I think we are obligated to at least explain to the patient why it would be in their best interest to stay, and at least try and figure out if there is some complaint or problem that could be remedied. Pts need to be informed. How you feel about them leaving AMA shouldn't matter, as long as you're professional. We all know some folks are a real PIA to take care of. Just make sure that there isn't something there that can be easily resolved.
Sometimes people are just fed up & angry. You know how awful it is in the hospital! ;-) Who can blame them for being mad? I had a pt one time who was NPO for almost 3 days waiting for a procedure that kept getting bumped due to trauma pts getting priority. (Why couldn't they have told us that they couldn't get to him one day so we could feed him? I don't know.)
But I know what you mean. I have a few patients whom I tried my darnest to please, and they were very unhappy & wanted to go AMA. After a certain point, you have to realize that they were informed of the risks/benefits & move on I guess. If you have been professional, and tried to address their concerns...then you did what you could. For some patients, nothing you do is going to be good enough, they don't want to contribute or discuss it & for those, let them make their informed decisons...be it going AMA or whatever. They are adults, after all.