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This paper analytically examines the ride-pooling market equilibrium under a single or multiple ride-pooling service(s) considering different pooling sizes (number of riders sharing the same vehicle) and endogenous congestion effect of the ride-pooling fleet, and investigates the optimal operation decisions of the service operator(s) regarding the pricing, fleet size, and pooling size in different administrative and market regimes. We firstly examine the scenario with a single ride-pooling service platform given a specific vehicle type for either profit-maximization or social welfare-maximization (different administrative regimes), and then extend the model to the scenario with two differentiated ride-pooling services (given different vehicle types), where the platforms may either compete or cooperate (different market regimes). We have the following major findings. First, the pooling size should be optimized to accommodate the congestion effect and thus simultaneously increase operator's profit and social welfare. The optimal pooling size balances the marginal pickup time, crowding cost and waiting time cost, and the optimal value is different in different market regimes. Second, the ride-pooling service operator will internalize the congestion effect of providing more vehicles to serve customers, which yields a larger ratio of waiting customers to vacant vehicles (larger demand–supply ratio). Under two differentiated ride-pooling services, in addition to the direct congestion effect of providing more vehicles, its effect on the other operator's decision will also be internalized by the service operator in concern. Our results also suggest that the competition between platforms tends to reduce the waiting customers-vacant vehicles ratio (demand–supply ratio), while the cooperation between or integration of two operators might yield a large ratio. Third, when the endogenous congestion effect of fleet size is zero, the operator's profit at social optimum will always be negative when the matching function exhibits increasing return to scale and the marginal operation cost is less than the average operation cost, which means that the social welfare seeking operator has to be subsidized. When the endogenous congestion effect of the fleet size is positive, the ride-pooling service operator may earn a positive profit at social optimum under certain conditions. This study enhances the understanding in relation to users' reactions to differentiated ride-pooling services, how such services interact with congestion, and how such services can be optimized in different market regimes. • Ride-pooling market equilibrium with flexible pooling size in the presence of congestion effect. • Optimization of pooling size, fleet size, and fare of ride-pooling service. • Differentiated ride-pooling services under competition/cooperation/integration market regimes. • Complex interactions among customer traveling behaviors, service operation decisions and road traffic congestion. [ABSTRACT FROM AUTHOR]

• 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]