On the clock of the combinatorial clock auction

Janssen, Maarten C. W.; Kasberger, Bernhard Article On the clock of the combinatorial clock auction Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Janssen, Maarten C. W.; Kasberger, Bernhard (2019) : On the clock of the combinatorial clock auction, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 14, Iss. 4, pp. 1271-1307, https://doi.org/10.3982/TE3203 This Version is available at: https://hdl.handle.net/10419/217095 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 14 (2019), 1271–1307 1555-7561/20191271 On the clock of the combinatorial clock auction Maarten Janssen Department of Economics, University of Vienna, National Research University Higher School of Economics Moscow, and CEPR Bernhard Kasberger The Queen’s College, University of Oxford The combinatorial clock auction (CCA) has frequently been used in recent spectrum auctions. It combines a dynamic clock phase and a one-off supplementary round. The winning allocation and the corresponding prices are determined by the Vickrey–Clarke–Groves rules. These rules should encourage truthful bidding, whereas the clock phase is intended to reveal information. We inquire into the role of the clock when bidders have lexicographic preferences for raising rivals’ costs. We show that in an efficient equilibrium, the clock cannot fully reveal bidders’ types. In the spirit of the ratchet effect, in the supplementary round competitors extract surplus from strong bidders whose type is revealed. We also show that if there is substantial room for information revelation, that is, if the uncertainty about the final allocation is large, all equilibria of the CCA are inefficient. Qualitative features of our equilibria are in line with evidence concerning bidding behavior in some recent CCAs. Keywords. Combinatorial auctions, spectrum auctions, spiteful bidding, raising rival’s cost, ratchet effect. JEL classification. D44, D47, L96. 1. Introduction In recent years, many regulators around the world have chosen the combinatorial clock auction (CCA) to allocate telecommunication spectrum. The CCA has partially replaced the older simultaneous ascending auction (SAA) for two reasons. First, in the SAA, bidders may strategically reduce demand. If it is relatively clear to the bidders what the final allocation of an auction is where bidders bid competitively, then they have an incentive to reach the same allocation at much lower prices (Grimm et al. 2003). The SAA Maarten Janssen: [email protected] Bernhard Kasberger: [email protected] Earlier versions of this paper were presented in Vienna (Workshop on Auction Design, 2016), Cardiff (QED Jamboree, 2015), Klagenfurt (NOeG, 2015), Istanbul (Conference on Economic Design, 2015), Montreal (11th World Congress of the Econometric Society, 2015), Munich (EARIE, 2015), and Cologne (Workshop on Auctions and Procurement Design, 2015). We thank audiences at these workshops and conferences, the two anonymous referees, Larry Ausubel, Oleg Baranov, Martin Bichler, Peter Cramton, Jon Levin, Paul Milgrom (and two of his graduate students), and Andy Skrzypacz for useful suggestions and comments. This research was supported by the Oesterreichische Nationalbank (Oesterreichische Nationalbank, Anniversary Fund, project number 15994). ©2019 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://econtheory.org.https://doi.org/10.3982/TE3203 1272 Janssen and Kasberger Theoretical Economics 14 (2019) provides bidders with the possibility of reaching such a noncompetitive outcome. The sophisticated design of the CCA should overcome this issue as it incorporates (i) a generalized second-price (Vickrey) rule providing bidders with an incentive to bid truthfully (Cramton 2013) and (ii) a clock phase that should facilitate “price and package discovery” (Ausubel et al. 2006). Second, in contrast to the SAA, bidders can express bids for packages in the CCA. Package bidding is deemed to be important as current spectrum auctions allocate multiple units where bidders may value a package of licenses more than the sum of the individual components. If that is the case, the SAA, but not the CCA, suffers from the well known exposure problem, i.e., at the end of the auction, bidders may end up with a few units at a price that is more than their value for these units. The focus of this paper is the first issue: is it the case that the CCA provides bidders with an incentive to bid truthfully and that the clock phase facilitates price and package discovery? The CCA is a dynamic version of the Vickrey–Clarke–Groves (VCG) mechanism and consists of two integrated phases.1In the first clock phase, bidders express their demand on packages at given prices in every round. If, for a certain good, demand is larger than supply in a given round, then the price for that good is increased in the next round. The clock phase ends when demand is not larger than supply for all the auctioned goods. Importantly, no goods are allocated and no prices are determined at the end of the clock phase. Instead, the clock phase imposes constraints on the bids that are allowed in the second, supplementary, phase. In that one-off sealed-bid phase, bidders can bid on as many additional packages as they like and they may raise bids on packages they have bid on in the clock phase. At the end of the supplementary phase, goods are allocated and prices are determined. The auctioneer uses all the bids from the clock phase and the supplementary phase to determine the value-maximizing combination of bids, while the Vickrey pricing rule determines the prices winners pay. Without the clock phase, the CCA reduces to the VCG mechanism. As the number of packages is an exponential function of the number of commodities, bidders in a VCG auction may need to consider bidding on a vast number of packages. In particular, if the uncertainty concerning competitors is large, bidders may have a fairly limited idea about the package they may eventually win and at which price. Through “price and package discovery,” the clock phase is meant to reveal this kind of information. Bidders can then focus their bidding in the supplementary round on the packages that may still be winning. Under standard preferences, truthful bidding in the clock and supplementary phase is indeed an equilibrium. If bidders bid truthfully, the outcome is efficient. However, truthful bidding is not a strict equilibrium, as bidders may be indifferent across many permissible bids in the supplementary round (Levin and Skrzypacz 2016). To eliminate the payment relevant indifferences, we consider bidders who ceteris paribus prefer outcomes where competitors pay more. We model this objective as a secondary dimension in a lexicographic way. 1In practice, there is a third phase: the assignment phase. In this phase, generic packages are allocated. We abstract away from this phase since it does not affect our analysis. Theoretical Economics 14 (2019) Combinatorial clock auction 1273 Our first main result is that if bidders have a lexicographic preference for raising rival’s costs, an efficient fully revealing equilibrium does not exist. This result implies that the CCA exhibits a fundamental trade-off between efficiency and information revelation in the clock phase. The trade-off follows from the fact that if bidders bid truthfully in the clock phase, the clock fully reveals information about the bidders’ types. Bidders would like to use this information to maximally raise the rival’s cost by