All this doctor can and should do is to recommend both to a transplant center to be evaluated for transplant. It's a very intense process consisting of a lot of medical testing, talking to psychiatrist, social worker, financial people, etc. The doctor would have no say over who gets a transplant. A board of doctors will go over each individual case and look at the results of all that testing and talking. They will then decide whether or not the patient can be put on the list.
Even if the person makes it this far and does get listed, the transplant doctors still have a lot of rules that are set up nationally under UNOS (United Network of Organ Sharing) that they must follow to determine who actually gets transplanted. The listed person is given a MELD (model for end stage liver disease) score that goes from 6-40. It is the results of several blood tests that shows how sick a person is with their liver disease. The person with the highest MELD score that is a match to the donor organ coming in will be called first. If there are more than one person waiting that has the same MELD score, it is then that the doctors are actually able to pick the one they feel needs the transplant most.
The drinker would not qualify for transplant until he proves that he has quit drinking and plan on staying that way. There will be extra rules he must follow to even begin to qualify. He must have at least 6 months of proven sobriety before they will do anything. Then if he manages to get through with that, he will have to pass his evaluation. Proven sobriety means he must go through a rehab or some type of program to find out why he drinks and how to live sober for the rest of his life. He must agree to submit to getting lab work done for alcohol/drug testing anytime the center feels like giving him one during the entire time he is under their care. If he ever tests positive for alcohol, they will immediately remove him from the list if he made it that far. He will have to submit to this testing all the way up to his transplant even if he is waiting years for one.
The single mother would have to pass the evaluation and wait like everyone else for her turn depending on her MELD score. If both the drinker and single mother are listed, there is no preference shown because he was a drinker. It might come into play if they are both called in for one organ and a decision has to be made as to who gets it if their MELD score are the same. The drinker would also have to quit smoking since they don't give transplants to smokers. If he is a life long smoker, he will probably have lung problems and may not qualify at all. Many people are turned down due to lung problems from smoking. They would have trouble surviving the surgery and be too great a risk.
Transplants are not done on a first come first serve basis. They are done on who is the sickest on the list. Someone might get their transplant a month after being accepted and someone else might have to wait many years depending on their MELD scores.