Efficient and strategy-proof multi-unit object allocation problems with money: decreasing incremental valuations with income effects

We consider the problem of allocating multiple units of an object and collecting payments. Each agent can receive multiple units, and his (consumption) bundle is a pair consisting of his quantity of the object and his payment. An agent’s preference over bundles is not necessarily quasi-linear. A class of preferences is called a domain. We show that if a sufficiently rich domain includes at least one preference exhibiting decreasing incremental valuations and either positive or negative income effects, then no rule satisfies efficiency, individual rationality, no subsidy for losers, and strategy-proofness. Our result demonstrates the necessity of certain assumptions for the existence of a rule satisfying the four properties in various environments-for instance, the quasi-linearity assumption is necessary for the classical existence result under quasi-linear preferences with non-increasing incremental valuations (Vickrey 1961).