User:IssaRice/Aumann's agreement theorem: Difference between revisions

Revision as of 23:23, 24 August 2018

Hal Finney's example

$\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ 6 & 7 & 8 & 9 & 10 & 11 & 12 \end{matrix}$

	1	2	3	4	5	6
1
2

$E = {ω \in Ω : Pr (A ∣ I (ω)) = q_{1} and Pr (A ∣ J (ω)) = q_{2}}$

One of the assumptions in the agreement theorem is that $E$ is common knowledge. This seems like a pretty strange requirement, since it seems like the posterior probability of $A$ can never change no matter what else the agents condition on in addition to $E$ . For example, what if we bring in agent 3 and make the posteriors common knowledge again?

What if we take $E^{'} = {ω \in Ω : Pr (A ∣ I (ω)) = q_{1}}$ and say that agent 1 knows $E^{'}$ ?

In the form of $E$ above, we can change $A$ to be any subset of $Ω$ and $q_{1}, q_{2}$ to be any numbers in $[0, 1]$ . We can also set the state of the world to be any $ω \in Ω$ . The agreement theorem says that as we vary these parameters, if we ever find that $(I \land J) (ω) \subset E$ , then we must have $q_{1} = q_{2}$ .

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References

↑ Tyrrell McAllister. "Aumann's agreement theorem". July 7, 2011.
↑ Wei Dai. "Probability Space & Aumann Agreement". December 10, 2009.
↑ Robert J. Aumann. "Agreeing to Disagree". November 1976.
↑ https://math.stackexchange.com/questions/303834/common-knowledge-and-concept-of-coarsening-partition
↑ John Geanakoplos. "Common Knowledge".

[1] Tyrrell McAllister. "Aumann's agreement theorem". July 7, 2011.

[2] Wei Dai. "Probability Space & Aumann Agreement". December 10, 2009.

[3] Robert J. Aumann. "Agreeing to Disagree". November 1976.

[4] ttps://math.stackexchange.com/questions/303834/common-knowledge-and-concept-of-coarsening-partition

[5] John Geanakoplos. "Common Knowledge".

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@@ Line 27: / Line 27: @@
 What if we take <math>E' = \{\omega \in \Omega : \Pr(A \mid I(\omega)) = q_1\}</math> and say that agent 1 knows <math>E'</math>?
+In the form of <math>E</math> above, we can change <math>A</math> to be any subset of <math>\Omega</math> and <math>q_1, q_2</math> to be any numbers in <math>[0,1]</math>. We can also set the state of the world to be any <math>\omega\in\Omega</math>. The agreement theorem says that as we vary these parameters, if we ever find that <math>(I\wedge J)(\omega) \subset E</math>, then we must have <math>q_1 = q_2</math>.
 <ref>Tyrrell McAllister. [https://web.archive.org/web/20110725162431/http://dl.dropbox.com/u/34639481/Aumann_agreement_theorem.pdf "Aumann's agreement theorem"]. July 7, 2011.</ref>