Exponential communication inefficiency of demand queries

In the problem of finding an efficient allocation when agents' utilities are privately known, we examine the effect of restricting attention to mechanisms using "demand queries," which ask agents to report an optimal allocation given a price Ust. We construct a combinatorial allocation problem with m items and two agents whose valuations he in a certain class, such that (i) efficiency can be obtained with a mechanism using O (m) bits, but (ii) any demand-query mechanism guaranteeing a higher efficiency than giving all items to one agent uses a number of queries that is exponential in m. The same is proven for any demand-query mechanism achieving an improvement in expected efficiency, for a constructed joint probabi hty distribution over agents' valuations from the class. These results cast doubt on the usefulness of such common combinatorial allocation mechanisms as "iterative auctions" and other "preference elicitation" mechanisms using demand queries, as well as "value queries" and "order queries" (which are easily replicated with demand queries in our setting).