The Criterion
If the driver were not a member of the group, but a hired driver, then the meaning of fairness would be clear: The professional driver charges a fixed price for every ride, and each day the people that show up split the price of the driver equally among them. When the driver is just one of the set of people that show up, this reasoning leads immediately to the following: If on a certain day, d people show up, each of them owes the driver 1/d of a ride. Therefore the desired number of times a participant should drive is the sum of the inverses of the number of people who showed up on the days the participant took part in the carpool.
Having defined the fair load (or share) on a participant, we should now ask whether all players can achieve this goal simultaneously. Assume for simplicity that the fair load on each player is an integer and suppose that we are given ahead of time for each day the set of people that will show up on that day. Is there a schedule such that every participant drives his or her fair share of the rides?
The answer and more scheduling questions