An equilibrium analysis of a core-selecting package auction with reserve prices

Abstract

This study analyzes the equilibrium of a core-selecting package auction under incomplete information. The ascending proxy auction of Ausubel and Milgrom (Front Theor Econ 1:1–42, 2002) is considered in a stylized environment with two goods, two local bidders, and one global bidder. Local bidders shade bids in the equilibrium because of the free-riding incentive. We examine the effect of reserve prices. We show that a reserve price for individual goods increases the equilibrium local bids, whereas they may be decreased by a reserve price for the package of goods. A flexible non-monotonic reserve price rule can improve allocative efficiency as well as seller revenue in the equilibrium.

Proofs

1.1 Proof of Theorem 1

Suppose $G(V)=V/2$. Suppose that there exists a symmetric equilibrium bidding function $\beta$ which is continuous and increasing in the interior of its range. Then, the first order condition (2) for bidder 1 yields

$$\begin{aligned} v_1-\beta (v_1) F(v_1) \le \int _{v>v_1} \beta (v) f(v) \mathrm {d}v, \end{aligned}$$

(18)

where equality holds if $\beta (v_1)>0$. Evaluating (18) at $v_1=1$ yields $\beta (1)=1$. Hence, (18) yields

$$\begin{aligned} \int _{v_1}^1 \{ \beta (v)f(v) + \beta '(v)F(v) -1 \} \mathrm {d}v \le \int _{v_1}^1 \beta (v)f(v) \mathrm {d} v, \end{aligned}$$

and thus,

$$\begin{aligned} \int _{v_1}^1 \{ \beta '(v)F(v) -1 \} \mathrm {d}v \le 0. \end{aligned}$$

(19)

Because (19) holds for all $v_1$, we have $\beta '(v_1) =1 / F(v_1)$. Using the initial condition $\beta ^0 (1)=1$, we have

$$\begin{aligned} \beta ^0 (v_1) = \max \left\{ 0, v_1 - \int _{v_1}^1 \frac{1-F(v)}{F(v)} \mathrm {d}v \right\} . \end{aligned}$$

(20)

The marginal payoff function is decreasing in $b_1$ and satisfies the second-order condition. Therefore, the bidding function (20) is equilibrium.$\square$

1.2 Proof of Theorem 2

Assume that bidder 2 takes a strategy $\beta$ such that is increasing in the interior of its range and jumps at most once (at $\hat{v}$). Let

$$\begin{aligned} \bar{h} (b_1,v_1) \equiv (v_1-b_1 ) - \int _{b_1 < \beta (s)} (\beta (s)-b_1) f(s)\mathrm {d}s \end{aligned}$$

(21)

and¹⁶

$$\begin{aligned} \underline{h} (b_1,v_1) \equiv (v_1-b_1)(1-F(r)) - \int _{b_1 < \beta (s)} (\beta (s)-b_1)f(s) \mathrm {d}s. \end{aligned}$$

(22)

Case 1. Suppose $R \le r$. Because bidder 1’s bid is $b_1 \ge r$, the first-order condition for bidder 1 is given by

$$\begin{aligned} \bar{h}( \beta (v_1),v_1)= (\le ) \ 0. \end{aligned}$$

Given that G is uniform, this is the same as in the no reserve price case. Therefore, we have the equilibrium bidding function $\beta ^{*} (v;r,R) = \max \{ r,\beta ^0 (v) \}$ for $v \ge r$.

Case 2. Suppose $R \ge 1$. Suppose that there exists a symmetric equilibrium bidding function $\beta$ which is continuous and increasing in the interior of its range. Because bidder 1’s bid is $b_1 \le R$, the first-order condition for bidder 1 is given by

$$\begin{aligned} \underline{h}( \beta (v_1),v_1) = (\le ) \ 0. \end{aligned}$$

(23)

By symmetry, (23) yields

$$\begin{aligned} (1-F(r)) v- (F(v) -F(r) )\beta (v) \le \int _{v}^1 \beta (s) f(s) \mathrm {d}s, \end{aligned}$$

