Optimal revenue-sharing mechanisms with seller commitment to ex-post effort

Abstract

This study considers the characterization and implementation of the optimal revenue-sharing mechanism when the seller’s ex-post effort affects the final outcome. In the optimal revenue-sharing mechanism with the seller’s full commitment to effort exertion, the seller commits to the first-best effort. Under the regularity condition guaranteed by the assumption of log-concave density, the optimal mechanism selects the bidder with the highest type, provided that the associated virtual surplus with the seller’s commitment to the first-best effort is positive. A first-price share auction with an appropriate reserve share and an effort commitment scheme can implement the optimal revenue-sharing mechanism. However, a second-price share auction might fail for implementation, such as in a case with two bidders and a uniform type distribution. Lastly, we introduce a sealed-bid share auction called the first-second price share auction, which can implement the optimal mechanism in dominant strategy.

Appendix

Proof of Lemma 1

First show “only if.” For any feasible revenue-sharing mechanism $(q,e, s) \in {\mathcal {M}}$, incentive compatibility implies that $Q_i(\theta _i)$ is nondecreasing. By the general envelope theorem (Milgrom and Segal 2002), incentive compatibility also implies

$$\begin{aligned} \begin{aligned} \pi _i(\theta _i)= \alpha _i(\theta _i)\theta _i-Q_i(\theta _i)k =\pi _i({\underline{\theta }}) + \int ^{{\theta }_i}_{{\underline{\theta }}}\alpha _i(\tau )d\tau . \end{aligned} \end{aligned}$$

(15)

Following Liu (2016), dividing (15) by $\theta _i^2$ and integrating both sides yield

$$\begin{aligned} \begin{aligned} \int ^{{\theta }_i}_{{\underline{\theta }}}\frac{\alpha _i(\theta ) }{\theta }d\theta - \int ^{{\theta }_i}_{{\underline{\theta }}}\frac{Q_i(\theta )k }{\theta ^2}d\theta =\int ^{{\theta }_i}_{{\underline{\theta }}}\frac{\pi _i({\underline{\theta }}) }{\theta ^2}d\theta +\int ^{{\theta }_i}_{{\underline{\theta }}}\frac{ \int ^{\theta }_{{\underline{\theta }}}\alpha _i(\tau )d\tau }{\theta ^2}d\theta , \end{aligned} \end{aligned}$$

then we have

$$\begin{aligned} \begin{aligned} \int ^{{\theta }_i}_{{\underline{\theta }}}\frac{Q_i(\theta )k }{\theta ^2}d\theta = \frac{\pi _i({\underline{\theta }})}{\theta _i} - \frac{\pi _i({\underline{\theta }})}{{\underline{\theta }}} +\frac{\int ^{\theta _i}_{{\underline{\theta }}}\alpha _i(\tau )d\tau }{\theta _i}. \end{aligned} \end{aligned}$$

(16)

Substituting $\int ^{\theta _i}_{{\underline{\theta }}}\alpha _i(\tau )d\tau$ from (15) into (16) yields

$$\begin{aligned} \begin{aligned} \pi _i(\theta _i)&= \frac{\pi _i({\underline{\theta }})\theta _i}{{\underline{\theta }}} +k\theta _i\int ^{{\theta }_i}_{{\underline{\theta }}}\frac{Q_i(\theta )}{\theta ^2}d\theta . \end{aligned} \end{aligned}$$

(17)

Furthermore, individual rationality requires $\pi _i({\underline{\theta }}) \ge 0$ for all i.

Now prove “if.” For any $\theta _i, \theta '_i \in [{\underline{\theta }}, {\overline{\theta }}]$, if $Q_i(\theta _i)$ is nondecreasing, then

