First Price Auctions with General Information Structures: Implications for Bidding and Revenue

Abstract

We explore the impact of private information in sealed bid first price auctions.For a given symmetric and arbitrarily correlated prior distribution over values, we characterize the lowest winning bid distribution that can arise across all information structures and equilibria. The information and equilibrium attaining this minimum leave bidders uncertain whether they will win or lose and indifferent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values.We report further analytic and computational characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has implications for the identification of value distributions from winning bid data and for the informationally robust comparison of alternative bidding mechanisms.

Robust Mechanisms Under Common Valuation

“…Standard auctions are able to extract the full surplus in large markets when bidders have one-dimensional signals that are symmetrically and smoothly distributed, as shown by Wilson (1977), Milgrom (1979), Pesendorfer and Swinkels (1997), Bali and Jackson (2002), among others. However, when one bidder has proprietary information and is strictly more informed about the value than all other bidders (Engelbrecht-Wiggans, Milgrom, and Weber (1983)), or when there is a resale market that prices the value at the maximum of everyone's signals (Bergemann, Brooks, andMorris (2017a, 2017c)), standard auctions typically fail to extract the full surplus even as the number of bidders goes to infinity. From the perspective of an auctioneer whose platform must accommodate diverse groups of bidders, it is useful to commit to an auction that can be used in a variety of situations, which saves the costs of having to customize an auction design to each specific situation.…”

“…The setup and methodology in my paper come from Bergemann, Brooks, and Morris (2017a), who analyzed the set of revenue and welfare outcomes in the first price auction as one varies the information structure. Subsequent to this paper and working with the same model, Bergemann, Brooks, and Morris (2017b) characterized an optimal mechanism that achieves the best possible revenue guarantee when there are two bidders and binary common values.…”

“…The setup and methodology in my paper come from Bergemann, Brooks, and Morris (2017a), who analyzed the set of revenue and welfare outcomes in the first price auction as one varies the information structure. Subsequent to this paper and working with the same model, Bergemann, Brooks, and Morris (2017b) characterized an optimal mechanism that achieves the best possible revenue guarantee when there are two bidders and binary common values. The optimal mechanism in Bergemann, Brooks, and Morris (2017b) shares some similar features with the exponential price auction: the bidder's message space is M i = [0 1] and can be interpreted as demands for fractions of the good; the allocation rule is piecewise linear over the messages; the payment rule is exponential in one's message and independent of the other's message when m 1 + m 2 < 1 (i.e., when the demands are compatible).…”

“…The optimal mechanism in Bergemann, Brooks, and Morris (2017b) shares some similar features with the exponential price auction: the bidder's message space is M i = [0 1] and can be interpreted as demands for fractions of the good; the allocation rule is piecewise linear over the messages; the payment rule is exponential in one's message and independent of the other's message when m 1 + m 2 < 1 (i.e., when the demands are compatible). Building on the techniques in Bergemann, Brooks, and Morris (2017b), Brooks and Du (2018) characterized an optimal mechanism for any finite number of bidders and an arbitrary distribution of common values. While the exponential price auction is not exactly optimal when N > 1, it has the merit of simplicity; in particular, its payment rule is much simpler and more intuitive than that of the mechanisms in Bergemann, Brooks, and Morris (2017b) and Brooks and Du (2018).…”

Credible Auctions: A Trilemma

“…Our paper contributes to the line of research that studies standard auction formats by relaxing various assumptions of the benchmark model (Milgrom and Weber (1982), Maskin and Riley (1984), Bulow, Huang, and Klemperer (1999), Fang and Morris (2006), Hafalir and Krishna (2008), Bergemann, Brooks, andMorris (2017, 2019)). While the usual approach is to compare the standard formats in terms of expected revenue, we instead characterize the standard formats with a few simple desiderata.…”