(24)

where equality holds if $\beta (v) > r$. Evaluating (24) at $v=1$ yields $\beta (1)=1$. Hence, (24) yields

$$\begin{aligned} \int _{v}^1 \{ (F(s)-F(r)) \beta ' (s) -(1-F(r)) \} \mathrm {d}s \le 0. \end{aligned}$$

(25)

Because (25) holds for all v, we have $\beta '(v) = \frac{1-F(r)}{F(v)-F(r)}$. Using the initial condition $\beta ^* (1)=1$, we have

$$\begin{aligned} \beta ^* (v) = \max \Bigl \{ r, v - \int _{v}^1 \frac{1-F(s)}{F(s)-F(r)} \mathrm {d}s \Bigr \}. \end{aligned}$$

(26)

The function $\underline{h}$ is decreasing in $b_1$ and satisfies the second-order condition. Therefore, the bidding function (26) is equilibrium.

Case 3. Suppose $r<R<1$ and $R \le 2r$. Suppose that bidder 2 takes a strategy $\beta$ such that $\exists \hat{v}$,

$$\begin{aligned} \beta (v_2) ={\left\{ \begin{array}{ll} \beta ^0 (v_2) &{} \text {if}\,\, v > \hat{v} \\ \underline{\beta } (v_2) &{} \text {if}\,\, r \le v< \hat{v} \end{array}\right. }. \end{aligned}$$

(27)

In the above strategy, $\beta ^0$ is given by (5), $\underline{\beta }$ is a nondecreasing function, and $\lim _{v \nearrow \hat{v}} \underline{\beta }(v) =\underline{\beta }(\hat{v}) <R \le \beta ^0 (\hat{v})$. Given this strategy, $\bar{h}$ and $\underline{h}$ is defined by (21) and (22). Then, bidder 1’s marginal payoff of increasing bid $b_1$ is given by

$$\begin{aligned} \frac{\partial }{\partial b_1} \pi _1 (b_1,v_1) = {\left\{ \begin{array}{ll} \frac{1}{2} \bar{h} (b_1,v_1) &{} \text {if}\,\, b_1 \ge R \\ \frac{1}{2} \underline{h} (b_1,v_1) &{} \text {if}\,\, r \le b_1 < R \end{array}\right. }. \end{aligned}$$

By inspection, both $\bar{h}$ and $\underline{h}$ are decreasing in $b_1$, and $\bar{h}(b_1,v_1) >\underline{h}(b_1,v_1)$ for all $v_1 >r$ and all $b_1 \in [r,v_1 )$. Therefore, given $v_1$, the optimal bid $b^*$ satisfies $b^* > R$ and $\bar{h} (b^*,v_1)=0$ if $\underline{h}(R,v_1) \ge 0$. In addition, the optimal bid satisfies $b^* <R$ and $\underline{h} (b^*,v_1) \le 0$ (in which equality holds for $b^*>r$,) if $\bar{h}(R,v_1) \le 0$.

Suppose that the optimal bid satisfies $b_1 \ge R$ and the upper first-order condition $\bar{h} =0$; i.e.,

$$\begin{aligned} v_1-b_1 -\int _{b_1 <\beta (s)} (\beta (s)-b_1) f(s)\mathrm {d}s =0. \end{aligned}$$

(28)

By Theorem 1, the symmetric bid $b_1 = \beta ^0 (v_1)$ satisfies the first-order condition.

Consider the case $\underline{h} (R ,v)<0 <\bar{h} (R,v)$. For such v, there are two locally optimal bids: one is $\beta ^0 (v)$ that satisfies $\bar{h} (\beta ^0 (v),v) =0$, and the other satisfies $\underline{h} (b,v) \le 0$. To show that bidder 2’s strategy $\beta$ forms an equilibrium, the two locally optimal bids yield the same expected payoff at the jump point $\hat{v}$. The upper locally optimal bid is $\beta ^0 (\hat{v})$. For now, we consider that the lower locally optimal bid satisfies $\underline{h}(b,\hat{v})=0$: i.e.,

$$\begin{aligned} (1-F(r))(\hat{v}-b) - \int _{b < \beta (s) } (\beta (s) -b)f(s) \mathrm {d}s =0. \end{aligned}$$

