Abstract: [Objective] Multiple companies coexist and compete with each other in the ride-sourcing market. The phenomenon of multihoming, wherein ride-sourcing drivers concurrently register and offer services across multiple platforms, is increasingly pervasive and poised to reshape the market configurations significantly. Facing a competitive market, drivers have incentives to be multi-homing. On one hand, multi-homing drivers expand the pool of service supply. On the other hand, multi-homing behaviours also intensify the competition on two sides. Which effect would dominate is uncertain and may vary with the returns to scale of the market. This raises the concerns that what is the optimal pricing strategies for the platforms under different returns to scale and what are the impacts of multi-homing behaviours of drivers on various stakeholders in the ride-sourcing market? [Method] We develop an analytical model to depict a duopoly ride-sourcing market in which the drivers could be multi-homing and the endogenous network externalities between customers and drivers are captured by a Cobb-Douglas matching function. The concept of a Nash equilibrium is employed to compare and analyse the equilibrium outcomes in various market configurations, i.e., drivers with single-homing and multi-homing behaviours, under different returns to scale of the market.Furthermore,a simplified model is proposed to describe the market with partial multi-homing drivers and numerical experiments are conducted to evaluate the market performance and welfare changes of all stakeholders. [Results] In both single-homing and multi-homing markets, the outcomes, e.g., supply, demand and prices at equilibrium are examined. It is found that the realized demand, vehicle supply are lower and the customer waiting time in the multi-homing market are higher than those in the single-homing one. Moreover, when the market exhibits the decreasing or constant returns to scale, the platform would charge less trip fare in the multi-homing market. The customers enjoy the lower price and suffer an inefficient market. The drivers are worse off owing to the reduced realized demand and lower wage per order offered by the platform at equilibrium. When the market displays the increasing returns to scale with larger elasticity of meeting rate, the platform might benefit via increasing the trip fare. In the market with partial multi-homing drivers, if the relative service supply of multi-homing drivers is high, i.e., multi-homing drivers provide almost the same service time for each platform as the single-homing drivers, a fraction of multi-homing drivers would achieve more orders and social welfare., especially in the market with increasing-returns-to-scale. [Conclusions] Our results illustrate the critical role the market configuration plays in realizing potential welfares in a ride-sourcing market. All drivers choosing to be multi-homing would lead to a disaster in the end. However, all stakeholders would be better-off when only a fraction of drivers chooses to be multi-homing. The insights on how the multi-homing behaviours of drivers should be regulated by the platforms or the government provided in this work is helpful for improving the benefits of all stakeholders.