Door-to-door transportation for elderly and disabled people is for many governments an important instrument to increase the mobility of this group of people. Many issues arise in the implementation of such a system, which is often modeled as the Dynamic Dial-a-Ride System with Time Windows (DDARPTW). One of those issues is that taxi companies try to maximize their profit by combining as many rides as possible. This leads to longer travel times, a measure that is expressed in the service quality of a ride. Our main contribution is a system in which multiple companies compete on service quality to increase the average service quality of the rides. We use an auction mechanism to assign rides to companies and an on-line optimization technique to insert assigned rides into current schedules. To determine an offer for announced requests, we allow companies to use knowledge about the distribution of future requests by the use of a Monte Carlo simulation.