Strategy-proofness versus symmetry in economies with an indivisible good and money

We consider the problem of allocating a single indivisible good among n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} agents when monetary transfers are allowed. We study the possibility of constructing strategy-proof, symmetric, and budget balanced mechanisms. We show that there is no strategy-proof, symmetric, and budget balanced mechanism (under the weak domain condition that the set of agent’s possible valuations includes at least n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document} common valuations). Moreover, this result implies that there is no strategy-proof, symmetric, and budget balanced mechanism (i) in the model where agents may have non-quasilinear preferences, and/or (ii) in the unit-demand model with n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} heterogeneous indivisible goods.