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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]