$$\begin{aligned} \begin{aligned}&\pi _i(\theta _i) -( \alpha _i( \theta '_i)\theta _i-Q_i( \theta '_i)k) \\ & = \theta _i\left( \frac{\pi _i({\underline{\theta }})}{{\underline{\theta }}}+\int ^{{\theta }_i}_{{\underline{\theta }}}\frac{Q_i(\theta )k }{\theta ^2}d\theta \right) - {\theta }_i\left( \frac{\pi _i({\underline{\theta }})}{{\underline{\theta }}} +\int ^{{ \theta }'_i}_{{\underline{\theta }}}\frac{Q_i(\theta )k }{\theta ^2}d\theta +\frac{Q_i(\theta '_i)k}{\theta _i'}\right) +Q_i( \theta '_i)k\\ & = -\theta _i\int ^{\theta '_i}_{{\theta }_i}\frac{Q_i(\theta )k }{\theta ^2}d\theta +\left( \frac{1}{\theta _i}-\frac{1}{\theta '_i}\right) {\theta }_iQ_i( \theta '_i)k\\ \ge&-\theta _iQ_i(\theta '_i)\int ^{\theta '_i}_{{\theta }_i}\frac{k }{\theta ^2}d\theta +\left( \frac{1}{\theta _i}-\frac{1}{\theta '_i}\right) {\theta }_iQ_i( \theta '_i)k\\ & = 0. \end{aligned} \end{aligned}$$

This implies that the revenue-sharing mechanism (q, e, s) is incentive compatible. If $\pi _i({\underline{\theta }}) \ge 0$, then $\frac{\pi _i(\theta _i)}{\theta _i} > \frac{\pi _i({\underline{\theta }})}{{\underline{\theta }}}\ge 0$, and thus $\pi _i(\theta _i) >0$ for all $\theta _i > {\underline{\theta }}$. $\square$

Proof of Lemma 2

Given that

$$\begin{aligned} \begin{aligned} J^*(\theta _i)&= \frac{\theta _i^2}{2 \uplambda }-\frac{k }{\theta _i}\cdot \frac{1-F(\theta _i)}{f(\theta _i)}\cdot \frac{\int ^{{\overline{\theta }}}_{{\theta }_i}\theta f(\theta ) d\theta }{\theta _i [1-F(\theta _i)]} - k \\&= \frac{\theta _i^2}{2 \uplambda }-\frac{k }{\theta _i}\cdot \frac{1-F(\theta _i)}{f(\theta _i)}\Big \{1 + \frac{\int ^{{\overline{\theta }}}_{{\theta }_i}[1- F(\theta )]d\theta }{\theta _i [1-F(\theta _i)]} \Big \} - k. \end{aligned} \end{aligned}$$

(18)

The log-concavity of density f implies the log-concavity of $1- F(\theta )$ and $\int _{\theta }^{{\overline{\theta }}}1-F(t)dt$, which further implies that the hazard rate $\frac{f(\theta )}{1-F(\theta )}$ and $\frac{1-F(\theta )}{\int _{\theta }^{{\overline{\theta }}}[1 - F(t)]dt}$ are both increasing in $\theta$ (see, Bagnoli and Bergstrom 2005). Therefore, $J^*(\theta )$ is strictly increasing over $[{\underline{\theta }}, {\overline{\theta }}]$. $\square$

Proof of Theorem 1

We first consider a relaxed problem as follows.

$$\begin{aligned} \begin{aligned} \max _{(q,s)}\;\;\;&\int _{{\varvec{\theta }}}\sum _{i=1}^nq_i({\varvec{\theta }})J^*(\theta _i)f({\varvec{\theta }})d{\varvec{\theta }} -\sum _{i=1}^n\frac{\pi _i({\underline{\theta }}){\mathbb {E}}[\theta _i]}{{\underline{\theta }}},\\ s.t. \;\;&\pi _i({\underline{\theta }}) \ge 0, \;\; \forall i \in N. \end{aligned} \end{aligned}$$

By Lemma 2, a revenue-sharing mechanism $(q^*,e^*, s^*)$ maximizes the seller’s expected payoff if and only if $\pi _i({\underline{\theta }})=0$ for each bidder i, and the associated optimal winning rule is

$$\begin{aligned} q_i^*({\varvec{\theta }})={\left\{ \begin{array}{ll} 1&{} \;\text {if}\; \theta _i > \max \{\max _{j\ne i}{\varvec{\theta }}_{-i}, \theta _r\}\\ 0 &{}\;\text {otherwise}, \end{array}\right. } \end{aligned}$$

