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Information acquisition during a descending auction


We compare the effects of information acquisition during a descending auction with its static counterpart, the first-price sealed-bid auction. In a framework with heterogeneous prior information, we show that an equilibrium with information acquisition exists in both auction formats. We show that everything else equal information acquisition is more desirable in the dynamic auction. Moreover, we characterize a set of parameter values where more information is acquired in the dynamic auction in equilibrium. If the costs of information acquisition are sufficiently low, the sealed-bid auction generates more revenue although the descending auction is more efficient.

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  1. For example, in spectrum auctions, corporate takeovers or procurement with unknown project cost information about ones valuation is costly.

  2. If information acquisition is not an issue, both formats are strategically equivalent as they can both be modeled as identical normal form games (Klemperer 2004; Milgrom 1989; Krishna 2009). Due to this apparent similarity with the first-price sealed-bid auction, the descending auction seems to have been somewhat neglected in the literature. For example, Ausubel and Crampton (2006) in their work for practitioners on “Dynamic auctions in procurement” do not consider the descending auction as a potential dynamic auction.

  3. The loss in revenue has some support from anecdotal evidence. In practice, descending auctions are used to sell goods fast, which, one may argue, is a way to prevent information acquisition and revenue loss. We thank one of the referees for bringing this point to our attention.

  4. After finishing this work, the authors were informed of the work by Miettinen (2010) who also analyzes information acquisition during a descending auction. As he considers bidders who are completely informed or completely uninformed, the effects described here, namely that bidders with different expected values have different interests in acquiring information, do not arise in his setup.

  5. See Bergemann and Välimäki (2002), Crémer et al. (2009), Hausch and Li (1993), Morath and Münster (2012), Persico (2000), and Shi (2011) among others.

  6. Assuming that \(c_{i}\) and \({\hat{v}}_{i}\) are private, information is a good approximation to most real-life application. We will comment on the technical implications of the assumption that the distributions of \(c_{i}\) and \({\hat{v}}_{i}\) are not degenerate in Subsect. 2.2.

  7. A similar result would hold if \(x_{i}\) was not independent of \({\hat{v}}\), i.e., \(x_{i}\sim L\left( x\left| {\hat{v}}\right. \right) \).

  8. See, e.g., Krishna (2009), p. 19.

  9. Let \(\bar{\mathcal{F }}\) denote the closure of \(\mathcal F \).

  10. The proof follows from the analysis by Rezende (2005). We comment on the differences to our aproach in the Appendix.

  11. For a proof of this version of Schauder’s theorem, see Cauty (2001).

  12. If one of the distributions is not absolutely continuous, \(T\circ R\) fails to be continuous and Schauder’s theorem is not applicable.

  13. We obtained some numerical results by directly iterating the operator \(T\circ R\) with different initial distributions \(F_{{\hat{v}}}\). The resulting fixed points were not instructing toward finding an analytical solution. Moreover, we derived a mixed strategy equilibrium for \(F_{{\hat{v}}}=U[0,1]\) and different costs of information \(c\). However, the results were not generalizable and very sensitive to the choice of \(c\).

  14. The existence of a fixed point can be established in exactly the same manner as in Proposition 2.

  15. Numerical results may be obtained by directly iterating the operators \(T\circ R\) and \(T_{s}\circ R_{s}\). Revenue and efficiency results may then be established directly for certain distributions and parameters. Such a numerical exercise, however, is beyond the scope of this paper.

  16. This follows from an application of the revenue equivalence theorem (see Krishna 2009, p. 66).

  17. Lemma 2 in the Appendix establishes that such an equilibrium always exists if the costs of information acquisition are sufficiently low.

  18. As stated above, a spread in the distribution of valuations leads to lower bids but more extreme valuations. The second effect predominates the first if the number of bidders grows large.

