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Core-selecting auctions with incomplete information

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Abstract

Core-selecting auctions were proposed as alternatives to the Vickrey–Clarke–Groves (VCG) mechanism for environments with complementarities. In this paper, we consider a simple incomplete-information model that allows correlations among bidders’ values. We perform a full equilibrium analysis of three core-selecting auction formats as applied to the “local-local-global” model. We show that seller revenues and efficiency from core-selecting auctions can improve as correlations among bidders’ values increase, producing outcomes that are closer to the true core than are the VCG outcomes. Thus, there may be a theoretical justification for policymakers to utilize core-selecting auctions rather than the VCG mechanism in certain environments.

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Notes

  1. The VCG mechanism was developed in the work of Vickrey (1961), Clarke (1971) and Groves (1973). Throughout this paper, we will use the terms “VCG mechanism” and “Vickrey auction” interchangeably.

  2. The core is the subset of allocations in payoff space that are feasible and unblocked by any coalition. When the auction outcome is not in the core, there exists a coalition of bidders willing to renegotiate the outcome with the seller, leading to instability. See Ausubel and Milgrom (2002, 2006) for these critiques and for a general characterization.

  3. These two auctions were the 2014 Canadian 700 MHz Auction ($5.27 billion in revenues) and the 2015 Canadian 2500 MHz Auction ($755 million in revenues). Note that the rules of these auctions included use of a core-selecting mechanism, so the actual outcomes were in the core relative to these bids. Also note that the bids from these auctions were disclosed after the auction on the Canadian regulator’s website, https://www.ic.gc.ca/eic/site/smt-gst.nsf/eng/h_sf01714.html, but to the authors’ knowledge, these have been the only combinatorial clock auctions with package bidding and regional licenses to date in which the regulator disclosed the bids.

  4. See Ausubel and Baranov (2019). Most prior spectrum auctions that utilized assignment rounds had used core-selecting auctions to determine the prices of bidders’ physical assignments. However, the FCC Broadcast Incentive Auction used the VCG mechanism, in order to minimize the extent to which revenues would be diverted from the main stage of the auction to the assignment stage.

  5. For examples of recent spectrum auctions that have used a core-selecting component, including the detailed auction rules and their results, see https://www.ofcom.org.uk/spectrum/spectrum-management/spectrum-awards/awards-archive/800mhz-2.6ghz and http://www.ic.gc.ca/eic/site/smt-gst.nsf/eng/h_sf10598.html. For discussions, see Cramton (2013) and Ausubel and Baranov (2017).

  6. “Bidder-optimal core-selecting auctions” are mechanisms that always choose allocations that minimize revenues within the set of core allocations and, hence, are optimal from the bidder’s viewpoint.

  7. To the best of our knowledge, the model first appeared in Krishna and Rosenthal (1996).

  8. Consider a reverse auction in which the auctioneer needs to repack three TV stations into two adjacent channels in a geographic area (and buy out any stations that cannot be repacked). There is one full-power (“global”) station, which creates both co-channel and adjacent-channel interference when broadcasting, so putting this station on either channel renders the other channel unusable. There are also two low-power (“local”) stations, which create only co-channel interference, so they can coexist on adjacent channels. Together, the three stations interact exactly as in the LLG model. However, the local bidders have more in common with each other than with the global bidder; indeed, the low-power stations are in the same industry as each other and are essentially in a different industry from the full-power station. Consequently, it is plausible that there may be a high degree of correlation between the two local bidders’ values, while very little correlation with the global bidder’s value.

  9. Consider an assignment phase that allocates four contiguous spectrum licenses, A, B, C and D, among three bidders. One (“global”) bidder has the right to receive two contiguous licenses and has a strong preferences for winning AB (perhaps, the global bidder already owns licenses for frequencies adjacent to A from below and winning AB would create a large contiguous segment). Two other (“local”) bidders have the right to receive one license each and have preferences for preventing the global bidder from obtaining AB. This can be accomplished by a local bidder winning either A or B. Thus, the three bidders interact as in the LLG model, where the winning side receives AB and the losing side gets CD. Furthermore, it is quite plausible that the assignment values of the local bidders (the benefit of depriving the global bidder of a large contiguous segment) are strongly correlated with each other while being relatively independent of the global bidder’s value.