(19)

for all i and ${\varvec{\theta }}$. Ties are randomly broken. As a result, bidder i’s expected winning probability

$$\begin{aligned} Q^*_i(\theta _i)= {\left\{ \begin{array}{ll} G(\theta _i)&{} \;\text {if}\; \theta _i \ge \theta _r\\ 0 &{}\;\text {otherwise} \end{array}\right. } \end{aligned}$$

is monotonic. Therefore, the allocation rule $q^*$ is also optimal for the seller’s original program (P). The optimal sharing rule $s^*$ is generally not unique, and can be constructed by Condition IC2 and $\pi _i({\underline{\theta }})=0$ for all i. $\square$

Proof of Proposition 1 and Corollary 1

We first show the monotonicity and upper-boundedness of $\xi$. We see that $\xi (\theta )$ is strictly increasing over $[\theta _r,{\overline{\theta }}]$, as for each $\theta \in (\theta _r,{\overline{\theta }})$,

$$\begin{aligned} \begin{aligned} \xi '(\theta ) & = \frac{\uplambda k}{\theta ^3}\Big [\Big ( \frac{\theta ^2 g(\theta )}{G^2(\theta )} + \frac{\theta }{G(\theta )}\Big )\int _{\theta _r}^{\theta }\frac{G(\tau )}{\tau ^2}d\tau + 1 \Big ]\\ >&0. \end{aligned} \end{aligned}$$

In addition, the upper-boundedness is obviously satisfied, since $\xi (\theta ) =1-\frac{\uplambda k}{\theta ^2}-\frac{\uplambda k}{G(\theta )\theta }\int _{\theta _r}^{\theta }\frac{G(\tau )}{\tau ^2}d\tau < 1$ for all $\theta \in [\theta _r, {\overline{\theta }}]$.

We then show that the first-price share auction with the reserve share $s_r$ and the effort commitment scheme $e^{I}$ admits a symmetric bidding equilibrium $(\beta ^I, \ldots , \beta ^I)$, in which $\beta ^I$ is given by

$$\begin{aligned} \beta ^{I}(\theta )={\left\{ \begin{array}{ll} \xi (\theta )&{} \;\;\text {if}\; \theta \ge \theta _r \\ \text {non-participation} &{}\;\;\text {otherwise}. \end{array}\right. } \end{aligned}$$

Given the other bidders following the bidding strategy $\beta ^{I}$, each bidder i with type $\theta _i$ obtains an expected winning probability given by

$$\begin{aligned} Q^{I}(b) = {\left\{ \begin{array}{ll} G(\beta ^{I-1}(b)) &{} \; \text {if}\;\; b\in [s_r, \xi ({\overline{\theta }}] \\ 1 &{} \; \text {if} \;b \in (\xi ({\overline{\theta }}), 1]\\ 0 &{} \; \text {if} \; b = \text {non-participation}, \end{array}\right. } \end{aligned}$$

and earns an expected payoff $\pi ^{I}(b, \theta _i)$ given by

$$\begin{aligned} \pi _i^{I}(b,\theta _i)= {\left\{ \begin{array}{ll} G\big (\xi ^{-1}(b)\big )[ \frac{(1-b)\theta _i\xi ^{-1}(b)}{\uplambda }-k] &{}\;\text {if} \; b \in [s_r, \xi ({\overline{\theta }})]\\ \frac{(1-b)\theta _i{\overline{\theta }}}{\uplambda }-k &{} \;\text {if} \; b \in [\xi ({\overline{\theta }}), 1]\\ 0, &{} \;\text {if}\; b = \text {non-participation} \end{array}\right. } \end{aligned}$$

(20)

from bidding $b \in \{\beta ^{I}(\theta )| \theta \in [{\underline{\theta }}, {\overline{\theta }}]\}$. By the expression of $\xi$, we then see that following the bidding strategy $\beta ^I$ yields bidder i a winning probability

$$\begin{aligned} Q^{I}(\beta ^{I}(\theta _i)) = {\left\{ \begin{array}{ll} G(\theta _i) &{} \; \text {if}\;\; \theta _i \in [\theta _r, {\overline{\theta }}] \\ 0 &{} \; \text {otherwise}, \end{array}\right. } \end{aligned}$$