  19. The maximal efficiency loss occurs when bidder \(i\) stops the clock right before bidder \(j\) acquired information, i.e., if \(v_{i}\) is equal to \({\hat{v}}_{j}+x^{*}(c_{j})\).

  20. We thank one of the referees for bringing this point to our attention.

  21. This follows from the straightforward comparison between the equilibrium conditions derived in this paper and in Rezende (2005)

  22. For static mechanisms, this issue was resolved by Shi (2011).


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Correspondence to Vitali Gretschko.

Additional information

We would like to thank Dirk Bergemann, Andreas Engel, Christian Hellwig, Wanda Mimra, Alexander Rasch, and seminar participants at Dortmund, Frankfurt, Regensburg, Berlin, and Yale University for helpful comments. The first version of this paper was written when the authors stayed at Yale University in 2007 and 2011. The authors thank the department of economics for its hospitality. Financial support from the German Science Foundation (DFG) through the research group “Design & Behavior” and from the Fulbright Commission is gratefully acknowledged. We also thank two anonymous referees for detailed and helpful comments that have greatly improved this paper.



1.1 Proof of Lemma 1


(i) and (iii) follow from the definition of \(\beta (v)\) as the optimal stopping time given the distribution of the bids of the competing bidders: A bidder who learns that her true valuation is \(v\) at some clock price larger than \(\beta (v)\) will accept the clock price in the descending auction once it reaches \(\beta (v)\). A bidder who never learns her true valuation stops the price clock at the optimal time given her available information, i.e., at \(\beta ({\hat{v}}).\) Concerning (ii), observe that \(\beta \) is increasing and the right-hand side of (1) is single peaked. Thus, a bidder who learns her true valuation will accept the clock price immediately, if the valuation is such that a bidder with this true valuation would have accepted the clock price before. \(\square \)

1.2 Proof of Proposition 1


Suppose a bidder with expected valuation \({\hat{v}}\) plans to acquire information when the clock price reaches \(\beta ({\hat{v}}+x^{\prime })\) for some \(x^{\prime }\) with \(-\Delta \le x^{\prime }\le \Delta \). If, at this point, this bidder learns that her true valuation is \(v={\hat{v}}+x\) with \(x\ge x^{\prime }\), she will accept the clock price immediately (Lemma 1 (ii)). If she learns that \(x<x^{\prime }\), she will wait until \(p=\beta ({\hat{v}}+x)\) before accepting the clock price if the auction has not ended before (Lemma 1 (iii)). Thus, her expected profit is given by

$$\begin{aligned}&H({\hat{v}}+x^{\prime })\left( \int \limits _{x^{\prime }}^{\Delta }({\hat{v}}+x-\beta ({\hat{v}}+x^{\prime }))dL(x)\right. \nonumber \\&\qquad +\left. \int \limits _{-\Delta }^{x^{\prime }}\frac{H({\hat{v}}+x)}{H({\hat{v}}+x^{\prime })}({\hat{v}}+x-\beta ({\hat{v}}+x))dL(x)-c\right) . \end{aligned}$$

The first term is the probability of reaching the clock price \(\beta ({\hat{v}}+x^{\prime })\) while the other terms are the expected profit conditional on reaching this clock price minus the costs of information. The first-order condition to the maximization problem given by (13) is

$$\begin{aligned}&h({\hat{v}}+x^{\prime })\int \limits _{x^{\prime }}^{\Delta }({\hat{v}}+x-\beta ({\hat{v}}+x^{\prime }))dL(x)\\&\quad +H({\hat{v}}+x^{\prime })\left( -({\hat{v}}+x^{\prime })l(x^{\prime })-\beta ^{\prime }({\hat{v}}+x^{\prime }))(1-L(x^{\prime }))+\beta ({\hat{v}}+x^{\prime }))l(x^{\prime })\right) \\&\quad +H({\hat{v}}+x^{\prime })({\hat{v}}+x^{\prime }-\beta ({\hat{v}}+x^{\prime }))l(x^{\prime })-h({\hat{v}}+x^{\prime })c=0. \end{aligned}$$

