  This paper 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 $\varepsilon$-Nash
equilibrium in classical mean-field games, the set of decentralized strategies
is shown to be a Nash equilibrium. For the infinite-horizon problem, a simple
condition is given for the solvability of the algebraic Riccati equation
arising from consensus. Furthermore, the social optimal control problem is
studied. Under a mild condition, the decentralized social optimal control and
the corresponding social cost are given.
