How to split the costs among travellers sharing a ride? Aligning system's optimum with users' equilibrium

How to form groups in a mobility system that offers shared rides, and how to split the costs within the travellers of a group, are non-trivial tasks, as two objectives conflict: 1) minimising the total costs of the system, and 2) making each user content with her assignment. Aligning both objectives is challenging, as users are not aware of the externalities induced to the rest of the system. In this paper, we propose protocols to share the costs within a ride so that optimal solutions can also constitute equilibria. To do this, we model the situation as a game. We show that the traditional notions of equilibrium in game theory (Nash and Strong) are not useful here, and prove that determining whether a Strong Equilibrium exists is an NP-Complete problem. Hence, we propose three alternative equilibrium notions (stronger than Nash and weaker than Strong), depending on how users can coordinate, that effectively represent stable ways to match the users. We then propose three cost-sharing protocols, for which the optimal solutions are an equilibrium for each of the mentioned intermediate notions of equilibrium. The game we study can be seen as a game-version of the well-known \textit{set cover problem}. Numerical simulations for Amsterdam reveal that our protocols can achieve stable solutions that are always close to the optimum, that there exists a trade-off between total users' costs and how equal do they distribute among them, and that having a central coordinator can have a large impact.