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 ε −3 to achieve a root-mean-square error of size ε. 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 δ > 0 that the proposed MLP approximation algorithm requires only a computational effort of order ε −(2+δ) to achieve a root-mean-square error of size ε.