This can be rewritten to give

$$\begin{aligned}&\left( h({\hat{v}}+x^{\prime })({\hat{v}}+x^{\prime })-h({\hat{v}}+x^{\prime })\beta ({\hat{v}}+x^{\prime })+H({\hat{v}}+x^{\prime })\beta ({\hat{v}}+x^{\prime })\right) (1-L(x^{\prime }))\nonumber \\&\qquad +h({\hat{v}}+x^{\prime })\left( \,\,\int \limits _{x^{\prime }}^{\Delta }(x-x^{\prime })dL(x)-c\right) =0. \end{aligned}$$

The equation \(h({\hat{v}}+x^{\prime })({\hat{v}}+x^{\prime })-h({\hat{v}}+x^{\prime })\beta ({\hat{v}}+x^{\prime })+H({\hat{v}}+x^{\prime })\beta ({\hat{v}}+x^{\prime })=0\) is the first-order condition for Problem (1). Thus, Eq. (14) reduces to

$$\begin{aligned} h({\hat{v}}+x^{\prime })\left( \,\,\int \limits _{x^{\prime }}^{\Delta }(x-x^{\prime })dL(x)-c\right) =0. \end{aligned}$$

Hence, the optimal \(x^{\prime }\) is given by

$$\begin{aligned} c=\int \limits _{x^{*}(c)}^{\Delta }(x-x^{*}(c))dL(x). \end{aligned}$$

The right-hand side is monotonously decreasing in \(x^{*}(c)\). Thus, the first-order condition is sufficient and \(x^{*}(c)\) uniquely determined. \(\square \)

1.3 Proof of Proposition 2


To make use of the Schauder fixed-point theorem, we have to establish that \(T\circ R\) is a continuous compact map from a convex subset of the normed vector space of continuous functions from \([-\Delta ,\bar{v}+\Delta ]\) to \([0,1]\) onto itself. More precisely, we will restrict our attention to the set \(\mathcal H =\left\{ H\in \bar{\mathcal{F }}\left| H(\epsilon )>0\ \forall \epsilon >0\right. \right\} \) and to the restriction of \(T\circ R\) to \(\mathcal H \). As \(\mathcal H \) is convex, we have to show that \(T\circ R\) is continuous on \(\mathcal H \) and that the image of \(T\circ R\) is contained in a compact subset of \(\mathcal H \). Hence, if \(T\circ R\) has a fixed point in \(\mathcal F \), it will be in \(\mathcal H \). To see that \(cl(T\circ R(\mathcal F ))\subset \mathcal H \), fix an \(\epsilon >0\). If all types \(({\hat{v}},c)\) with \({\hat{v}}\le \epsilon \) remain uninformed with probability one, then for any \(H\in \mathcal F ,\, T\circ R(H)(\epsilon )\ge F_{{\hat{v}}}(\epsilon )\). Note that \(F_{{\hat{v}}}(\epsilon )>0\) as \(F_{{\hat{v}}}\) is absolutely continuous and its density is bounded away from zero. If on the other hand, a subset of types \(A\subset \left\{ ({\hat{v}},c)\left| {\hat{v}}\le \epsilon \right. \right\} \) with \(\text {Prob}(A)>0\) acquires information, then

$$\begin{aligned}&T\circ R(H)(\epsilon )\ge L(0)\text {Prob}(A\left| \left\{ ({\hat{v}},c)\left| {\hat{v}}\le \epsilon \right. \right. \right\} )\\&\quad +\text {Prob}(A^{c}\left| \left\{ ({\hat{v}},c)\left| {\hat{v}}\le \epsilon \right. \right. \right\} )F_{{\hat{v}}}(\epsilon )\ge \min \{L(0),F_{{\hat{v}}}\}>0 \end{aligned}$$