  10. In the former case, local bidder i derives positive utility \(v_i\) from winning either item, and the global bidder exhibits classic increasing returns to scale. In the latter case, there are two heterogeneous items, e.g., East and West; local bidder 1 obtains positive utility only from East, local bidder 2 obtains positive utility only from West and the global bidder views East and West as perfect complements.

  11. Consider \(x \,> \, y \, > \, z\) and let \(\Psi (.,.)\) denote the joint probability of \(v_i\) and \(v_j\). Then \((y, y) \, \vee \, (x,z) \, = \, (x,y)\) and \((y, y) \, \wedge \, (x,z) \, = \, (y,z)\), but \(\Psi (x,y)\cdot \Psi (y,z) \, < \, \Psi (y,y) \cdot \Psi (x,z)\), contradicting the affiliation inequality.

  12. Consider a symmetric first-price auction with two bidders whose values are correlated in the same way. If one bidder knows that the other bidder has the same value with a positive probability, her best response fails to exist.

  13. See Ott and Beck (2013) for the analysis of the LLG model when bidders are allowed to submit bids on bundles that include unwanted items.

  14. The nearest-bid rule can be articulated in the LLG model as follows. In case of winning, each local bidder pays her bid minus a discount. The amount of the discount is half of the “money left on the table”, i.e. \(\frac{1}{2} [b_1 + b_2 - B]\). When bids are too different, the amount of the discount can exceed the bid amount, in which case the bidder pays zero.

  15. Interestingly, Bosshard et al. (2018) have demonstrated in a general setting that the nearest-VCG rule does not satisfy the condition that a bidder’s payment is always non-decreasing in its own bid, while other core-selecting rules (including the proxy rule and the nearest-bid rule) do satisfy this condition.

  16. The quality of our numerical solutions is validated using true solutions for \(\gamma = 0\). The average absolute difference between two solutions is \(4.7e-06\) for the proxy rule, \(7.9e-08\) for the nearest-VCG Rule, and \(1.1e-0.6\) for the nearest-bid rule in this case.

  17. See Kagel and Levin (1986) for a classic example of a “corner inference” problem in the auction literature. There, the value \(x_0\) of an item is drawn from a uniform distribution on \([\underline{x}, \bar{x}]\), and the signal \(x_i\) of each bidder i is drawn from a uniform distribution on \([x_0 - \epsilon , x_0 + \epsilon ]\). A corner inference is required whenever \(x_i \, < \, \underline{x} \, + \, \epsilon \,\) or \(\,x_i \, > \, \bar{x} \, - \, \epsilon \). In the current paper’s constant weights model, a corner inference is required, for example, if \(v_i \, = \, 0\). Then, it must be the case that \(s \, = \, 0\) and the range for \(v_j|v_i\) is \([0, \, (1 \, - \, \omega )\,\bar{v}]\). In contrast, the range for \(\,v_j|v_i\,\) in the Bernoulli weights model is always \([0, \, \bar{v} ]\) as long as \(\gamma \, < \, 1\).

  18. To see this, consider \(\sigma \rightarrow 0\). Then \(\sigma /(1 \, + \, \sigma ) \, \rightarrow \, 0\) while \(\sigma /(1 \, + \, \sigma \, - \, 2^{-\sigma }) \, \rightarrow \, 1/(1 \, + \, ln(2)) \, \approx \, 0.59\). For the nearest-bid rule, the first-price effect dominates. In contrast, the first-price effect has limited impact on the nearest-VCG rule and dissipates for low \(\sigma \).

  19. In the same setting, Sano (2012) reports the same fully asymmetric equilibria for a dynamic version of the proxy rule.

  20. We are grateful to an anonymous referee for providing this critique.

  21. Indeed, our Bernoulli weights model has already been implemented in the experimental design of Levkun et al. (2018).

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Correspondence to Oleg Baranov.

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Lawrence M. Ausubel – I am chairman of and have an ownership interest in Power Auctions LLC, a consultancy that provides auction design and software implementation services to governments, non-governmental organizations and commercial enterprises, and I have received fees from Power Auctions exceeding US$10,000 in the past three years. Power Auctions may be considered an “interested” or a “relevant” party to the research reported in this paper. Oleg Baranov – I am an academic associate of Power Auctions LLC, a consultancy that provides auction design and software implementation services to governments, non-governmental organizations and commercial enterprises, and I have received fees from Power Auctions exceeding US$10,000 in the past three years. Power Auctions may be considered an “interested” or a “relevant” party to the research reported in this paper.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We gratefully acknowledge the support of National Science Foundation Grant SES-09-24773. We are also grateful to Peter Cramton, Paul Milgrom, John Rust, Alex Teytelboym and Dan Vincent for valuable conversations, to the editor and two anonymous referees for their suggestions, and to seminar participants at the University of Maryland, Carnegie Mellon University, the Econometric Society World Congress, the NBER Market Design Working Group Meeting, the University of Colorado, the New Economic School and the INFORMS Annual Meetings for helpful comments. All errors are our own.

