I made a mistake again. As described above, complete only pseudodominates incomplete.
But this is easily patched with the trick described in the OP. So we need the choice complete to make two changes to the downstream decisions. First, change decision 1 to always choose up (as before), second, change the distribution of Decision 2 to {1−q(1−p), q(1−p)}, because this keeps the probability of B constant. Fixed diagram:
Now the lottery for complete is {B: q(1−p), A+: 1−q(1−p), A:0}, and the lottery for incomplete is {B: q(1−p), A+: pq, A:(1−q)}. So overall, there is a pure shift of probability from A to A+. [Edit 23/7: hilariously, I still had the probabilities wrong, so fixed them, again].
I made a mistake again. As described above, complete only pseudodominates incomplete.
But this is easily patched with the trick described in the OP. So we need the choice complete to make two changes to the downstream decisions. First, change decision 1 to always choose up (as before), second, change the distribution of Decision 2 to {1−q(1−p), q(1−p)}, because this keeps the probability of B constant. Fixed diagram:
Now the lottery for complete is {B: q(1−p), A+: 1−q(1−p), A:0}, and the lottery for incomplete is {B: q(1−p), A+: pq, A:(1−q)}. So overall, there is a pure shift of probability from A to A+.
[Edit 23/7: hilariously, I still had the probabilities wrong, so fixed them, again].