as all distributions are absolutely continuous. It follows \(cl(T\circ R)\subset \mathcal H \). To apply Schauder’s fixed-point theorem, it remains to show that \(T\circ R_\mathcal{H }\) is continuous and its image is relatively compact. We start by establishing that \(T(\mathcal B )\) is relatively compact. A set is relatively compact if it is a subset of a compact set. We apply the Arizola-Ascoli theorem (see, e.g., Rudin 1987), which states that a subset of continuous functions on \([-\Delta ,\bar{v}+\Delta ]\) is compact if and only if it is pointwise bounded, closed, and equicontinuous. As all distribution functions are pointwise bounded in the uniform norm, it remains to verify that \(T(\mathcal B )\) is equicontinuous. Fix an \(\epsilon >0\). To show equicontinuity, it is sufficient to show that for all \(F\in T(\mathcal B )\) and for all \(v\in [-\Delta ,\bar{v}+\Delta ]\), there exists a \(\delta >0\) such that \(F(v+\delta )-F(v)<\epsilon \). For any \(F\in T(\mathcal B )\), it holds

$$\begin{aligned} F(v+\delta )-F(v)&= \int \limits _{B}\text {Prob}\left[ v\le {\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \le v+\delta \left| {\hat{v}},c\right. \right] dF_{c}dF_{{\hat{v}}}\\&+\int \limits _{B^{c}}\text {Prob}\left[ v\le {\hat{v}}\le v+\delta \left| {\hat{v}},c\right. \right] dF_{c}dF_{{\hat{v}}}\\&= \text {Prob}[B]\text {Prob}\left[ v\le {\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \le v+\delta \left| B\right. \right] \\&+\text {Prob}[B^{c}]\text {Prob}\left[ v\le {\hat{v}}\le v+\delta \left| B^{c}\right. \right] \\&\le \text {Prob}[{\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \in [v+\delta ,v]]+\text {Prob}[{\hat{v}}\in [v+\delta ,v]]\\&\le \text {Prob}[{\hat{v}}+x^{*}(c)\in [v+\delta ,v]]+\text {Prob}[{\hat{v}}+x\in [v+\delta ,v]]\\&+\text {Prob}[{\hat{v}}\in [v+\delta ,v]] \end{aligned}$$

From (15) we know that \(x^{*}(c)\) is a monotone function of \(c\). As the distribution of \(c\) is absolutely continuous, it follows that the distribution of \(x^{*}(c)\) is absolutely continuous. Hence, as a convolution of absolutely continuous distributions, \({\hat{v}}+x^{*}(c)\) and \({\hat{v}}+x\) are also distributed with absolutely continuous distribution functions. It follows that there exists a \(\delta >0\) such that

$$\begin{aligned}&\text {Prob}[{\hat{v}}+x^{*}(c)\in [v+\delta ,v]]\le \frac{\epsilon }{3},\\&\quad \text {Prob}[{\hat{v}}+x\in [v+\delta ,v]]\le \frac{\epsilon }{3},\\&\qquad \text {and}\,\text {Prob}[{\hat{v}}\in [v+\delta ,v]]\le \frac{\epsilon }{3} \end{aligned}$$

for all \(v\in [-\Delta ,\bar{v}+\Delta ]\). As \(\delta \) only depends on \(\epsilon \) and not on \(F\), we can conclude that \(T(\mathcal B )\) is equicontinuous. To see that \(T\circ R_\mathcal{H }\) is continuous, take a sequence \((F_{k})_{k\in \mathbb N }\in \mathcal H \) with \(\lim _{k\rightarrow \infty }F_{k}=F\) uniformly. It follows \(\lim _{k\rightarrow \infty }r_{F_{k}^{n-1}}=r_{F^{n-1}}\) pointwise. Expression (3) can be rewritten as

$$\begin{aligned} T(R(F))(v)&= \int \mathbf 1 _{\{r_{F^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)\Bigr [\text {Prob}\left[ {\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \le v\left| {\hat{v}},c\right. \right] \\&\quad -\mathbf 1 _{\{{\hat{v}}\le v\}}({\hat{v}},c)\Bigr ]+\mathbf 1 _{\{{\hat{v}}\le v\}}({\hat{v}},c)dF_{c}dF_{{\hat{v}}}. \end{aligned}$$

