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GTFLAT, as a game theory-based add-on, addresses an important research question: How can a federated learning algorithm achieve better performance and training efficiency by setting more effective adaptive weights for averaging in the model aggregation phase? The main objectives for the ideal method of answering the question are: (1) empowering federated learning algorithms to reach better performance in fewer communication rounds, notably in the face of heterogeneous scenarios, and last but not least, (2) being easy to use alongside the state-of-the-art federated learning algorithms as a new module. To this end, GTFLAT models the averaging task as a strategic game among active users. Then it proposes a systematic solution based on the population game and evolutionary dynamics to find the equilibrium. In contrast with existing approaches that impose the weights on the participants, GTFLAT concludes a self-enforcement agreement among clients in a way that none of them is motivated to deviate from it individually. The results reveal that, on average, using GTFLAT increases the top-1 test accuracy by 1.38%, while it needs 21.06% fewer communication rounds to reach the accuracy.