The emergency of federated learning (FL) enables distributed data owners to collaboratively build a global model without sharing their raw data, which creates a new business chance for building data market. However, in practical FL scenarios, the hardware conditions and data resources of the participant clients can vary significantly, leading to different positive/negative effects on the FL performance, where the client selection problem becomes crucial. To this end, we propose <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AUCTION</i> , an <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</u> utomated and q <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">U</u> ality-aware <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</u> lient selec <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">TION</u> framework for efficient FL, which can evaluate the learning quality of clients and select them automatically with quality-awareness for a given FL task within a limited budget. To design <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AUCTION</i> , multiple factors such as data size, data quality, and learning budget that can affect the learning performance should be properly balanced. It is nontrivial since their impacts on the FL model are intricate and unquantifiable. Therefore, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AUCTION</i> is designed to encode the client selection policy into a neural network and employ reinforcement learning to automatically learn client selection policies based on the observed client status and feedback rewards quantified by the federated learning performance. In particular, the policy network is built upon an encoder-decoder deep neural network with an attention mechanism, which can adapt to dynamic changes of the number of candidate clients and make sequential client selection actions to reduce the learning space significantly. Extensive experiments are carried out based on real-world datasets and well-known learning models to demonstrate the efficiency, robustness, and scalability of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AUCTION</i> .