Optimal auctions with information acquisition

TL;DR: In this article, the optimal auction design in a private value setting with endogenous information gathering was studied and it was shown that the optimal monopoly price is always lower than the standard monopoly price.

About: This article is published in Games and Economic Behavior.The article was published on 2012-03-01 and is currently open access. It has received 66 citations till now. The article focuses on the topics: Reservation price & Common value auction.

Summary

1 Introduction

• The purpose of this paper is to study how a seller should design the selling mechanism when information acquisition is endogenous and costly for buyers.
• Suppose a consumer tries a newly opened restaurant and finds the food spicy.
• If two signals are rotation-ordered, then the two distributions of posterior estimates generated by these signals cross each other only once.
• Since the buyer always prefers a low reserve price, it may seem at first glance that a lower reserve price always gives the buyer a higher incentive to gather information.

2 Related Literature

• This paper is related to the growing literature on information and mechanism design, extending the principal-agent model with information acquisition to a multi-agent setting.
• The authors model shares a similar information structure and a focus on the interim participation constraint, though the authors incorporate strategic interactions among bidders.
• 13 A third strand of related literature studies optimal auctions when the seller controls either the access to information sources or the timing of information acquisition.
• These models (hereafter referred to as "entry models") impose an ex-ante participation constraint, so the buyers' information decision is essentially an entry decision.
• 13 Bergemann and Välimäki (2002) also study information acquisition and mechanism design, but their focus is efficient mechanisms.

3.1 The Information Structure

• Consistency selling mechanism in which the seller charges a positive entry fee and extracts the full surplus from buyers.
• 16 Cremer, Spiegel, and Zheng (2007) also analyze optimal auctions where buyers can acquire information prior to participation, but the seller, rather than the buyer, pays the information cost.
• 17 The analysis can be extended to a multi-unit setting where each buyer has a unit demand.

3.2 Timing

• Figure 1 summarizes the timing of the game:.
• The solution concept is Bayesian Nash equilibrium.

3.3 Mechanisms

• This suggests that the design problem here is multi-dimensional and could potentially be very complicated.
• Furthermore, the seller cannot screen the two pieces of information separately.
• Therefore, the posterior estimate v i is the only variable that the seller can use to screen different buyers.

Definition 1 (Rotation Order)

• Two distributions ordered in terms of rotation cross only once: the distribution with lower α i crosses the distribution with higher α i from below.
• As shown below, the rotation order implies second-order stochastic dominance.
• The reverse is not true: two distributions ordered in terms of second-order stochastic dominance can cross each other more than once.
• Following Blackwell (1951 Blackwell ( ) (1953)) , the authors say that one signal is more informative than the other if a decision-maker can achieve a higher expected utility when basing a decision on the realization of the more informative signal.

4 Optimal Auctions with One Bidder and Gaussian Specification

• The authors start with a simple model with one buyer.
• With endogenous information, their logic still applies and a posted price is optimal.
• 18 Therefore, the authors can also reinterpret the seller's optimization problem as a monopoly pricing problem with endogenous information.
• Then the authors analyze the seller's information preferences and demonstrate that the seller would prefer a more informed buyer if and only if the monopoly price is above the mean valuation.
• For the purposes of illustration, the authors focus on a special but important rotation-ordered information structure: the Gaussian specification, though it is straightforward to extend the analysis to general rotation-ordered information structures.

4.1 Gaussian Specification

• Lowering β has the consequence that the prior distribution becomes more spread out, yielding more potential gains from information acquisition.
• The first part, α, is the endowed signal precision; the second term γ i is the additional precision obtained by investing in information acquisition.
• As shown in the next section, after incorporating the information acquisition constraint, the seller's objective function will be the Lagrangian specified in (10).
• Therefore, their proof of the optimality of the posted price mechanism still applies here.

It immediately follows that

• The following two graphs capture the relationship between two distributions of the posterior estimate with different information choices.
• The left graph in Figure 2 shows that the density of the posterior estimate with a more informative signal is more dispersed than the one with a less informative signal.
• The right graph shows that the distribution with a less informative signal crosses the distribution with a more informative one from below at the mean valuation.

