Information acquisition during a descending auction

Abstract

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.

Appendix

1.1 Proof of Lemma 1

Proofs

(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

Proof

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

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

(13)

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

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

(14)

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

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

(15)

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

Proof

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,

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

(16)

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.

Proof

$r_{H}^{s}({\hat{v}},0)>0$ 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

Proof

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

Proof

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

Proof

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. $\square$