which is weakly increasing in $\theta _i$, and an expected payoff

$$\begin{aligned} \pi _i^{I}(\beta ^{I}(\theta _i), \theta _i)= {\left\{ \begin{array}{ll} k \theta _i \int _{\theta _r}^{\theta _i}\frac{G(\theta )}{\theta ^2}d\theta &{} \;\text {if} \; \theta _i \in [\theta _r, {\overline{\theta }}]\\ 0 &{} \;\text {otherwise}, \end{array}\right. } \end{aligned}$$

if the other bidders adopt the same bidding strategy. By the revelation principle and Lemma 1, it holds that

$$\begin{aligned} \pi _i^{I}(\beta ^{I}(\theta _i), \theta _i) \ge \pi _i^{I}(b,\theta _i), \end{aligned}$$

for all $b \in \big \{\beta ^{I}(\theta )| \theta \in [{\underline{\theta }}, {\overline{\theta }}]/\{\theta _i\}\big \}$; that is, bidder i with type $\theta _i$ has no incentive to deviate from bidding $\beta ^{I}(\theta _i)$ to other types’ bids specified by the strategy $\beta ^{I}$. Moreover, we have that

$$\begin{aligned} \pi _i^{I}(\beta ^{I}(\theta _i), \theta _i) \ge \pi _i^{I}(\beta ^{I}({\overline{\theta }}),\theta _i) > \pi _i^{I}(b,\theta _i) \end{aligned}$$

for all $\theta _i$ and all $b \in (\beta ^{I}({\overline{\theta }}), 1]$. Therefore, the strategy profile $(\beta ^I, \ldots , \beta ^{I})$ is a symmetric bidding equilibrium of the first-price share auction with the reserve price $s_r$ and the effort commitment scheme $e^{I}$. In this equilibrium, the bidder with the highest type wins, provided that it is not less than the threshold type $\theta _r$, and each bidder i earns an expected equilibrium payoff that is the same as that in in the optimal mechanism $(q^*, e^*, s^*)$. Hence, we conclude that the proposed first-price share auction can implement the optimal revenue-sharing mechanism. $\square$

The derivation of equation (12). Suppose that the second-price share auction with the reserve share $s_r$ and an appropriate effort commitment scheme $e^{II}$ can implement the optimal mechanism, then it should admit a symmetric bidding equilibrium $(\beta ^{II}, \ldots , \beta ^{II})$, in which participating bidders follow a strictly increasing bidding strategy $\gamma$. Accordingly, the effort commitment scheme satisfies $e^{II}(b_w)=\frac{\gamma ^{-1}(b_w)}{\uplambda }$ for all $b_w\in \{ \gamma (\theta )| \theta \in [\theta _r, {\overline{\theta }}] \}$. Bidding a share $b \in \{ \gamma (\theta )| \theta \in [\theta _r, {\overline{\theta }}]$ yields bidder i with type $\theta _i$ an expected payoff $\pi ^{II}(b,\theta _i)$ given by

$$\begin{aligned} \pi _i^{II}(b,\theta _i)= \Big [G(\theta _r)(1-s_r)+\int _{\theta _r}^{\gamma ^{-1}(b)}(1-\gamma (\theta )) dG(\theta )\Big ]\cdot \frac{\gamma ^{-1}(b)\theta _i}{\uplambda }- k G(\gamma ^{-1}(b)). \end{aligned}$$

Furthermore, the equilibrium payoff of bidder i with type $\theta _i\in [\theta _r,\overline{\theta }]$ is given by

$$\begin{aligned} \begin{aligned} \pi _i^{II}(\gamma (\theta _i), \theta _i)& = \Big [G(\theta _r)(1-s_r) + \int _{\theta _r}^{\theta _i} \big (1-\gamma (\tau )) dG(\tau )\Big ]\frac{\theta _i^2}{\uplambda }- G(\theta _i)k\\ & = k \theta _i \int _{ \theta _r}^{\theta _i}\frac{ G(\tau )}{\tau ^2} d\tau \\ & = \pi _i^{I}(\xi (\theta _i),\theta _i) \\ & = G(\theta _i)\big [(1-\xi (\theta _i))\frac{\theta _i^2}{\uplambda } - k\big ], \end{aligned} \end{aligned}$$