Proofs

Proofs

Proof of Lemma 1

A bid b by bidder i is pivotal if, for any \(\epsilon > 0\), a bid \(b+\epsilon \) yields bidder i a non-empty set of items, while a bid of \(b-\epsilon \) yields bidder i the empty set. If bidder i’s bid, b, is pivotal, then the incremental surplus contributed by bidder i is zero. By the specification of the VCG mechanism, bidder i’s payoff in the mechanism equals zero. Consequently, bidder i’s payment in the VCG mechanism is \(p^{VCG} = b\). But then in any core-selecting auction, the payment of bidder i is at least its VCG payment \((p^{CS} \ge p^{VCG})\) (otherwise it is blocked by coalition of other bidders and the seller) and at most the bid amount \((p^{CS} \le b)\) (otherwise it is blocked by bidder i alone) . Consequently, bidder i’s payment must be \(p^{CS} = b\). \(\square \)

Proof of Lemma 2

This follows a standard argument for the global bidder, who always pays \(b_1+b_2\) when she wins. For a local bidder, her payment is nondecreasing in her own bid due to regularity conditions. Then overbidding her value can only harm the local bidder. \(\square \)

Proof of Proposition 1

The optimality condition is given by: \( \frac{\partial \pi _i(b_i,v_i)}{\partial b_i} = v_i \, \phi _i(b_i,v_i) - \frac{\partial P_i(b_i,v_i)}{\partial b_i} \le 0\) (with equality when \(b_i > 0\)). Then equation (11) follows since \(\phi _i(b_i,v_i) = \int _{v_j} f(v_j|v_i) g(b_i+\beta _j(v_j)) dv_j\) and \(\frac{\partial P_i(b_i,v_i)}{\partial b_i} = MP_i(b_i,v_i) + b_i\, \phi _i(b_i,v_i)\) due to regularity conditions and the pivotal pricing property which ensures that \(p_i(b_i,\beta _j(v_j),b_i+\beta _j(v_j)) = b_i\). \(\square \)

Proof of Theorem 1

By Proposition 1, the first-order conditions are given by (11). For all pricing rules, \(\phi _i(b_i,v_i) = 1/\bar{u}\). Local bidder j follows strategy \(\beta _j(v_j) \le v_j\) which is strictly increasing on interval \([\hat{v},\bar{v}]\) and is equal to zero on \([0,\hat{v}]\) where \(\hat{v} \ge 0\). (a): For the proxy rule,

$$\begin{aligned} \frac{MP_i(b_i,v_i)}{\phi _i(b_i,v_i)} = \left\{ \begin{array}{lll} \gamma [\beta _j(v_i)-b_i] + (1-\gamma ) \int \limits _{ \min (\beta _j^{-1}(b_i))}^{\bar{v}} \, \,[\beta _j(v_j)-b_i] f(v_j) dv_j &{} \quad \text {if} &{} \; b_i < \beta _j(v_i) \\ (1-\gamma ) \int \limits _{\beta _j^{-1}(b_i)}^{\bar{v}} [\beta _j(v_j)-b_i] f(v_j) dv_j &{} \quad \text {if} &{} \; b_i \ge \beta _j(v_i) \\ \end{array} \right. \end{aligned}$$

