Suppose we play Rock, Paper, Scissors but the outcome isn't deterministic and guaranteed. If you play Rock and I play Scissors, you will only beat me with some probability in $[0,1]$. Likewise for the other non-mirror possibilities. A mirrored outcome (e.g. both players play Rock) results in a draw.
What is the optimal strategy for playing this variant of the game?
Secondly, instead suppose that the mirrored outcomes (Rock plays against Rock) will result in a win for one player with a probability in $[0,1]$, exactly like the non-mirrored outcomes. Draws don't exist in this variation.
What is the optimal strategy for playing this variant of the game?
In both variations, it occurs to me that the win probabilities for a given player can be represented as a 3x3 matrix, with the diagonal as 0.5 in the first variant (even though a draw isn't a win for either player, but I think this should work), and diagonals in [0, 1] in the second variant.
In both variants, assume information symmetry and agreement on the probabilities between both players.
Not homework. The application is Starcraft meta-gaming in a context where the race pick isn't known up front and both players are strong at all the races.
These are my thoughts on a possible computational approach. Barring empirical data on actual tendencies, we could start with an assumption of your opponent's tendencies (say, [1, 0, 0], or [0.5, 0.4, 0.1]), and then find the optimal frequencies ([Rock %, Paper %, Scissors%]) to beat it via optimization. Iterate - keep finding the optimal frequencies to beat those from your last optimization step until you converge. I have no idea if this would converge on non-exploitable frequencies (I believe not), but it's the best I can think of right now. It should converge on a non-exploitable strategy in the trivial case where one option always dominates (e.g. Rock beats everything, we should converge to [1, 0, 0] frequencies), which is one nice property of this method. Another nice property of this method is that we should converge to [1/3, 1/3, 1/3] in the traditional Rock, Paper, Scissors game.