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Demand responsive transportation has the potential to provide efficient public door-to-door transport with a high quality. In currently implemented systems in the Netherlands, however, we observe a decrease in the quality of service (QoS), expressed in longer travel times for the customers. Currently, generally one transport company is responsible for transporting all customers located in a specified geographic zone. In general it is known that when multiple companies compete on costs, the price for customers decreases. In this paper, we investigate whether a similar result can be achieved when competing on quality instead. To arrive at some first conclusions, we set up a multiagent environment to simulate the assignment of rides to companies through an auction on QoS, and the insertion of allocated rides in the companies' schedules using online optimization. Our results reveal that this set-up improves the quality of the service offered to the customers at moderately higher costs.