Solidarity for public goods under single-peaked preferences: characterizing target set correspondences

We consider the problem of choosing a set of locations of a public good on the real line R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document} when agents have single-peaked preferences over points. We ordinally extend preferences over compact subsets of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}, and extend the results of Ching and Thomson (1996), Vohra (1999), and Klaus (2001) to choice correspondences. We show that efficiency and replacement-dominance characterize the class of target point functions (Corollary 2) while efficiency and population-monotonicity characterize the class of target set correspondences (Theorem 1).