Strategy-proof and symmetric allocation of an indivisible good

We consider economies with a single indivisible good and money. We study the possibility of constructing strategy-proof, symmetric, and budget balanced mechanisms. We show three impossibility results on restricted domains: there is no strategy-proof, symmetric, and budget balanced mechanism satisfying either (i) equal compensation, (ii) normal compensation, or (iii) individual rationality. Moreover, the first result is strengthened by replacing symmetry with weak symmetry. In contrast, we show that our sequential mechanisms satisfy strategy-proofness, weak symmetry, budget balance, and the other three axioms on much more restricted domains.