4.2 Marginal Value of Information to the Buyer

• The following proposition shows how the marginal value of information to the buyer varies with respect to the reserve price.
• With some algebra, the authors can show that EQUATION.
• The following two graphs illustrate the buyer's gain from more information.
• On the other hand, when r ≤ µ, according to expression (5), the buyer's gain from more information is the shaded area in the right graph.
• Hence, the solution to the buyer's maximization problem will be unique, and the buyer's information choice will be decreasing in the information cost c (see Proposition 4 below).

4.3 Seller's Pricing Decision

• For the seller, she chooses r and equilibrium α * to maximize her revenue.
• The buyer's information choice is unobservable to the seller , and the seller sets r to align the buyer's incentives to her own.
• The standard way to solve this problem, the so-called first-order approach, is to assume that the second-order condition of the agent's maximization problem is satisfied, and use the first-order condition to replace the incentive constraint.
• The authors will assume the second-order condition is satisfied for now, and discuss it in detail at the end of this subsection.
• Let λ be the Lagrangian multiplier for the constraint.

Definition 3 (Standard Reserve Price)

• The standard reserve price r α is the optimal reserve price when the buyer's signal α is exogenous.
• Third, with endogenous information acquisition, a price increase will affect the buyer's incentive to acquire information, thereby the probability of trade (term C).
• The authors conclude this subsection by presenting sufficient conditions for the second-order condition of the buyer's maximization problem to be satisfied.
• These conditions are stronger than necessary and are not very restrictive.
• Note that more than 95% of the normal density is within two standard deviations of the mean.

5 Optimal Auctions with Many Bidders

• In the single-bidder model, the strategic interaction among bidders is absent, so the simple posted price mechanism is optimal.
• Specifically, the authors show that: 1) A bidder's incentives to acquire information increase as the reserve price moves toward the rotation point;.
• 2) If the authors restrict attention to the symmetric mechanism that induces all bidders to acquire the same level of information, standard auctions with an adjusted reserve price are optimal;.
• 3) If information decision is discrete, then the optimal selling mechanism reduces the level of price discrimination against strong bidders compared to the case with exogenous information;.
• One insight that cannot be carried over from the one-bidder case, however, concerns the seller's information preferences.

5.1 Marginal Value of Information to the Buyer

• It is well-known (Myerson (1981) , and Rochet (1987) ) that the incentive compatibility constraint (IC) is equivalent to the following two conditions: EQUATION and EQUATION ).
• Let r i denote the personalized reserve price for bidder i in the optimal auction.
• Then the two distributions associated with two different α i 's must cross at least twice.
• Theorem 1 generalizes Proposition 2 to a setting with many bidders and rotation-ordered information structures.

5.2 Characterization of Symmetric Optimal Auctions

• If the firstorder approach is valid, the authors can can replace bidder i's optimization problem by (8).
• Finally, the seller has to give bidders not only incentives to acquire information, but also incentives to tell the truth, i.e., their model is a mixed model with moral hazard and adverse selection.
• This restriction helps reduce the system of first-order conditions to a single equation (8).
• Let λ denote the Lagrangian multiplier for the (IA) constraint, and write the Lagrangian for the seller's maximization problem as EQUATION ) Both the rotation order assumption and the regularity assumption are mild assumptions.

Lemma 3 (All Assumptions Hold for the Two Leading Examples)

• Note that Assumption 1 does not imply that the underlying distribution F is symmetric.
• For the truth-or-noise technology, the underlying distribution F could be convex or concave, but the rotation point is still µ.
• In order to characterize the symmetric optimal auction, the authors first need to identify the seller's information preferences, that is, the sign of the Lagrangian multiplier λ for the (IA) constraint.
• It turns out that this is the most difficult part of the analysis.
• The authors use the technique in Rogerson 23 Indeed, the monotonicity assumption, together with the mean-preserving property of their information structures, implies rotation order.

5.3 Asymmetric Mechanisms with Discrete Information Acquisition

• The authors assume information acquisition is discrete, so one can drop both the symmetric restriction and the first-order approach.
• That is, the mechanism has to ensure that the first m bidders have incentives to acquire information and the remaining (n − m) bidders have incentives not to acquire information.
• Interestingly, endogenous information acquisition reduces the level of ex-post discrimination against "strong" bidders, as shown in the following theorem.
• Moreover, the mechanism reduces the level of price discrimination against "strong" bidders if signals are rotation-ordered.
• The first part of the theorem shows that the optimal rule for adjusting reserve price identified in Theorem 2 is still valid in this discrete setting.

