# Stable Matching of Users in a Ridesharing Model
## 5 Conclusions and Future Work  
This paper studies the stable ride-matching problem, an allocation process of ridesharing users in which no user prefers another different from the assigned one. With the aim to address this problem, a new ridesharing model using a specific pairwise utility function is introduced. This utility function is based on two fundamental aspects of the ridesharing service: travel savings and the reputation of users. While travel savings consider some spatial and temporal characteristics of the trip, the users' reputation is calculated based on community members' positive and negative opinions. In particular, the reputation system proposed here is a modified version of the well-known EigenTrust algorithm, where each user evaluates a service based on punctuality, cordiality, satisfaction, and trust. A small case study was developed to show how and whether the proposed ridesharing model applies in practice. The case study focuses on public transportation users in the Metropolitan Zone of Guadalajara, the third largest city in Mexico, who are willing to improve their travel options through a ridesharing service. Although using public transportation is commonly seen as an option to reduce traffic congestion, resulting data shows that ridesharing services provide a higher utility in terms of cost and time savings (commuting public services could take hours in Mexico). In this vein, a model suitable to these contexts could motivate the use of ridesharing services, increasing the occupancy rates of vehicles, one of the significant factors behind road traffic congestion. However, mitigating traffic congestion is regarded as a result rather than a factor. The proposed ridesharing model is used to introduce an algorithm designed to produce stable matching solutions in ridesharing systems. This algorithm consists of an iterated version of the Gale-Shapley algorithm, in which restrictions about the equality in the size of the sets and the assumption of complete and strict preferences are relaxed through the concept of a 'waiting list'. Experimental simulations were conducted by considering a benchmarking of 2880 random bipartite graphs as test cases. Experimental results show that the proposed algorithm always converges to a stable matching solution for all test cases. An experimental price of stability indicates a level of degradation of 7.63% in the social welfare due to the stable solutions. This study may be extended in several directions for future work. One consists of adjusting the proposed ridesharing model to incorporate trips with deviation, where no common proper subroute exists between passenger and driver. This adjustment would be in addition to the three supported types of trips: identical, inclusive, and partial. Another interesting direction would be to implement a reputation system that considers not only the service's evaluation but also the users' identity. Finally, a critical question is whether this ridesharing model can reduce traffic congestion. It would be interesting to evaluate if this is possible considering a critical mass of shareable trips in terms of specific factors like trip density, city scale, and demand heterogeneity, among others. Conceptualization, D.F.-D. and C.H.-B. ; methodology, D.F.-D., C.H.-B. and M.G.S.-C.; software, D.F.-D., C.H.-B. and M.G.S.-C.; validation, M.G.S.-C. and J.E.M.-S.; writing--original draft preparation, D.F.-D. and M.G.S.-C.; supervision, J.A.T.-S. and I.E.E.-C.; project administration, D.F.-D. All authors have read and agreed to the published version of the manuscript. This research was funded by the Tecnologico Nacional de Mexico grant numbers 8288.20-P and 11078.21-P.  
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