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

## TL;DR

This paper develops a novel mathematical framework and algorithm for fair cost allocation in ridesharing, utilizing cooperative game theory and the nucleolus concept to ensure equitable distribution among participants.

## Contribution

It introduces a new algorithm to efficiently compute the nucleolus in ridesharing games, addressing computational challenges and demonstrating effectiveness in autonomous vehicle systems.

## Key findings

- The algorithm computes the nucleolus with only 1.6% of coalition constraints.
- Approximate nucleolus closely matches the actual nucleolus when the core is empty.
- The method is effective for systems with non-empty cores, ensuring fairness.

## Abstract

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.