placing bids in the supplementary phase on large packages that they know cannot be winning. The stronger their competitor, the more they can raise their price. The rules of the CCA are such that bidders are only able to raise the rival’s cost if they expand demand in the clock phase, as this relaxes the constraints on the supplementary phase bids. Predicting that the clock phase eventually will fully reveal information, bidders can expand demand in the early phase of the clock without the risk of affecting the final allocation. Knowing that the competitor is able and inclined to raise their cost if their types are fully revealed, stronger bidders have an incentive to pool with weaker types in the clock phase. This result is best understood from the perspective of the ratchet effect known from the dynamic principal–agent literature (Laffont and Tirole 1988). In that literature, an agent may have an incentive not to reveal his type to a principal if the principal could use that information to extract more surplus from the agent in future interactions. In our case, knowing the competitor is strong, a bidder (by bidding more aggressively in the supplementary phase) may increase the price the competitor has to pay beyond what it would be if the competitor’s type were unknown. Rationally anticipating this exploitation, stronger bidders prefer to pool with weaker types. The intuition for our first main result differs in two dimensions from the traditional ratchet effect. First, unlike the principal–agent model, the roles of bidders in an auction are symmetric to one another so that each bidder is both the object of and the initiator of surplus extraction. Second, the extent to which bidders can raise the rival’s cost in the supplementary round is not exogenously given, but endogenously determined by their behavior in the clock phase. Thus, bidders will only be able to raise the rival’s cost if they expand demand in the clock phase. The result that fully revealing efficient equilibria do not exist does not rule out the existence of efficient equilibria. Even with lexicographic preferences for raising rival’s cost, efficient equilibria exist. We present examples of efficient equilibria, where to be able to raise rival’s cost, bidders demand the full supply (even if prices are such that truthful bidding would tell them to demand less).2The demand expansion phase ends with a sudden switch to truthful bidding. In one type of equilibrium, the clock stops immediately when all bidders drop demand. In such an equilibrium, there is no price or package discovery whatsoever. This clock phase development allows all bidders to bid their true marginal values in the supplementary round. As a result, the final allocation is efficient. We show that any efficient equilibrium of the CCA involves this type of demand expansion in the clock phase. 2This is in line with, for example, the Austrian 2013 auction, where (as we mention below) bidders were bidding very aggressively in the clock phase. 1274 Janssen and Kasberger Theoretical Economics 14 (2019) Our second main result is that if the uncertainty concerning the competitor’s type is sufficiently large, all equilibria of the CCA are inefficient. Efficiency and the high uncertainty require that weak bidders drop out at relatively high prices. Due to the spite motive, some strong bidders expand demand prior to these dropout prices. When the clock does not end, a relatively strong bidder infers from the continuation of the clock that the competitor is not too weak. This type of learning creates the opportunity for the strong bidder to make the supplementary round behavior conditional on the price at which the clock phase stops. Knowing the competitor is not too weak, the strong bidder can raise the rival’s cost more without running the risk of winning an inferior share. Consequently, some types have an incentive to obfuscate their type and do this by reducing demand toward the end of the clock phase. This demand reduction rules out expressing true marginal values for all shares in the supplementary round, resulting in an inefficient final allocation. As we also show that the static VCG mechanism always has efficient equilibria, we claim that it is the clock phase that creates this inefficiency.3 Ausubel (2004, p. 1452) states that “the auctions literature has provided us with two fundamental prescriptions guiding effective auction design”: first, “the winner’s price should depend solely on opposing participants’ bids—as in the sealed-bid, second-price auction—so that each participant has full incentive to reveal truthfully her value for the good. Second, an auction should be structured in an open fashion that maximizes the information made available to each participant at the time she places her bids.” Our results show that following these two prescriptions can be at the expense of generating efficient outcomes in multi-unit auctions where bidders have a weak incentive to raise rival’s costs. If efficiency is preserved, then the information that is revealed through the open format is fairly limited. The lexicographic modeling of bidders’ preference for raising the rival’s costs implies that if two bidding strategies yield the same expected surplus to a bidder, the bidder chooses the strategy where the rival pays more.4The motivation to raise rival’s costs may arise from (i) principal–agent issues within a firm (bidder)5or from (ii) the fact that (in spectrum auctions) bidders face weaker competitors in the market after an auction if competitors have paid more for their licenses. If firm A makes firm B pay more for spectrum, B’s credit rating may fall and its cost of capital may go up, weakening its strategic position. Milgrom (2004)andCramton and Ockenfels (2017) mention fairness as a reason why bidders may want to raise rival’s costs. 3Note that as we do not present an alternative auction model that is clearly better than the CCA (or the SAA), the CCA cannot be fully discarded on these grounds. Nevertheless, it is important to understand that the CCA rules can be gamed and this may have consequences. 4The analysis with lexicographic preferences provides a robustness check on the equilibria under standard preferences: equilibria under our preferences are also equilibria under standard preferences, but the reverse does not necessarily hold true. 5In spectrum auctions, given the considerable uncertainty concerning future technological developments and uptake of data services, it is difficult for bidders to evaluate what the spectrum is worth. Valuations are highly subjective. Accordingly, if a bidder wants to have a more objective evaluation measure of his bidding team’s performance, it might be better to evaluate performance relative to other bidders than relative to his own uncertain and subjective valuation. Theoretical Economics 14 (2019) Combinatorial clock auction 1275 The motivation to raise rival’s costs motive has become a concern in designing auctions.6After the 2013 auction, the Austrian regulator RTR attributed the high revenue to overly aggressive behavior by bidders: during the clock phase, bidders were bidding very aggressively, and the majority of the supplementary bids were on very large packages that had a low probability of winning but played a crucial role in determining other bidders’ prices. The fact that payments in the Austrian auction were essentially the same as the final clock prices is a clear signal of aggressive bidding, as with Vickrey pricing and “downward sloping demand” one would not expect marginal and average prices to be identical. The observed behavior, however, is reminiscent of the equilibria we describe. Moreover, the British regulator Ofcom (2014, p. 38, 6.73–6.77) explicitly mentions the possibility of price driving by placing “risk-free bids” in the supplementary phase as a problematic aspect of the CCA. Some of the potential bidders’ responses share this concern (e.g., BT 2015). None of these arguments for raising rival’s costs implies that bidders should have a lexicographic preference for doing so; lexicographic preferences are, however, a useful modeling approach to inquire into the robustness of the results of the CCA to slight changes in assumptions regarding preferences. This is the first paper that provides a full equilibrium analysis of the CCA when bidders have a lexicographic preference for raising rival’s costs. The most closely related paper is Levin and Skrzypacz (2016). They put forward a sequence of three related models in which, as in our study, two players compete for a perfectly divisible good in the CCA. In some parts of their analysis, they also consider spiteful bidders. In a first model where bidders have standard preferences, Levin and Skrzypacz (2016) elegantly uncover the existence of multiple equilibria due to a key indifference condition. Both bidders use linear proxy clock demand functions so that the clock ends with market clearing. The activity rules then permit a specific range of supplementary bids that are all such that the final clock allocation is the final allocation. As the activity rules fix the allocation, the VCG pricing scheme makes bidders with standard preferences indifferent across all supplementary bids. How bidders resolve the indifference impacts optimal clock behavior, leading to a multiplicity of equilibria. This indifference partly motivates Levin and Skrzypacz (2016) to consider spiteful lexicographic preferences in their next two models. In their second model, they (exogenously) restrict one bidder to linear proxy strategies. It is, however, not clear why one of the ex ante symmetric bidders would prefer to restrict himself and take this disadvantageous role. In their online Appendix, Levin and Skrzypacz (2016)discussathird model with two predatory bidders. This model is closest to the model we analyze in our paper. In that third model, Levin and Skrzypacz (2016) have bidders using linear proxy strategies in the clock phase, but weaken the constraints on supplementary bids implied by this clock behavior and the activity rules. Technically, they achieve this by introducing an exogenous parameter that measures how much bidders violate the activity rules. Importantly, such bidding behavior violates the rules of the CCA (see Figure 1 for more detail). 6See, e.g., (i) Levin and Skrzypacz (2016) on the outcome of the Swiss auction and the discussion on why Sunrise paid much more for comparable spectrum than other bidders, and (ii) Ofcom (2012, p. 122, point 7.9) in response to an earlier consultation on the U.K. spectrum auction in 2013. 1276 Janssen and Kasberger Theoretical Economics 14 (2019) In contrast, the focus of our paper is on how bidders behave in the clock phase so that they are able, within the rules of the CCA, to weaken the constraints of the activity rule and submit spiteful supplementary bids. We show that this is not innocuous, as our results are qualitatively and quantitatively different from the findings of Levin and Skrzypacz (2016). First, where Levin and Skrzypacz (2016) conclude that equilibria are inefficient, we show that efficient equilibria can exist if the uncertainty concerning bidders’ types is not too large. In any efficient equilibrium, bidders endogenously relax the constraints of the activity rule through demand expansion in the clock phase. In this way, the clock does not perfectly reveal bidders’ types and may end with excess supply so that the last clock round does not fix the final allocation. In the supplementary phase, bidders then have a strong incentive to bid true marginal values on possible final shares. Second, where we observe inefficient equilibria for large uncertainty, the source of inefficiency is very different from that in Levin and Skrzypacz (2016), where the source of inefficiency is the best response to exogenously distorted marginal prices; the inefficiency in our model derives from the incentives of strong bidders to obfuscate their types (as in the literature on the ratchet effect in the dynamic principal–agent literature) to avoid being exploited in the supplementary round. While we consider the interaction between the clock and the supplementary phase, Janssen and Karamychev (2016) focus only on the supplementary phase of the CCA. Assuming a particular clock phase behavior, they show how the supplementary phase can be solved using iterative elimination of dominated strategies, resulting in bidders raising rival’s costs without running the risk of winning undesired packages. The current paper analyzes the equilibrium properties of both stages of the CCA, i.e., the entire game. A variant of the CCA was first suggested by Ausubel et al. (2006) and further developed in Cramton (2013). Ausubel and Baranov (2014) discuss the evolution of the CCA. Gretschkoetal.(2017) discuss why bidding can be complicated in a CCA. Bichler et al. (2013) report experimental evidence on the CCA and present a simple example in which one bidder submits a spiteful bid. The rest of this paper is organized as follows. Section 2 describes the auction rules and the model. Section 3 proves our first main result that there do not exist efficient equilibria of the CCA where the clock phase fully reveals bidders’ types. Section 4 presents our second main result, namely that if the uncertainty concerning the competitor’s type is large, the CCA does not have efficient equilibria. Both sections present general propositions and illustrate the main results through examples of equilibria. The examples also show that the nonexistence of equilibria that satisfy certain properties is not due to a general nonexistence of equilibria. Section 5 analyzes the VCG mechanism as a benchmark. We show that under standard preferences, iterated elimination of weakly dominated strategies always results in an efficient outcome, but it leaves the bids of weak types on large shares undetermined. Lexicographic preferences impose that these bids are chosen to raise rival’s costs. Section 6 concludes with a discussion where we also consider the relevance of our paper for interpreting real-world auctions. Most proofs are provided in the Appendix. Theoretical Economics 14 (2019) Combinatorial clock auction 1277 2. Auction rules and the model We consider auctions where two bidders compete to get a share xi∈[01]of a unit of a divisible good. Throughout the paper, when a bidder has label i=12, the other bidder’s label is j=3−i. As the VCG auction is an important part of the CCA, we first describe the rules of the VCG auction before we go into the details of the CCA. After presenting the auction rules, we describe our assumptions regarding each bidder’s preferences. VCG rules In the VCG auction, all bidders simultaneously submit their bids for all shares, that is, bidder isubmits a bidding function Si:[01]→R+. Bidders cannot bid a positive amount on getting nothing, i.e., Si(0)=0. Subsequently, the auctioneer chooses the allocation x=(x1x2)that maximizes the sum of bids, i.e., x∈arg maxxS1(x1)+ S2(x2)such that x1+x2≤1and xi≥0for i=12. If two or more allocations solve the maximization problem, the auctioneer implements the allocation that minimizes the distance to the allocation (1/21/2). Bidder ireceives share xiand pays the VCG price maxySj(y) −Sj(xj), i.e., the opportunity cost he (reportedly) imposes on the other bidder. When there is no possibility of confusion, we sometimes drop subscripts. Hence, with strictly increasing bidding functions, the final allocation is (x 1−x) and bidder ihas to pay Sj(1)−Sj(1−x). Throughout the paper, we use the bid on the full supply to raise the rival’s costs. CCA rules The CCA is an auction with two stages. In the first stage, the clock phase, the auctioneer successively increases the price of the good and bidders report demands. The second, supplementary stage is a VCG auction where, in addition to the rules specified above, the bids are subject to so-called activity rules that are described below. Put differently, the clock phase elicits a demand function, whereas in the supplementary phase, bidders submit an inverse demand function. Activity rules aim for the consistency of the two functions. As explained in the Introduction, the rationale of the clock phase is price and package discovery, while VCG pricing should incentivize truthful bidding (Cramton 2013). The supplementary phase is meant to avoid some units remaining unsold and to allow bidders to express their preferences better. At each point of time in the clock phase, the auctioneer announces a price and bidders report the share they demand at current prices. If aggregate demand is larger than supply, the price is increased. The clock ends as soon as there is no excess demand so that the clock can end with market clearing or excess supply. Importantly, bidders are not allowed to increase their demand during the clock phase. We model this in the following way. The clock phase begins at an initial price p0=0. The clock price is increased continuously as long as there is excess demand. Bidder i’s action in the clock phase is a weakly decreasing demand function xi:R+→[01]that maps prices to demand. The clock phase stops at ˜ pif excess demand is smaller than or equal to zero, i.e., if x1(˜ p) +x2(˜ p) ≤1. In the main part of the paper, we analyze a CCA where bidders do not receive any information concerning aggregate demand in the clock phase.7Hence, each bidder can 7Real-world CCAs have used different regimes concerning information disclosure in the clock phase. In one regime, bidders are only informed about the fact that there is still excess demand and that the clock 1278 Janssen and Kasberger Theoretical Economics 14 (2019) only condition his demand on the price, but not on his rival’s previous demand. This assumption facilitates the formal analysis of the auction. We also comment, however, on an information policy where the last clock round demands are announced. At the end of Section 4, we present an example of an inefficient equilibrium under this information policy. In the supplementary phase, bidders can submit bids on all possible shares, that is, they submit bidding functions Si:[01]→R+. Given the supplementary bids, the auctioneer uses the same rules as described above for the VCG mechanism to compute the final allocation and individual CCA prices.8 Importantly, the CCA has activity rules linking the clock and the supplementary phase by translating the clock demand behavior into constraints on the supplemental bids. More specifically, the supplementary bidding function Simust satisfy three types of constraints. First, clock bids remain valid, i.e., if bidder idemanded xat clock price p, then it has to be the case that Si(x) ≥p·x. Unlike for all other bids, there are no further constraints for the bid on the final clock round demand. Second, the so-called final cap rule requires that supplementary bids satisfy the axiom of revealed preference with respect to the final clock round demand, i.e., Si(x) ≤Si(˜ xi)+˜ p(x −˜ xi),x= ˜ xi,where ˜ pis the final clock round price and ˜ xiis bidder i’s demand in the final clock round. Finally, the relative cap requires that if in the clock phase bidder idemanded xat a price p,then for any x>x, bidder icannot express an incremental bid for xin the supplementary round that is larger than p, i.e., Si(x)≤Si(x) +p(x−x).Abidonxcannot be larger than the area under the expressed clock demand curve.9 The intuitive rationale behind these three activity rules is as follows. The first rule requiring that clock bids remain valid is a minimal requirement to make clock bids meaningful. The final cap rule guarantees that if the clock ends with market clearing, the final clock allocation is the final allocation. As bidders do not know in advance when is the last clock round, this rule encourages bidders to bid truthfully in the clock.10 Finally, the relative cap rule motivates bidders when choosing between two different packages to bid according to their relative preference evaluated at the current round prices. Because of the second price rule, it is considered that bidders have an incentive to bid value on all possible packages in the supplementary round. The final cap and relative cap are such that by bidding truthfully in the clock, bidders can bid value in the supplementary phase. phase continues. In another regime, bidders are informed about aggregate demand in every clock round. The first regime was used in the first part of the Austrian auction and seems to be favored if collusion between bidders might be an issue. In the consultation document on the award of the 2.3 GHz and 3.4 GHz bands, Ofcom (2014) proposed using either the CCA or the SAA without demand disclosure. In a reaction for Hutchinson 3G, Power Auctions LLC (2015) claims that a dynamic auction with no demand disclosure is basically a sealed-bid auction. We show, however, that even without demand disclosure, the equilibrium outcomes that can be sustained in a CCA differ from the outcomes of the VCG. 8We do not consider the “core-selecting” elements in the pricing rule of real-world CCAs (see, e.g., Day and Milgrom 2008,Day and Cramton 2012, and Erdil and Klemperer 2010,aswellasGoeree and Lien 2016 and Ausubel and Baranov 2019). 9Levin and Skrzypacz (2016) provide a figure that illustrates the activity rules. 10Below, we formally define what it means to bid truthfully in the clock. Theoretical Economics 14 (2019) Combinatorial clock auction 1285 full supply. It is clear that after such a deviation, the efficient allocation is implemented. To make this a nonprofitable deviation, it should be the case that bidder ipays a price that is not smaller than the price he has to pay after the clock ends with market clearing and truthful bidding. Note, however, that the postdeviation CCA price cannot be higher, as bidder jalready fully raises the CCA price on the equilibrium path. Bidder j must, therefore, bid true marginal values on [xj(˜ p)xj]when the clock ends with excess supply at ˜ pwith demands (x∗ i˜ xj). Consequently, as in the proof of Proposition 1,bidder ican demand 1 for prices less than ˜ pand demand truthfully at ˜ p, where truthful demand is xi(˜ p) < x∗(θiθj), and bid true marginal values in the subsequent supplementary phase. The deviation increases the rival’s cost without affecting the allocation and his own payment. The next proposition states a property of any efficient equilibrium, namely that the clock cannot stop at very low prices and that weak bidders expand demand at some stage of