Signal Orderings Based on Dispersion and the Supply of Private Information in Auctions

This paper provides a novel approach to ordering signals based on the property that more informative signals lead to greater variability of conditional expectations. We define two nested information criteria (supermodular precision and integral precision) by combining this approach with two variability orders (dispersive and convex orders). We relate precision criteria with orderings based on the value of information to a decision maker. We then use precision to study the incentives of an auctioneer to supply private information. Using integral precision, we obtain two results: (i) a more precise signal yields a more efficient allocation, (ii) the auctioneer provides less than the efficient level of information. Supermodular precision allows us to extend the previous analysis to the case in which supplying information is costly and to obtain an additional finding, (iii) there is a complementarity between information and competition, so that both the socially efficient and the auctioneer's optimal choice of precision increase with the number of bidders.