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What's the relationship among universal type space, Aumann's semantic knowledge model and Samet's syntactic knowledge model?

Here's my confusion:

Regarding universal type space and the model in Aumann's Agreeing to disagree, I used to think they're really the same thing from two different angles. The type space construction is a bottom-up construction. It starts with all the payoff relevant parameters from players' point of view, and generates a set of states of world in the end. Aumann's model is top-down. The set of states of world and players' knowledge as partitions are exogenous. Interesting results are derived by imposing some consistency rules.

It seems to me, there's a one-to-one correspondence between a player's type in the former and a cell( or atom, block) in the selfsame player's partition in the latter, because a player's strategy has to be measurable with respect to them in these settings.

But after having a skimming of an unpublished paper by Robert Simon,The Common Prior Assumption in Belief Spaces: An Example, I found it's not the case.

In that paper, he actually imposed a partition on Merten and Zamir's type space(he claimed so) for each player, so there's no such correspondence.

Another thing that is weird is that it seems to me he's actually not working on on a belief space. It looks like he's working on a model in Dov Samet's Ignoring ignorance and Agreeing to Disagree, which is homeomorphic to a Cantor set, a set which is generated by assigning truth values on underlying propositions and the use of non-repeated knowledge operators of the same player. It seems to me Dov Samet's model is not homeomorphic to Merten and Zamir's type space, because we might need to add operators of belief from probability 0 to probability 1 in Dov Samet's model to do that, and the generated set of state of state of world will have a cardinality strictly larger than a Cantor set.

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There is no "universal Aumann model", as shown in Heifetz & Samet, GEB 1998, "Knowledge Spaces with Arbitrarily High Rank ", even though there is a universal type space.

On a less technical level, Aumanns model does not allow for wrong beliefs. A generalization, so called Kripke structures, do however.

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  • $\begingroup$ Thank you for your answer. So what about Samet's model? $\endgroup$ – Metta World Peace May 11 '15 at 11:25
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    $\begingroup$ I haven't read that paper carefully, but one can obtain universal spaces in an Aumann type of model if one imposes measurability restrictions on events. This is shown in Meier 2008, "Universal knowledge–belief structures", GEB. The resulting model is not equivalent to the Mertens-Zamir type space though. Universal type spaces always have a product structure, but this is usually not compatible with knowledge, if knowledge is taken to imply truth. $\endgroup$ – Michael Greinecker May 11 '15 at 11:39

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