  In designing dynamic shared service systems that incentivize customers to opt
for shared rather than exclusive service, the traditional notion of individual
rationality may be insufficient, as a customer's estimated utility could
fluctuate arbitrarily during their time in the shared system, as long as their
realized utility at service completion is not worse than that for exclusive
service. In this work, within a model that explicitly considers the
"inconvenience costs" incurred by customers due to sharing, we introduce the
notion of sequential individual rationality (SIR) that requires that the
disutility of existing customers is nonincreasing as the system state changes
due to new customer arrivals. Next, under SIR, we observe that cost sharing can
also be viewed as benefit sharing, which inspires a natural definition of
sequential fairness (SF) - the total incremental benefit due to a new customer
is shared among existing customers in proportion to the incremental
inconvenience suffered.
  We demonstrate the effectiveness of these notions by applying them to a
ridesharing system, where unexpected detours to pick up subsequent passengers
inconvenience the existing passengers. Imposing SIR and SF reveals interesting
and surprising results, including: (a) natural limits on the incremental
detours permissible, (b) exact characterization of "SIR-feasible" routes, which
boast sublinear upper and lower bounds on the fractional detours, (c) exact
characterization of sequentially fair cost sharing schemes, which includes a
strong requirement that passengers must compensate each other for the detour
inconveniences that they cause, and (d) new algorithmic problems related to and
motivated by SIR.
