The deputy (or Cap'n as I like to sometimes call him/her) is a member of the class who gets a "WINNER" medal and gets as many special privileges as you are comfortable granting. I give my deputy a wide leash to leave the room whenever he/she wants, settle disputes, and make important decisions in the classroom. The point is to give the office enough power to make it desirable.
The students come up with non-market ways in which we could allocate this very scarce resource and then vote as a class to see who should be selected. Foot races, spelling bees, rock/paper/scissor tourneys, raffles, and trasketball have all given me deputies over the years. Once the deputy is picked, we launch into a discussion of real world examples of non-market allocation. Kids learn econ and I acquire a big time ally that the students themselves had a hand in picking. I believe Michael Scott would call that a win-win-win situation.