Presumably, both players make their move without knowledge of the other player’s move, like in Paper Rock Scissors. In this case, a strategy isn’t just “Pick red” or the like; it’s a set of probabilities for each (such as “Red 70%, White 10%, Blue 20%”). If there were a single optimum color for you, then your opponent could figure out what it is, and always pick the color that has a negative expectation for you, but if you’re picking somewhat randomly, your opponent wouldn’t be able to do that.

The trick for finding the optimum set of probabilities is that usually, when you use the optimum set of probabilities, your expected payoff is the same no matter what your opponent does. Otherwise, your opponent’s optimal strategy would be to always pick the one that was best for him, and you would adjust your strategy to take advantage of his simple optimum.

For this particular case, let R, W, and B be the probabilities of you picking each of the choices, in your optimal strategy. In that case, R + W + B = 1, since those are all the choices, and the payoff for each of your opponent’s choices are the same: P[sub]R[/sub] = 0*R + 6*W + 2*B, P[sub]W[/sub] = -2*R + 4*W + 3*B, and P[sub]B[/sub] = 3*R -3*W -4*B, and P[sub]R[/sub] = P[sub]W[/sub] = P[sub]B[/sub] . We now have a system of four equations in four unknowns (the three probabilities and the payoff), which can be solved by any of the standard algebraic methods.