Strategy-proof rules in object allocation problems with hard budget constraints and income effects

We consider the single-object allocation problem with monetary transfers. Agents may have hard budgets and their utility functions may exhibit income effects. When hard budget constraints are present, it is known that efficiency and strategy-proofness are incompatible along with individual rationality and no subsidy. Our objective is to clarify what forms of partial efficiency are compatible with strategy-proofness alongside individual rationality and no subsidy for losers. To do so we decompose efficiency into two properties: constrained efficiency and no wastage. First, we show that some of threshold-price rules are the only rules that satisfy individual rationality, no subsidy for losers, and strategy-proofness. Next, we show that some of truncated Vickrey rules with endogenous reserve prices are the only rules that satisfy constrained efficiency, individual rationality, no subsidy for losers, and strategy-proofness. Finally, we show that step-wise threshold-price rules generally satisfy no wastage, individual rationality, no subsidy, and strategy-proofness, and that they are the only rules satisfying these properties in the two-agent case. Whether these three types of rules satisfy strategy-proofness, constrained efficiency, and no wastage depends critically on the structure of the tie-breaking rule. We identify what structures are necessary and sufficient for these rules to satisfy the corresponding properties.