  We generalize the multilevel Monte Carlo (MLMC) method of Giles to the
simulation of systems of particles that interact via a mean field. When the
number of particles is large, these systems are described by a McKean-Vlasov
process - a stochastic differential equation (SDE) whose coefficients depend on
expectations of the solution as well as pathwise data. In contrast to standard
MLMC, the new method uses mean field estimates at coarse levels to inform the
fine level computations. Using techniques from the theory of propagation of
chaos, we prove convergence and complexity results for the algorithm in a
special case. We find that the new method achieves $L^1$ error of size
$\varepsilon$ with $O(\varepsilon^{-2} (\log \varepsilon)^5)$ complexity, in
contrast to the $O(\varepsilon^{-3})$ complexity of standard methods. We also
prove a variance scaling result that strongly suggests similar performance of
the algorithm in a more general context. We present numerical examples from
applications and observe the expected behavior in each case.