(21)

where the second equality holds from the general envelope theorem (Milgrom and Segal 2002) and the zero-payoff argument for the threshold type $\theta _r$, and the third equality is obtained from Eq. (20). From Eq. (21), the expression of $\gamma$ can be derived out as

$$\begin{aligned} \begin{aligned} \gamma (\theta )&=1-\frac{\uplambda k}{\theta ^2}+\frac{\uplambda k}{\theta ^2 g(\theta )}\left[ \frac{G(\theta )}{\theta }+\int _{ \theta _r}^{\theta } \frac{G(\tau )}{\tau ^2}d\tau \right] \\&= \xi (\theta ) + \frac{G(\theta )}{g(\theta )}\xi '(\theta ). \end{aligned} \end{aligned}$$

(22)

Note that the equilibrium bid for the threshold type may be higher than the reserve share, as $\lim \nolimits _{\theta \rightarrow \theta _r}\gamma (\theta )=1-\frac{\uplambda k}{\theta ^2}+\frac{\uplambda k}{\theta ^2 g(\theta )}\frac{G(\theta _r)}{\theta _r}> s_r$. Hence, the prescribed equilibrium bidding strategy $\beta ^{II}$ satisfies:

$$\begin{aligned} \beta ^{II}(\theta )={\left\{ \begin{array}{ll} b_r \in [s_r, \gamma (\theta _r)]&{} \;\;\text {if}\; \theta = \theta _r\\ \gamma (\theta )&{} \;\;\text {if}\; \theta > \theta _r \\ \text {non-participation} &{}\;\;\text {otherwise}. \end{array}\right. } \end{aligned}$$

(23)

Proof of Proposition 2

The proof proceeds in two steps.

Step 1: We show that $\beta ^{12}$ is a weakly dominant strategy for each bidder under the share auction ${\mathcal {A}}^{12}$. Fix a bidder i. Let $b_{(1,-i)}$ denote the highest bid of the bidders other than i. If no other bidder participates, set $b_{(1,-i)} = s_r$. Bidding $b_i$ yields bidder i with $\theta _i$ a payoff given by

$$\begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i) = {\left\{ \begin{array}{ll} 0 &\text {if} \;\; b_i < b_{(1, -i)}\\ \theta _i \cdot \sqrt{(1- b_{(1, -i)})(1-b_i)}\cdot e^{12}(b_i) -k & \text {if} \;\;b_i> b_{(1, -i)} \\ \frac{1}{1 + m(b_{(1,-i)})}\big (\theta _i\cdot \sqrt{(1- b_{(1, -i)})(1-b_i)}\cdot e^{12}(b_i) -k\big ) & \text {if} \;\;b_i = b_{(1, -i)}, \end{array}\right. } \end{aligned}$$

where $m(b_{(1,-i)}) = |\{j \ne i|b_j = b_{(1, -i)}\}| \ge 1$ is the cardinality of the set of bidder i’s opponents with the highest bid.

It holds that

$$\begin{aligned} \sqrt{(1-b_i)} e^{12}(b_i) {\left\{ \begin{array}{ll} = \sqrt{\frac{k}{\uplambda }} &{}\text {if} \;\; b_i \in [s_r, \beta ^{12}(\overline{\theta })]\\ < \sqrt{\frac{k}{\uplambda }} &{}\text {if} \;\;b_i \in (\beta ^{12}(\overline{\theta }),1]. \end{array}\right. } \end{aligned}$$

(24)

We then claim that for $\theta _i \in [\theta _r, {\overline{\theta }}]$, bidding $\beta ^{12}(\theta _i)$ weakly dominates submitting any other bid $b_i\in [s_r, 1]$ and not participating. To prove it, we consider three cases as follows.

Case 1: $b_i>\beta ^{12}(\theta _i)$. Bidding $b_i$ is weakly dominated by bidding $\beta ^{12}(\theta _i)$, since:

(i) If $\beta ^{12}(\theta _i)<b_{(1, -i)}< b_i$, then

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i)&=\theta _i \cdot \sqrt{(1- b_{(1, -i)})(1-b_i)}\cdot e^{12}(b_i) -k\\&\le \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k \\&<0\\&=\pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i), \end{aligned} \end{aligned}$$

where the first inequality is due to Condition (24), which applies to (i), (ii), (iv) and (v) in this case.

