Strategy-proof Allocation of Indivisible Goods when Preferences are Single-peaked

We consider assignment problems where heterogeneous indivisible goods are to be assigned to individuals so that each individual receives at most one good. Individuals have single-peaked preferences over the goods. In this setting, first we show that there is no strategy-proof, non-bossy, Pareto efficient, and strongly pairwise reallocation-proof assignment rule on a minimally rich single-peaked domain when there are at least three individuals and at least three objects in the market. Next, we characterize all strategy-proof, Pareto efficient, top-envy-proof, non-bossy, and pairwise reallocation-proof assignment rules on a minimally rich single-peaked domain as hierarchical exchange rules. We additionally show that strategy-proofness and non-bossiness together are equivalent to group strategy-proofness on a minimally rich single-peaked domain, and every hierarchical exchange rule satisfies group-wise reallocation-proofness on a minimally rich single-peaked domain.