[Paper Review] The Costs and Benefits of Sharing: Sequential Individual Rationality and Sequential Fairness

Ragavendran Gopalakrishnan, Koyel Mukherjee+1 more|arXiv (Cornell University)|Jul 25, 2016

Transportation and Mobility Innovations4 citations

TL;DR

This paper introduces sequential individual rationality (SIR) and sequential fairness (SF) to ensure dynamic fairness in shared service systems like ridesharing, where passengers' disutility must not increase due to new arrivals. It proves that SIR-feasible routes are equivalent to Hamiltonian paths, and SF requires passengers to compensate each other for detour-induced inconvenience, leading to NP-hard optimization problems with sublinear detour bounds.

ABSTRACT

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.

Motivation & Objective

• To address dynamic fairness in shared services where individual utility can fluctuate due to new arrivals.
• To formalize sequential individual rationality (SIR), ensuring existing customers' disutility does not increase upon new arrivals.
• To define sequential fairness (SF), where benefits from new customers are shared proportionally to the inconvenience caused.
• To apply these concepts to ridesharing, modeling detours as inconvenience costs.
• To characterize algorithmically the set of SIR-feasible and SF-compliant routes and cost-sharing schemes.

Proposed method

• Introduces a disutility model combining monetary cost and inconvenience cost from shared usage.
• Defines sequential individual rationality (SIR) as a constraint requiring disutility to be nonincreasing with each new arrival.
• Proposes sequential fairness (SF) as a benefit-sharing mechanism proportional to incremental inconvenience caused by new customers.
• Reduces the SIR-feasibility problem to finding Hamiltonian paths in a graph, establishing NP-hardness.
• Constructs a metric-based reduction from Metric-Path-TSP to Opt-SIR-Route, proving NP-hardness of optimal SIR-compliant route selection.
• Uses integral flow decomposition to model path-disjoint routing in network reductions, linking to minimum-cost flow problems.

Experimental results

Research questions

• RQ1 What conditions ensure that existing passengers in a ridesharing system do not experience increasing disutility due to new passenger arrivals?
• RQ2 How can fairness be enforced not just at the end of service, but dynamically throughout the service duration?
• RQ3 What are the structural and algorithmic properties of routes that satisfy sequential individual rationality (SIR)?
• RQ4 Can sequential fairness be formalized such that the benefit from a new passenger is shared proportionally to the inconvenience they cause?
• RQ5 What is the computational complexity of finding optimal SIR-compliant and SF-compliant routes in ridesharing systems?

Key findings

• SIR-feasible routes in ridesharing correspond exactly to Hamiltonian paths in an underlying graph, establishing a one-to-one correspondence.
• The fractional detour for any SIR-feasible route is bounded between sublinear lower and upper bounds, specifically O(1/j) for the j-th leg.
• Sequential fairness requires that passengers compensate each other for detour-induced inconvenience, leading to a strong interdependence in cost-sharing schemes.
• The problem of finding an optimal SIR-compliant route is NP-hard, as it reduces to the Metric-Path-TSP problem.
• All n! possible routes in a constructed instance of Opt-SIR-Route can be SIR-feasible under appropriate distance and cost settings, highlighting structural flexibility.
• The reduction from Hamiltonian path to SIR-Feasibility confirms that determining SIR-feasibility is NP-complete.