Use New York Lotto Quick Draw (like Keno). Schedule the lotto to take place on a certain day (like the day after we're full). Before the lotto day, assign everyone a time slot between 00:00 and 23:56. Draws occur every 4 minutes. Lowest combined total gets the top pick, second lowest gets second pick, etc. Ties? Broken by 1-2-3 beats 1-2-4, etc.