the clock phase. In combination with the next subsection where we construct an efficient equilibrium, this proposition is of interest as it shows that, in contrast to Levin and Skrzypacz (2016), bidders do not necessarily want to reduce demand in the face of a competitor with a spite motive. The example of an efficient equilibrium also shows that the fact that efficient clock-separating equilibria do not exist in either information regime does not mean that efficient equilibria do not exist in general. Proposition 2. In any efficient equilibrium, a bidder will not demand ˆ xi≤xiat prices p<min{u(1θ)ui(ˆ xi)}. The smallest final clock price ˜ pis strictly larger than min{u(1θ)u(1/2θ)}and some types expand demand for some prices p< ˜ p. The argument is as follows. Suppose to the contrary that in an efficient equilibrium, bidder idemands ˆ xi≤xiat clock price p<min{u(1θ)ui(ˆ xi)}. We distinguish two cases. First, suppose ˆ xiis possibly an efficient share, i.e., xi≤ˆ xi≤xi. Efficiency of equilibrium requires that bidders bid true marginal values on possible efficient shares. However, this is not feasible for bidder i, as, independent of whether the clock ends at por continues, the relative cap imposes that the supplementary bids for ˆ ximust satisfy si(ˆ xi)≤p<u i(ˆ xi). Hence, in an efficient equilibrium where bidders reduce demand at these low prices, it must be that ˆ xi<x i. Second, we argue that in an ex post equilibrium, the highest types do not want to implement the efficient allocation if their competitor reduces demand to ˆ xi<x i.Consider bidder jwith type θj=θ. In an efficient ex post equilibrium, in the supplementary phase in which bidders iand jmeet, bidder jmust prefer winning the efficient share x∗ j over 1−ˆ xi, i.e., Uj(1−ˆ xi)−max ySi(y) +Si(ˆ xi)≤Uj1−x∗ i−max ySi(y) +Six∗ i As x∗ i>ˆ xi, the relative cap implies that Si(x∗ i)≤Si(ˆ xi)+p(1−x∗ j−ˆ xi)so that the above inequality implies Uj(1−ˆ xi)−Ujx∗ j≤p1−ˆ xi−x∗ j 1286 Janssen and Kasberger Theoretical Economics 14 (2019) However, as p≤u(1θ), this inequality cannot hold, i.e., the strongest types of bidder j strictly prefer winning 1−ˆ xiover the efficient share. Given this argument and the impossibility of fully revealing equilibria, it is clear that the clock cannot stop at a price ˜ p≤min{u(1θ)u(1/2θ)}. As at least one bidder must demand less than 1/2for the clock to end, it is clear that at these relatively low clock prices this bidder reduces demand and to a quantity smaller than xi, which we have just shown is not possible in an efficient equilibrium. It is then also easy to see that some bidders would want to expand demand at some prices p≤˜ p. Doing so, while keeping fixed the rest of the clock phase bidding, allows the bidder to raise the rival’s cost in the supplementary round in a way that does not risk winning these bids, as explained above. 3.1 Efficient equilibria in the quadratic utility model Proposition 1 rules out fully revealing efficient equilibria. In this subsection, we present an example where bidders have quadratic utility functions. The example shows that (i) equilibria exist, (ii) equilibria can be efficient, and (iii) what equilibria with demand expansion in the first phase of the clock may look like. Thus, the previous propositions have economic content and are not due to a lack of equilibrium existence. The example is also useful to understand the main intuition behind the second main result presented in the next section. The equilibrium is clock semi-separating as the bidding in the clock phase might reveal some information about the rival’s type. Both bidders expand demand by bidding on the full supply until a threshold clock price ˜ p>u(1/2θ). At prices larger than the threshold price, bidders bid truthfully. In accordance with the previous propositions, as ˜ p>u(1/2θ)and bidders bid truthfully for prices p≥˜ p, there are (at least) some low types for which there is pooling behavior in the clock phase. The threshold price plays a crucial role in the equilibrium construction, and we will identify constraints on it for this type of equilibrium to exist. Clock behavior As bidders have quadratic utility functions, the above described clock behavior with extreme demand expansion at prices p< ˜ pand bidding according to true marginal values for p≥˜ pgives clock demand xi(p) =⎧ ⎪ ⎨ ⎪ ⎩ 1if p< ˜ p maxθi−p σ0if p≥˜ p Figure 3 illustrates the two possible ways in which the clock can end in equilibrium: the clock ends either (i) with excess supply at ˜ por (ii) with market clearing at p∗>˜ p.The figure shows bidder 1’s clock demand function (the dashed line) and 1−x2(p),theresidual supply function faced by bidder 1(the solid line). The dotted shaded (line shaded) area between the two curves at price pindicates excess demand (supply). In the two plots, bidder 1’s demand is the same. Bidder 2’s type determines whether the left or the right figure applies to the clock phase. If bidder 2’s type is sufficiently low, aggregate demand at ˜ pis smaller than the available supply (Figure 3(a)). Conversely, when bidder Theoretical Economics 14 (2019) Combinatorial clock auction 1287 Figure 3. Clock behavior in the semi-separating equilibrium. Figure 4. Information revelation in the semi-separating equilibrium for ˜ p=(θ +θ−σ)/2. 2’s type is high, there may be excess demand at ˜ p(Figure 3(b)). Clearly, we must have ˜ p<θ−σ/2so that the highest types demand more than half of the supply at ˜ p.Atp> ˜ p, bidders bid truthfully and the clock eventually ends with market clearing at p∗>˜ p. Bidders update their prior about the other bidder as the clock proceeds. Figure 4 summarizes the information revelation during the course of the clock phase. The square depicts all possible type profiles. The clock ends at ˜ pfor type profiles in the gray area, i.e., if θi+θj≤2˜ p+σ. Hence, bidder iwith type θiinfers from the clock ending at ˜ pthat j’s type is at most 2˜ p+σ−θi, i.e., (˜ p) =[θ2˜ p+σ−θi]. If the types are such that the clock does not stop at ˜ p, the parallel diagonal lines reflect the combination of types for which the clock ends at p> ˜ p. For each such p, the clock ends with market clearing for types (θiθj)such that (θi−p)/σ +(θj−p)/σ =1, yielding the lines θj=2p+σ−θi. As the clock proceeds, the diagonal line in Figure 4 shifts to the northeast. If the clock does not stop at ˜ p, bidder iknows that at any p> ˜ p, the lowest possible type of the other bidder is 2p+σ−θi. Observing the final clock price p∗, each type θicorrectly infers the rival’s type (p∗)={2p∗+σ−θi}in the candidate equilibrium. 