Monte Carlo simulation of the atomic configurational behavior of Co-Pt nanoparticles is performed in wide temperature-composition-size ranges. The simulation is based on bulk and local (inhomogeneous) cluster expansions calculated from first principles. A sharp drop of equilibrium L1 0 order is predicted for small equiatomic particles at low temperatures. This drop is explained by the interplay of two effects. First, the strong Pt and Co segregation to the first and second surface layers, respectively (core/Co/Pt or "onion2shell" profile) cause a depletion of Pt atoms within the core of small Co-Pt particles. Second, L1 0 is an adaptive structure on the fcc-restricted ground-state phase diagram of Co-Pt. Nevertheless, we show that a remarkable degree of L1 0 order can be restored by tuning the total composition of small Co-Pt particles.