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I have to write a short paper(about 20 pages) and prepare a presentation (about 1 hour) for an exam on Game Theory (it is a general, introductory course). I've looked up some things on the internet about the intersections of game theory and logic, and it seems to me that the main link between the two is the concept of determinacy in descriptive set theory.

Now, the topic feels really interesting, but I fear that it may be too complicated on the side of logic, either for myself or for the audience, even though I have already taken an introductory course on set theory (up to cofinality and Von Neuman hierarchy, no Large Cardinals nor forcing), one on model theory, and know a few things on ultrafilters.

My main worry however is that it might be related to game theory only tangentially since it seems to me that non-cooperative games enter the discussion only as a way to visualize logical issues, and I don't know if some substantial results from game theory are used for this subject.

So I need some reference on this kind of question in order to make up my mind and prepare the presentation if the topic is in fact suited.

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  • $\begingroup$ Economic or combinatorial game theory? (You're using one tag for each.) $\endgroup$
    – Théophile
    Apr 26, 2021 at 15:43
  • $\begingroup$ @Théophile The course included both economic and combinatorial topics, but I should say mostly economic. $\endgroup$ Apr 26, 2021 at 16:16

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I believe that the shortest path from games to logic via determinacy is the failure of the “infinitely long“ De Morgan law.

Namely, we all know that for $X\subseteq A^n$ ($n$ even). $$ \neg\bigl(\forall a_0 \exists a_1 \forall a_2 \exists a_3\dots \forall a_n : (a_0,\dots,a_n) \in X\bigr) $$ is equivalent to $$ \exists a_0 \forall a_1 \exists a_2 \forall a_3 \dots \exists a_n : \neg((a_0,\dots,a_n) \in X). $$ But if there is an infinite sequence of quantifiers, this is simply not true. A counterexample is given by a Bernstein subset $X$ of $A^{\mathbb{N}}$, which is not determined. This infinite De Morgan law is exactly determinacy.

I recommend any bibliography that allows you to fill in the gaps above (for instance, Kechris' Classical Descriptive Set Theory). You only need to know that a winning strategy for either player gives you a perfect subset of $X$ or its complement, and the definition and construction of a Bernstein set.

If you read Spanish, check out my blog post about this.

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  • $\begingroup$ I enjoyed the blog post :). Very well written. $\endgroup$ Apr 30, 2023 at 10:21
  • $\begingroup$ @ElchananSolomon Gracias! $\endgroup$ Apr 30, 2023 at 13:50

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