The Costs and Benefits of Ridesharing: Sequential Individual Rationality and Sequential Fairness.

We introduce a cost sharing framework for ridesharing that explicitly takes into account the of passengers due to detours. We introduce the notion of sequential individual rationality (SIR) that requires that the disutility of existing passengers is non-increasing as additional passengers are picked up, and show that these constraints induce a natural limit on the incremental detours permissible as the ride progresses. We provide an exact characterization of all routes for which there exists some cost sharing scheme that is SIR on that route, and under these constraints, for realistic scenarios, we also show a $Theta(sqrt{n})$ upper bound and a $Theta(log n)$ lower bound on the total detour experienced by the passengers as a fraction of the direct distance to their destination. Next, we observe that under any budget-balanced cost sharing scheme that is SIR on a route, the total amount by which the passengersu0027 disutilities decrease (which can be viewed as the total incremental benefit due to ridesharing) is a constant. This observation inspires a dual notion of viewing cost sharing schemes as benefit sharing schemes, under which we introduce a natural definition of sequential fairness-the total incremental benefit due to the addition of a new passenger is (partly) shared among the existing passengers in proportion to the incremental inconvenience costs they suffer. We then provide an exact characterization of sequentially fair cost sharing schemes, which brings out several useful structural properties, including a strong requirement that passengers must compensate each other for the detour inconveniences that they cause. Finally, we conclude with an extended discussion of new algorithmic problems related to and motivated by SIR, and important future work.