Efficient and strategy-proof multi-unit object allocation with money: (Non)decreasing marginal valuations without quasi-linearity

We consider the problem of allocating multiple units of an indivisible object among agents and collecting payments. Each agent can receive multiple units of the object, and his (consumption) bundle is a pair of the units he receives and his payment. An agent's preference over bundles may be non-quasi-linear, which accommodates income effects or soft budget constraints. We show that the generalized Vickrey rule is the only rule satisfying efficiency, strategy-proofness, individual rationality, and no subsidy for losers on rich domains with nondecreasing marginal valuations. We further show that if a domain is minimally rich and includes an arbitrary preference exhibiting both decreasing marginal valuations and a positive income effect, then no rule satisfies the same four properties. Our results suggest that in non-quasi-linear environments, the design of an efficient multi-unit auction mechanism is possible only when agents have nondecreasing marginal valuations.