To add to this (probably silly) post - I was drawn to this problem for the reasons that Eliezer considers it the interesting case. In this form, it's a variant on the halting problem / completeness problem. Each of these can't be solved because some cases turn into an infinitely deep recursive nesting. Any purported algorithm to predict whether another algorithm will halt can't solve the case where it's analysing itself analysing itself analysing itself etc. Hence my rather tongue-in-cheek response.

And my response hits the same problem in a different form. When it meets itself, it gets into an infinite loop as each side tries to run the other. To solve it, at some level you need a 'cooperate' to be introduced in order to terminate the loop. And doing that breaks the symmetry of the situation.