The question of whether the PhD applicant should disclose what other schools they are applying to has already been discussed here. There does not appear to be any decisive conclusion to this, however. The issue also infringes on the ethical issues of lying and bluffing.
Assume that the applicant wants to fill this question in the application. Given that most of the applications ask for 3-5 other institutions and some (most?) PhD students are applying to 10-15 departments that are spread out in terms of prestige, this leaves some degree of freedom. If the goal is to increase the chances of an offer, this creates an optimization problem:
- Applying to a top-level department, what other institutions should one list? (e.g. top-level or mediocre ones)
- Applying to a mediocre department, what other institutions should one list?
I assume that in the first case, it does not matter but I have no idea about the latter. Some have suggested that listing top-level institutions would lower the chances of admission. On the other hand, listing these prestigious institutions could create a good and positive association with the application (anchoring bias).