The question you ought to ask yourself is what information is getting conveyed by the 4th letter than isn't already mentioned in the first 3 letters?
Does anyone have any advice about whether or not it's a good idea to add a 4th referee for the universities that give you the option?
I know that Duke recommends that you don't unless you really feel it's necessary which is fair enough, but others like Chicago, Cornell and Brown give you the option of adding up to 5 referees and don't seem to recommend against it.
Is it likely to be an advantage to provide 4 referees for schools like Chicago etc, or is it definitely better to stick to 3 for all of them?
Certain university application forms let you choose 5 referees for the sake of flexibility. There are other departments/degrees where it's common to have letters from work experience or teaching.
In PhD economics, virtually every applicant only has 2-3 letters. You should assume Duke's attitude about 4th letters also applies to other departments mentioned, even if they don't state it explicitly.
If an applicant submits 4 letters, adcoms may (1) ask staff to filter out the least relevant letter; (2) throw away a letter randomly; (3) take the 4th letter into account.
Option (1) would be neutral for you, option (2) would be bad, and option (3) is likely only a small benefit, at best.
You'll also have to justify asking a referee to only write letters for 2-3 programs.
In short, I don't think most applicants should submit a 4th letter anywhere.
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