User:IssaRice/Aumann's agreement theorem - Revision history

IssaRice: /* References */

2020-01-29T01:45:45Z

References

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	<references/>		<references/>

			things to read if i ever come back to this:

			* https://web.archive.org/web/20110725162431/http://dl.dropbox.com/u/34639481/Aumann_agreement_theorem.pdf
			* https://www.lesswrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement
			* https://wiki.lesswrong.com/wiki/Aumann's_agreement_theorem
			* http://www.ma.huji.ac.il/~raumann/pdf/Agreeing%20to%20Disagree.pdf

IssaRice: /* Join and meet */

2020-01-14T03:32:22Z

Join and meet

← Older revision		Revision as of 03:32, 14 January 2020
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	this tells us things like <math>a \leq b \iff a\wedge b = a \iff a \vee b = b</math>, and <math>a \wedge b \leq a \leq a \vee b</math>.		this tells us things like <math>a \leq b \iff a\wedge b = a \iff a \vee b = b</math>, and <math>a \wedge b \leq a \leq a \vee b</math>.

	But what this ''doesn't'' tell you is what <math>\leq</math> should mean, because order theory works with many different kinds of partial orders. so we need to do one of two things:		But what this ''doesn't'' tell you is what <math>\leq</math> should mean (or alternatively what <math>\wedge</math> or <math>\vee</math> should mean), because order theory works with many different kinds of partial orders. Specifying any one of <math>\leq</math>, <math>\wedge</math>, or <math>\vee</math> "locks in" the meanings of the other two. so we need to do one of two things:

	* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.		* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.

IssaRice: /* Join and meet */

2020-01-14T03:27:05Z

Join and meet

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	* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.		* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.
	* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write Hasse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], which is to work by analogy with the poset of sets (where we've already decided that <math>\leq</math> means <math>\subseteq</math> rather than <math>\supseteq</math>).		* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write Hasse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], which is to work by analogy with the poset of sets (where we've already decided that <math>\leq</math> means <math>\subseteq</math> rather than <math>\supseteq</math>). Just to give one example, [https://books.google.com/books?id=t5r79vZ9ogoC&ppis=_c&lpg=PA249&dq=join%20meet%20partition&pg=PA249#v=onepage&q=join%20meet%20partition&f=false this book] defines meet as the coarsest common refinement.

	see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].		see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

IssaRice: /* Join and meet */

2020-01-14T01:35:20Z

Join and meet

← Older revision		Revision as of 01:35, 14 January 2020
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	* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.		* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.
	* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write Hasse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], by ~~looking at~~ the poset of sets.		* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write Hasse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], which is to work by analogy with the poset of sets (where we've already decided that <math>\leq</math> means <math>\subseteq</math> rather than <math>\supseteq</math>).

	see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].		see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

IssaRice: /* Join and meet */

2020-01-14T01:32:47Z

Join and meet

← Older revision		Revision as of 01:32, 14 January 2020
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	* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.		* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.
	* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write ~~hesse~~ diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], by looking at the poset of sets.		* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write Hasse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], by looking at the poset of sets.

	see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].		see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

IssaRice: /* Join and meet */

2020-01-14T01:31:19Z

Join and meet

← Older revision		Revision as of 01:31, 14 January 2020
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	But what this ''doesn't'' tell you is what <math>\leq</math> should mean, because order theory works with many different kinds of partial orders. so we need to do one of two things:		But what this ''doesn't'' tell you is what <math>\leq</math> should mean, because order theory works with many different kinds of partial orders. so we need to do one of two things:

	* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>		* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>. This is the approach that Aumann took in his original paper.
	* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement".		* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement". This is the approach that many wikipedia pages implicitly take (e.g. they write hesse diagrams of partitions with the finer partitions on the bottom). This one also makes sense because <math>\wedge = \cap</math> and <math>\vee = \cup</math>, and "join" becomes the operation that takes the union of the partitions (and then merges them to form a valid partition) and "meet" becomes the operation that takes all possible intersections between the blocks in the existing partitions. tyrrell mcallister gives another reason [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/FpfrnJeyAr8x5oAy7 here], by looking at the poset of sets.

	see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].		see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

IssaRice: /* Join and meet */

2020-01-14T01:24:04Z

Join and meet

← Older revision		Revision as of 01:24, 14 January 2020
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	this tells us things like <math>a \leq b \iff a\wedge b = a \iff a \vee b = b</math>, and <math>a \wedge b \leq a \leq a \vee b</math>.		this tells us things like <math>a \leq b \iff a\wedge b = a \iff a \vee b = b</math>, and <math>a \wedge b \leq a \leq a \vee b</math>.

