Fair cost allocation for ridesharing services - modeling, mathematical programming and an algorithm to find the nucleolus

This paper addresses one of the most challenging issues in designing an efficient and sustainable ridesharing service: ridesharing market design. We formulate it as a fair cost allocation problem through the lens of the cooperative game theory. A special property of the cooperative ridesharing game is that its characteristic function values are calculated by solving an optimization problem. Several concepts of fairness are investigated and special attention is paid to a solution concept named nucleolus, which aims to minimize the maximum dissatisfaction in the system. Due to its computational intractability, we break the problem into a master-subproblem structure and two subproblems are developed to generate constraints for the master problem. We propose a coalition generation procedure to find the nucleolus and approximate nucleolus of the game. Experimental results showed that when the game has a non-empty core, in the approximate nucleolus scheme the coalitions are computed only when it is necessary and the approximate procedure produces the actual nucleolus. And when the game has an empty core, the approximate nucleolus is close to the actual one. Regardless of the emptiness of the game, our algorithm needs to generate only a small fraction (1.6%) of the total coalition constraints to compute the approximate nucleolus. The proposed model and results nicely fit systems operated by autonomous vehicles.

💡 Research Summary

The paper tackles the problem of allocating the total travel cost generated by a ridesharing service in a way that is perceived as fair by all participants. The authors model the situation as a cooperative game in which each subset (coalition) of riders has an associated cost equal to the minimum‑cost route that serves exactly those riders while respecting pickup‑and‑delivery precedence constraints. This route‑finding subproblem is a Traveling Salesman Problem with Pickup‑and‑Delivery (TSP‑PD), which is NP‑hard; consequently, the characteristic function of the game cannot be pre‑computed for all 2ⁿ coalitions.

Two solution concepts from cooperative game theory are examined: the core and the nucleolus. The core consists of cost‑allocation vectors that (i) exactly cover the total system cost and (ii) give every coalition a cost no larger than its stand‑alone optimal cost. When the core is non‑empty, any allocation inside it guarantees that no group of riders has an incentive to deviate. The nucleolus refines this idea by minimizing the lexicographically ordered vector of excesses (the difference between a coalition’s stand‑alone cost and the sum of its members’ allocated payments). In other words, it seeks the allocation that makes the most dissatisfied coalition as satisfied as possible, thereby providing the strongest notion of fairness and stability, even when the core is empty.

Because directly enumerating all coalitions is infeasible, the authors design a master‑subproblem algorithm with constraint generation. The master problem is a linear program that contains only those coalition constraints (core defining inequalities, CDIs) that have been identified so far. Two subproblems are solved iteratively:

1. Non‑profitable or infeasible coalition detection – using the propositions that any coalition whose stand‑alone cost exceeds the sum of its proper sub‑coalitions (non‑profitable) or that violates vehicle capacity (infeasible) cannot define a binding core constraint. These coalitions are therefore ignored, dramatically reducing the number of constraints that must ever be considered.

2. Maximum‑excess coalition identification – given the current allocation from the master LP, an integer program searches for the coalition with the largest excess (i.e., the most dissatisfied group). If this excess is negative, the allocation already lies in the core; otherwise, the identified coalition is added to the master LP as a new constraint, and the LP is re‑solved.

The process repeats until the excess vector cannot be improved, at which point the allocation is the nucleolus (or an approximation thereof). The algorithm therefore only generates a tiny fraction of the 2ⁿ possible constraints. Empirical tests on randomly generated instances with 8–12 riders show that on average only 1.6 % of all possible coalitions need to be created before convergence, and the total computation time is on the order of a few hundredths of a second. When the core is non‑empty, the procedure often discovers the exact nucleolus; when the core is empty, the obtained “approximate nucleolus” is shown to be very close to the true nucleolus.

The paper also proves that the characteristic function of the ridesharing game is monotone and sub‑additive, guaranteeing that larger coalitions never increase per‑rider cost and that forming a coalition is beneficial whenever feasible. A proposition is given that any optimal solution to the ridesharing routing problem must consist solely of profitable routes, allowing the column‑generation formulation of the routing problem to discard non‑profitable columns a priori.

Overall contributions are:

• A novel cooperative‑game formulation of ridesharing cost allocation where each coalition’s cost is defined via an embedded TSP‑PD.
• Theoretical results that prune unnecessary coalition constraints (non‑profitable and infeasible coalitions) from the core definition.
• A practical master‑subproblem algorithm that computes the nucleolus (or a high‑quality approximation) by generating only a minuscule subset of coalition constraints.
• Extensive computational experiments demonstrating scalability and near‑real‑time performance, suggesting applicability to autonomous‑vehicle fleets that must re‑allocate costs dynamically.

Future work suggested includes extending the model to dynamic, time‑varying rider arrivals, heterogeneous vehicle capacities, stochastic travel times, and integrating non‑linear cost components such as congestion pricing. The methodology could also be adapted to other shared‑resource systems (e.g., bike‑sharing, electric‑vehicle charging) where fair cost allocation is critical for long‑term system stability.