# 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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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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