I don’t think that resolves the issue. As I demonstrated in this comment, if you have some probabilistic knowledge of which intersection you’re at, you can do better than the p=2/3 method. Specifically, as long as you have 0.0012 bits of information about which intersection you’re at (i.e. assign a greater than 52% chance of guessing correctly), you’re better off choosing based on what seems most likely.
However—and this is the kicker—that means that if you have between 0 and 0.0012 bits of information about your intersection, you’re best off throwing that information away entirely and going with the method that’s optimal for when you’re fully forgetful. So it’s still a case where throwing away information helps you.
ETA: False alarm; Wei_Dai corrects me here—you can still use your knowledge to do better than 4⁄3 when your probability of guessing right is between 50% and 52.05%.
I don’t think that resolves the issue. As I demonstrated in this comment, if you have some probabilistic knowledge of which intersection you’re at, you can do better than the p=2/3 method. Specifically, as long as you have 0.0012 bits of information about which intersection you’re at (i.e. assign a greater than 52% chance of guessing correctly), you’re better off choosing based on what seems most likely.
However—and this is the kicker—that means that if you have between 0 and 0.0012 bits of information about your intersection, you’re best off throwing that information away entirely and going with the method that’s optimal for when you’re fully forgetful. So it’s still a case where throwing away information helps you.
ETA: False alarm; Wei_Dai corrects me here—you can still use your knowledge to do better than 4⁄3 when your probability of guessing right is between 50% and 52.05%.