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Suppose two perfect game theorists play rock paper scissors. Scoring is slightly different:
- 2 points when beating scissors with rock
- 1 point when beating rock with paper
- 1 point when beating paper with scissors
In case of a tie or loss, 0 points are awarded.
Assuming both players play an infinite number of games and both play optimally, what random distribution would give the highest expected score per game?
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5
The expected value of each throw is EV=2P(R)Q(S) + P(P)Q(R) + P(S)Q(P) where P(x) is the probability that we choose x as our throw and Q(x) is the probability our opponent chooses x as their throw. I take our opponent playing optimally to mean that they will be able to exploit our strategy to maximize their EV, so what we’d really like is that the expected value of each of our throws to be the same regardless of what our opponent throws.
This is done by setting 2P(R) = P(P) = P(S), or a 20% chance of throwing rock, and a 40% chance of throwing each of paper and scissors. This simplifies our EV equation to EV = .4(Q(S)+Q(R)+Q(P)) = .4
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