Revenue-superior variants of the second-price auction

Abstract

With two bidders, one strong and one weak, the introduction of at least some degree of anonymous “pay-your-bid” in the payment rule of the second-price auction behoves any risk-neutral seller who, while possibly efficiency minded, cares about revenues. This can be achieved by adding to the winner’s payment a uniform proportion of his own bid, as in Güth and van Damme’s auction, or by having bidders receive a uniform proportion of the losing bid, as in Goeree and Offerman’s Amsterdam auction, or even by selling uniform toeholds to the bidders prior to the auction. We demonstrate one-to-one relations between the equilibria of these auctions and of first-price auctions. By assuming a power relation between the bidders’ value cumulative or decumulative functions, we obtain explicit expressions for the first-order effects of the pay-your-bid rule.

Appendices

Appendix 1

1.1 Proof of Theorem 1

Proof of (i): The proof of the necessity of the characterization (i) proceeds along similar lines as the proof in Lebrun (1997, 1999) of the characterization in Theorem 1 in Lebrun (1999) of the equilibria of the FPA. We relegate it to the online supplements.

It is weakly dominated to submit a bid strictly above one’s value. It is unprofitable to submit a bid strictly below $c$ when one’s value is strictly above $c$ or to submit a bid strictly above the common maximum bid $\eta \left( k\right)$. The sufficiency of the system (1) of FOCs then follows from the strict monotonicity of the strategies and the nonnegativity of the cross second-order partial derivative, equal to $\frac{\hbox {d}}{\hbox {d}b}F_{j}\left( \alpha _{j}\left( b\right) \right)$, of any bidder $i$’s payoff with respect to his own bid $b$ in $\left( r,\eta \left( k\right) \right)$ and own value.

Proof of (ii): From Theorem 1 in Lebrun (1999), the differential system and boundary conditions in (i) are the same as those that characterize the equilibria of the FPA where the value distributions are $F_{1}^{1/k},...,F_{n}^{1/k}$ and (ii) follows. $\square$

1.2 Proof of Corollary 1

Proof of (i): The existence follows from Theorem 1 (ii) above and Lebrun (1999) (or other papers on the FPA, such as Lebrun 1997 or Maskin and Riley 2000).³⁰ Strict log-concavity of $F_{i}$ and strict log-concavity of $F_{i}^{1/k}$ are equivalent. Uniqueness then follows from Lebrun (2006).

Proof of (ii): The formula is a direct consequence of Theorem 1 (ii) above and the envelope theorem or Myerson (1981).

Proof of (iii): From $F_{2}^{1/k}=F_{1}^{l/k}$, $F_{2}^{1/k}$ reverse-hazard-rate dominates $F_{1}^{1/k}$. From Corollary 3 (ii) in Lebrun (1999), we then have $\beta _{1}\left( .;k\right) \ge \beta _{2}\left( .;k\right)$ or, equivalently, $\alpha _{2}\left( .;k\right) \ge \alpha _{1}\left( .;k\right)$. The equations (1) then immediately imply $\frac{\hbox {d}}{\hbox {d}b}\ln F_{2}\alpha _{2}\left( b;k\right) \ge \frac{\hbox {d}}{\hbox {d}b}\ln F_{1}\alpha _{1}\left( b;k\right)$ and (iii.1) is proved.

Assume $l>1$. If there existed $b$ in $\left( r,\eta \left( k\right) \right)$ such that $\alpha _{1}\left( b;k\right) =\alpha _{2}\left( b;k\right)$, we would have from (1) $l\alpha _{2}^{\prime }\left( b;k\right) \rho _{1}\left( \alpha _{2}\left( b;k\right) \right) =\frac{\hbox {d}}{\hbox {d}b} \ln F_{2}\left( \alpha _{2}\left( b;k\right) \right) =\frac{\hbox {d}}{\hbox {d}b}\ln F_{1}\left( \alpha _{1}\left( b;k\right) \right) =\alpha _{1}^{\prime }\left( b;k\right) \rho _{1}\left( \alpha _{1}\left( b;k\right) \right)$. Consequently, we would have $\alpha _{2}^{\prime }\left( b;k\right) <\alpha _{1}^{\prime }\left( b;k\right)$ and $\alpha _{2}\left( .;k\right)$ would be strictly larger than $\alpha _{1}\left( .;k\right)$ to the right of $b$, which would contradict (iii.1). No such $b$ then exists and we have proved (iii.2). $\square$

Lemma 1

Uniform upper bound on the rate of convergence of the weak bidder’s strategy:

$$\begin{aligned} \overline{\lim }_{k\rightarrow 0}\max _{v\in \left[ c,d\right] }\left( \max \left( \frac{v-\beta _{1}\left( v;k\right) }{k},\frac{\alpha _{1}\left( v;k\right) -v}{k}\right) -\rho _{1}\left( v\right) ^{-1}\right) \le 0. \end{aligned}$$

Proof

From Corollary 1 (iii), we have for $w\le v$:

$$\begin{aligned} \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( v;k\right) \right) }\le \left( \frac{F_{1}\left( w\right) }{ F_{1}\left( v\right) }\right) . \end{aligned}$$

Consequently, from Corollary 1 (ii) we find:

$$\begin{aligned} 0\le \frac{v-\beta _{1}\left( v;k\right) }{k}\le \frac{1}{k} \int _{c}^{v}\left( \frac{F_{1}\left( w\right) }{F_{1}\left( v\right) } \right) ^{1/k}\hbox {d}w, \end{aligned}$$

(11)

from which it follows:

$$\begin{aligned} 0&\le \overline{\lim }_{k\rightarrow 0}\frac{v-\beta _{1}\left( v;k\right) }{k} \\&\le \lim _{k\rightarrow 0}\frac{1}{k}\int _{c}^{v}\left( \frac{F_{1}\left( w\right) }{F_{1}\left( v\right) }\right) ^{1/k}\hbox {d}w \\&= \frac{F_{1}\left( v\right) }{f_{1}\left( v\right) }, \end{aligned}$$

where the equality is from Lemma 2 (i) below.

That the upper bound is uniform over $\left[ c,d\right]$, that is,

$$\begin{aligned} \overline{\lim }_{k\rightarrow 0}\max _{v\in \left[ c,d\right] }\left( \frac{ v-\beta _{1}\left( v;k\right) }{k}-\frac{F_{1}\left( v\right) }{f_{1}\left( v\right) }\right) \le 0, \end{aligned}$$

(12)

is a consequence of Lemma 2 (ii) (here, we use our assumption of bounded elasticity $\varepsilon _{1}$). We have proved the statement about $\beta _{1}\left( v;k\right)$.

