Consider two college roommates, one who smokes and one who does not. The smoker wishes to smoke in the room, and the nonsmoker dislikes smoking in the room. Suppose the smoker would be willing to pay $500 to be allowed to smoke in the room during the semester, and the nonsmoker would be willing to pay $600 to keep the room smoke-free. What should happen in the socially optimal outcome

The socially optimal outcome is that there will be no smoking in the room.

Explanation:

The above answer is  based on the willingness of the nonsmoker to ensure that no smoking happens in the room.  For instance, the nonsmoker can pay off the smoker, paying $600 to dissuade him from smoking in the room. On the contrary, the smoker is only willing to cough out $500, which is less than $600, in order to smoke in the room. Under social optimality, the nonsmoker wins this game.