(ii) If $b_{(1, -i)}= b_i$, then

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i)&\le \frac{1}{1 + m(b_{(1,-i)})}\left( \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k\right) \\&<0\\&=\pi ^{12}_i(\beta ^{12}(\theta _i), b_{(1,-i)}; \theta _i). \end{aligned} \end{aligned}$$

(iii) If $b_{(1, -i)}> b_i$, he loses and obtains the same zero payoff as bidding $\beta ^{12}(\theta _i)$.

(iv) If $b_{(1, -i)} < \beta ^{12}(\theta _i)$, then

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i) \le&\theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k \\ & = \pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i). \end{aligned} \end{aligned}$$

(v) If $b_{(1,-i)} = \beta ^{12}(\theta _i)$, his payoff is

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i) \le&\theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k \\ & = 0\\ & = \pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i). \end{aligned} \end{aligned}$$

Case 2: $b_i<\beta ^{12}(\theta _i)$. Bidding $\beta ^{12}(\theta _i)$ is weakly better than bidding $b_i$, since:

(i) If $\beta ^{12}(\theta _i)\ge b_{(1, -i)}>b_i$, then bidder i loses and obtains zero payoff that is worse off than bidding $\beta ^{12}(\theta _i)$, i.e.,

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i)&=0\\&<\theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k\\&=\pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i). \end{aligned} \end{aligned}$$

(ii) If $b_{(1, -i)}>\beta ^{12}(\theta _i)$, then bidder i loses and obtains the same zero payoff as bidding $\beta ^{12}(\theta _i)$.

(iii) If $b_{(1, -i)}= b_i$, then bidder i obtains payoff

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i)&=\frac{1}{1 + m(b_{(1,-i)})}\left( \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k\right) \\&< \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k\\&=\pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i). \end{aligned} \end{aligned}$$

(iv) If $b_{(1, -i)}<b_i$, then bidder i obtains the same payoff as bidding $\beta ^{12}(\theta _i)$, i.e.,

$$\begin{aligned} \begin{aligned} \pi _i(b_i, b_{(1, -i)};\theta _i) & = \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1- b_{(1, -i)})} -k \\ & = \pi _i(\beta ^{12}(\theta _i), b_{(1, -i)};\theta _i). \end{aligned} \end{aligned}$$

Case 3: Non-participation. Obviously, not participating that yields zero payoff is weakly dominated by $\beta ^{12}(\theta _i)$.

Consequently, for type $\theta _i \in [\theta _r, {\overline{\theta }}]$, bidding $\beta ^{12}(\theta _i)$ is weakly better than submitting any other bid $b_i \ne \beta ^{12}(\theta _i)$ and not participating.

In addition, we have that for $\theta _i \in [{\underline{\theta }}, \theta _r)$, participating and making a bid $b_i \in [s_r, 1]$ yields him a non-positive payoff upon winning, since

$$\begin{aligned} \theta _i\sqrt{(1- b_{(1,-i)})(1-b_i)}e(b_i) - k \le \theta _i \cdot \sqrt{\frac{k}{\uplambda }} \cdot \sqrt{(1 - b_{(1,-i)})} -k \le \left( \frac{\theta _i}{\theta _r}-1\right) \cdot k <0. \end{aligned}$$

The above demonstration shows that $\beta ^{12}$ is a weakly dominant strategy for each bidder under the first-second share auction with reserve price $s_r$ and effort exertion $e^{12}$.

Step 2: we consider the implementation. If all bidders adopt the weakly dominant strategy $\beta ^{12}$, then the winning bidder if there is any is the bidder with the highest type (denoted by $\theta _w$), which is not less than $\theta _r$, and the seller’s effort exertion equals to $\frac{\theta _w}{\uplambda }$. Recall that the seller’s expected payoff in the optimal mechanism is determined by the optimal allocation rule and the first-best effort commitment with respect to the winner’s type (see Theorem 1). We then conclude that the proposed share auction ${\mathcal {A}}^{12}$ can implement the optimal mechanism in dominant strategy. $\square$