Let $\varepsilon$ and $\gamma$ be arbitrary strictly positive numbers. From (12), there exists $k^{\prime }>0$ such that for all $0<k<k^{\prime }$ we have:

$$\begin{aligned} \frac{v-\beta _{1}\left( v;k\right) }{k}\le \frac{F_{1}\left( v\right) }{ f_{1}\left( v\right) }+\varepsilon , \end{aligned}$$

for all $v$ in $\left[ c,d\right]$. Applying the inequality above to $v=\alpha _{1}\left( b;k\right)$, for all $b$ in $\left[ c,d\right]$, we find:³¹

$$\begin{aligned} 0&\le \frac{\alpha _{1}\left( b;k\right) -b}{k} \nonumber \\&\le \frac{\alpha _{1}\left( b;k\right) -\beta _{1}\left( \alpha _{1}\left( b;k\right) ;k\right) }{k} \nonumber \\&\le \frac{F_{1}\left( \alpha _{1}\left( b;k\right) \right) }{f_{1}\left( \alpha _{1}\left( b;k\right) \right) }+\varepsilon . \end{aligned}$$

(13)

As $\rho _{1}^{-1}\left( v\right) =F_{1}\left( v\right) /f_{1}\left( v\right)$ is bounded (it tends towards zero when $v$ tends towards $c$), we obtain that $\alpha _{1}\left( b;k\right)$ tends towards $b$ uniformly over $\left[ c,d\right]$ when $k$ tends towards zero and the statement about $\alpha _{1}\left( b;k\right)$ then follows from (13). $\square$

Lemma 2

Technical lemma: Let $G$ be a continuous function over $\left[ c,d\right]$ that is continuously differentiable and strictly positive over $\left( c,d\right]$ with a strictly positive derivative $g$ over this semi-open interval. Then:

• (i) For all $v$ in $\left( c,d\right]$, we have:

  $$\begin{aligned} \lim _{l\rightarrow +\infty }l\int _{c}^{v}\left( \frac{G\left( w\right) }{ G\left( v\right) }\right) ^{l}\hbox {d}w=\frac{G\left( v\right) }{g\left( v\right) }. \end{aligned}$$

• (ii) For all $v$ in $\left( c,d\right]$, we have:

  $$\begin{aligned} \lim _{l\rightarrow +\infty }\int _{c}^{v}\left( \frac{G\left( w\right) }{ G\left( v\right) }\right) ^{l}\hbox {d}w=0, \end{aligned}$$

  and the convergence is uniform in $v$ over $\left( c,d\right]$ if $G$ is log-concave over an interval $\left[ c,c+\eta \right]$, with $\eta >0$, $G\left( c\right) =0$, $g$ is continuously differentiable over $\left( c,d\right]$ and its elasticity $\varepsilon$ with respect to $G$ is bounded from below over $\left[ c,d\right]$.

Proof

See the online supplements. $\square$

1.3 Proof of Theorem 2

From Lemma 1 above, $\beta _{1}\left( v;k\right)$ and $\alpha _{1}\left( v;k\right)$ are continuous in $k$ uniformly with respect to $v$ in $\left[ c,d\right]$. The joint continuity at all such points follows.

We next prove the statement about $\beta _{2}$ and $\alpha _{2}$. For all $v>c$, $\beta _{2}\left( v;k\right)$ must be optimal for bidder $2$ with value $v$ and in particular better than the bid $v$. After integration by parts, his expected payoff $\left( 1-k\right) \int _{c}^{v}\left( v-b\right) dF_{1}\left( \alpha _{1}\left( b\right) \right)$ if he submits $v$ is:

$$\begin{aligned} \left( 1-k\right) \int _{c}^{v}F_{1}\left( \alpha _{1}\left( b;k\right) \right) db \end{aligned}$$

and if he submits $\beta _{2}\left( v;k\right)$:

$$\begin{aligned}&k\left( v-\beta _{2}\left( v;k\right) \right) F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v;k\right) ;k\right) \right) \\&\qquad +\,\left( 1-k\right) \int _{c}^{\beta _{2}\left( v;k\right) }\left( v-b\right) dF_{1}\left( \alpha _{1}\left( b;k\right) \right) \\&\quad =\left( v-\beta _{2}\left( v;k\right) \right) F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v;k\right) ;k\right) \right) +\left( 1-k\right) \int _{c}^{\beta _{2}\left( v;k\right) }F_{1}\left( \alpha _{1}\left( b;k\right) \right) db. \end{aligned}$$

Consequently, we must have:

$$\begin{aligned}&\left( v-\beta _{2}\left( v;k\right) \right) F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v;k\right) ;k\right) \right) \\&\quad =\int _{\beta _{2}\left( v;k\right) }^{v}F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v;k\right) ;k\right) \right) db \\&\quad \ge \left( 1-k\right) \int _{\beta _{2}\left( v;k\right) }^{v}F_{1}\left( \alpha _{1}\left( b;k\right) \right) db. \end{aligned}$$

and then:

$$\begin{aligned}&k\left( d-c\right) \nonumber \\&\quad \ge k\int _{\beta _{2}\left( v;k\right) }^{v}F_{1}\left( \alpha _{1}\left( b;k\right) \right) db \nonumber \\&\quad \ge \int _{\beta _{2}\left( v;k\right) }^{v}\left( F_{1}\left( \alpha _{1}\left( b;k\right) \right) -F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v;k\right) ;k\right) \right) \right) db \nonumber \\&\quad \ge 0. \end{aligned}$$

(14)

Let $\left( v_{l},k_{l}\right) _{l\ge 1}$ be a sequence tending towards $\left( v,0\right)$ with $v$ in $\left( c,d\right]$ and such that $k_{l}>0$, for all $l$. Suppose $\beta _{2}\left( v_{l};k_{l}\right)$ does not tend towards $v$. Extracting a subsequence if necessary, we may assume that it tends towards $b^{\prime }\ne v$. As $\beta _{2}\left( v_{l};k_{l}\right) \le v_{l}$, we must have $b^{\prime }<v$. From (14) applied to $\left( v_{l},k_{l}\right)$, we have:

$$\begin{aligned}&k_{l}\left( d-c\right) \\&\quad \ge \int _{\beta _{2}\left( v_{l};k_{l}\right) }^{v_{l}}\left( F_{1}\left( \alpha _{1}\left( b;k_{l}\right) \right) -F_{1}\left( \alpha _{1}\left( \beta _{2}\left( v_{l};k_{l}\right) ;k_{l}\right) \right) \right) db \\&\quad \ge 0. \end{aligned}$$

Making $l$ tends towards $+\infty$ and using the joint continuity of $\alpha _{1}\left( b;k\right)$, we find:

$$\begin{aligned} \int _{b^{\prime }}^{v}\left( F_{1}\left( b\right) -F_{1}\left( b^{\prime }\right) \right) db=0, \end{aligned}$$

which is impossible. We have proved that $\beta _{2}\left( v;k\right)$ is continuous at $\left( v;0\right)$ jointly in $\left( v;k\right)$, for all $v>c$. The inequalities $c\le \beta _{2}\left( v;k\right) \le v$ imply that $\beta _{2}\left( v;k\right)$ tends towards $c$ if $\left( v;k\right)$ tends towards $\left( c;0\right)$, and we also obtain the continuity at $\left( c;0\right)$.

