Player Two plays X with probability 2/3 and Y with probability 1/3.

Should that be the other way round?

As written, player 1 expects to score 3(1-ϵ) + 2(ϵ/3) + (2ϵ/3)(6/3) = 3 - 3ϵ + 2ϵ/3 + 4ϵ/3 = 3-ϵ (assuming I haven't made a dumb error there), and so would do better by unilaterally switching to the pure strategy A.

But if player 2 plays Y with probability 2/3 and X with probability 1/3, player 1 can expect to score 3(1-ϵ) + 2(ϵ/3) + (2ϵ/3)(12/3) = 3 - 3ϵ + 2ϵ/3 + 8ϵ/3 = 3 + ϵ/3, which beats the pure strategy A.

Edit: no, ignore me, I forgot that the whole point of proper equilibrium is that ϵ is an arbitrary parameter imposed from outside and assumed nonzero. Player 1 isn't allowed to set it to zero. [Slaps self on forehead.]