In federated learning (FL), local models are trained independently by clients, local model parameters are shared with a global aggregator or server, and then the updated model is used to initialize the next round of local training. FL and its variants have become synonymous with privacy-preserving distributed machine learning. However, most FL methods have maximization of model accuracy as their sole objective, and rarely are the clients’ needs and constraints considered. In this paper, we consider that clients have differing performance expectations and resource constraints, and we assume local data quality can be improved at a cost. In this light, we treat FL in the training phase as a game in satisfaction form that seeks to satisfy all clients’ expectations. We propose two novel FL methods, a deep reinforcement learning method and a stochastic method, that embrace this design approach. We also account for the scenario where certain clients can adjust their actions even after being satisfied, by introducing probabilistic parameters in both of our methods. The experimental results demonstrate that our proposed methods converge quickly to a lower cost solution than competing methods. Furthermore, it was found that the probabilistic parameters facilitate the attainment of satisfaction equilibria (SE), addressing scenarios where reaching SEs may be challenging within the confines of traditional games in satisfaction form.