0_1wedgePi^0_1$-determinacy.




• A very fine-grained analysis has been conducted by Nemoto, e.g. here.



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  At the higher determinacy levels, there is a tight analysis by Montalban/Shore (see e.g. this paper); it's a bit technical, however, due to their proof that no true $Sigma^1_4$ sentence can imply $Delta^1_2$-CA$_0$ (in particular, full $Sigma^1_1$ determinacy doesn't imply $Delta^1_2$-CA$_0$), which renders straightforward reversals impossible.

  
  
  
  • In particular, they show that $(i)$ for each $n$, Z$_2$ proves $n$-$Pi^0_3$ determinacy, but $(ii)$ $Delta^0_4$-determinacy isn't provable in Z$_2$ (this refuted an earlier claim by Martin).
  
  



• An astronomically less important, but still in my mind neat, example (and plugging my own work): determinacy for Banach-Mazur games for Borel subspaces of Baire space is equivalent to ATR$_0$ and to Banach-Mazur determinacy for analytic subspaces (I got the lower bounds, the upper bounds being essentially due to Steel); meanwhile, determinacy for $Sigma^1_2$-Banach-Mazur games is independent of ZFC, so there could be some really cool stuff here (but I haven't been able to tease it out).

Moving from determinacy to equilibria, I'm less familiar with this but Yamazaki/Peng/Peng showed that Glicksberg's theorem ("every continuous game has a mixed Nash equilibrium") is equivalent to ACA$_0$. See also Weiguang Peng's thesis.

In general, equilibria seem to have not been studied as much as determinacy principles in reverse math; I suspect this is because of the hugely important role determinacy principles play in set theory.

Possibly also of interest, but not strictly reverse math:

• The complexity-theoretic difficulty of finding Nash equilibria has been studied by several people, e.g. Daskalakis/Goldberg/Papadimitriou.




• Equilibria have been studied from the perspective of Weirauch reducibility by Pauly.




• Tanaka looked at equilibria in a constructive context.