I was the one who downvoted the parent, because you criticized Wei Dai's correct solution by arguing about a different problem than the one you agreed to several comments upstream

I didn't change the problem; I pointed out that he hadn't been appropriately representing the existing problem when trying to generalize it to partial information. Having previously agreed with his (incorrect) assumptions in no way obligates me to persist in my error, especially when the exchange makes it clear!

his solution to the original problem is unambiguously correct. (I suspect, but haven't checked, that a mixed strategy beats the pure one on your problem setup as well.)

Which original problem? (If the ABM problem as stated, then my solution is gives the same p=2/3 result. If it's the partial knowledge variant, Wei_Dei doesn't have an unambiguously correct solution when he fails to include the possibility of picking Y at X and X at Y like he did for the reverse.) Further, I do have the mixed strategy dominating -- but only up to r = 61%. Feel free to find an optimum where one of p and q is not 1 or 0 while r is greater than 61%.

Yes, you'd get a different solution if you assumed that the random variable for information gave independent readings at X and Y, instead of being engineered for maximum correlation.

That wasn't the reason for our different solutions.

I simply downvoted rather than commented, because (a) I was feeling tired and (b) your mistake seemed pretty clear to me.

Well, I hope you're no longer tired, and you can check my approach one more time.