1. You are playing rock, paper, scissors with a friend. Each
time you win, your friend gives you a dollar. Each time you lose,
you pay your friend a dollar. Both of you play optimally with a
mixed strategy of playing rock, paper, and scissors exactly
one-third of the time each in a completely unpredictable way.
Suddenly, a third person enters the room and offers to pay your
friend (not you) 50 cents each time your friend throws a rock....