TITLE:  I know that you know... 
QUESTION [28 upvotes]: A bit unsure if the following vague question has enough mathematical content to be suitable upon here. In the case, please feel free to close it.
In several circumstances of competition, a particular situation of partial information occurs, usually described as "I know that you know that I know... something". We may distinguish a whole hierarchy of more and more complicate situations closer and closer to a complete information. E.g. :

$I_0$: I know $X$, but you don't know that I know.
$I_1$: I know $X$, you know that I know, but I don't know that you know that I know.
$I_1$: I know $X$, you know that I know, I know that you know that I know, but you don't know that I know that you know that I know.
.... &c.   

For small values of $k$, I can imagine simple situations where passing from $I_k$ to $I_ {k+1}$ really makes a difference (for instance: you are Grandma Duck, and $X$ is : "you left a cherry pie to cool on the window ledge". Clearly, $I_0$ is quite agreeable position; $I_1$ may lead to an unpleasant end (for me); $I_2$ leaves me some hope, if I behave well, and so on). But, I can't imagine how passing from $I_6$ to $I_7$ may affect my strategy, or Grandma's. 

Are there situations, real or 
  factitious, concrete or abstract,  where $I_k$
  implies a different strategy than
  $I_{k+1}$ for the competitors? What
  about $I_{\omega}$ and, more generally,
  $I_\alpha$ for an  ordinal $\alpha$
  (suitably defined by induction)? How these situations are modeled mathematically?

REPLY [2 votes]: A good source for this sort of reasoning is:
J.-J. Meyer and W. van der Hoek, Epistemic Logic for AI and Computer Science, 
Number 41 in Cambridge Tracts in Theoretical Computer Science, Cambridge 
University Press, 1995.
The question of common knowledge, and of the algorithmics of epistemic modal logics, provides a lot of good material on this sort of problem.  (It is good fun to try the mathematical side of this in Popularisation activities.  I used Muddy Children with various groups with a lot of laughter and appreciation of the mathematical models.)