1288 Janssen and Kasberger Theoretical Economics 14 (2019) Supplementary bids The supplementary bidding functions depend on whether the clock ends at ˜ por at p∗.Iftheclockendsat ˜ p, bidder ibids in the supplementary phase according to S˜ p i(x) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0for x< ˜ xi Ui(x) for ˜ xi≤x<1 minUi(xi)+U(1−xi) Ui(˜ xi)+˜ p(1−˜ xi)for x=1 (1) where ˜ xiis bidder i’s truthful demand at ˜ p.13 Each bidder bids true utility on shares that might be obtained given the clock behavior and submits a spiteful bid on 1, which will be discussed below. If the clock ends at p∗>˜ p, bidder iuses the bidding function Sp∗ i(x) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ui(x) for x≤˜ xi Ui(˜ xi)for ˜ xi<x<1 Ui(˜ xi)+˜ p(1−˜ xi)for x=1 (2) One difference between the two bidding functions is that bidder ibids true marginal values on (efficient) shares higher than ˜ xiafter ˜ p,butnotafterp∗. We see below that this difference prevents jfrom further expanding demand in the clock phase. Another difference is, as explained below, the bid on the full supply. It is straightforward to check that the supplementary bidding functions implement the efficient allocation and that they satisfy the activity rules given the stipulated clock behavior. We now argue that these supplementary bidding functions are optimal from the perspective of raising the rival’s cost in that bidders want to raise their rival’s cost as much as possible without running the risk of winning a bid inadvertently. Whatever bidders bid on their last clock round share in the supplementary round, the relative cap implies they can maximally bid ˜ p(1−˜ xi)more on the full supply if the clock ends at p∗or ˜ p. However, when the clock phase ends at ˜ pand their bid on the entire supply is more than Si(xi)−Si(˜ xi)+S(1−xi)higher than their bid on ˜ xi, they run the risk of winning the full supply if the rival bidder’s type is low. In the candidate equilibrium, this means that bidders do not want to bid more than Ui(xi)+U(1−xi)on the full supply. As a result, when the clock ends at ˜ p, bidders bid Si(1)=min{Ui(xi)+U(1−xi)Ui(˜ xi)+˜ p(1−˜ xi)}. Now we consider the clock ending at p∗>˜ p. Observing the clock ended at p∗, bidders update their belief about the rival bidder’s type and believe that the clock ended with market clearing. Due to the final cap rule, bidders believe that the final clock allocation is also the final allocation and maximally raise their bid on the full supply, i.e., for x∈ [xi(p∗) ˜ xi]the relative cap si(x) ≤ui(x) holds with equality and Si(1)=Si(˜ xi)+˜ p(1−˜ xi). For later reference, it is useful to consider how Ui(xi)+U(1−xi)and Ui(˜ xi)+˜ p(1− ˜ xi)depend on a bidder’s type. Both expressions are represented in Figure 5.Itturnsout that there is a cutoff type ˆ θ( ˜ p) =˜ p(2+√2)−θ(1+√2)+σ 13Bidding 0on shares x< ˜ xisimplifies the proof that no bidder has an incentive to deviate. Theoretical Economics 14 (2019) Combinatorial clock auction 1289 θˆ θ( ˜ p) θ 2U(1 2) Ui(xi)+U(1−xi) Ui(˜ xi)+˜ p(1−˜ xi) Figure 5. Constraints on the supplementary bid for the full supply. such that Ui(xi)+U(1−xi)<U i(˜ xi)+˜ p(1−˜ xi)if and only if θi<ˆ θ( ˜ p).14 As a result, (only) bidders with a low type bid S˜ p i(1)=Ui(xi)+U(1−xi). Equilibrium constraints on clock behavior We now determine the restrictions on ˜ p such that no bidder has an incentive to deviate in the clock phase. First, bidders should acquire positive utility from bidding. As the minimal value of the efficient allocation is attained if both bidders are of the lowest possible type, it is sufficient to require that U(1/2)≥˜ p/2, which is equivalent to θ−σ/4≥˜ p. Second, it should not be the case that bidder iwants to reduce his demand in theclockphasetopreventrivaljfrom raising the price ihas to pay if jsuccessively learns bidder i’s type. To this end, define ˜ θ( ˜ p) =2˜ p+σ−θto be the highest type for which the clock always ends at ˜ p. Suppose now that ˆ θ( ˜ p) > max{θ˜ θ( ˜ p)}so that there exists a type θj∈[˜ θ( ˜ p) ˆ θ( ˜ p)) for which the clock phase does not necessarily stop at ˜ pand the bidders’ bid on the full supply is contingent on the final clock price. If the clock stops at ˜ p,theybidUj(xj)+U(1−xj)as they do not want to risk winning the full supply. If the clock ends at a higher price, due to market clearing in the last clock round, they can safely bid Uj(˜ xj)+˜ p(1−˜ xj). Thus, for types in the interval [˜ θ( ˜ p) ˆ θ( ˜ p)), the clock not stopping at ˜ pmakes their bid on the full supply jump discretely by Uj(˜ xj)+˜ p(1−˜ xj)−(Uj(xj)+U(1−xj)). Knowing this, it is profitable for some types higher than ˜ θ( ˜ p)—for whom the clock does not definitely stop—to reduce demand at ˜ pto be certain to end the clock. Thus, ˆ θ( ˜ p) > max{θ˜ θ( ˜ p)}cannotbepartof an equilibrium. Alternatively, if ˜ θ( ˜ p) ≥ˆ θ( ˜ p) or if ˆ θ( ˜ p) ≤θ, then the supplementary bids of all types θj>˜ θ( ˜ p)—for which the clock phase possibly continues at prices p> ˜ p—are independent of the final clock price, so that the bidders cannot raise their supplementary bids after obtaining information through the final clock price. For this case, we show that there is no incentive for demand reduction. Suppose type θi>˜ θ( ˜ p) reduces demand at ˜ p.Letθj>2˜ p+σ−θibe such that the clock ends at ˜ punder i’s demand reduction, although it would continue under truthful bidding. Bidder jbids 0 on x∗ jif the 14Formally, the equation Ui(xi)+U(1−xi)=Ui(˜ xi)+˜ p(1−˜ xi)has a second root ˆ θ2(˜ p) =˜ p(2−√2)+ σ−θ(1−√2). As in all equilibria we consider, we have that ˜ p≥min(θ −σ/2θ−σ/4),soitiseasytosee that ˆ θ2(˜ p) > θand that we effectively only have to consider ˆ θ( ˜ p). 1290 Janssen and Kasberger Theoretical Economics 14 (2019) clock ends at ˜ p,asx∗ j<˜ xj. As the efficient allocation cannot be implemented, the demand reduction leads to a decrease in bidder i’s primary utility. Therefore, equilibrium requires that ˜ θ( ˜ p) ≥ˆ θ( ˜ p) or ˆ θ( ˜ p) ≤θ. Third, we should also make sure that bidders do not have an incentive to expand demand further than is stipulated in the candidate equilibrium strategies by deviating and demanding more than ˜ xi(˜ p) until p> ˜ p. To this end, we first argue that for all types θi< θ, the clock phase must stop at ˜ pwith positive probability in equilibrium. From the candidate equilibrium strategies, it is clear that if the clock can end for the highest possible type θ, it can also end for all other types. The reason the clock must possibly stop for all types is that if a bidder with type θknows that the clock will certainly not end under truthful bidding, then he prefers to continue demanding the full supply. To see this, recall that to have a semi-separating equilibrium, it should be the case that ˜ x( ˜ pθ) > 1/2, which implies that ˜ p<θ−σ/2. Given the definition of ˜ θ( ˜ p) and the constraints derived in the previous paragraph, this implies that ˆ θ( ˜ p) < θ. Thus, in the candidate equilibrium strategy, the activity rules restrain bidder θfrom fully raising the rival’s cost. Continuing bidding on the entire supply would allow further raising the rival’s cost without affecting the final allocation (and the price he pays). To make sure that it is possible for the clock to end along the equilibrium path for all types θi< θ, it should be the case that ˜ p≥(θ+θ−σ)/2=u(x θ) =u(xθ). Next, we argue that expanding demand at ˜ presults in a decrease of expected surplus, as there is a positive probability that the clock ends by bidding truthfully at ˜ pfor all types. To see this, we use the difference between the supplementary bidding strategies S˜ p i(x) and Sp∗ i(x) in (1)and(2), respectively. If bidder jwere to bid truthfully, the clock would stop at ˜ pfor all types θi≤2˜ p+σ−θj. Importantly, the aggregate clock demand at ˜ pis arbitrarily close to 1if the competitor’s type is just below 2˜ p+σ−θj.Iftheclock ends at ˜ p,wehavex∗ j>˜ xj, so that the supplementary bidding function (1) guarantees that bidder jgets the efficient share at a price min{Ui(xi)+U(1−xi)Ui(˜ xi)+˜ p(1− ˜ xi)}−Ui(x∗ i). Consider then that bidder jexpands demand at ˜ p. In that case, there exist some types θijust below 2˜ p+σ−θjfor which the clock ends at a higher price than ˜ p. Given that the supplementary bid strategy of these types changes from (1)to(2), bidder jgets at most a utility of Uj(1−˜ xi)−[Ui(˜ xi)+˜ p(1−˜ xi)−Ui(˜ xi)].As1−˜ xi>x ∗ jfor some types θi, bidder jis better off not