Then the best-response of bidder i is to bid zero when \(v_i \le \gamma \beta _j(v_i) + (1-\gamma ) E[\beta _j(v_j)]\) and bid a positive amount for larger \(v_i\). In the symmetric equilibrium, \(\beta (v) = 0\) for \(v \le \hat{v}\) and \(\beta (v) = \tilde{\beta }(v)\) for \(v > \hat{v}\) where \(\hat{v} = (1-\gamma ) E[\beta (v)]\) and \( v - \tilde{\beta }(v) \ \equiv \ (1-\gamma ) \left[ \int _v^{\bar{v}} \tilde{\beta }(v_j) f(v_j)dv_j - \tilde{\beta }(v)(1-F(v)) \right] \) for all \(v \in [\hat{v},\bar{v}]\) which is equivalent to (13). (b): For the nearest-VCG rule, \( \frac{MP_i(b_i,v_i)}{\phi _i(b_i,v_i)} \ = \ \frac{\gamma }{2} \beta _j(v_i) + \frac{1-\gamma }{2} \int _{0}^{\bar{v}} \beta _j(v_j) f(v_j) dv_j\). Then the best-response of bidder i is to bid zero when \(v_i \le 0.5[\gamma \beta _j(v_i) + (1-\gamma ) E[\beta _j(v_j)]]\) and bid a positive amount for larger \(v_i\). In the symmetric equilibrium, \(\beta (v) = 0\) for \(v \le \hat{v}\) and \(\beta (v) = \tilde{\beta }(v)\) for \(v > \hat{v}\) where \(\hat{v} = 0.5(1-\gamma ) E[\beta (v)]\) and \(v - \tilde{\beta }(v) \ \equiv \ \frac{1}{2} \bigl [ \gamma \tilde{\beta }(v) + (1-\gamma )E[\tilde{\beta }(v_j)] \bigr ]\) for all \(v \in [\hat{v},\bar{v}]\). It follows that \(\tilde{\beta }'(v) = \frac{2}{2+\gamma }\) and \(\tilde{\beta }(v) = \frac{2}{2+\gamma }(v-\hat{v})\) on the interval \([\hat{v},\bar{v}]\) where \(\hat{v} \ = \ \frac{1-\gamma }{2+\gamma } \int _{\hat{v}}^{\bar{v}} (v_j-\hat{v})f(v_j)dv_j\) which is equivalent to (15). The equation has a unique solution \(\hat{v}\) on \((0,\bar{v})\) which is strictly decreasing with \(\gamma \). (c): For the nearest-bid rule,

$$\begin{aligned} \frac{MP_i(b_i,v_i)}{\phi _i(b_i,v_i)} = \left\{ \begin{array}{lll} \gamma b_i + (1-\gamma ) \left[ \int \limits _{0}^{\beta _j^{-1}(b_i)} \beta _j(v_j) f(v_j) dv_j + \int \limits _{\beta _j^{-1}(b_i)}^{\bar{v}} b_i f(v_j) dv_j \right] &{} \quad \text {if} &{} \; b_i < \beta _j(v_i) \\ \gamma \beta _j(v_i) + (1-\gamma ) \left[ \int \limits _{0}^{\beta _j^{-1}(b_i)} \beta _j(v_j) f(v_j) dv_j + \int \limits _{\beta _j^{-1}(b_i)}^{\bar{v}} b_i f(v_j) dv_j \right] &{} \quad \text {if} &{} \; b_i \ge \beta _j(v_i) \end{array} \right. \end{aligned}$$

Then the best response of bidder i is to bid \(b_i>0\) for any \(v_i \in (0,\bar{v})\). In the symmetric equilibrium, \( v - \beta (v) \equiv \gamma \beta (v) + (1-\gamma ) \left[ \int _{0}^{v} \beta (v_j) f(v_j) dv_j + \beta (v)(1-F(v)) \right] \) for all \(v \in [0,\bar{v}]\) which is equivalent to (16). \(\square \)

Proof of Corollary 2

(a): For the proxy rule, \(\beta _{\gamma }(v) \le \beta _{\gamma '}(v)\) for all \(v \in [0, \bar{v}]\) follows since \(\tilde{\beta }_{\gamma }(\bar{v})=\tilde{\beta }_{\gamma '}(\bar{v})\) and \(\tilde{\beta }'_{\gamma }(v) \ge \tilde{\beta }'_{\gamma '}(v)\). (b): For the nearest-VCG rule, \(\hat{v}(\gamma )\) is strictly decreasing function. Due to linearity of the bidding function with slope \(\frac{2}{2+\gamma }\), \(\beta _{\gamma }(v) \le \beta _{\gamma '}(v)\) for \(v \in [0,\tilde{v}]\) and \(\beta _{\gamma }(v) \ge \beta _{\gamma '}(v)\) for \(v \in [\tilde{v},\bar{v}]\) where \(\tilde{v} \in (\hat{v}(\gamma ),\bar{v}]\). If \(\tilde{v}=\bar{v}\), then \(MP^{\gamma }_i(b_i,\bar{v}) < MP^{\gamma '}_i(b_i,\bar{v})\) and the best-response of bidder i with \(v_i=\bar{v}\) should drop under \(\gamma '\) which is a contradiction. It follows that \(\tilde{v} \in (\hat{v}(\gamma ),\bar{v})\). (c): For the nearest-bid rule, \(\beta _{\gamma }(v) \ge \beta _{\gamma '}(v)\) for all \(v \in [0, \bar{v}]\) follows since \(\beta _{\gamma }(0)=\beta _{\gamma '}(0)\) and \(\beta '_{\gamma }(v) \ge \beta '_{\gamma '}(v)\). \(\square \)