5.4 Informational Efficiency

• Theorem 3 states that standard auctions with an adjusted reserve price are optimal when the authors restrict the equilibrium to be symmetric.
• This subsection will examine the informational efficiency of standard auctions and obtain a slightly more general results that can apply to optimal symmetric auctions.
• The difference between the social and individual gain from acquiring information is EQUATION.
• Thus, the authors have proved the following result.
• Proposition 5 (Informational Efficiency) Suppose Assumption 1 holds.

5.5 Discussion

• The rotation order ranks different information structures by comparing the distributions of the posterior estimates.
• In contrast, most existing information orders (for example, Lehmann (1988) ) impose restrictions on the prior or posterior distributions of underlying states and signals.
• Assumption 1 restricts the rotation point to be the mean valuation.
• Thus, their analysis may remain valid even when the second-order condition of the bidders' maximization problem fails.
• The first term ω i represents the individual idiosyncratic valuation and is unknown ex-ante.

6 Conclusion

• The mechanism design literature studies how carefully designed mechanisms can be used to elicit agents' private information in order to achieve a desired goal.
• This distinguishes their model from papers studying information acquisition in fixed auction formats.
• The authors can rewrite the first-order condition for the buyer's maximization problem as EQUATION Applying implicit function theorem to (19), they can show that ∂α ∂r EQUATION ).
• It is easy to see that the other two assumptions are satisfied as well.
• This condition is analogous to the CDFC (convexity of the distribution function condition) in the principal-agent literature, which requires that the distribution function of output be convex in the action the agent takes (Mirrlees (1999), and Rogerson (1985) ). 30 For their two leading information structures, the authors will provide sufficient conditions under which the first-order approach is valid.

Frequently asked questions

Q1. What are the contributions in "Optimal auctions with information acquisition∗" ?

This paper studies optimal auction design in a private value setting with endogenous information acquisition. The authors then show that under certain conditions, standard auctions with a reserve price remain optimal, but the optimal reserve price lies between the mean valuation and the standard reserve price in Myerson ( 1981 ). The authors provide sufficient conditions under which the value of information to the seller is positive, and also characterize the necessary and sufficient conditions under which equilibrium information acquisition in private value auctions is socially excessive. First, the authors develop a general framework for modeling information acquisition when a seller wants to sell an object to one of several potential buyers who can each gather information about their valuations prior to participation. 

Q2. What is the purpose of the subscript i of the expectation operator?

The subscript α∗−i of the expectation operator is to remind the readers that the expectation depends on the information choice α∗−i of bidder i’s opponents. 

Q3. What is the way to characterize the optimal selling mechanism?

If the authors assume that information acquisition is discrete rather than continuous, however, the authors can characterize the optimal selling mechanism without the symmetric restriction, as shown in the following subsection. 

Q4. What order of distributions are used to rank the informativeness of signals?

The resulting family of distributions of the posterior estimates with different signals are rotation-ordered3 – the information order the authors use to rank the informativeness of signals. 

Q5. What can be the general framework the authors develop to model information acquisition in a private value setting?

The general framework the authors develop to model information acquisition in a private value setting can also apply to mechanism design problems when agents can make investment prior to the auction. 

Q6. What is the effect of increased information acquisition on the seller’s revenue?

Since an increase in information leads to an increase in the dispersion of buyers’ valuation estimates, increased information acquisition has two competing effects on the seller’s revenue. 

Q7. What is the simplest way to solve the buyer’s optimization problem?

the authors can replace the buyer’s optimization problem with the first-order condition, and rewrite the seller’s optimization problem as20max r,α∗ r (1−Hα∗ (r)) s.t. : − ∫ ∞ r ∂Hα∗ (vi) ∂α∗ dvi − c = 0. [λ] 

Q8. What is the first order condition of the buyer’s maximization problem?

Proposition 4 (Validity of the First Order Approach) If r ∈ [µ− 2σ (α) , µ+ 2σ (α)] and α ≥ β, the second-order condition of the buyer’s maximization problem is satisfied.