Proving the continuity of $\alpha _{2}$ from the result we have already established is straightforward.

At this point, the joint continuity of $\varphi$ at all $\left( v;k\right)$ is immediate.³² $\square$

Appendix 2

From (1), the rates of change with respect to $k$ of the direct and inverse bidding functions are related to the derivatives of these functions with respect to value and bid. Although from Theorem 2, $\varphi \left( .;k\right)$ approaches the identity function, the value of its derivative does not approach one everywhere. Indeed, dividing the second equation in (1) by the first, we find:

$$\begin{aligned} \frac{\hbox {d}\ln F_{2}\left( \alpha _{2}\left( b;k\right) \right) }{\hbox {d}\ln F_{1}\left( \alpha _{1}\left( b;k\right) \right) }=\frac{\alpha _{2}\left( b;k\right) -b}{\alpha _{1}\left( b;k\right) -b}; \end{aligned}$$

(15)

which combined with power relation and the condition (3), implies:

$$\begin{aligned} \varphi ^{\prime }\left( d;k\right) =\frac{1}{l}. \end{aligned}$$

Thus, when the values are differently distributed, $\varphi ^{\prime }\left( d;k\right)$ stays away from one.

To acquire information about the behaviour of the derivative of $\varphi$, we differentiate (15) and, after rearranging, we find:

$$\begin{aligned}&\frac{\hbox {d}}{\hbox {d}b}\ln \frac{\hbox {d}\ln F_{2}\left( \alpha _{2}\left( b;k\right) \right) }{\hbox {d}\ln F_{1}\left( \alpha _{1}\left( b;k\right) \right) } \nonumber \\&\quad =\frac{\left( \alpha _{2}\left( b;k\right) +k\frac{F_{2}\left( \alpha _{2}\left( b;k\right) \right) }{f_{2}\left( \alpha _{2}\left( b;k\right) \right) }\right) -\left( \alpha _{1}\left( b;k\right) +k\frac{F_{1}\left( \alpha _{1}\left( b;k\right) \right) }{f_{1}\left( \alpha _{1}\left( b;k\right) \right) }\right) }{\left( \alpha _{1}\left( b;k\right) -b\right) \left( \alpha _{2}\left( b;k\right) -b\right) } \nonumber \\&\quad =\frac{\gamma _{1}\left( \alpha _{2}\left( b;k\right) ;k\right) -\gamma _{2}\left( \alpha _{1}\left( b;k\right) ;k\right) }{\left( \alpha _{1}\left( b;k\right) -b\right) \left( \alpha _{2}\left( b;k\right) -b\right) }, \end{aligned}$$

(16)

where the functions $\gamma _{1},\gamma _{2}$ are defined as follows:

$$\begin{aligned} \gamma _{i}\left( v;k\right) =v+\frac{k}{\rho _{j}\left( v\right) }, \end{aligned}$$

for all $v$ in $\left( c,d\right]$ and $i,j=1,2$, $i\ne j$. Thus, $\gamma _{i}\left( .;k\right)$ would be bidder $i$’s inverse bidding function if bidder $j$ bid truthfully. The following properties of these functions are simple to prove.

Lemma 3

There exist $\zeta ,\mu >0$ such that $\gamma _{1}\left( .;.\right)$, $\gamma _{2}\left( .;.\right)$ can be continuously extended to $\left[ c,d+\mu \right) \times \left( -\zeta ,\zeta \right)$ in such a way that, for $i\ne j$:

• (i) $\gamma _{i}\left( c;k\right) =c$, for all $k$ in $\left( -\zeta ,\zeta \right)$;

• (ii) Over $\left( c,d+\mu \right) \times \left( -\zeta ,\zeta \right)$: $\gamma _{i}\left( .;.\right)$ is continuously differentiable; the derivative $\gamma _{i}^{\prime }\left( .;.\right)$ with respect to $b$ is strictly positive and bounded away from zero; $\frac{\partial }{\partial k}\gamma _{i}\left( .;.\right)$ is strictly positive and bounded from above; and $\gamma _{i}\left( d+\mu ;k\right) >d$, for all $k$ in $\left( -\zeta ,\zeta \right)$.

• (iii) Over $\left( c,d\right] \times \left( -\zeta .\zeta \right)$: $\frac{\partial }{\partial k}\gamma _{i}^{-1}\left( .;.\right)$ is strictly negative, bounded from below, and equal to $-\left( \gamma _{i}^{\prime }\left( \gamma _{i}^{-1}\left( v;k\right) ;k\right) \rho _{j}\left( \gamma _{i}^{-1}\left( v;k\right) \right) \right) ^{-1}$.

Proof

See the online supplements.

The sign of the expression (16) does not depend directly on $b$: $\ln F_{2}\left( \varphi \left( v;k\right) \right)$ is concave or convex with respect to $\ln F_{1}\left( v\right)$ depending on whether $\gamma _{1}\left( \varphi \left( v;k\right) ;k\right)$ is smaller or larger than $\gamma _{2}\left( v;k\right)$. We are lead to consider the functions $\gamma _{1}^{-1}\left( \gamma _{2}\left( .;k\right) ;k\right) ,\varphi \left( .;k\right)$, and the identity function in the space of logarithms of cumulative probabilities. We denote the new functions $\Psi \left( .;k\right) ,\Phi \left( .;k\right)$, and $\Lambda$ and define them formally below. The linearity of $\Lambda$ follows immediately from power relation.

