Sure, you can break symmetry (in this case making the decision makers not identical because they have different random number generators available), but the remaining symmetry means identical mixes must be chosen, and so a mixed strategy would only be chosen if it maximize his value for both people cooperatively.

Maybe it's a bit subtle that they said clones and I said identical decision makers; I'm letting you fill in the gap for how much clones may diverge and how much that matters.