Question

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. What is your friend’s optimal strategy now? What is your optimal strategy?

Answer 1

The payoff table for the game:

		FRIEND
		Rock	Paper	Scissors
ME	Rock	0, 0	-1, 1	1, -1
Paper	1, -1	0, 0	-1, 1
Scissors	-1, 1	1, -1	0, 0

As we see, this is a zero sum game. Here, the strategy is to play rock, paper or scissors with probbility 1/3 for each of the players.  

When a third perosn enters, Friend gets $0.50 each time he throws Rock. So, the new payoff table will be:

		FRIEND
		Rock	Paper	Scissors
ME	Rock	0, 0.50	-1, 1	1, -1
Paper	1, -0.50	0, 0	-1, 1
Scissors	-1, 1.50	1, -1	0, 0

Now my Frend's optimal strategy is to play Rock. He will get a higher payoff against whatever I play compared to earlier. And if I play Scissors when he plays Rock, his payoff is optimal.  My optimal strategy is to play Paper. Because I know Friend will play Rock and my optimal strategy will be Paper against his Rock.