A line ride-sharing service is supplied along a given roadway path by an operator that matches Users (riders) and Agents (drivers), under specific protocol that involves price schedule on both the U and A sides, waiting policy on either side and transaction times. The resulting time and money items add up over trip legs of Run / Stop / Transaction / Wait, yielding trip time and money cost depending on the service role, A or U, compared to Non-commitment, called role N for Neutral. The article brings about a traffic model of people involvement in the service. Service conditions of frequency φ and average number of users per car run ω are key factors of the time and money features of the alternative roles A, U and N. Individual choice of role is modeled as a rational behavior of maximizing the individual utility at the trip level. Aggregation over trip-makers according to a multinomial logit discrete choice model yields the respective role flows (y_A,y_U,y_N), which in turn determine the macroscopic factors (φ,ω). Traffic equilibrium is defined as a balance condition between the “supplied flows” and the “demanded flows” of the three roles. A computational scheme is provided, with graphical interpretation in the (ω,φ) plane, leading to properties of existence and uniqueness of equilibrium. A numerical experiment is conducted, exhibiting the influence of the demand volume on the model outcomes.