Definitions

Let $k$ be in $\left( 0,\zeta \right)$, where $\zeta >0$ is as in Lemma 3 above. Let $\Psi \left( .;k\right) ,\Phi \left( .;k\right)$, and $\Lambda$ be the following functions:

• (i)
  $$\begin{aligned} \Psi \left( u;k\right) =\ln F_{2}\left( \gamma _{1}^{-1}\left( \gamma _{2}\left( F_{1}^{-1}\left( \exp u\right) ;k\right) ;k\right) \right) , \end{aligned}$$

  for all $u<\ln F_{1}\left( x\left( k\right) \right)$, where $x\left( k\right) \in \left( c,d\right]$ is defined as follows:

  $$\begin{aligned} x\left( k\right) =\gamma _{2}^{-1}\left( \gamma _{1}\left( d;k\right) ;k\right) ; \end{aligned}$$

  and

  $$\begin{aligned} \Psi \left( u;k\right) =0, \end{aligned}$$

  for all $u$ in $\left[ \ln F_{1}\left( x\left( k\right) \right) ,0\right]$;

• (ii)
  $$\begin{aligned} \Phi \left( u;k\right) =\ln F_{2}\left( \varphi \left( F_{1}^{-1}\left( \exp u\right) ;k\right) \right) , \end{aligned}$$

  for all $u\le 0$;

• (iii)
  $$\begin{aligned} \Lambda \left( u\right) =lu, \end{aligned}$$

  for all $u\le 0$.

As $\rho _{2}=l\rho _{1}$, we have $\gamma _{2}\left( .;k\right) \ge$ $\gamma _{1}\left( .;k\right)$ and $x\left( k\right)$ in (i) above is well defined. All functions above are continuous. The function $\Psi \left( .;k\right)$ is strictly increasing over $\left( -\infty ,\ln F_{1}\left( x\left( k\right) \right) \right)$ and equal to zero over $\left[ \ln F_{1}\left( x\left( k\right) \right) ,0\right]$. The function $\Phi \left( .;k\right)$ strictly increases from $-\infty$ to zero.

Figure 3 displays a possible configuration of the graphs of these functions. From Corollary 1 (iii), $\Phi \left( .;k\right)$ is at least equal to $\Lambda$, and strictly above it over the interior of its definition domain if $l>1$. Because $\Psi \left( .;k\right)$ reaches zero already at $\ln F_{1}\left( x\left( k\right) \right)$ and $\Phi \left( .;k\right)$ only at zero, $\Psi \left( .;k\right)$ is above $\Phi \left( .;k\right)$ in the neighbourhood of the origin. If $l>1$, $\gamma _{2}\left( .;k\right) >\gamma _{1}\left( .;k\right)$ and $\Psi \left( .;k\right)$ is strictly above $\Lambda$. As we already inferred from (16), the position of $\Phi \left( .;k\right)$ relative to $\Psi \left( .;k\right)$ determines the direction of its concavity. Thus, Lemma 4 below holds true.

Fig. 3

Possible configuration of the functions $\Psi$, $\Phi$, and $\Lambda$ when $l>1$

Lemma 4

If $l>1$:

• (i) $\Psi \left( u;k\right) >\Lambda \left( u\right)$ , for all $u\le 0$;

• (ii) $\Psi \left( u;k\right) >\Phi \left( u;k\right)$ , for all $u$ in $\left( \ln F_{1}\left( x\left( k\right) \right) ,0\right)$.

• (iii) $\Phi \left( u;k\right) >\Lambda \left( u\right)$ , for all $u<0$;

• (iv) $\Phi ^{\prime \prime }\left( u;k\right) >$ ($<$; $=$) $0$ if and only if $\Phi \left( u;k\right) >$ ($<$; $=$) $\Psi \left( u;k\right)$ , for all $u<0$.

From the explicit expression in the definition of $\Psi \left( .;k\right)$, it is simple to prove that it and its derivative tend towards $\Lambda$ and its derivative. From its definition and Theorem 2, $\Phi \left( .;k\right)$ tends towards $\Lambda$. The downward-pointing arrows in Figure 3 above represent these convergences.

From Lemma 4, we formally prove in the online supplements that the derivative of $\Phi \left( .;k\right)$ converges towards the derivative of $\Lambda$ compactly over $\left( -\infty ,0\right)$, that is, uniformly over every compact subinterval $K$ of $\left( -\infty ,0\right)$. The main ideas of the proof are as follows. From the convergence of $\Psi \left( .;k\right)$ and its derivative, there exists $k^{\prime }$ such $\Psi ^{\prime }\left( .;k\right)$ is close to $l$ over $K$ and smaller or only possibly slightly larger than $l$ over $\left[ \min K,0\right]$, for all $0<k<k^{\prime }$. If, at some point³³ $s$ in $K$ and for small $k$, the derivative of $\Phi \left( .;k\right)$ was further above $l$ than $\Psi ^{\prime }\left( .;k\right)$ is over $\left[ \min K,0\right]$, $\Phi \left( .;k\right)$ could not at the same time tend towards $\Lambda$, remain above it, and be strictly increasing. For example, if $\Phi ^{\prime }\left( .;k\right)$ at a point $s$ was further above $l$ than $\Psi ^{\prime }\left( .;k\right)$ is over $\left[ \min K,0\right]$ and if $\Phi \left( s;k\right)$ was not smaller than $\Psi \left( s;k\right)$, $\Phi \left( .;k\right)$ would be convex at $s$ (from Lemma 4 (iv)), its derivative would even be higher and $\Phi \left( .;k\right)$ would hence never meet $\Psi \left( .;k\right)$ to the right of $s$. As depicted in Fig. 4, $\Phi \left( .;k\right)$ would then be equal to zero to the left of $\ln F_{1}\left( x\left( k\right) \right)$; something which is impossible as $\Phi \left( u;k\right)$ vanishes only at $u=0$.

Fig. 4

Ruling out $\Phi ^{\prime }\left( .;k\right)$ further above $l$ than $\Psi ^{\prime }\left( .;k\right)$ is while $\Phi \left( .;k\right)$ is not smaller than $\Psi \left( .;k\right)$

If $\Phi ^{\prime }\left( .;k\right)$ at a point $s$ was again further above $l$ than $\Psi ^{\prime }\left( .;k\right)$ is over $\left[ \min K,0 \right]$, but if $\Phi \left( s;k\right)$ was now less than or equal to $\Psi \left( s;k\right)$, $\Phi \left( .;k\right)$ would be concave at $s$ (from Lemma 4 (iv)), its derivative would be higher and hence $\Phi \left( .;k\right)$ would never meet $\Psi \left( .;k\right)$ and would remain concave to the left of $s$. However, as the derivative of $\Lambda$ is the constant $l$, a derivative of $\Phi \left( .;k\right)$ uniformly bounded away from $l$ to the left of $s$ is incompatible with $\Phi \left( .;k\right)$ tending everywhere towards $\Lambda$. For an illustration of this case, see Fig. 5. Ruling out $\Phi ^{\prime }\left( .;k\right)$ further below $l$ than $\Psi ^{\prime }\left( .;k\right)$ is over $K$ proceeds along similar lines.

