Communication and E¢ ciency in Auctions

We study auctions under restricted communication. Bidders have valuations in a compact interval, but can only report one of a …nite number of messages. We provide a characterization of welfare as well as revenue maximizing auctions in the symmetric independent private values case when bidders report simultaneously. We show that the seller choosing how to allocate the …xed communication capacity allocates it evenly over the bidders. The optimal auction is asymmetric, contrasting the symmetry of optimal auctions when communication is not restricted. The analysis is extended to the case of multiple identical objects and single unit demand. Finally, we characterize welfare maximizing auctions under restricted communication when two bidders report sequentially and there are only two periods of reporting. These auctions can be thought of as a two step procedure. In the …rst step, the …rst bidder chooses a price from a …nite menu of prices. In the second step the object is o¤ered to the second bidder at a higher price. If the second bidder accepts it he receives the object and pays the o¤ered price, otherwise the …rst bidder receives the object at the price he chose.