This work is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward-backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the asymptotic-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. Comparison with classical results of mean-field games is also discussed.
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