Fig. 4: Transition from low to high ride-sharing adoption.

a Phase diagram of the fraction of shared rides \({S}_{{\rm{share}}}/S\) for different relative importance of financial and inconvenience incentives ϵ/ζ. Ride-sharing is adopted dominantly if the financial discount fully compensates the expected inconvenience, ϵ/ζ > 1 (high-sharing, dark blue). Otherwise, the total number of shared ride requests saturates and the overall adoption of ride-sharing decreases with increasing number of users S (low-sharing, compare Fig. 3a). In the limit of infinitely many requests S → ∞ the transition becomes discontinuous (see Supplementary Note 3). b Qualitatively different sharing behavior emerges for different relative incentives ϵ/ζ (compare red lines in panel a). When ϵ/ζ > 1 all users request shared rides (\({S}_{{\rm{share}}}=S\), dark blue triangles). When ϵ/ζ < 1, the system is in a low-sharing regime where users request shared rides at low numbers of users S but the number of shared ride requests saturates and becomes constant at high S (\({S}_{{\rm{share}}}<S\), light green triangles). c Hybrid states of high- and low-sharing adoption may emerge if users with heterogeneous preferences ϵ/ζ mix and interact. A fraction of users (for whom ϵ/ζ > 1) is in the high sharing regime (blue). The others (green, for whom ϵ/ζ < 1) decrease their ride-sharing adoption as the overall demand increases, consistent with the prediction for homogeneous user preferences (panel b). Macroscopically, the system exhibits partial ride-sharing adoption (gray). d The superposition of different realizations of this partial ride-sharing adoption represents the expected outcome in a city with multiple origins, each with heterogeneous preference distributions and demand (see Methods for parameters and Supplementary Note 4 for simulation details). While the macroscopic state suggests partial ride-sharing adoption, individual origins and user groups split into a mix of low- and high- sharing states, following the fundamental adoption regimes from the basic model.