common knowledge and concept of coarsening partition

Here is a proof of the equivalence between my definition and Aumann's for "common knowledge". I'm assuming some familiarity with set partitions. Aumann's definition is in terms of the Kolmogorov model of probability. In particular, a proposition is identified with the set of possible worlds in which the proposition is true. Let P₁ be that partition of the possible worlds such that two worlds share the same block in P₁ if and only if I condition on the same body of knowledge when computing posterior probabilities in the two worlds. Let P₂ be the analogous partition of the possible worlds for you. For each world w, let P₁(w) denote the block in my partition containing w, and let P₂(w) be the block in your partition containing w. Let P denote the finest common coarsening of our respective partitions**, and let P(w) be the block of P containing w. (from http://lesswrong.com/r/discussion/lw/6je/an_explanation_of_aumanns_agreement_theorem/4hc5)

So, what exactly is common coarsening of partitions? And here what does it mean? Is it just union of two partition maps?

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2 Answers

Fix a base set $X$ and let $P$ and $Q$ be partitions of $X$.

$P$ is finer than $Q$ (or equivalently, $Q$ is coarser than $P$) if every set in $P$ is a subset of a set in $Q$. The partition where all the subsets have size 1 is the finest possible partition. The partition where the entire original set lies in only one set is the coarsest possible partition.

The finest common coarsening of $P$ and $Q$ is the finest partition $R$ such that $R$ is coarser than both $P$ and $Q$. (There is a dual notion, called the coarsest common refinement, which is the coarsest $R$ that is finer than both $P$ and $Q$.)

For example, if $$X = \{1,2,3,4,5,6,7\}$$ $$P = \{\{1,3,4\}, \{2,5\}, \{6,7\}\}$$ $$Q = \{\{1,2,5\},\{3,4\}, \{6\}, \{7\}\}$$ then the finest common coarsening of $P$ and $Q$ is $\{\{1,2,3,4,5\},\{6,7\}\}$ while the coarsest common refinement of $P$ and $Q$ is $\{\{1\}, \{2,5\}, \{3,4\}, \{6\}, \{7\}\}$.

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Ted's answer on common coarsening is correct. I'll try to brief Aumann's agreement theorem which is basically an application of his original formulation of common knowledge in terms of partitions(compared with David Lewis).

On one hand, Aumann(1976) assumed a common prior for all possible worlds, which assigns probabilities to all events conceivable. Moreover,common prior is known by two players, and the event they know the common prior is also known by them, and the event they know they know …… ad infinitum. On the other hand, two players' difference of knowledge are modeled in the same fashion as two people at different positions seeing the whole picture from afar, i.e. some states of world can not be distinguishable from others, while some others can. When one particular state of world, as a full specification of events of interest obtain, people receive some signal which can be utilized to define an equivalence relation on states of world. That's where partitions come from.

Noticeably, each players' partition are also common knowledge, even though it's not necessarily known by his opponent which atom that contains the true state. Events that are common knowledge are those contains at least one atom of common coarsening of their partitions. The point is, even though his opponent may not know which atom contains the true state, at least he know the candidate atoms are those that can derive the same posterior probability and those in the one atom of common coarsening of their partitions. In fact, the latter is a subset of the former, since the posterior is common knowledge as assumed. Thus posterior probabilities derived by two players' partition must be the same as that derived by common coarsening of their partitions, which shows that two players posterior probabilities must coincide.

Added:Why not have a look at Aumann's classic, and Geanakoplos' survay among others?

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