(29)

Symmetry requires that (29) holds with $b= \underline{\beta } (\hat{v})$. Because $\underline{\beta } (\hat{v}) < \beta (s)$ implies $s > \hat{v}$, (29) yields

$$\begin{aligned} (1-F(r))\hat{v} -(F(\hat{v})-F(r)) \underline{\beta } (\hat{v}) - \int _{\hat{v}}^{1} \beta (s)f(s)\mathrm {d}s =0. \end{aligned}$$

(30)

Because $\beta (s) =\beta ^0 (s)$ for $s >\hat{v}$ and using

$$\begin{aligned} \hat{v} -\beta ^0 (\hat{v}) F(\hat{v}) =\int _{\hat{v}}^1 \beta ^0 (s)f(s)\mathrm {d}s, \end{aligned}$$

(31)

we have

$$\begin{aligned} (1-F(r))\hat{v} -(F(\hat{v})-F(r)) \underline{\beta } (\hat{v}) - (\hat{v}-\beta ^0 (\hat{v}) F(\hat{v})) =0, \end{aligned}$$

(32)

which yields

$$\begin{aligned} \underline{\beta } (\hat{v}) = \frac{F(\hat{v}) \beta ^0 (\hat{v}) - F(r) \hat{v} }{F(\hat{v})-F(r)} . \end{aligned}$$

(33)

Suppose that $\hat{v}$ satisfies $\underline{\beta }(\hat{v})<R <\beta ^0 (\hat{v})$. Because $\underline{\beta }(\hat{v})$ and $\beta ^0 (\hat{v})$ give the same expected payoff, $\hat{v}$ is specified by

$$\begin{aligned}&\pi _1 (\beta ^0 (\hat{v}),\hat{v}) = \pi _1 (\underline{\beta } (\hat{v}), \hat{v}) \nonumber \\&\quad \Leftrightarrow \int _{\hat{\beta }(\hat{v})}^{R} \underline{h} (b,\hat{v})\mathrm {d}b +\int _{R}^{\beta ^0 (\hat{v})} \bar{h} (b,\hat{v})\mathrm {d}b=0. \end{aligned}$$

(34)

Since the light hand side of (34) is increasing in $\hat{v}$, (34) has at most one solution.

Case 3 (a). There exists the solution $\hat{v}$ of (34) and $r \le \hat{\beta }(\hat{v})<R <\beta ^0 (\hat{v})$.

Define $\hat{v}$ as the solution, and consider that bidder 2’s strategy is such that $\beta (v_2) = \beta ^0 (v_2)$ for $v_2 >\hat{v}$ and (33). Suppose $v_1<\hat{v}$. Consider that local bidders take a symmetric strategy derived from the local optimum condition $\underline{h}( \underline{\beta } (v),v) \le 0$. By symmetry, we have

$$\begin{aligned} (1-F(r))-(F(v)-F(r)) \underline{\beta } (v) - \int _{v}^{1}\beta (s)f(s)\mathrm {d}s \le 0. \end{aligned}$$

(35)

Using (32), (35) yields

$$\begin{aligned} \int _{v}^{\hat{v}} \bigl \{ (F(s)-F(r)) \underline{\beta }' (s)-(1-F(r)) \bigr \} \mathrm {d}s \le 0 . \end{aligned}$$

(36)

Thus, the locally optimal bidding function is given by $\underline{\beta }'(v) =\frac{1-F(r)}{F(v)-F(r)}$ with the initial condition (33). Therefore, the symmetric bidding function $\beta$ is specified by (10).

