Every day one of the children gets to be Apple of the Day. We borrowed this from our friends at the Brooklyn Apple Academy.
The Apple acts as a tie-breaker and takes on some grown-up responsibilities like leading the way on field trips and blowing out the candle after circle time.
The system is known as rotating chairmanship and it is part of many democratic institutions. For example, the Presidency of the European Council is a rotating chairmanship.
Now, whenever there's a conflict (who gets to press the elevator button?) the children have an obvious way to resolve it without needing grown-up referees.