  We discuss and compare two methods of investigations for the asymptotic
regime of stochastic differential games with a finite number of players as the
number of players tends to the infinity. These two methods differ in the order
in which optimization and passage to the limit are performed. When optimizing
first, the asymptotic problem is usually referred to as a mean-field game.
Otherwise, it reads as an optimization problem over controlled dynamics of
McKean-Vlasov type. Both problems lead to the analysis of forward-backward
stochastic differential equations, the coefficients of which depend on the
marginal distributions of the solutions. We explain the difference between the
nature and solutions to the two approaches by investigating the corresponding
forward-backward systems. General results are stated and specific examples are
treated, especially when cost functionals are of linear-quadratic type.