We have Lemma 5 below.³⁴

Fig. 5

Ruling out $\Phi ^{\prime }\left( .;k\right)$ further above $l$ than $\Psi ^{\prime }\left( .;k\right)$ is while $\Phi \left( .;k\right)$ is not larger than $\Psi \left( .;k\right)$

Lemma 5

If the function spaces below are endowed with the compact-open topology (the topology of uniform convergence over every compact subset), we have:

• (i) $\lim _{k\rightarrow 0}x\left( k\right) =d$;

• (ii) $\Psi ^{-1}\left( .;k\right)$ tends towards $\Lambda ^{-1}$ in $C^{1}\left( \mathbb {R}_{-},\mathbb {R}\right)$;

• (iii) $\ \Phi \left( .;k\right)$ tends towards $\Lambda$ in $C^{1}\left( \left( -\infty ,0\right) ,\mathbb {R}\right)$;

• (iv) $\overline{\lim }_{\left( s;k\right) \rightarrow \left( 0;0\right) }\Phi ^{\prime }\left( s;k\right) \le l$;

• (v) $\overline{\lim }_{\left( s;k\right) \rightarrow \left( -\infty ;0\right) }\Psi ^{\prime }\left( s;k\right) , \overline{\lim }_{\left( s;k\right) \rightarrow \left( -\infty ;0\right) }\Phi ^{\prime }\left( s;k\right) \le l$. .

Proof

See the online supplements.³⁵ $\square$

Thanks to Lemma 5, we can prove that, away from the extremities of the value interval, the slope of $\varphi \left( .;k\right)$ tends towards one. We list this and other useful consequences from Lemma 5 about the limits of derivatives in Theorem 6 below.

Theorem 6

Convergence of the derivatives with respect to value or bid in the 2-k-PA: Extend the derivatives $\beta _{i}^{\prime }\left( .;k\right)$, $\alpha _{i}^{\prime }\left( .;k\right) ,\varphi ^{\prime }\left( .;k\right)$ to $k=0$ as follows:

$$\begin{aligned} \beta _{i}^{\prime }\left( v;0\right) =\alpha _{i}^{\prime }\left( v;0\right) =\varphi ^{\prime }\left( v;0\right) =1, \end{aligned}$$

for all $v$ in $\left( c,d\right)$ and all $i=1,2$. Then:³⁶

• (i) $\beta _{i}^{\prime }\left( v;k\right) ,\alpha _{i}^{\prime }\left( v;k\right) ,\varphi ^{\prime }\left( v;k\right)$ are continuous jointly in both variables at $\left( v;0\right)$, for all $v$ in $\left( c,d\right)$;

• (ii) $\overline{\lim }_{\left( v;k\right) \rightarrow \left( d;0\right) }\varphi ^{\prime }\left( v;k\right) \le 1$;

• (iii) $\overline{\lim }_{\left( v;k\right) \rightarrow \left( c;0\right) }\frac{\hbox {d}\ln F_{2}\left( \varphi \left( v;k\right) \right) }{\hbox {d}\ln F_{1}\left( v\right) }\le l$.

Proof of (i):

• (a) For all $u$ in $\left( -\infty ,0\right)$, the first equality below follows from Lemma 5 (iii):

  $$\begin{aligned} l&= \lim _{\left( s,k\right) \rightarrow \left( u,0\right) }\Phi ^{\prime }\left( s;k\right) \\&= \lim _{\left( s,k\right) \rightarrow \left( u,0\right) }\frac{lf\left( \varphi \left( F^{-1}\left( \exp s\right) ;k\right) \right) }{F\left( \varphi \left( F^{-1}\left( \exp s\right) ;k\right) \right) }\frac{\exp s}{ f\left( F^{-1}\left( \exp s\right) \right) }\varphi ^{\prime }\left( F^{-1}\left( \exp s\right) ;k\right) \\&= l\lim _{\left( s,k\right) \rightarrow \left( u,0\right) }\varphi ^{\prime }\left( F^{-1}\left( \exp s\right) ;k\right) . \end{aligned}$$

  The second equality follows from the definition of $\Phi$ and the third from the joint continuity, according to Theorem 2, of $\varphi \left( v;k\right)$. Consequently, $\lim _{\left( s,k\right) \rightarrow \left( u,0\right) }\varphi ^{\prime }\left( F^{-1}\left( \exp s\right) ;k\right)$, or equivalently $\lim _{\left( v,k\right) \rightarrow \left( w,0\right) }\varphi ^{\prime }\left( v;k\right)$, where $w\!=\!F^{-1}\left( \exp u\right) \!\in \! \left( c,d\right)$, exists and is equal to $1$.

• (b) Let $v$ be in $\left( c,d\right)$ . From Corollary 1 (ii), we have $\beta _{1}\left( u;k\right) =u-\int _{c}^{u}\left( \frac{ F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\hbox {d}w$, for all $u$ in $\left( c,d\right)$. Differentiating this equation, we find:

  $$\begin{aligned} \beta _{1}^{\prime }\left( u;k\right)&= \frac{1}{k}\int _{c}^{u}\left( \frac{ F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\hbox {d}w\frac{f_{2}\left( \varphi \left( u;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\varphi ^{\prime }\left( u;k\right) \nonumber \\&= \frac{1}{k}\int _{c}^{e}\left( \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\hbox {d}w \frac{f_{2}\left( \varphi \left( u;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\varphi ^{\prime }\left( u;k\right) \nonumber \\&+\,\int _{e}^{u}\frac{F_{2}\left( \varphi \left( w;k\right) \right) }{\varphi ^{\prime }\left( w;k\right) f_{2}\left( \varphi \left( w;k\right) \right) } d\left( \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k} \nonumber \\&\cdot \,\frac{f_{2}\left( \varphi \left( u;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\varphi ^{\prime }\left( u;k\right) , \end{aligned}$$

  (17)

  where $e$ is a fixed number strictly between $c$ and $v$. The first term in the RHS of the last equality above tends towards zero if $\left( u;k\right)$ tends towards $\left( v;0\right)$. In fact, we have:

  $$\begin{aligned} 0&\le \frac{1}{k}\int _{c}^{e}\left( \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\hbox {d}w\frac{f_{2}\left( \varphi \left( u;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\varphi ^{\prime }\left( u;k\right) \\&\le \frac{1}{k}\left( \frac{F_{2}\left( \varphi \left( e;k\right) \right) }{ F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\frac{ f_{2}\left( \varphi \left( u;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\varphi ^{\prime }\left( u;k\right) , \end{aligned}$$

  and, from Theorem 2 and what we have just proved in (a) above, $\varphi \left( u;k\right)$ and $\varphi ^{\prime }\left( u;k\right)$ are jointly continuous in $\left( u;k\right)$ at $\left( v;0\right)$.

