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In this paper we develop a dynamic continuous solution to the clustering problem of data characterized by a mixture of K distributions, where K is given a priori. The proposed solution resorts to game theory tools, in particular mean field games and can be interpreted as the continuous version of a generalized Expectation-Maximization (GEM) algorithm. The main contributions of this paper are twofold: first, we prove that the proposed solution is a GEM algorithm; second, we derive closed-form solution for a Gaussian mixture model and show that the proposed algorithm converges exponentially fast to a maximum of the log-likelihood function, improving significantly over the state of the art. We conclude the paper by presenting simulation results for the Gaussian case that indicate better performance of the proposed algorithm in term of speed of convergence and with respect to the overlap problem.