We consider the problem of choosing the location of a public facility either (a) on a tree network or (b) in a Euclidean space.
(a) Ching and Thomson (1996) characterize the class of status quo solutions on a tree network by Pareto optimality and population-monotonicity. Using Vohra's (1998) characterization of solutions that satisfy Pareto optimality and replacement-domination, we give a short proof of the previous characterization and show that it also holds on the domain of symmetric preferences.
(b) For the similar problem of choosing the location of a public facility in a Euclidean space, it turns out that none of the status quo solutions satisfies replacement-domination or population monotonicity (or strategyproofness). Therefore, instead of Pareto optimality, we consider unanimity and characterize the class of coordinatewise status quo solutions by unanimity, strategy-proofness, and either population-monotonicity or replacement-domination.