Testwiki:Reference desk/Archives/Mathematics/2023 October 21

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October 21

Under which conditions does the Aumann theorem hold?

This is about the aumann theorem. Under which requirements does it hold? Which premises must be in craft for the theorem to be proved? 2A02:8071:60A0:92E0:2D70:2361:BBD3:2FC6 (talk) 19:03, 21 October 2023 (UTC)

The question is not entirely clear. The statement of the theorem is of the form “if X holds, Y holds”. Is the question about a further characterization of the antecedent X? Or is it about Aumann's set-theoretic model of common knowledge? Or is it about conditions in the real world that ensure this model is an adequate description of the relevant aspects of reality? The latter question falls outside the scope of mathematics.

Could you explain a bit more about comman knowledge and how it is defined? — Preceding unsigned comment added by 2A02:8071:60A0:92E0:F0DC:D8BC:3C8D:36F2 (talk) 21:41, 23 October 2023 (UTC)

Have you read the article Common knowledge? Using the notation of the article on epistemic modal logic, we can give a recursively specified property:

$C_G\phi ⟺\mathrm{\forall }i\in G[K_i(\phi \wedge C_G\phi )].$

In words, it is common knowledge among all agents in group $G$ that $\phi$ holds if, and only if, each of these agents knows not only that $\phi$ holds, but also that all agents in the group know that this is common knowledge in the group.  --Lambiam 08:58, 24 October 2023 (UTC)