deviating. All of the above constraints can be jointly satisfied for a variety of final clock prices. For example, we can set ˜ p=(θ+θ−σ)/2as we did in Figure 4. At this price, the highest type for which the clock definitely ends at the threshold price is ˜ θ( ˜ p) =θand equilibrium exists whenever ˆ θ( ˜ p) ≤θ,orθ−θ≤(√2−1)σ. Discussion We conclude that a semi-separating equilibrium as discussed above exists if the uncertainty concerning the competitor’s type, measured by θ−θ,isnottoolarge. This equilibrium is efficient, as all bidders bid their true marginal utilities on possibly efficient shares in the supplementary phase and other bids are such that the winning bid combination is in this range of possibly efficient shares. Thus, there are efficient equilibria with some information revelation, where low types pool and high types are constrained by the activity rule so that they cannot exploit new information to raise their rival’s cost. Theoretical Economics 14 (2019) Combinatorial clock auction 1291 The semi-separating equilibrium is noteworthy as it shows that even if bidders know that the competitor is raising their cost in the supplementary phase, they do not reduce demand in the clock phase. This is in contrast to Levin and Skrzypacz (2016), who restrict bidders to linear proxy strategies and show that bidders will engage in demand reduction in the clock phase, assuming (against the auction rules) that a demand reduction strategy in the clock phase does not affect the ability to raise a rival’s cost. The example also shows that, in contrast to what some observers of the CCA have argued, the clock phase may well end up with excess supply, while bidders are still able to raise their rival’s cost.15 A variant of this equilibrium occurs if the clock stops at ˜ pfor all type profiles. In such a clock-pooling equilibrium, the clock does not reveal any information. There are three constraints on the clock-pooling price ˜ p. First, the truthful demand at ˜ pof the strongest bidder should be smaller than half the supply for the clock to end. Second, no bidder should have an incentive to further expand demand because ˜ pis such that the final cap and relative cap do not restrict the spiteful bid. Third, as before, the weakest type should still derive nonnegative utility. The constraints on ˜ pspecify a tighter upper bound on the range θ−θfor such a clock-pooling equilibrium to exist. 4. Nonexistence of efficient equilibria with large uncertainty So far we have seen that efficient clock-revealing equilibria do not exist, but that efficient equilibria may nevertheless exist even if bidders are spiteful. The example presented in the previous section constructs an efficient equilibrium where the uncertainty concerning the final allocations, measured by θ−θ, is relatively small. In this section, we consider auctions where the ex ante uncertainty concerning the final allocations is relatively large and, consequently, information revelation might be more important. Our second main result shows, however, that the CCA does not have efficient equilibria when the uncertainty about the other bidder’s type is sufficiently large. To simplify the proof, we consider (type-) symmetric equilibria, that is, equilibria where identical types use identical strategies.16,17 15See, e.g., Levin and Skrzypacz (2016, Remark 2, p. 2542), where they observe that “[i]f we allowed bidder 2to create excess supply at the end of the clock phase, she could increase bidder 1paymentevenmore.... Such extreme predatory behavior is even more difficult to execute and even more risky for player 2than what we describe. Moreover, analyzing equilibria in this case is difficult, so we maintain the assumption that player 2is not allowed to create excess supply in the clock phase.” Similarly, Kroemer et al. (2016, p. 38) observe that “[i]n recent spectrum auction implementations, the regulator decided not to reveal excess supply in the last round, in order to make spiteful bidding risky. It depends on the market specifics, if this risk is high enough to eliminate spiteful bidding.” The British regulator Ofcom (2015, A8.48 p. 16) also writes in a similar vein when they consider the Austrian 2013 CCA outcome: “We also noted that at the end of the clock rounds there was an excess supply of 2 ×10 MHz in each of the 900 MHz and 1800 MHz bands.... This further suggests a possible reason why bidders may have considered price driving in the supplementarybidstobeariskystrategy....” 16We believe that asymmetric efficient equilibria do not exist either, but a formal proof would require checking many different cases. 17The proof of this result can also be used to show that efficient equilibria do not exist if the lowest type θdoes not value the good at all, i.e., u(xθ)=0. 1292 Janssen and Kasberger Theoretical Economics 14 (2019) Proposition 3. Let u(1θ) > u(0θ). Due to the high ex ante uncertainty about the final allocation, no symmetric efficient equilibrium exists. Importantly, if the uncertainty concerning bidders’ types is substantial, all equilibria of the CCA, which is a dynamic implementation of the VCG auction, are inefficient. The next section shows that no matter how large this uncertainty is, the VCG auction always has efficient equilibria. It is, therefore, the information that is transmitted during the clock phase that may destroy efficiency (even if little information is provided). The result demonstrates that blending two well meant auction design principles (the second-price principle and an open format) may have unintended consequences. The main intuition for this result can be developed by combining different arguments that we have previously developed. From Proposition 2, we know that in any efficient equilibrium, some types will expand demand. Because of the relative cap and the large type space, the clock must last long in an efficient equilibrium if one bidder is sufficiently strong. The long duration gives some relatively strong types the possibility of weakening the constraints of the activity rules by expanding demand. Because of the high uncertainty, the highest and the lowest types cannot pool at a threshold price in an efficient equilibrium. Consequently, the clock not ending when low types drop out reveals to strong types that their competitor is strong. The spiteful bid of high types then jumps discretely in any supplementary round that follows a longer duration of the clock. Other types will reduce their clock phase demand in anticipation of this behavior, which—given the activity rules—necessarily leads to inefficiencies. This argument is developed in more detail with the following notation. Note that if u(0θ)<u(1θ),thereexistsatypeθ>θsuch that u(0θ)=u(1θ) and that the lowest possible efficient share of all types in [θθ]is 0. Likewise, there is a type θ< θ such that u(1θ)=u(0θ), so that the largest possible efficient share of all types in [θ θ]is the full supply. We first argue that in any efficient equilibrium, when the lowest type θmeets a type above θ, the final clock price must be u(0θ), which is the clock price at which the lowest type drops out of the auction under truthful bidding. Because of the high uncertainty, the truthful demand of types above θis the full supply at this price. If the final clock price pwas smaller for such a type profile, then at least one of the bidders would have reduced demand, as for these prices u(1θ)>u(0θ)>pfor all types θ> θ,sothatif none of them would have reduced demand, then aggregate demand