Skip to main content

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

Notes

  1. See Ausubel and Milgrom (2006a) for a discussion of the theoretical drawbacks of the VCG mechanism.

  2. Core-selecting auctions are not unique. See Erdil and Klemperer (2010); Day and Cramton (2012), and Lamy (2012) for designs of particular core-selecting auctions.

  3. See Cramton (2013) and Ausubel and Baranov (2014) for recent use of core-selecting auctions in spectrum-license allocations around the world.

  4. The terms local bidder and global bidder are drawn from a story of geographically divided spectrum licenses. Suppose that the spectrum licenses are divided into two regions, named A and B. In each region, a regional firm operates a mobile phone service. Each regional firm only needs a license to operate in its own region; therefore, it is regarded as a local bidder. In contrast, a national firm that operates throughout the country would need both licenses; therefore, it is regarded as the global bidder.

  5. Reserve prices also prevent participation by speculators.

  6. In this interpretation, reserve prices are non-anonymous (or discriminatory) rather than non-linear. However, because each bidder is supposed to submit only one bid in the LLG model, we do not distinguish between non-linear and non-anonymous reserve prices.

  7. The same ascending auction algorithm is proposed by Parkes and Ungar (2000).

  8. Ausubel and Baranov (2010) permit a special class of correlations between local bidders’ values.

  9. Ties are broken randomly when \(b_1+b_2=b_3\).

  10. See Ausubel and Milgrom (2002, 2006b) for a precise and general definition of the ascending proxy auction.

  11. The formal description of the reserve bidder rule is slightly different. See Day and Cramton (2012) for details.

  12. The analysis in this section is based on an earlier working paper, Sano (2010).

  13. It is also difficult to have a reasonable condition that induces \(\beta ^* (v_i;r) \ge \beta ^0 (v_i)\) for all \(v_i\).

  14. The author thanks anonymous Reviewer 1 for pointing out the relation between the current model and bilateral trade model.

  15. For the derivation of the second best mechanism, see Myerson and Satterthwaite (1983) and Guth and Hellwig (1986). Although Guth and Hellwig consider pure public goods, it is straightforward to arrange the mechanism to excludable public goods.

  16. Both \(\bar{h}\) and \(\underline{h}\) depend on r and R through \(\beta (v_2)\).

References

  • Ausubel LM, Baranov O (2010) Core-selecting auctions with incomplete information. Working paper, University of Maryland

  • Ausubel LM, Baranov O (2014) Market design and the evolution of the combinatorial clock auction. Am Econ Rev 104:446–451

    Article  Google Scholar 

  • Ausubel LM, Baranov O (2017) A practical guide to the combinatorial clock auction. Econ J 127:F334–F350

    Article  Google Scholar 

  • Ausubel LM, Cramton P (2004) Vickrey auction with reserve pricing. Econ Theory 23:493–505

    Article  Google Scholar 

  • Ausubel LM, Milgrom P (2002) Ascending auctions with package bidding. Front Theor Econ 1:1–42

    Article  Google Scholar 

  • Ausubel LM, Milgrom P (2006a) The lovely but lonely vickrey auction. In: Cramton P, Shoham Y, Steinberg R (eds) Combinatorial auctions. MIT Press, Cambridge, pp 17–40

    Google Scholar 

  • Ausubel LM, Milgrom P (2006b) Ascending proxy auctions. In: Cramton P, Shoham Y, Steinberg R (eds) Combinatorial auctions. MIT Press, Cambridge, pp 79–98

    Google Scholar 

  • Beck M, Ott M (2013) Nash equilibria of sealed-bid package auctions. Working paper, RWTH Aachen University

  • Clarke EH (1971) Multipart pricing of public goods. Public Choice 11:17–33

    Article  Google Scholar 

  • Cramton P (2013) Spectrum auction design. Rev Ind Organ 42:161–190

    Article  Google Scholar 

  • Day R, Cramton P (2012) Quadratic core-selecting payment rules for combinatorial auctions. Oper Res 60:588–603

    Article  Google Scholar 

  • Day R, Milgrom P (2008) Core-selecting package auctions. Int J Game Theory 36:393–407

    Article  Google Scholar 

  • Erdil A, Klemperer P (2010) A new payment rule for core-selecting package auctions. J Eur Econ Assoc 8:537–547

    Article  Google Scholar 

  • Goeree JK, Lien Y (2016) On the impossibility of core-selecting package auctions. Theor Econ 11:41–52

    Article  Google Scholar 

  • Groves T (1973) Incentives in teams. Econometrica 41:617–631

    Article  Google Scholar 

  • Guth W, Hellwig M (1986) The private supply of a public good. J Econ Suppl 5:121–159

    Article  Google Scholar 

  • Hafalir IE, Yektas H (2015) Core deviation minimizing auctions. Int J Game Thery 44:367–376

    Article  Google Scholar 

  • Hatfield J, Milgrom P (2005) Matching with contracts. Am Econ Rev 95:913–935

    Article  Google Scholar 

  • Kelso A, Crawford VP (1982) Job matching, coalition formation, and gross substitutes. Econometrica 50:1483–1504

    Article  Google Scholar 

  • Lamy L (2012) On minimal ascending auctions with payment discounts. Games Econ Behav 75:990–999

    Article  Google Scholar 

  • Ledyard J (2007) Optimal combinatoric auctions with single-minded bidders. In: Proceedings of the 8th ACM conference on electronic commerce, pp 237–242

  • Myerson R (1981) Optimal auction design. Math Oper Res 6:58–73

    Article  Google Scholar 

  • Myerson R, Satterthwaite A (1983) Efficient mechanisms for bilateral trading. J Econ Theory 29:265–281

    Article  Google Scholar 

  • Parkes DC, Ungar LH (2000) Iterative combinatorial auctions: theory and practice. In: Proceedings of the 17th national conference on artificial intelligence (AAAI-2000), pp 74–81

  • Sano R (2010) An equilibrium analysis of a package auction with single-minded bidders. Working paper, available at SSRN: http://ssrn.com/abstract=1594471. Accessed 20 June 2018.

  • Sano R (2011) Incentives in core-selecting auctions with single-minded bidders. Games Econ Behav 72:602–606

    Article  Google Scholar 

  • Sano R (2012) Non-bidding equilibrium of an ascending core-selecting auction. Games Econ Behav 74:637–650

    Article  Google Scholar 

  • Vickrey W (1961) Counterspeculation, auctions, and competitive sealed tenders. J Finance 16:8–37

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ryuji Sano.

Additional information

I gratefully acknowledge the associate editor and two anonymous reviewers for their insightful comments and suggestions. I am also grateful to Hitoshi Matsushima, Michihiro Kandori, Makoto Hanazono, Jun Nakabayashi, and Daniel Marszalec for their helpful comments. This research was supported by the Japan Society for the Promotion of Sciences (JSPS) KAKENHI 25780132 and 15H03346.

Proofs

Proofs

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.\(\square \)

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)

andFootnote 16

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

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 \(MW(r^*) =0\). \(\square \)

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. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sano, R. An equilibrium analysis of a core-selecting package auction with reserve prices. Rev Econ Design 22, 101–122 (2018). https://doi.org/10.1007/s10058-018-0212-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10058-018-0212-5

Keywords

  • Core-selecting auction
  • Package auction
  • Ascending proxy auction
  • Reserve price

JEL Classification

  • D44
  • D47