Exponential Communication Ine¢ ciency of Demand Queries

Nisan and Segal (forthcoming) show that when verifying the e¢ ciency of a combinatorial allocation, without increasing the communication burden one can restrict attention to announcing a price equilibrium. We …nd that a parallel conjecture for deterministic communication fails: We demonstrate a class of valuations for which the restriction to “demand queries,” which ask agents to report their preferred allocations at given (possibly nonlinear) allocation prices, brings about an exponential blowup in the communication burden of …nding an e¢ cient allocation. Namely, for this class, an e¢ cient mechanism exists that uses a number of bits that is proportional to the number of objects, but any demand-query mechanism that achieves e¢ ciency, or even any improvement upon the “dictatorial”allocation of all the objects to one agent, must use an exponential number of demand queries. We also show a parallel average-case result: We construct a joint probability distribution over the agents’valuations from this class for which any improvement in the expected surplus over the dictatorial allocation requires using an exponential expected number of demand queries. Our results bring into question the usefulness of combinatorial auction mechanisms such as “iterative auctions,”and other “preference elicitation” mechanisms that use demand queries or value queries. Of course, the results still open the possibility that demand-query mechanisms work well on some classes of valuations, or on average for some “realistic”probability distributions of valuations, but such cases are yet to be characterized.