Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.
If you win with Rock, you get \$9. Your opponent loses the \$9.
If you win with Paper, you get \$3. Your opponent loses the \$3.
If you win with Scissors, you get \$5. Your opponent loses the \$5.
If you tie, you get $0
My first intuition would be that you should play Rock with a probability of
9/(9+3+5), Paper with
3/(9+3+5)and Scissors with
5/(9+3+5)however this seems wrong, as it doesn't take into consideration the risk you expose yourself to (if you play
Paper, you have an upside of \$3 but a downside of \$5).
So I put the question to you, in such a game -- what is the ideal strategy.
Edit: By "ideal" strategy, I mean playing against an adversarial player who knows your strategy.