From the fact that the indicator function (\(\mathbf 1 _{ \{ \}}\)) is bounded by one together with the properties of the uniform norm, we obtain that

$$\begin{aligned}&\left\| T(R(F_{k}))(v)\right. - \left. T(R(F))(v)\right\| _{\infty }\\&\quad = \left\| \int \Bigr (\mathbf 1 _{\{r_{F_{k}^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)-\mathbf 1 _{\{r_{F^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)\Bigr )\right. \\&\qquad \times \left. \Bigr [\text {Prob}\left[ {\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \le v\left| {\hat{v}},c\right. \right] -\mathbf 1 _{\{{\hat{v}}\le v\}}({\hat{v}},c)\Bigr ]dF_{c}dF_{{\hat{v}}}\right\| _{\infty }\\&\quad \le \left\| \int \Bigr (\mathbf 1 _{\{r_{F_{k}^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)-\mathbf 1 _{\{r_{F^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)\Bigr )\right. \\&\qquad \times \left\| \left. \Bigr [\text {Prob}\left[ {\hat{v}}+\min \left\{ x,x^{*}(c)\right\} \le v\left| {\hat{v}},c\right. \right] -\mathbf 1 _{\{{\hat{v}}\le v\}}({\hat{v}},c)\Bigr ]\right\| _{\infty }dF_{c}dF_{{\hat{v}}}\right\| _{\infty }\\&\quad \le \left\| \int \Bigr (\mathbf 1 _{\{r_{F_{k}^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)-\mathbf 1 _{\{r_{F^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)\Bigr )dF_{c}dF_{{\hat{v}}}\right\| _{\infty }. \end{aligned}$$

Independent of \(v,\, \int \left\| \Bigr (\mathbf 1 _{\{r_{F_{k}^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)-\mathbf 1 _{\{r_{F^{n-1}}({\hat{v}},c)\ge 0\}}({\hat{v}},c)\Bigr )\right\| _{\infty }dF_{c}dF_{{\hat{v}}}\) converges to zero whenever \(\text {Prob}\left[ \left\{ r_{F^{n-1}}({\hat{v}},c)=0\right\} \right] =0\). It follows that the convergence is uniform as long as bidders are indifferent with respect to buying information with probability zero. Hence, to complete the proof, we have to show that \(\text {Prob}\left[ \left\{ r_{F^{n-1}}({\hat{v}},c)=0\right\} \right] =0\) for all \(F\in \mathcal H \). For this purpose, we show that \(F({\hat{v}}+x^{*}(c))r_{F^{n-1}}({\hat{v}},c)\) is strictly increasing in \(x^{*}(c)\) (and thus strictly decreasing in \(c\)) whenever \(r_{F^{n-1}}({\hat{v}},c)=0\). It then follows that for each \({\hat{v}}\), there is at most one \(c\) such that \(r_{F^{n-1}}({\hat{v}},c)=0\). Thus, \(\text {Prob}\left[ \left\{ r_{F^{n-1}}({\hat{v}},c)=0\right\} \right] =0\). Indeed,

$$\begin{aligned}&\frac{\partial }{\partial x^{*}}\left( (1-L(x^{*}))\int \limits _{{\hat{v}}}^{{\hat{v}}+x^{*}}F^{n-1}(s)ds\right. -\left. \int \limits _{-\Delta }^{x^{*}}\left( \int \limits _{{\hat{v}}+x}^{{\hat{v}}}F^{n-1}(s)ds\right) dL(x)\right) \nonumber \\&\quad =(1-L(x^{*}))F^{n-1}({\hat{v}}+x^{*}). \end{aligned}$$

