# Stable Matching of Users in a Ridesharing Model  
[PERSON]  
1 Department of Systems and Computation, Tecnologico Nacional de Mexico/Instituto Tecnologico de Cd. Guzman, Ciudad Guzman 49100, Mexico; [EMAIL_ADDRESS] (D.F.-D.); [EMAIL_ADDRESS] (C.H.-B. ); [EMAIL_ADDRESS] (J.E.M.-S.)  
[PERSON]  
1 Department of Systems and Computation, Tecnologico Nacional de Mexico/Instituto Tecnologico de Cd. Guzman, Ciudad Guzman 49100, Mexico; [EMAIL_ADDRESS] (D.F.-D.); [EMAIL_ADDRESS] (C.H.-B. ); [EMAIL_ADDRESS] (J.E.M.-S.)  
[PERSON]  
1  
[PERSON]  
1  
[PERSON]. +++ ==WARNING: Truncated because of repetitions==  
++  
and drivers) seeking to share a trip implies defining a relationship between them. Establishing this relationship of users based on their preferences is known as the ride-matching problem [15, 16]. Current ridesharing systems provided by companies such as Uber ([[https://www.uber.com/](https://www.uber.com/)]([https://www.uber.com/](https://www.uber.com/)), accessed on 25 July 2022), Flinc ([[https://flinc.org/](https://flinc.org/)]([https://flinc.org/](https://flinc.org/)), accessed on 25 July 2022), and Lyft ([[http://www.lyft.me](http://www.lyft.me)]([http://www.lyft.me](http://www.lyft.me)), accessed on 25 July 2022) are based on a simple matching algorithm that allocates users only based on their traveling route [17, 18]. According to [18, 19], these commercial systems employ a user-matching scheme limited by two main aspects: (1) the driver's destination depends on the passenger's destination; (2) the cost of the service is defined by a provider based on distances and times, without considering the cost-sharing of both the passenger and the driver. According to [20], the extant literature on the ride-matching problem focuses mainly on settings considering ridesharing optimization, dynamic ridesharing pricing, and cost-sharing schemes. Some works regarding ridesharing optimization address issues about vehicle routes and the improvement of ridesharing matching rates [13, 21]. For example, [PERSON] et al. [22] studied the bounded rational users' travel decision-making from a perspective of a multi-objective problem. In particular, they sought to maximize the number of users willing to share a trip while minimizing the total vehicle usage cost. On the other hand, research on dynamic ridesharing pricing commonly addresses several problems with the pricing and dispatching rules governing the ridesharing platforms [23]. A particular concern in these works is that trips may be mispriced relative to each other, especially when spatial imbalance and temporal variation of supply and demand incentivize drivers [24]. Finally, other research efforts study cost-sharing schemes about allocating cost savings as an important factor in the adoption of ridesharing by the users [25, 26, 27]. In [25], [PERSON] et al. proposed a simple cost-sharing scheme based on a double auction mechanism involving the passenger and the driver. Their solution approach includes a particle swarm optimization algorithm to maximize cost savings while respecting timing constraints. In [26], a study to compare metaheuristic algorithms intended to optimize monetary incentives in ridesharing systems is performed. A theoretical study on the comparison of cost-sharing schemes in ridesharing literature is proposed in [27] to provide a guideline for characterizing the performance of proportional methods. Nevertheless, compared with the current research on these topics, a limited number of works on the ride-matching problem deal with stable matching (a matching in which no user prefers another different from the assigned one). Formally, the _stable_ matching problem represents a pairwise relationship \(M\) between users of two disjoint sets, passengers and drivers, where no user prefers to other one different from the current assigned [28]. Given two passengers, \(p_{1}\) and \(p_{2}\), and two drivers, \(d_{1}\) and \(d_{2}\), where the assigned pairs \((p_{1},d_{2})\) and \((p_{2},d_{1})\) are part of a matching \(M\).