  We may consider $\left( \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}$ in the second term as a probability distribution over the couples $\left( w;t\right)$ that is the product of two distributions: the distribution over $\left[ e,u\right]$ whose cumulative function is $\left( \frac{F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}$ and hence that has a mass point at $e$; and the degenerate distribution at $k$. That is, the second term is $\mathop {\displaystyle \iint }\limits _{\left( e,u\right] \times \left[ 0,1\right] }\frac{F_{2}\left( \varphi \left( w;t\right) \right) }{\varphi ^{\prime }\left( w;t\right) f_{2}\left( \varphi \left( w;t\right) \right) }d\left( \left( \frac{ F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k}\times \delta _{\left\{ k\right\} }\left( t\right) \right)$. As $\left( u;k\right)$ tends towards $\left( v;0\right)$, this product distribution tends weakly towards the distribution concentrated at $\left( v;0\right)$. From Theorem 2 and (a) above, the integrand is a continuous function of $\left( u;t\right)$. Consequently, we have:

  $$\begin{aligned}&\lim _{\left( u;k\right) \rightarrow \left( v;0\right) }\int _{e}^{u}\frac{ F_{2}\left( \varphi \left( w;k\right) \right) }{\varphi ^{\prime }\left( w;k\right) f_{2}\left( \varphi \left( w;k\right) \right) }d\left( \frac{ F_{2}\left( \varphi \left( w;k\right) \right) }{F_{2}\left( \varphi \left( u;k\right) \right) }\right) ^{1/k} \\&\quad =\frac{F_{2}\left( v\right) }{f_{2}\left( v\right) }. \end{aligned}$$

  (17) then implies $\lim _{\left( u;k\right) \rightarrow \left( v;0\right) }\beta _{1}^{\prime }\left( u;k\right) =1$, for all $v$ in $\left( c,d\right)$.

• (c) Let $b$ be in $\left( c,d\right)$. For all $k$ and all $\widetilde{b}$ such that $c<\widetilde{b}<\beta \left( d;k\right) =\eta \left( k\right)$, we obviously have $\alpha _{1}^{\prime }\left( \widetilde{ b};k\right) =1/\beta _{1}^{\prime }\left( \alpha _{1}\left( \widetilde{b} ;k\right) ;k\right) .$ (b) above then implies $\lim _{\left( \widetilde{b} ;k\right) \rightarrow \left( b;0\right) }\alpha _{1}^{\prime }\left( \widetilde{b};k\right) =1$. From the identity $\alpha _{2}\left( \widetilde{b};k\right) =\varphi \left( \alpha _{1}\left( \widetilde{b} ;k\right) ;k\right)$, we have $\alpha _{2}^{\prime }\left( \widetilde{b} ;k\right) =\varphi ^{\prime }\left( \alpha _{1}\left( \widetilde{b};k\right) ;k\right) \alpha _{1}^{\prime }\left( \widetilde{b};k\right)$ and consequently $\lim _{\left( \widetilde{b};k\right) \rightarrow \left( b;0\right) }\alpha _{2}^{\prime }\left( \widetilde{b};k\right) =1$. Finally, from $\beta _{2}^{\prime }\left( u;k\right) =1/\alpha _{2}^{\prime }\left( \beta _{2}\left( u;k\right) ;k\right)$, for all $u$ in $\left( c,d\right)$ and $k$, we have $\lim _{\left( u;k\right) \rightarrow \left( v;0\right) }\beta _{2}^{\prime }\left( u;k\right) =1$, for all $v$ in $\left( c,d\right)$.

Proof of (ii): The proof of (ii) from Lemma 5 (iv) is similar to (a) in the proof of (i) above.

Proof of (iii): Proceeding again as in (a) in the proof of (i) above, (iii) is an immediate consequence of Lemma 5 (v). $\square$

Proof of Theorem 3 (i): Let $v$ be in $\left( c,d\right)$. From (1), we have $\left( \alpha _{i}\left( b;k\right) -b\right) /k=F_{j}\left( \alpha _{j}\left( b;k\right) \right) /\left( f_{j}\left( \alpha _{j}\left( b;k\right) \right) \alpha _{j}^{\prime }\left( b;k\right) \right)$, for all $b$ close enough to $v$ and all $k$ small enough. Letting $\left( b;k\right)$ tend towards $\left( v;0\right)$ and using Theorem 6 (i) and Theorem 2, we find the statement in (i) about $\alpha _{i}$.

We have $\left( u-\beta _{i}\left( u;k\right) \right) /k=\left( \alpha _{i}\left( \beta _{i}\left( u;k\right) ;k\right) -\beta _{i}\left( u;k\right) \right) /k$, for all $u$ close enough to $v$ and $k$ small enough. The statement in (i) about $\beta _{i}$ then follows from the statement about $\alpha _{i}$ and from Theorem 2.

From (15), we have:

$$\begin{aligned}&\frac{\varphi \left( u;k\right) -u}{k}\nonumber \\&\quad =\frac{u-\beta _{1}\left( u;k\right) }{k}\left\{ \frac{d\ln F_{2}\left( \varphi \left( u;k\right) \right) }{d\ln F_{1}\left( u\right) }-1\right\} \end{aligned}$$

(18)

$$\begin{aligned}&\quad =\frac{u-\beta _{1}\left( u;k\right) }{k}\left\{ l\varphi ^{\prime }\left( u;k\right) \frac{f_{1}\left( \varphi \left( u;k\right) \right) F_{1}\left( u\right) }{F_{1}\left( \varphi \left( u;k\right) \right) f_{1}\left( u\right) }-1\right\} . \end{aligned}$$

(19)

Letting $\left( u;k\right)$ tend towards $\left( v;0\right)$ and using the statement about $\beta _{1}$ and Theorem 6 (i) and Theorem 2, we obtain the statement in (i) about $\varphi$. $\square$

Lemma 6

Uniform upper bound on the rate of convergence of $\mathbf {\varphi }$: $\overline{\lim }_{k\rightarrow 0}\max _{u\in \left[ c,d\right] }\left( \frac{\varphi \left( u;k\right) -u}{k}-\left( l-1\right) \rho _{1}\left( u\right) ^{-1}\right) \le 0$.