If \(x^{*}(c)\le 0\), then \(r_{F^{n-1}}({\hat{v}},c)<0\). If \(x^{*}(c)>0\), then the right-hand side of Eq. (16) is strictly positive for all \(F\in \mathcal H \). We have established that \(T\circ R\) is a continuous compact map from the convex and closed subset \(\mathcal H \) of \(\bar{\mathcal{F }}\) onto itself. By Schauder’s fixed-point theorem, there exists a fixed point of \(T\circ R\) and thereby an equilibrium of the descending auction. \(\square \)

1.4 Comparison of proofs with Rezende (2005)

Rezende (2005) decomposed the best reply to the bidding strategies of the bidders and the information acquisition decision in the context of the English auction. He used the best-response strategies to define an operator over the set of all continuous distribution functions and then used Schauder’s fixed-point theorem to prove the existence of a pure-strategy equilibrium.

Our proof also uses the approach of decomposing the bidding strategy and the information acquisition strategies and defining a best-response operator over the set of all continuous distribution functions. The main difference is that the decomposition of the bidding and the information acquisition strategy for the descending auction is different from the English auction as no dominant bidding strategies exist. Moreover, Rezende (2005) has to make the assumption that the distribution of costs \(F_{c}\) has a mass point at \(0\) to ensure that at least some types will acquire information. We do not need this assumption due to the fact that in the descending auction, the types close to \((0,0)\) always have an incentive to acquire information if \({\underline{\mathrm{c}}}\) is small enough (see Proposition 4).

1.5 Existence of equilibria with information acquisition

Lemma 2

For each \(n,\, F_{{\hat{v}}}\), and \(L(x)\), there exists a \(\bar{c}\) such that for all \(F_{c}\) with a support of \([\underline{c},\bar{c}]\), \(B=[0,\bar{v}]\times [\underline{c},\bar{c}]\) in the sealed-bid and in the descending auction in equilibrium. In the descending auction, bidders only acquire information if the price is sufficiently low and if the object is still available.


\(r_{H}^{s}({\hat{v}},0)>0\) and \(r_{H}({\hat{v}},0)>0\) for all \({\hat{v}}\in [0,\bar{v}]\) and all \(H\in \mathcal F \). Moreover, \(r_{H}^{s}\) and \(r_{H}\) are continuous in \(c\). Hence, there exists a \(\bar{c}\) such that all types from \(\bar{B}{:=}[0,\bar{v}]\times [\underline{c},\bar{c}]\) acquire information in equilibrium, i.e., \(T(\bar{B})\) is a fixed point of \(T\circ R\) and \(T_{s}(\bar{B})\) is a fixed point of \(T_{s}\circ R_{s}\). \(\square \)

1.6 Proof of Proposition 3


To establish the result, we have to establish that for \(H=F^{n-1}\), \(r_{H}({\hat{v}},c)\ge r_{H}^{s}({\hat{v}},c)\), i.e., we have to show that

$$\begin{aligned}&\frac{1}{H({\hat{v}}+x^{*}(c))}\left[ (1-L(x^{*}(c)))\int \limits _{{\hat{v}}}^{{\hat{v}}+x^{*}(c)}H(s)ds\right. -\left. \int \limits _{-\Delta }^{x^{*}(c)}\left( \int \limits _{{\hat{v}}+x}^{{\hat{v}}}H(s)ds\right) dL(x)\right] \\&\quad \ge \int \limits _{-\Delta }^{\Delta }\int \limits _{{\hat{v}}}^{{\hat{v}}+x}H(s)dsdL(x)-c. \end{aligned}$$

or equivalently

$$\begin{aligned}&(1-L(x^{*}(c)))\int \limits _{{\hat{v}}}^{{\hat{v}}+x^{*}(c)}H(s)ds+\int \limits _{-\Delta }^{x^{*}(c)}\left( \int \limits _{{\hat{v}}}^{{\hat{v}}+x}H(s)ds\right) dL(x)\\&\quad \ge \int \limits _{-\Delta }^{\Delta }\int \limits _{{\hat{v}}}^{{\hat{v}}+x}H(s)dsdL(x)-c. \end{aligned}$$

