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? 
> 
> 
> 



 I know that you know... 
My wife and I have a standing agreement where I pick up our son Horatio from school and she picks up our daughter Hypatia. 


One day, because I knew I would be near Hypatia's school, it was convenient to swap duties. I emailed her a message, "I'll pick up Hypatia today, and you get Horatio. Please confirm; otherwise it is as usual." She texted me back, "Let's do it. Let me know if you get this message, so that I know we're really on." I left her a voicemail, "OK, we're definitely on for the swap! ....as long as I know you get this message." She emailed me back, "Got the message. We're on! But let me know that you get this message so I can count on you." You see, without confirmation she couldn't be sure that I knew she had gotten my earlier confirmation of her acknowledgement of my first message, and she may have worried that the plan to swap was consequently off. 


And so on *ad infinitum*......


How truly frustrating it was for us that at no stage of our conversation could we seem to know for certain that the other person had all the necessary information to ensure that the plan would be implemented! The result, of course, since we had time to exchange only at most a finite number of messages, was that the only rational course of action was for us each to abandon the plan to swap: we both independently decided just to pick up the usual child. 


To see that this was rational, observe that clearly the first message needed to have been confirmed in order for the plan to be implemented properly. Furthermore, if the $n$-th message need not have been confirmed, then it wasn't important to know that it had been received and the algorithm should have worked whether or not it was received, meaning that it didn't actually need to have been sent. So by induction, no number of confirmations suffices to implement the common knowledge that we both needed, namely, that we had each agreed to make the swap.


See also the [two generals problem](http://en.wikipedia.org/wiki/Two_Generals%27_Problem). 