Finally, we verify that the derived locally optimal bid is globally optimal. Let $\bar{\Pi } (v) \equiv \pi _1 (\beta ^0 (v),v)$ be the expected payoff when bidder 1 bids $\beta ^0$ satisfying $\bar{h}(\beta ^0 (v),v) =0$. Similarly, $\underline{\Pi } (v) \equiv \pi _1 (\underline{\beta } (v),v)$ is the expected payoff when bidder 1 bids $\underline{\beta }$ satisfying $\underline{h}(\underline{\beta } (v),v) \le 0$. By construction, $\bar{\Pi } (\hat{v}) =\underline{\Pi }(\hat{v})$. By the envelope theorem, $\bar{\Pi }' (v) =\Phi (\beta ^0 (v))$ and $\underline{\Pi }' (v) = \Phi (\underline{\beta }(v))$ where $\Phi$ is winning probability of bidder 1. Because $\beta ^0 (v) >\underline{\beta } (v)$, we have $\bar{\Pi } (v) > \underline{\Pi }(v)$ for $v > \hat{v}$, and $\bar{\Pi } (v) < \underline{\Pi } (v)$ for $v< \hat{v}$. Therefore, strategy (10) is an equilibrium strategy.

Case 3 (b). There exists no $\hat{v}$ that satisfies both (34) and $r \le \underline{\beta }(\hat{v})<R<\beta ^0 (\hat{v})$.

Now define $\tilde{v}$ such that

$$\begin{aligned}&\pi _1 (\beta ^0 (\tilde{v}) ,\tilde{v}) = \pi _1 (r,\tilde{v}) \nonumber \\&\quad \Leftrightarrow \int _{r}^{R} \underline{h} (b, \tilde{v}) \mathrm {d}b + \int _{R}^{\beta ^0 (\tilde{v})} \bar{h} (b, \tilde{v}) \mathrm {d}b=0 \end{aligned}$$

(37)

The light hand side of (37) is increasing in $\tilde{v}$, and (37) has a unique solution $\tilde{v} > ( \beta ^0 )^{-1} (R)$.

Let bidder 2’s strategy be

$$\begin{aligned} \beta _2 (v) ={\left\{ \begin{array}{ll} \beta ^0 (v) &{} \text {if}\,\, v >\tilde{v} \\ r &{} \text {if}\,\, r \le v \le \tilde{v} \end{array}\right. }. \end{aligned}$$

(38)

Given $\beta _2$, it is locally optimal for bidder 1 to bid $\beta ^0 (v_1)$ for $v > \tilde{v}$. By definition of $\tilde{v}$, both $\beta ^0 (\tilde{v})$ and r are optimal and indifferent for $v_1 =\tilde{v}$. For $v_1 < \tilde{v}$, bidding r is locally optimal because $\underline{h} (r,v) <0$.

Finally, we verify that the derived strategy is globally optimal. Let $\bar{\Pi } (v) \equiv \pi _1 (\beta ^0 (v),v)$ be the expected payoff when bidder 1 bids $\beta ^0 (v)$ satisfying $\bar{h}(\beta ^0 (v) ,v) =0$. Let $\underline{\Pi } (v) \equiv \pi _1 (r,v)$. By construction, $\bar{\Pi } (\tilde{v}) =\underline{\Pi }(\tilde{v})$. The envelope theorem implies $\bar{\Pi }' (v) =\Phi (\beta ^0 (v))$ and $\underline{\Pi }' (v) = \Phi (r)$. Hence, $\bar{\Pi } (v) > \underline{\Pi }(v)$ for $v > \tilde{v}$, and $\bar{\Pi } (v) < \underline{\Pi } (v)$ for $v< \tilde{v}$. Therefore, strategy (11) is an equilibrium strategy. $\square$

1.3 Proof of Theorem 3

Suppose Assumption 1 holds. By Theorem 2, it is clear that the expected social welfare increases by decreasing the package reserve price R for any given r. Hence, $R=0$.