Proof

Let $\gamma$ and $\varepsilon$ be arbitrary strictly positive numbers. From (18) and (19), Theorem 3 (i), which we have just proved, Lemma 1, and Theorem 6 (i, ii, iii), there exists $k^{\prime }>0$ such that for all $0<k<k^{\prime }$ and $u$ in $\left[ c,d\right]$, we have:

$$\begin{aligned}&\frac{\varphi \left( u;k\right) -u}{k} \\&\quad \le \left( \frac{F_{1}\left( u\right) }{f_{1}\left( u\right) } +\varepsilon \right) \left( l-1+\varepsilon \right) \\&\quad \le \frac{F_{1}\left( u\right) }{f_{1}\left( u\right) }\left( l-1\right) +\varepsilon \left( M+l-1+\varepsilon \right) , \end{aligned}$$

where $M$ is the maximum of $F_{1}/f_{1}$. Consequently, $\overline{\lim } _{k\rightarrow 0}\sup _{u\in \left[ c,d\right] }\left( \frac{\varphi \left( u;k\right) -u}{k}-\right. \left. \left( l-1\right) \frac{F_{1}\left( u\right) }{f_{1}\left( u\right) }\right)$ $\le \varepsilon \left( M+l-1+\varepsilon \right)$ and the lemma follows by making $\varepsilon$ tend towards zero. $\square$

Proof of Theorem 3 (ii): From Corollary 1 (iii), bidder 1 bids more aggressively and the differences $\Delta ER\left( k\right)$ and $\Delta ES\left( k\right)$ between the expected revenues and total surpluses from the 2-k-PA and the SPA are as follows:

$$\begin{aligned} \Delta ER\left( k\right)&= \int _{c}^{d}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( \omega _{1}\left( v_{1}\right) -\omega _{2}\left( v_{2}\right) \right) dF_{2}\left( v_{2}\right) dF_{1}\left( v_{1}\right) \\ \Delta ES\left( k\right)&= \int _{c}^{d}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( v_{1}-v_{2}\right) dF_{2}\left( v_{2}\right) dF_{1}\left( v_{1}\right) . \end{aligned}$$

The rate of change of the expected revenues can then be written as:

$$\begin{aligned} \frac{\Delta ER\left( k\right) }{k}=\int _{c}^{d}\frac{1}{k} \int _{v_{1}}^{\varphi \left( v_{1};k\right) }\Delta \left( v_{1},v_{2}\right) \hbox {d}v_{2}\hbox {d}v_{1}, \end{aligned}$$

(20)

where:

$$\begin{aligned} \Delta \left( v_{1},v_{2}\right)&= \left( \omega _{1}\left( v_{1}\right) -\omega _{2}\left( v_{2}\right) \right) f_{2}\left( v_{2}\right) f_{1}\left( v_{1}\right) \\&= \left( v_{1}f_{1}\left( v_{1}\right) -\left( 1-F_{1}\left( v_{1}\right) \right) \right) f_{2}\left( v_{2}\right) \\&-\,\left( v_{2}f_{2}\left( v_{2}\right) -\left( 1-F_{2}\left( v_{2}\right) \right) \right) f_{1}\left( v_{1}\right) , \end{aligned}$$

which is continuous over $\left[ c,d\right] ^{2}$. Let $K$ be the maximum of $\left| \Delta \left( v_{1},v_{2}\right) \right|$ over this square. From Lemma 6 above, there exists $k^{\prime }>0$ such that, for all $0<k<k^{\prime }$ and all $u$ in $\left( c,d\right)$, we have: $0\le \left( \varphi \left( u;k\right) -u\right) /k\le 1+\left( l-1\right) F_{1}\left( u\right) /f_{1}\left( u\right)$. As $F_{1}\left( u\right) /f_{1}\left( u\right)$ is bounded, there exists $L$ such that $0\le \left( \varphi \left( u;k\right) -u\right) /k\le L$, for all $u$ in $\left[ c,d \right]$ and $0<k<k^{\prime }$. For all $0<k<k^{\prime }$, we then have:

$$\begin{aligned} \left| \frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\Delta \left( v_{1},v_{2}\right) \hbox {d}v_{2}\right|&\le K\left( \frac{\varphi \left( v_{1};k\right) -v_{1}}{k}\right) \\&\le KL, \end{aligned}$$

for all $v_{1}$ in $\left[ c,d\right]$, and the integrand in the RHS of (20) is bounded.

From the previous paragraph, if we can find the almost everywhere pointwise limit of the integrand of (20), we will be entitled to apply the Lebesgue dominated convergence theorem. Let $v_{1}$ be in $\left( c,d\right)$. The integrand of (20) at $v_{1}$ can be rewritten as:

$$\begin{aligned}&\frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\Delta \left( v_{1},v_{2}\right) \hbox {d}v_{2} \\&\quad =\frac{\varphi \left( v_{1};k\right) -v_{1}}{k}\Delta \left( v_{1},v_{1}\right) +\frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( \Delta \left( v_{1},v_{2}\right) -\Delta \left( v_{1},v_{1}\right) \right) \hbox {d}v_{2}. \end{aligned}$$

Let $\varepsilon$ be an arbitrary strictly positive number. By continuity of $\Delta \left( v_{1},v_{2}\right)$ over $\left[ c,d\right] ^{2}$, there exists $\xi >0$ such that $\left| \Delta \left( v_{1},v_{2}\right) -\Delta \left( v_{1},v_{1}\right) \right| <\varepsilon$ for all $v_{2}$ such that $\left| v_{1}-v_{2}\right| <\xi$. From Theorem 2, there exists $k^{\prime \prime }$, which we may assume smaller than $k^{\prime }$, such that, for all $0<k<k^{\prime \prime }$, we have $\varphi \left( v_{1};k\right) -v_{1}<\xi$. We then obtain:

$$\begin{aligned}&\left| \frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( \Delta \left( v_{1},v_{2}\right) -\Delta \left( v_{1},v_{1}\right) \right) \hbox {d}v_{2}\right| \\&\quad \le \varepsilon \left( \frac{\varphi \left( v_{1};k\right) -v_{1}}{k}\right) \\&\quad \le \varepsilon L, \end{aligned}$$

for all $0<k<k^{\prime \prime }$. As $\varepsilon$ was arbitrary, we have proved

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( \Delta \left( v_{1},v_{2}\right) -\Delta \left( v_{1},v_{1}\right) \right) \hbox {d}v_{2}=0. \end{aligned}$$

From Theorem 3 (i), we also have $\lim _{k\rightarrow 0}\frac{\varphi \left( v_{1};k\right) -v_{1}}{k}=\frac{l-1}{l}\frac{F_{1}\left( v_{1}\right) }{ f_{1}\left( v_{1}\right) }$ and we have proved:

$$\begin{aligned}&\lim _{k\rightarrow +\infty }\frac{1}{k}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\Delta \left( v_{1},v_{2}\right) \hbox {d}v_{2} \\&\quad =\frac{l-1}{l}\frac{F_{1}\left( v_{1}\right) }{f_{1}\left( v_{1}\right) } \Delta \left( v_{1},v_{1}\right) \\&\quad =\frac{l-1}{l}\left( \omega _{1}\left( v_{1}\right) -\omega _{2}\left( v_{1}\right) \right) F_{1}\left( v_{1}\right) f_{2}\left( v_{1}\right) \\&\quad =\left( l-1\right) \left( \sigma _{2}\left( v_{1}\right) ^{-1}-\sigma _{1}\left( v_{1}\right) ^{-1}\right) F_{1}\left( v_{1}\right) ^{l}f_{1}\left( v_{1}\right) \end{aligned}$$

