Meet of two partitions (in the context of this post) is the finest common coarsening of those partitions.
Consider the coarsening relation on the set of all partitions of the given set. Partition A is a coarsening of partition B if A can be obtained by “lumping together” some of the elements of B. Now, for this order, a “meet” of two partitions X and Y is a partition Z such that
Z is a coarsening of X, and it is a coarsening of Y
Z is the finest such partition, that is for any other Z’ that is a coarsening of both X and Y, Z’ is also a coarsening of Z.
Meet of two partitions is the finest common coarsening of those partitions.
Under the usages familiar to me, the common coarsening is the join, not the meet. That’s how “join” is used on the Wikipedia page for set partitions. Using “meet” to mean “common refinement” is the usage that makes sense to me in the context of the proof in the OP. [ETA: I’ve been corrected on this point; see below.]
Of course, what you call “meet” or “join” depends on which way you decide to direct the partial order on partitions. Unfortunately, it looks like both possibilities are floating around as conventions.
It is not difficult to see that the common knowledge accessibility function [...] corresponds to the finest common coarsening of the partitions [...], which is the finitary characterization of common knowledge also given by Aumann in the 1976 article.
The idea is that the partitions define what each agent is able to discern, so no refinement of what a given agent can discern is possible (unless you perform additional communication). Aumann’s agreement theorem is about a condition for when the agents already agree, without any additional discussion between them.
Hmm. Then I am in a state of confusion much like Psy-Kosh’s. These opposing convention aren’t helping, but, at any rate, I evidently need to study this more closely.
It was confusing for me too, which is why I gave an imperative definition: first form the union of I and J, then merge any overlapping elements. Did that not help?
It should have. The fault is certainly mine. I skimmed your definition too lightly because you were defining a technical term (“meet”) in a context (partitions) where I was already familiar with the term, but I hadn’t suspected that it had any other usages than the one I knew.
The term “meet” would correspond to considering a coarser partition as “less” than a finer partition, which is natural enough if you see partitions as representing “precision of knowledge”. The coarser partition is able to discern less. Greatest lower bound is usually called “meet”.
It’s always called that, but the greatest lower bound and the least upper bound switch places if you switch the direction of the partial order. And there’s a lot of literature on set partitions in which finer partitions are lower in the poset. (That’s the convention used in the Wikipedia page on set partitions.)
The justification for taking the meet to be a refinement is that refinements correspond to intersections of partition elements, and intersections are meets in the poset of sets. So the terminology carries over from the poset of sets to the poset of set partitions in a way that appeals to the mathematician’s aesthetic.
But I can see the justification for the opposite convention when you’re talking about precision of knowledge.
Ah, thanks. In that case… wouldn’t the meet of A and B often end up being the entire space?
For that matter, why this coarsening operation rather than the set of all the possible pairwise intersections between members of I and members of J?
ie, why coarsening instead if “fineing” (what’s the appropriate word there anyways?)
When two rationalists exchange information, shouldn’t their conclusions then sometimes be finer rather than coarser since they have, well, each gained information they didn’t have previously?
When two rationalists exchange all information, their new partition is the ‘join’ of the two old partitions, where the join is the “coarsest common fining”. If you plot omega as the rectangle with corners at (-1,-1) and (1,1) and the initial partitions are the x axis for agent A and the Y axis for agent B, then they share information and ‘join’ and then their common partition separates all 4 quadrants.
“common knowledge” is the set of questions that they can both answer before sharing information. This is the ‘meet’ which is the coarsest common fining. In the previous example, there is no information that they both share, so the meet becomes the whole quadrant.
If you extend omega down to y = −2 and modify the original partitions to both fence off this new piece on its own, then the join would be the original four squares plus this lower rectangle, while the meet would be the square from (-1,1) to (1,1) plus this lower rectangle (since they now have this as common knowledge).
wait, what? is it coarsest common fining or finest common coarsening that we’re interested in here?
And isn’t common knowledge the set of questions that not only they can both answer, but that they both know that both can answer, and both know that both know, etc etc etc?
Actually, maybe I need to reread this a bit more, but now am more confused.
Actually, on rereading, I think I’m starting to get the idea about meet and common knowledge (given that before exchanging info, they do know each other’s partitioning, but not which particular partition the other has observed to be the current one).