Consider the marginal welfare of increasing r. On the one hand, the positive effect emerges when $V_3 \approx \beta ^{*} (v_1;r,0) +\beta ^{*} (v_2;r,0)$ and at least one of the local bidders submit(s) r. Let $\hat{v}(r)$ be the maximum value with which a local bidder submits r in equilibrium. Then, the positive effect is given by

$$\begin{aligned} \begin{aligned}&\int _r^{\hat{v}(r)} \int _{r}^{\hat{v}(r)} (v_1+v_2-2r)g(2r)f(v_2)f(v_1)\mathrm {d}v_2\mathrm {d}v_1 \\&\quad + 2 \int _r^{\hat{v}(r)} \int _{\hat{v}(r)}^1 \bigl ( v_1+v_2-(r+\beta ^{*} (v_2) ) \bigr ) g(r+\beta ^{*} (v_2) )f(v_2)f(v_1) \mathrm {d}v_2\mathrm {d}v_1 \\&\quad + 2 \int _0^r \int _{r}^{\hat{v}(r)} (v_1-r) g(r) f(v_1)f(v_2)\mathrm {d}v_2 \mathrm {d}v_1 \end{aligned} \end{aligned}$$

(39)

On the other hand, the negative effect of increasing r is that a local bidder decides not to submit a bid when $v_i \approx r$, which is given by

$$\begin{aligned} \begin{aligned}&- 2f(r) \int _r^1 \int _{\beta ^{*} (v)}^{r+\beta ^{*} (v)} (r+v -V_3) g(V_3)f(v) \mathrm {d}V_3\mathrm {d}v \\&\quad - 2f(r) \int _r^1 \int _0^{\beta ^{*} (v)} rg(V_3)f(v)\mathrm {d}V_3 \mathrm {d}v \\&\quad - 2f(r) \int _0^r \int _0^r (r-V_3) g(V_3)f(v) \mathrm {d}V_3\mathrm {d}v. \end{aligned} \end{aligned}$$

(40)

Using Assumption 1 and after some calculations, the marginal welfare MW(r) of increasing r, the sum of (39) and (40), is rearranged as

$$\begin{aligned} MW(r) = p(r) \Bigl ( \mu (r) -r + \int _{\hat{v}(r)}^{1} (v-\beta ^{*} (v;r))f(v) \mathrm {d}v \Bigr ) - \frac{rf(r)}{2} \Bigl ( r + 2 \int _{r}^{1} vf(v) \mathrm {d}v \Bigr ) , \end{aligned}$$

(41)

where $p(r) = \Pr \{ \beta ^{*} (v;r)=r \}$ and $\mu (r) = E[v| \beta ^{*} (v;r)=r ]$.

Because $MW(0) >0$, it is immediate that the socially optimal reserve price is strictly positive $r^* >0$. When F is uniform, we have $\hat{v}(r)=\mathrm {e}^{r-1}$, $p(r) = \mathrm {e}^{r-1}-r$, and $\mu (r) = \frac{ \mathrm {e}^{r-1}+r}{2}$. Substituting them into (41), we solve $MW(r^*) =0$. $\square$

1.4 Proof of Lemma 2

Suppose that both F and G are uniform distributions. Hence, $J_F (v_i) =2v_i -1$ and $J_G (V_3) =2V_3-2$. By the standard argument of Myerson (1981) and Ledyard (2007), the optimal allocation rule is the efficient allocation rule in terms of virtual values. That is, goods are not allocated if $v_i < 1/2$ for local bidders, or if $V_3 < 1$ for the global bidder. When all the bidders have positive virtual values, then local bidders obtain goods if and only if

$$\begin{aligned} J_F(v_1) +J_F (v_2 ) \ge J_G (V_3) \Leftrightarrow v_1 +v_2 \ge V_3, \end{aligned}$$

i.e., the optimal allocation is efficient. When only bidders 1 and 3 have positive virtual valuations, then bidder 1 obtains good A if and only if

$$\begin{aligned} J_F(v_1) \ge J_G (V_3) \Leftrightarrow v_1 + \frac{1}{2} \ge V_3. \end{aligned}$$

Hence, the optimal allocation rule is identical to the efficient allocation rule with a fictitious bid $\hat{r}_B=1/2$. Similarly, when only bidders 2 and 3 have positive virtual valuations, the optimal allocation rule is identical to the efficient allocation rule with a fictitious bid $\hat{r}_A=1/2$. Given two fictitious bids $\hat{r}_A =\hat{r}_B =1/2$, bidder 3 needs to have at least $V_3 \ge 1$ to win. Hence, we do not have to impose an additional reserve price on bidder 3. $\square$