From the Lebesgue theorem of dominated convergence, $\frac{d_{r}}{dk} ER\left( 0\right) =\lim _{k\rightarrow _{>}0}\frac{\Delta ER\left( k\right) }{ k}$ exists and (4) holds true. That $\frac{d_{r}}{dk}ES\left( 0\right)$ exists and satisfies (5) can be proved similarly. If $F_{1},F_{2}$ are different, we have $l>1$ and $\sigma _{2}<\sigma _{1}$ over $\left( c,d\right)$. The strict positivity of $\frac{d_{r}}{dk}ER\left( 0\right)$ follows from (4). $\square$

Appendix 3

1.1 Proof of Theorem 5

Proof of (i): It follows directly from Theorem 3 (i) and the previous theorem. For example, from Theorem 3 (i), we have $\lim _{\left( \widetilde{b};k\right) \rightarrow \left( \widetilde{v};0\right) }\frac{ \widetilde{\alpha }_{i}\left( \widetilde{b};k\right) -\widetilde{b}}{k}= \widetilde{\rho }_{j}\left( \widetilde{v}\right) ^{-1}$. However, through the change of variables $\widetilde{b}=c+d-b$, $\widetilde{v}=c+d-v$, we have $\widetilde{\alpha }_{i}\left( \widetilde{b};k\right) -\widetilde{b} =b-\alpha _{i}\left( b\right)$, $\widetilde{\rho }_{j}\left( \widetilde{v} \right) =\frac{\widetilde{f}_{j}\left( \widetilde{v}\right) }{\widetilde{F} _{j}\left( \widetilde{v}\right) }=\frac{f_{j}\left( v\right) }{1-F_{j}\left( v\right) }=\sigma _{j}\left( v\right)$ and the result follows. The second inequality is proved similarly.

From Theorem 3 (i), we have $\lim _{\left( \widetilde{u},k\right) \rightarrow \left( \widetilde{v},0\right) }\frac{\widetilde{\varphi }\left( \widetilde{u} ;k\right) -\widetilde{u}}{k}=\frac{l-1}{l}\widetilde{\rho }_{2}\left( \widetilde{v}\right) ^{-1}$, where $\widetilde{\varphi }=\widetilde{\alpha } _{1}\widetilde{\beta }_{2}$ (as bidder 2 is the weak bidder for the transformed distributions). Compounding this equality with the joint continuity of $\widetilde{\varphi }^{-1}\left( \widetilde{u};k\right)$, from Theorem 2, we find $\lim _{\left( \widetilde{u},k\right) \rightarrow \left( \widetilde{v},0\right) }\frac{\widetilde{u}-\widetilde{\varphi } ^{-1}\left( \widetilde{u};k\right) }{k}=\frac{l-1}{l}\widetilde{\rho } _{2}\left( \widetilde{v}\right) ^{-1}$. The last statement of (i) then follows by changing the variables as above.

Proof of (ii):

• (a) From Lemma 6, we have

  $\overline{\lim }_{k\rightarrow 0}\max _{\widetilde{u}\in \left[ c,d\right] }\left( \frac{\widetilde{u}-\widetilde{\varphi }^{-1}\left( \widetilde{u};k\right) }{k}-\left( l-1\right) \widetilde{\rho }_{2}\left( \widetilde{\varphi }^{-1}\left( \widetilde{u};k\right) \right) ^{-1}\right) \le 0$, with $\widetilde{\varphi }=\widetilde{\alpha }_{1}\widetilde{\beta } _{2}$, and, from the joint continuity of $\widetilde{\varphi }^{-1}\left( \widetilde{u};k\right)$, we then have

  $\overline{\lim }_{k\rightarrow 0}\max _{\widetilde{u}\in \left[ c,d\right] }\left( \frac{\widetilde{u}-\widetilde{\varphi }^{-1}\left( \widetilde{u} ;k\right) }{k}-\left( l-1\right) \widetilde{\rho }_{2}\left( \widetilde{u} \right) ^{-1}\right) \le 0$. Changing the variables, we obtain $\overline{ \lim }_{k\rightarrow 0}\max _{u\in \left[ c,d\right] }\left( \frac{\varphi \left( u;k\right) -u}{k}-l\left( l-1\right) \sigma _{1}\left( u\right) ^{-1}\right) \le 0$.

• (b) As the expected payoff of either bidder with value $c$ is the premium $k\left( \theta \left( k\right) -c\right)$ he receives, the differences $\Delta ER^{A}\left( k\right)$ and $\Delta ES^{A}\left( k\right)$ between the expected revenues and total surpluses from the AA and the SPA are as follows:

  $$\begin{aligned} \Delta ER^{A}\left( k\right)&= \int _{c}^{d}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( \omega _{1}\left( v_{1}\right) \!-\!\omega _{2}\left( v_{2}\right) \right) dF_{2}\left( v_{2}\right) dF_{1}\left( v_{1}\right) \!-\!2k\left( \theta \left( k\right) \!-\!c\right) , \\ \Delta ES^{A}\left( k\right)&= \int _{c}^{d}\int _{v_{1}}^{\varphi \left( v_{1};k\right) }\left( v_{1}-v_{2}\right) dF_{2}\left( v_{2}\right) dF_{1}\left( v_{1}\right) . \end{aligned}$$

  The rate of change of the expected revenues can then be written as:

  $$\begin{aligned} \frac{\Delta ER^{A}\left( k\right) }{k}=\int _{c}^{d}\frac{1}{k} \int _{v_{1}}^{\varphi \left( v_{1};k\right) }\Delta \left( v_{1},v_{2}\right) \hbox {d}v_{2}\hbox {d}v_{1}-2\left( \theta \left( k\right) -c\right) , \end{aligned}$$

  where $\Delta \left( v_{1},v_{2}\right)$ is defined as in the proof of Theorem 3. From Theorem 2, $\theta \left( k\right) =c+d-\widetilde{\beta } _{i}\left( d;k\right)$ tends towards $c$ as $k$ tends towards zero. The rest of the proof proceeds as the proof of Theorem 3 and appeals to (i) and (a) above. $\square$