A number of producers of heterogeneous goods with heterogeneous costs compete in prices. When producers know their own production costs and consumers know their values, consumer surplus and total surplus are aligned: the information structure and equilibrium that maximize consumer surplus also maximize total surplus. We report when alignment extends to the case where either consumers are uncertain about their own values or producers are uncertain about their own costs, and we also give examples showing when it does not. Less information for either producers or consumers may intensify competition in a way that benefits consumers but results in inefficient production.
We study price discrimination in a market in which two ﬁrms engage in Bertrand competition. Some consumers are contested by both ﬁrms, and other consumers are “captive” to one of the ﬁrms. The market can be divided into segments, which have diﬀerent relative shares of captive and contested consumers. It is shown that the revenue-maximizing segmentation involves dividing the market into “nested” markets, where exactly one ﬁrm may have captive consumers.
Consider a market with identical ﬁrms oﬀering a homogeneous good. A consumer obtains price quotes from a subset of ﬁrms and buys from the ﬁrm oﬀering the lowest price. The “price count” is the number of ﬁrms from which the consumer obtains a quote. For any given ex ante distribution of the price count, we derive a tight upper bound (under ﬁrst-order stochastic dominance) on the equilibrium distribution of sales prices. The bound holds across all models of ﬁrms’ common-prior higher-order beliefs about the price count, including the extreme cases of full information (ﬁrms know the price count) and no information (ﬁrms only know the ex ante distribution of the price count). A qualitative implication of our results is that a small ex ante probability that the price count is equal to one can lead to a large increase in the expected price. The bound also applies in a large class of models where the price count distribution is endogenously determined, including models of simultaneous and sequential consumer search.
We describe a methodology for making counterfactual predictions in settings where the information held by strategic agents and the distribution of payoﬀ-relevant states of the world are unknown. The analyst observes behavior assumed to be rationalized by a Bayesian model, in which agents maximize expected utility, given partial and diﬀerential information about the state. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of the state and agents’ information about the state are held ﬁxed. When the data and the desired counterfactual prediction pertain to environments with ﬁnitely many states, players, and actions, the counterfactual prediction is described by ﬁnitely many linear inequalities, even though the latent parameter, the information structure, is inﬁnite dimensional.
We characterize revenue maximizing mechanisms in a common value environment where the value of the object is equal to the highest of bidders’ independent signals. If the object is optimally sold with probability one, then the optimal mechanism is simply a posted price, with the highest price such that every type of every bidder is willing to buy the object. A suﬀicient condition for the posted price to be optimal among all mechanisms is that there is at least one potential bidder who is omitted from the auction. If the object is optimally sold with probability less than one, then optimal mechanisms skew the allocation towards bidders with lower signals. This can be implemented via a modiﬁed Vickrey auction, where there is a random reserve price for just the high bidder. The resulting allocation induces a “winner’s blessing,” whereby the expected value conditional on winning is higher than the unconditional expectation. By contrast, standard auctions that allocate to the bidder with the highest signal (e.g., the ﬁrst-price, second-price or English auctions) deliver lower revenue because of the winner’s curse generated by the allocation rule. Our qualitative results extend to more general common value environments where the winner’s curse is large.
In a recent paper, [Bergemann et al. 2017a], we derive results about equilibrium behavior in the ﬁrst-price auction that hold across all common-prior information structures. The purpose of this letter is to give an informal introduction into the results. At the end we oﬀer a brief discussion of related work.
We study auction design when bidders have a pure common value equal to the maximum of their independent signals. In the revenue maximizing mechanism, each bidder makes a payment that is independent of his signal and the allocation discriminates in favor of bidders with lower signals. We provide a necessary and suﬀicient condition under which the optimal mechanism reduces to a posted price under which all bidders are equally likely to get the good. This model of pure common values can equivalently be interpreted as model of resale: the bidders have independent private values at the auction stage, and the winner of the auction can make a take-it-or-leave-it-oﬀer in the secondary market under complete information.
A single unit of a good is to be sold by auction to one of two buyers. The good has either a high value or a low value, with known prior probabilities. The designer of the auction knows the prior over values but is uncertain about the correct model of the buyers’ beliefs. The designer evaluates a given auction design by the lowest expected revenue that would be generated across all models of buyers’ information that are consistent with the common prior and across all Bayesian equilibria. An optimal auction for such a seller is constructed, as is a worst-case model of buyers’ information. The theory generates upper bounds on the seller’s optimal payoﬀ for general many-player and common-value models.
We characterize revenue maximizing auctions when the bidders are intermediaries who wish to resell the good. The bidders have diﬀerential information about their common resale opportunities: each bidder privately observes an independent draw of a resale opportunity, and the highest signal is a suﬀicient statistic for the value of winning the good. If the good must be sold, then the optimal mechanism is simply a posted price at which all bidders are willing to purchase the good, and all bidders are equally likely to be allocated the good, irrespective of their signals. If the seller can keep the good, then under the optimal mechanism, all bidders make the same expected payment and have the same expected probability of receiving the good, independent of the signal. Conditional on the good being sold, the allocation discriminates in favor of bidders with lower signals. In some cases, the optimal mechanism again reduces to a posted price. The model provides a foundation for posted prices in multi-agent screening problems.
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 indiﬀerent between their equilibrium bids and all higher bids. Our results provide lower bounds for bids and revenue with asymmetric distributions over values. We also report further characterizations of revenue and bidder surplus including upper bounds on revenue. Our work has implications for the identiﬁcation of value distributions from data on winning bids and for the informationally robust comparison of alternative bidding mechanisms.
We study how the outcomes of a private-value ﬁrst price auction can vary with bidders’ information, for a ﬁxed distribution of private values. In a two bidder, two value, setting, we characterize all combinations of bidder surplus and revenue that can arise, and identify the information structure that minimizes revenue. The extremal information structure that minimizes revenue entails each bidder observing a noisy and correlated signal about the other bidder’s value.
In the general environment with many bidders and many values, we characterize the minimum bidder surplus of each bidder and maximum revenue across all information structures. The extremal information structure that simultaneously attains these bounds entails an eﬀicient allocation, bidders knowing whether they will win or lose, losers bidding their true value and winners being induced to bid high by partial information about the highest losing bid. Our analysis uses a linear algebraic characterization of equilibria across all information structures, and we report simulations of properties of the set of all equilibria.
We analyze the welfare consequences of a monopolist having additional information about consumers’ tastes, beyond the prior distribution; the additional information can be used to charge diﬀerent prices to diﬀerent segments of the market, i.e., carry out “third degree price discrimination.” We show that the segmentation and pricing induced by the additional information can achieve every combination of consumer and producer surplus such that: (i) consumer surplus is non-negative, (ii) producer surplus is at least as high as proﬁts under the uniform monopoly price, and (iii) total surplus does not exceed the surplus generated by eﬀicient trade.