Meet of two partitions (in the context of this post) is the finest common coarsening of those partitions.
Consider the coarsening relation on the set of all partitions of the given set. Partition A is a coarsening of partition B if A can be obtained by “lumping together” some of the elements of B. Now, for this order, a “meet” of two partitions X and Y is a partition Z such that
Z is a coarsening of X, and it is a coarsening of Y
Z is the finest such partition, that is for any other Z’ that is a coarsening of both X and Y, Z’ is also a coarsening of Z.
Under the usages familiar to me, the common coarsening is the join, not the meet. That’s how “join” is used on the Wikipedia page for set partitions. Using “meet” to mean “common refinement” is the usage that makes sense to me in the context of the proof in the OP. [ETA: I’ve been corrected on this point; see below.]
Of course, what you call “meet” or “join” depends on which way you decide to direct the partial order on partitions. Unfortunately, it looks like both possibilities are floating around as conventions.
See for example on Wikipedia: Common knowledge (logic)
The idea is that the partitions define what each agent is able to discern, so no refinement of what a given agent can discern is possible (unless you perform additional communication). Aumann’s agreement theorem is about a condition for when the agents already agree, without any additional discussion between them.
Hmm. Then I am in a state of confusion much like Psy-Kosh’s. These opposing convention aren’t helping, but, at any rate, I evidently need to study this more closely.
It was confusing for me too, which is why I gave an imperative definition: first form the union of I and J, then merge any overlapping elements. Did that not help?
It should have. The fault is certainly mine. I skimmed your definition too lightly because you were defining a technical term (“meet”) in a context (partitions) where I was already familiar with the term, but I hadn’t suspected that it had any other usages than the one I knew.
The term “meet” would correspond to considering a coarser partition as “less” than a finer partition, which is natural enough if you see partitions as representing “precision of knowledge”. The coarser partition is able to discern less. Greatest lower bound is usually called “meet”.
It’s always called that, but the greatest lower bound and the least upper bound switch places if you switch the direction of the partial order. And there’s a lot of literature on set partitions in which finer partitions are lower in the poset. (That’s the convention used in the Wikipedia page on set partitions.)
The justification for taking the meet to be a refinement is that refinements correspond to intersections of partition elements, and intersections are meets in the poset of sets. So the terminology carries over from the poset of sets to the poset of set partitions in a way that appeals to the mathematician’s aesthetic.
But I can see the justification for the opposite convention when you’re talking about precision of knowledge.
Ah, thanks. In that case… wouldn’t the meet of A and B often end up being the entire space?
For that matter, why this coarsening operation rather than the set of all the possible pairwise intersections between members of I and members of J?
ie, why coarsening instead if “fineing” (what’s the appropriate word there anyways?)
When two rationalists exchange information, shouldn’t their conclusions then sometimes be finer rather than coarser since they have, well, each gained information they didn’t have previously?
If I’ve got this right...
When two rationalists exchange all information, their new partition is the ‘join’ of the two old partitions, where the join is the “coarsest common fining”. If you plot omega as the rectangle with corners at (-1,-1) and (1,1) and the initial partitions are the x axis for agent A and the Y axis for agent B, then they share information and ‘join’ and then their common partition separates all 4 quadrants.
“common knowledge” is the set of questions that they can both answer before sharing information. This is the ‘meet’ which is the coarsest common fining. In the previous example, there is no information that they both share, so the meet becomes the whole quadrant.
If you extend omega down to y = −2 and modify the original partitions to both fence off this new piece on its own, then the join would be the original four squares plus this lower rectangle, while the meet would be the square from (-1,1) to (1,1) plus this lower rectangle (since they now have this as common knowledge).
Does this help?
wait, what? is it coarsest common fining or finest common coarsening that we’re interested in here?
And isn’t common knowledge the set of questions that not only they can both answer, but that they both know that both can answer, and both know that both know, etc etc etc?
Actually, maybe I need to reread this a bit more, but now am more confused.
Actually, on rereading, I think I’m starting to get the idea about meet and common knowledge (given that before exchanging info, they do know each other’s partitioning, but not which particular partition the other has observed to be the current one).
Thanks!