Proof of Corollary 3

(a): For the proxy rule, \(\beta _{F_1}(v) \le \beta _{F_2}(v)\) for all \(v \in [0, \bar{v}]\) follows since \(\tilde{\beta }_{F_1}(\bar{v})=\tilde{\beta }_{F_2}(\bar{v})\) and \(\tilde{\beta }'_{F_1}(v) \ge \tilde{\beta }'_{F_2}(v)\). (b): For the nearest-VCG rule, \(\beta _{F_1}(v) \le \beta _{F_2}(v)\) for all \(v \in [0, \bar{v}]\) follows since \(\hat{v}_{F_2} \le \hat{v}_{F_1}\). (c): For the nearest-bid rule, \(\beta _{F_1}(v) \le \beta _{F_2}(v)\) for all \(v \in [0, \bar{v}]\) follows since \(\beta _{F_1}(0)=\beta _{F_2}(0)\) and \(\beta '_{F_1}(v) \le \beta '_{F_2}(v)\). \(\square \)

Proof of Proposition 2

For the proxy rule, the expected marginal payment of bidder i is \(MP_i(b_i,v) = G(\beta _j(v)+b_i) - G(2b_i)\) when \(b_i \le \beta _j(v)\) and \(MP_i(b_i,v) = 0\) when \(b_i > \beta _j(v)\). If a symmetric equilibrium exists, \(MP_i(\beta (v_i),v_i) = 0\). But then \(\beta (v) = v\) by Proposition 1 since \(\beta (v)>0\) for any \(v>0\). \(\square \)

Proof of Proposition 3

In a symmetric equilibrium, \(\beta (v)>0\) for any \(v>0\). Then \((v_i - \beta (v_i))g(2\beta (v_i)) = MP_i(\beta (v_i),v_i)\) for all pricing rules. Then (28) follows since

\(MP^{Proxy}_i(\beta (v_i),v_i) = 0\), strictly less than \(MP^{N-VCG}_i(\beta (v_i),v_i) = 1/2[G(2\beta (v)) - G(\beta (v))]\) which is in turn strictly less than \(MP^{N-BID}_i(\beta (v_i),v_i) = 1/2[G(2\beta (v))]\). \(\square \)

Proof of Theorem 2

By Proposition 1, the first-order conditions are given by (11). For all pricing rules, \(\phi _i(b_i,v_i) = g(b_i+\beta (v_i))\). In a symmetric equilibrium, \(\beta (v)>0\) for any \(v>0\). (a): For the proxy rule, the expected marginal payment of bidder i is \(MP_i(b_i,v) = G(\beta _j(v)+b_i) - G(2b_i)\) when \(b_i \le \beta _j(v)\) and \(MP_i(b_i,v) = 0\) when \(b_i > \beta _j(v)\). In a symmetric equilibrium \(\beta (v) = v\) by Proposition 2. In an asymmetric equilibrium where \(\beta _i(v)<\beta _j(v)\), bider j must be bidding truthfully. If \(\beta _j(v)=v\), then the best response of bidder i is to bid \(b_i = v\) since \((v-b_i)\,g(v+b_i) \ \ge \ G(v+b_i) - G(2b_i)\) (with a strict sign for all \(b_i<v\)) for all \(b_i \in [0,v]\). The last inequality follows for \(G(u) = (u/\bar{u})^{\sigma }\) since it is equivalent to \(h(x) = x^{\sigma } + \sigma (1-x) \ge 1\) where \(x = \frac{2b_i}{v+b_i} \in [0,1]\). It is satisfied if and only if \(\sigma > 1\) since \(h(0)=\sigma \), \(h(1)=1\) and \(h'(x) < 0\) for all \(x \in [0,1)\). (b): For the nearest-VCG rule, the expected marginal payment of bidder i is given by \(MP_i(b_i,v) = \frac{1}{2} \bigl [ G(\beta _j(v)+b_i) - G(b_i) \bigr ]\). A fully asymmetric equilibrium cannot exist for \(\sigma > 1\) since \(2vg(v)>G(v)\). For a partially asymmetric equilibrium where \(b_j(v)=x\,\beta _i(v)\) with \(x \in (0,1)\) and \(G(u) = (u/\bar{u})^{\sigma }\), according to the first-order conditions (11),

