... Let's do that!

Simple model: A plays A, B and C with probabilities a, b, and c, with the constraint that each must be above the trembling probability t (=p/3 using the p above). (Two doesn't tremble for simplicity's sake)

Two picks X with probability x and Y with probability (1-x).

So their expected utilities are:

One: 3a + 2b+6c(1-x)

Two: 2b(1-x) + cx = 2*b + (c - 2b) x

It seems pretty clear that One wants b to be as low as possible (either a or c will always be better), so we can set b=t.

So One's utility is (constant) - 3c+6c -6cx

So One wants c to maximize (1-2x)c, and Two wants x to maximize (c-2t)c

The Nash equilibrium is at 1-2x=0 and c-2t=0, so c=2t and x=0.5

So in other words, if One's hand can tremble than he should also sometimes deliberately pick C to make it twice as likely as B, and Two should flip a coin.

(and as t converges towards 0, we do indeed get One always picking A)