so some interesting results:
if you know your opponent will always attack with a certain constant probability on each of their turns, no matter what that probability is, you should:
* hold on your first turn if you're A
* attack on every other turn
(implicit here is that both players will attack if they've attacked on the previous turn. I've always been assuming that ftr. maybe I should simplify the setup somehow so that's not necessary)
but, critically, if you're P2 and you know that's P1's strategy, you should now hold on your first turn and attack every other turn
but, if P1 knows that's P2's strategy, P1 should attack on every turn
and if P2 knows that's P1's strategy, P2 should attack on every turn
and if P1 knows that's P2's strategy, then they should go back to the first strategy in this message