  We consider ordinary differential equations (ODEs) which involve expectations
of a random variable. These ODEs are special cases of McKean-Vlasov stochastic
differential equations (SDEs). A plain vanilla Monte Carlo approximation method
for such ODEs requires a computational cost of order $\varepsilon^{-3}$ to
achieve a root-mean-square error of size $\varepsilon$. In this work we adapt
recently introduced full history recursive multilevel Picard (MLP) algorithms
to reduce this computational complexity. Our main result shows for every
$\delta>0$ that the proposed MLP approximation algorithm requires only a
computational effort of order $\varepsilon^{-(2+\delta)}$ to achieve a
root-mean-square error of size $\varepsilon$.