$$\begin{aligned} \beta _i(v) \ = \ \frac{2\,\sigma }{2\sigma +1+x-(1+x)^{(1-\sigma )}}\, v \quad \beta _j(v) = \frac{2\,\sigma \,x}{2\sigma +1+x-(1+x)^{(1-\sigma )}}\, v \end{aligned}$$

where \(2\,\sigma \,(1-x) \ = \ (1+x)^{(1-\sigma )}(1-x^{\sigma })\). It can be shown that the above system of equations does not admit solutions such that \(x<1\) for \(\sigma \ge 1\). The symmetric equilibrium is given by \(\beta (v) = \frac{\sigma }{1+\sigma - 2^{-\sigma }}\,v\) (plug \(x=1\)). This equilibrium exists when \(2\sigma (v-b_i) \ge \Psi (b_i)\) for all \(b_i \in [0,\beta (v)]\) and \(2\sigma (v-b_i) \le \Psi (b_i)\) for all \( b_i \in [\beta (v),v]\) where \(\Psi (b_i) = (b_i + \beta (v)) - \frac{b_i^{\sigma }}{(\beta (v)+b_i)^{\sigma -1}}\). For \(\sigma \ge 1\), \(\Psi ''(b_i) \le 0\). Then \(\Psi '(b_i)\) is a decreasing function that is positive at \(b_i=v\), implying that \(\Psi '(b_i) \ge 0\) for all \(b_i \in [0,v]\). Then \(\Psi (b_i)\) is increasing on [0, v] and the symmetric equilibrium exists. (c): For the nearest-bid rule, the expected marginal payment of bidder i is given by

$$\begin{aligned} MP_i(b_i,v) = \left\{ \begin{array}{lll} \frac{1}{2} \bigl [ G(\beta _j(v)+b_i) - G(\beta _j(v)-b_i) \bigr ] &{} \quad \text {if} &{} \; 0 \le b_i \le \beta _j(v) \\ \frac{1}{2} \bigl [ G(\beta _j(v)+b_i) - G(b_i - \beta _j(v)) \bigr ] &{} \quad \text {if} &{} \; \beta _j(v) < b_i \\ \end{array} \right. \end{aligned}$$

Note that \(b_i=0\) is never a best response when \(v>0\) since then \(MP_i(0,v)=0\) and \(vg(\beta (v)+v)>0\). Now suppose that \(0<\beta _j(v)<\beta _i(v)\). Then \(MP_i(\beta _i(v),v) = MP_j(\beta _j(v),v)\) and \(\beta _j(v)=\beta _i(v)\) by first-order conditions (11). Thus, there are no asymmetric equilibria for this payment rule. In a symmetric equilibrium \(b_i = \beta (v)\) and \(MP_i(\beta (v_i),v) = G(2\beta (v))/2\). Then, for \(G(u) = (u/\bar{u})^{\sigma }\), according to the first-order conditions (11), \(\beta (v) \ = \ \frac{\sigma }{1+\sigma }\,v\). This equilibrium exists when \(2\sigma (v-b_i) \ge \Psi (b_i)\) for all \(b_i \in [0,\beta (v)]\) and \(2\sigma (v-b_i) \le \Psi (b_i)\) for all \( b_i \in [\beta (v),v]\) where \(\Psi (b_i) = (b_i + \beta (v)) - (\beta (v)-b_i) \left[ \frac{\beta (v)-b_i}{\beta (v)+b_i} \right] ^{\sigma -1}\). The first inequality is satisfied for any \(\sigma >0\) since the left-hand side is strictly decreasing, the right-hand side is strictly increasing and the inequality is still satisfied at \(b_i=\beta (v)\). The second inequality is satisfied for \(\sigma \ge 1\). \(\square \)

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Ausubel, L.M., Baranov, O. Core-selecting auctions with incomplete information. Int J Game Theory 49, 251–273 (2020). https://doi.org/10.1007/s00182-019-00691-3

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