I just lost against myself. But on the other hand, I just won against myself. Fascinating.
Interesting question. Just a quick mistake in point 4 before tackling it. You forgot to mention that they also gain a win if they beat the person below them. In the statement it says they win if they draw and every other combination apart from a draw is a loss.
My usual Maths work: a few examples, thinking a bit about the results, relating the results to my current knowledge and trying to use knowledge I already have (not too much!).
I noticed the summation from 1 to n in the results here, used that formula, and then that's it.
Sorry for poor non-mathematical explanation. Hope I get a prize for my answer by the way.
Ah, but if your opponent is aware of all that, they will specifically pick the OTHER choice so as to confuse you.
Although if he is aware that you are aware that he is aware, then perhaps he will pick the choice that he should traditionally pick because you expect him not to pick it!
^^ dizzying intellect