			But what this ''doesn't'' tell you is what <math>\leq</math> should mean, because order theory works with many different kinds of partial orders. so we need to do one of two things:

			* decide that <math>\leq</math> means "is coarser than". This is equivalent to deciding that "meet" means "finest common coarsening", because we need <math>P \wedge Q \leq P,Q</math>
			* decide that <math>\leq</math> means "is finer than". This is equivalent to deciding that "meet" means "coarsest common refinement".

	see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].		see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

IssaRice: /* Join and meet */

2020-01-14T01:19:53Z

Join and meet

← Older revision		Revision as of 01:19, 14 January 2020
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	TODO aumann assumes the join of the partitions consists of nonempty events. it would be good to show where in his proof this assumption is used.		TODO aumann assumes the join of the partitions consists of nonempty events. it would be good to show where in his proof this assumption is used.

			--

			warning: there are actually two conventions for what "meet" and "join" can mean. this happens because if you have two partitions <math>P,Q</math>, and P is coarser than Q (i.e. Q is finer than P), do you write this as <math>P \leq Q</math> ("P coarser than Q") or <math>Q \leq P</math> ("Q finer than P")?

			from order theory, we already have the convention that:

			* <math>\wedge</math> = inf = meet
			* <math>\vee</math> = sup = join

			this tells us things like <math>a \leq b \iff a\wedge b = a \iff a \vee b = b</math>, and <math>a \wedge b \leq a \leq a \vee b</math>.

			see more discussion about this [https://www.greaterwrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement/comment/k8Mv2BT7MJtbsA52Z here].

	==Common knowledge==		==Common knowledge==

IssaRice: /* Mechanics of reaching agreement */

2018-11-13T21:53:12Z

Mechanics of reaching agreement

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	Given <math>I_0</math> and <math>J_0</math>, can we write <math>I_1, J_1</math> in general? It seems like what we want is <math>I_0 \vee J_0</math>, but then <math>I_1 = J_1</math>. Is this okay? It seems like the convergence then happens in a single step? Maybe the examples I'm considering are too simple. Aumann's second coin toss example seems to have more than one exchange, so I should study that in detail. [http://www.polemarchakis.org/a16-cdf.pdf] uses the auxiliary sets <math>a_t, b_t</math> to track updates rather than changing the information partitions.		Given <math>I_0</math> and <math>J_0</math>, can we write <math>I_1, J_1</math> in general? It seems like what we want is <math>I_0 \vee J_0</math>, but then <math>I_1 = J_1</math>. Is this okay? It seems like the convergence then happens in a single step? Maybe the examples I'm considering are too simple. Aumann's second coin toss example seems to have more than one exchange, so I should study that in detail. [http://www.polemarchakis.org/a16-cdf.pdf] uses the auxiliary sets <math>a_t, b_t</math> to track updates rather than changing the information partitions.

	There is a paper called something like "we can't disagree forever", which might be similar to the above. Ok, I just checked [http://www.polemarchakis.org/a16-cdf.pdf ~~the paper],~~ and it looks like this is exactly what the paper is about.		There is a paper called something like "we can't disagree forever", which might be similar to the above. Ok, I just checked the paper,<ref>http://www.polemarchakis.org/a16-cdf.pdf</ref> and it looks like this is exactly what the paper is about.

	==Hal Finney's example==		==Hal Finney's example==

IssaRice: /* Join and meet */

2018-08-27T18:16:49Z

Join and meet

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	* mind melding = join		* mind melding = join
	* common knowledge = meet		* common knowledge = meet

			In fact, [http://www.polemarchakis.org/a16-cdf.pdf] gives an example where directly exchanging the information leads to a different equilibrium than exchanging posteriors back and forth. So common knowledge and mind melding can't be the same thing.

	TODO aumann assumes the join of the partitions consists of nonempty events. it would be good to show where in his proof this assumption is used.		TODO aumann assumes the join of the partitions consists of nonempty events. it would be good to show where in his proof this assumption is used.