Blimix

Problem 2 is relatively easy, IMO. Since the bank is paying all the bonuses, we want every play to result in the maximum $9.00 payout for a rock-breaks-scissors victory. The best way to ensure this is to play cooperatively, with each player alternately playing rock and scissors. Thus, if I start with rock and you play scissors, the next round I will play scissors and you will play rock. Assuming that both players are smart enough to figure this out, it should be easy to set up this cooperative (if boring) mode of play even if verbal communication isn't allowed - at worst, it would take a few rounds to get the rhythm going.