Sequential dictatorship rules in multi-unit object assignment problems with money

We study consistency in multi-unit object allocation problems with money. Objects are identical and each agent has a multi-demand and quasi-linear preferences. We consider the class of weak object monotonic preferences and that of single-peaked preferences. We first show that on those domains, if a rule satisfies consistency, strategyproofness, individual rationality, no subsidy, non-wasteful tie-breaking, and minimal tradability, then it is a sequential dictatorship rule. Since not all sequential dictatorship rule are strategy-proof and consistent, we then focus on a specific class of sequential dictatorship rules which we call sequential dictatorship rules with lowest tie-breaking. On the weakly object monotonic domain, when the reservation prices are increasing in the number of objects, sequential dictatorship rules with lowest tie-breaking satisfy consistency and independence of unallocated objects if and only if there is a common priority ordering for more than one object and this is an acyclic ordering of the priority ordering for one object. We also show that this condition is a necessary and sufficient condition for a sequential dictatorship rule with lowest tie-breaking to satisfy consistency and independence of unallocated objects on the single-peaked domain.