Multi-unit Object Allocation Problems with Money for (Non)Decreasing Incremental Valuations: Impossibility and Characterization Theorems

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 the units he receives and his payment. An agent’s preference over bundles may not be quasi-linear. A class of preferences is rich if it includes all quasi-linear preferences with constant incremental valuations. We show that for an odd number of units, if a class of preferences is rich and includes at least one preference exhibiting both decreasing incremental valuations and either positive or negative income effects, then no rule satisfies efficiency, individual rationality, no subsidy for losers, and strategy-proofness. In contrast, for an even number of units, the existence of a rule satisfying the four properties depends on the size of the income effects. We further show that if a rich class of preferences includes only preferences that exhibit nondecreasing incremental valuations, then the generalized Vickrey rule (Saitoh and Serizawa, 2008; Sakai, 2008) is the only rule satisfying the four properties. Our results suggest that (i) there a rule satisfying the four properties “almost” only when preferences exhibit nondecreasing incremental valuations, and (ii) it depends not only on the properties of preferences such as nondecreasing incremental valuations, but also on other characteristics of the environment such as the number of units.