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?