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• Describe equilibrium of ride-sourcing market with a number of competitive platforms. • Quantify the price (efficiency gain or loss) of competition and fragmentation. • Examine trends of platform profits and social welfare towards fragmentation level. • Discuss market scenarios with increasing, constant, and decreasing returns to scale. • Our model covers monopoly, duopoly, oligopoly, and perfect competition. This paper proposes a general model for describing the equilibrium state of a ride-sourcing market with an arbitrary number of platforms competing with each other. As the number of platforms increases, the market changes from monopoly to duopoly, oligopoly, and finally perfect competition, bringing about two different effects on system efficiency. On the one hand, as in other service markets, competition in the ride-sourcing markets prevents a monopolist platform from extracting excessive profit by distorting its operating strategies from socially efficient levels. On the other hand, competition between platforms leads to market fragmentation, thereby increasing matching frictions and passengers' waiting time. To well characterize these two opposite driven forces, we develop a game-theoretical model to find out the Nash equilibrium solutions of a competitive ride-sourcing market, at which no platform can increase its profits by unilaterally changing its own strategy. Then we try to quantify the price (efficiency gain or loss) of competition and fragmentation by establishing an upper bound of the inefficiency ratio, i.e., the ratio of social welfare under a social optimum to social welfare under a competitive Nash equilibrium. We show that the results of market equilibrium, including the inefficiency ratio, are jointly governed by the degree of market fragmentation and competition among platforms. In particular, we find that some key market measures, such as consumer surplus, platform profits, social welfare, display diverse trends of changes with respect to the number of platforms, as the on-demand matching between passengers and drivers exhibits increasing, constant, and decreasing returns to scale. [ABSTRACT FROM AUTHOR]

• Nash and Stackelberg models to capture the interactions of DNO and SEV operator. • A best response-based algorithm is proposed to solve the bilevel problem. • Supply-demand equilibrium existence is proved and uniqueness is numerically tested. • Cruising traffic is increased with medium travel demand while reduced in other cases. This paper considers a group of shared electric vehicles (SEV) that provide transport and energy services through a ride-sourcing platform. Under the novel business model, the transport and power networks are coupled and the resulting market equilibrium depends on the pricing strategies of the distribution network operator (DNO) and the SEV platform operator. To capture the interaction between the DNO and the SEV platform operator, we develop two market competition models, including (a) a Stackelberg competition model where the DNO makes the decision first and the SEV operator reacts accordingly, and (b) a Nash competition model where the DNO and the SEV operator make decisions simultaneously. The strategic interactions of the DNO and the SEV platform operator are analyzed over a coupled transport and power network, which captures the routing and discharging behaviors of EV drivers, the matching process between passengers and drivers, and the corresponding traveler and power flows. We analytically prove that the supply and demand equilibrium between travelers and EV drivers over the transport network always exists. We also numerically validate that the equilibrium solution between the SEV operator and the DNO is stable and unique. By varying the transport demands, we show that the SEV platform operator, the DNO and EV drivers always benefit from the discharge business even though passengers may be slightly worse off. Surprisingly, we find that integrating the discharging service will have diverse impacts on the cruising traffic depending on the demand levels of the transport service: under low demand level, it will reduce the cruising traffic; under medium demand level, it will increase the cruising traffic; under high demand level, it has no impact on the cruising traffic. [ABSTRACT FROM AUTHOR]