Observe that

$$\begin{aligned}&(1-L(x^{*}(c)))\int \limits _{{\hat{v}}}^{{\hat{v}}+x^{*}(c)}H(s)ds =\int \limits _{x^{*}(c)}^{\Delta }\left( \int \limits _{{\hat{v}}}^{{\hat{v}}+x^{*}(c)}H(s)ds\right) dL(x)\\&\qquad =\int \limits _{x^{*}(c)}^{\Delta }\left( \,\int \limits _{{\hat{v}}}^{{\hat{v}}+x}H(s)ds\right) dL(x)-\int \limits _{x^{*}(c)}^{\Delta }\left( \,\,\int \limits _{{\hat{v}}+x^{*}(c)}^{{\hat{v}}+x}H(s)ds\right) dL(x). \end{aligned}$$

Since \(0\le H\le 1\), the last term

$$\begin{aligned} \int \limits _{x^{*}(c)}^{\Delta }\left( \,\,\int \limits _{{\hat{v}}+x^{*}(c)}^{{\hat{v}}+x}H(s)ds\right) dL(x)<\int \limits _{x^{*}(c)}^{\Delta }\left( \,\,\int \limits _{{\hat{v}}+x^{*}(c)}^{{\hat{v}}+x}ds\right) dL(x)=c. \end{aligned}$$

Hence, the result. \(\square \)

1.7 Proof of Proposition 4


Ad (i): if no information is acquired in the first-price auction, the distribution of valuations is just \(F^{*}(v)=F_{{\hat{v}}}(v)\) and the value of information is given by

$$\begin{aligned} \int \limits _{-\Delta }^{\Delta }\int \limits _{{\hat{v}}}^{{\hat{v}}+x}F_{{\hat{v}}}(s)^{n-1}dsdL(x)\le F_{{\hat{v}}}({\hat{v}}+\Delta )^{n-1}\int \limits _{0}^{\Delta }xdL(x)-\int \limits _{-\Delta }^{0}\int \limits _{{\hat{v}}+x}^{{\hat{v}}}F_{{\hat{v}}}(s)^{n-1}dsdL(x). \end{aligned}$$

The inequality is due to the fact that \(F_{{\hat{v}}}(v)^{n-1}\) is increasing. Define \(\underline{c}\) as

$$\begin{aligned} \underline{c}{:=}\max _{{\hat{v}}}F_{{\hat{v}}}({\hat{v}}+\Delta )^{n-1}\int \limits _{0}^{\Delta }xdL(x)-\int \limits _{-\Delta }^{0}\int \limits _{{\hat{v}}+x}^{{\hat{v}}}F_{{\hat{v}}}(s)^{n-1}dsdL(x). \end{aligned}$$

From the fact that \(F_{{\hat{v}}}\) is absolutely continuous and the fact that \(F_{{\hat{v}}}(\Delta )<1\), it follows that

$$\begin{aligned} \underline{c}<\int \limits _{0}^{\Delta }xdL(x). \end{aligned}$$

Hence, there exists a \(\tilde{c}\) such that \(x^{*}(c)>0\) for all \(c\in [\underline{c},\tilde{c}]\). It follows that \(r_{F_{{\hat{v}}}(v)^{n-1}}(0,\underline{c})>0\), i.e., the best reply of a bidder with type \((0,\underline{c})\) is to acquire information in the descending auction. As \(F_{{\hat{v}}}\) is absolutely continuous and \([\underline{c},\tilde{c}]\) is of strictly positive length, it follows that Prob\([\left\{ ({\hat{v}},c)\left| r_{F_{{\hat{v}}}(v)^{n-1}}({\hat{v}},c)>0\right. \right\} ]>0\), i.e., a set of types with positive measure are strictly better off by acquiring information in the descending auction. Hence, not acquiring information is not an equilibrium of the Dutch auction.

