With advances in digitization and automation, autonomous mobility on demand services have the potential to disrupt the future mobility system landscape. Ridepooling services in particular can both decrease land consumption by reducing the need for parking and increase transportation efficiency by increasing the average vehicle occupancy. Nevertheless, because ridepooling services require a sufficient user base for pooling to take effect, their performance can suffer if multiple operators offer such a service and must split the demand. This study presents a simulation framework for evaluating the impact of competition and cooperation among multiple ridepooling providers. Two different kinds of interaction via a broker platform are compared with the base cases of a single monopolistic operator and two independent operators with divided demand. In the first, the broker presents trip offers from all operators to customers (similar to a mobility-as-a-service platform), who can then freely choose an operator. In the second, a regulated broker platform can manipulate operator offers with the goal of shifting the customer-operator assignment from a user equilibrium towards a system optimum. To model adoptions of the service design depending on the different interaction scenario, a game setting is introduced. Within alternating turns between operators, operators can adapt parameters of their service (fleet size and objective function) to maximize profit. Results for a case study based on Manhattan taxi data, show that operators generate the highest profit in the broker setting while operating the largest fleet. Additionally, pooling efficiency can nearly be maintained compared to a single operator. The regulated competition benefits not only operators (profit) and cities (increased pooling efficiency), but customers also experience higher service rate, though they need to accept slightly increased waiting and travel time due to increased pooling efficiency. Contrarily, when users can decide freely, the lowest pooling efficiency and operator profit is observed.