I agree with the above that psychology may be a better approach vs psychiatry in this matter.
Psychiatry focuses on treating mental illness through a medical model. Psychology focuses more on the human mind as a whole, and has more educational aspects within in devoted to attaching values to the reward & aversion aspects someone may face.
Is it something that doctors use to treat patients? If not, it's never going to be focused as a fellowship.
I have seen psychologists use it. Where I worked last year (and this year but as a fellow), specific patients who were not following the rules were sometimes given a behavioral management plan. In that plan, a psychologist wrote down a series of rewards someone could experience if they followed the rules, and this was based on the specific person's needs.
This of course brings in other problems. E.g. if you give a behaviorally problematic patient internet access as a reward for following the rules, what about those patients who followed the rules since the beginning? Do they also get internet access (no they don't.)
Punishing patients is considered illegal in Ohio-which IMHO actually was not a good thing. When I say "punishment" I only mean methods which schools provide such as detention, a "time out" where a patient would be restricted to their room a few hours a day, etc. Several patients I felt could've benefited from a "time out" were they could sit in their room for a few hours for bad behavior. In fact that is more humane IMHO than forced medication, especially when several of the patients who acted out did not so because they were manic, or psychotic, but because they were just being immature. Nope--you could only give rewards, which were not given to patients who followed the rules from the beginning. For that reason, the patients in this facility who will be here for years, some of then intentionally do bad things so they can get the rewards. I know of one patient who every few months intentionally stirs things up so a psychologist will make a plan that gives him rewards. He gets the rewards for a few months, then when he's about to lose them because he's considered a behavioral management success--he stirs things up again.
I have never seen anyone attach specific units of currency to specific people.
A psychiatrist can do the above, just that all the psychiatrists I work with would rather have the psychologist do it. I have also unfortunately seen plenty of psychiatrists who pretty much forget everything outside of psychopharmacology. Such a psychiatrist would not be well equiped to make a behavioral management plan. (Point of info: I am not advocating that psychiatrists limit themselves in this manner, just that this is what several on their own choose to do). When one runs a psychiatrist institution with several psychologists & psychiatrists, its easier from a management perspective to only make the psychologists do this type of work.