I think you can just compute the Nash Equilibria.

Things don't feel so simple to me. (A, X) is a Nash equilibrium (and the only pure strategy NE for this game), but is nonetheless unsatisfactory to me; if player 1 compares that pure strategy against the mixed strategy proposed by Wei_Dai, they'll choose to play Wei_Dai's strategy instead. Nash equilibrium doesn't seem to be a strong enough requirement ("solution concept") to force a plausible-looking solution. [Edit: oops, disregard this paragraph. I misinterpreted Wei_Dai's solution so switching to it from the NE pure equilibrium won't actually get player A a better payoff.]

(I also tried computing the mixed strategy NE by finding the player 1 move probabilities that maximized their expected return, but obtained a contradiction! Maybe I screwed up the maths.)