Top-left panel: Like Fig. 1 but un-normalised. Note that at early times, the MW and M31 had a very similar mass growth history, diverging from z ∼ 4. The larger scatter in the present-day M31 mass reflects the larger (observational) uncertainties on its mass from W24. Top-right panel: Same but for MW-mass and M31-mass galaxies from an unconstrained simulation. Bottom panels: Comparisons of the MWs and M31s from the constrained simulations with isolated MW- or M31-mass haloes.