The success of Federated Learning (FL) depends on the quantity and quality of the data owners (DOs) as well as their motivation to join FL model training. Reputation-based FL participant selection methods have been proposed. However, they still face the challenges of the cold start problem and potential selection bias towards highly reputable DOs. Such a bias can result in lower reputation DOs being prematurely excluded from future FL training rounds, thereby reducing the diversity of training data and the generalizability of the resulting models. To address these challenges, we propose the Gradual Participant Selection scheme for Auction-based Federated Learning (GPS-AFL). Unlike existing AFL incentive mechanisms which generally assume that all DOs required for an FL task must be selected in one go, GPS-AFL gradually selects the required DOs over multiple rounds of training as more information is revealed through repeated interactions. It is designed to strike a balance between cost saving and performance enhancement, while mitigating the drawbacks of selection bias in reputation-based FL. Extensive experiments based on real-world datasets demonstrate the significant advantages of GPS-AFL, which reduces costs by 33.65% and improved total utility by 2.91%, on average compared to the best-performing state-of-the-art approach.