Efficiency and consistency for locating multiple public facilities

In the problem of locating multiple public facilities studied by Barberà and Beviá [Self-selection consistent functions, J. Econ. Theory 105 (2002) 263–277; Locating public facilities by majority: stability, consistency and group formation, Games Econ. Behav. 56 (2006) 185–200], we offer simple necessary and sufficient conditions for efficiency, decentralizability of efficient decisions in a game of community division and local public goods provision, and a constructive algorithm for efficient and consistent decisions.