Ad (ii): if no information is being acquired in an equilibrium of the Dutch auction, the distribution of valuations is just \(F^{*}(v)=F_{{\hat{v}}}(v)\) and \(r_{F_{{\hat{v}}}(v)^{n-1}}<0\) for all \(v\in [0,\bar{v}]\) and all \(c\in [\underline{c},\bar{c}]\). From Proposition 3, it follows that \(r_{F_{{\hat{v}}}(v)^{n-1}}^{s}<0\). Hence, not acquiring information is also an equilibrium of the sealed-bid auction.

Ad (iii): if all bidders acquire information in an equilibrium of the first-price auction, the distribution of valuations is \(F_{{\hat{v}}}*L(v)\) and \(r_{F_{{\hat{v}}}*L(v)^{n-1}}^{s}\ge 0\) for all \(v\in [0,\bar{v}]\) and all \(c\in [\underline{c},\bar{c}]\). From Proposition 3, it follows that \(r_{F_{{\hat{v}}}*L(v)^{n-1}}\ge 0\). Hence, every bidder acquires information with positive probability in an equilibrium of the descending auction. \(\square \)

1.8 Proof of Proposition 7


Take any realizations of the types \(({\hat{v}}_{1},c_{1}),\ldots ,({\hat{v}}_{n},c_{n})\) and the true valuations \(v_{1},\ldots ,v_{n}\). Order the bidders such that \({\hat{v}}_{1}+x^{*}(c_{1})\ge \ldots \ge {\hat{v}}_{n}+x^{*}(c_{n})\). The efficiency loss in the first-price auction in the case where every bidder acquires information is just the sum of the costs \(\sum _{i=1}^{n}c_{i}\). Now consider the descending auction. Suppose bidder \(1\) to bidder \(m\le n\) acquire information in equilibrium. The allocation will be efficient for these \(m\) bidders in the sense that if the object is bought by one of the bidders, it will be the bidder with \(v_{i}\ge \max _{j\in \{1..m\}}v_{j}\). If \(m=n\), then the efficiency is the same as in the first-price auction. If \(m<n\), an allocative inefficiency may arise if a bidder \(i\in \{1,..m\}\) stops the clock, but a bidder \(j\in \{m+1,\ldots ,n\}\) has a higher true valuation for the object. This can only happen if \({\hat{v}}_{j}+x^{*}(c_{j})<v_{i}\) and \(v_{j}>v_{i}\). The expected efficiency loss is then at most \({\hat{v}}_{j}+\text {Prob}[x_{j}>x^{*}(c_{j})]E[x_{j}|x_{j}>x^{*}(c_{j})]-({\hat{v}}_{i}+x_{i})\le {\hat{v}}_{j}+\text {Prob}[x_{j}>x^{*}(c_{j})]E[x_{j}|x_{j}>x^{*}(c_{j})]-{\hat{v}}_{j}-x^{*}(c_{j})\le \text {Prob}[x_{j}>x^{*}(c_{j})]E[x_{j}-x^{*}(c_{j})|x_{j}>x^{*}(c_{j})]=c_{j}\). Hence, the efficiency loss from misallocating the good is less than the costs of information acquisition for the bidder that remained uninformed. It follows that the efficiency in the descending auction is at least as large as in the first-price auction. \(\square \)

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Gretschko, V., Wambach, A. Information acquisition during a descending auction. Econ Theory 55, 731–751 (2014).

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  • Descending auction
  • Dutch auction
  • First-price sealed-bid auction
  • Information acquisition

JEL Classification

  • D44
  • D82
  • D83