would be larger than supply. Given the restrictions imposed by the relative cap, these bidders could not bid marginal utilities on all possibly efficient shares in the supplementary round (Proposition 2). The final clock price for such a type combination cannot be larger than u(0θ)either, as this would imply that θand some marginally larger types have excessively expanded demand. Demand expansion at these high prices and the requirement that supplementary bids must be at least as high as clock bids lead to the bidder necessarily winning too much or too little. Thus, the clock must end at u(0θ)for these type profiles. Second, we show that in any efficient equilibrium, types marginally larger than θwill demand truthfully at clock prices slightly larger than u(0θ). By the same reasoning as in Theoretical Economics 14 (2019) Combinatorial clock auction 1293 the previous paragraph, the clock cannot end later than uj(0)for θj>θthrough demand expansion. Bidders also cannot reduce demand as the relative cap then prevents them from bidding true marginal values on efficient shares, which is necessary for ex post efficiency. As a result, they must bid truthfully at these prices. Given this behavior of types just above θ, we next argue that similar to the reasoning in Proposition 1, types just below θfind it optimal to maximally expand demand in the clock phase for prices p<u(0θ). The reason is that by maximally expanding demand, the clock cannot end at prices p<u(0θ), while they know that in an efficient equilibrium, the clock will continue for them if the rival is sufficiently strong. Expanding demand in the clock implies they discretely increase their supplementary bid on the full supply if the clock stops at a price larger than u(0θ)compared to the situation where the clock stops at u(0θ)and they find it optimal to do so. In anticipation of this behavior, some types in (θθ]will then find it profitable, however, to reduce their demand at u(0θ), ending the clock prematurely to prevent the rival from further raising their costs. Such behavior is inconsistent with efficiency, however, showing that there is no efficient equilibrium. 4.1 Inefficient equilibria in the quadratic utility model This subsection uses the quadratic utility model to present an example of an inefficient equilibrium that illustrates the kind of equilibria that may exist when efficient equilibria do not exist. Thus, the example underlines that Proposition 3 is not due to a general nonexistence of equilibrium. In our example, the final allocation is almost surely inefficient as bidders win either 0,1/2, or the full supply. For simplicity, we consider a CCA in which bidders are informed about aggregate demand in the final clock round. Let σ<θ−θ≤3/2·σ. We have three threshold prices ˜ p1<˜ p2<˜ p3, and partition the type space into [θθ1),[θ1θ2),and[θ2 θ].Callθ3=θand θ0=θ.LetI∈{123}be the index for type θi∈[θI−1θI), where the interval is closed for I=3. Define Janalogously for a bidder with type θj. We choose the prices such that U1/2θI−1−˜ pI/2=0 and choose the cutoff types so that U1/2θI−˜ pI/2=U1θI−˜ pI Thus, at price ˜ pI,typeθI−1is indifferent between dropping demand to 0and bidding for half of the supply, while type θIis indifferent between bidding for half of the supply and the full supply. For θ, the equality can be an inequality so that the left-hand side is larger than the right-hand side. When bidders have quadratic utility functions, we have ˜ pI=θ−σ/4+(I −1)·σ/2and θI=θ+I·σ/2. 1294 Janssen and Kasberger Theoretical Economics 14 (2019) Strategies In the clock phase, type θi∈[θI−1θI)follows the clock demand function xi(p) =1for p< ˜ pI 0for p≥˜ pI Hence, the clock ends with both bidders demanding 0 if I=J, and with one demanding the full supply and the other demanding 0 if I=J. The supplementary bids depend on aggregate demand in the final clock round. If the clock ends with aggregate demand of 0 at the final clock price ˜ p,thenθibids S˜ pI ix|xi(˜ p) +xj(˜ p) =0=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˜ p/2for x=1/2 ˜ pfor x=1 0else If aggregate demand in the last clock round is positive, then the supplementary bidding function is given by S˜ pI ix|xi(˜ p) +xj(˜ p) > 0=˜ pfor x=1 0else The supplementary bidding functions clearly satisfy the constraints of the activity rules, and the final cap and the relative cap are binding for strictly positive bids. The difference between the two bidding functions is that in the former case, a positive bid on 1/2is submitted, whereas in the latter case, no such bid is made. No incentive to deviate If bidders belong to the same partition group, i.e., I=J,then both win half of the supply at the CCA price ˜ pI/2. The construction of ˜ pIand the cutoff types make it clear that these bidders prefer this outcome over winning the full supply at a price of ˜ pI. Bidders could win x>1/2by deviating to demanding xat the final clock price ˜ pI(instead of 0). The CCA price for xwould then be ˜ pI. It is clear that this gives less surplus than the full supply, which in turn is a worse outcome than winning 1/2. Bidder icould also win a share 0<x<1/2at the CCA price of ˜ pI/2by deviating in the supplementary phase that follows ˜ pIand zero aggregate demand. Again, this leads to a lower surplus than winning 1/2. Next consider the case where θiand θjare such that I<J.Theclockendsat ˜ pIwith market clearing. Bidder jwins the full supply at the CCA price ˜ pI. The construction of the prices and cutoff types is such that the stronger bidder prefers winning 1at CCA price ˜ pIover 1/2at the CCA price ˜ pI/2. The lowest price at which bidder icould win a positive amount is ˜ pJ, which is so high that bidder iwould incur a loss. Hence, bidders do not have an incentive to deviate, even if they know the competitor’s type. Hence, the proposed strategies form an ex post equilibrium. There are also no profitable deviations in the spite dimension of the preferences. If a bidder demanded x>0at ˜ pI,thenhe would lose the possibility of winning 1/2at a price at which he makes a positive surplus. Hence, a bidder cannot further raise his rival’s costs without decreasing his own expected surplus. Theoretical Economics 14 (2019) Combinatorial clock auction 1301 utility on [01/2].AsSˆ p(0)=0, this implies bidding true utility, i.e., Sˆ p j(x) =Uj(x) for x∈[01/2]. We now show that the efficient allocation (1/21/2)is not implemented if both bidders have type θj.Letδ>0be small enough so that ˆ p−δ>u j(0)and let x=xj(ˆ p−δ) > 1/2. For the equilibrium to be efficient, we must have 2Sˆ p j(1/2)=2Uj(1/2)≥Sˆ p j(x) +Sˆ p j(1−x) ≥(ˆ p−δ)x +Uj(1−x) where we use truthful bidding on possible efficient shares and the fact that clock bids remain valid. The inequality simplifies to Uj(1/2)+1/2 1−x uj(y)dy ≥(ˆ p−δ)x ⇒uj(0)/2+uj(1−x)(x −1/2)>(ˆ p−δ)(1/2+x−1/2) The inequality is false, however, as ˆ p−δ>u j(0)≥uj(1−x). Claim 3. Let u(0θ)<u(1θ). In any symmetric efficient equilibrium, there is an >0 such that for all θi∈[θ− θ], we have τ(θiθ)=p=u(0θ). Proof. Suppose to the contrary that there is a symmetric efficient equilibrium such that for all >0,thereisatypeθi∈[θ− θ]with τ(θiθ)= u(0θ). There are three exhaustive cases to consider. First, Claim 2 shows that τ(θiθ)>u(0θ)cannot occur for any θiin a symmetric efficient equilibrium, because if it happened for θi,itwould also happen for θin a monotone equilibrium. Second, we show that it is impossible that τ(θiθ)<u(0θ)for any θi∈[θθ]. The third case considers types just below θ. It is straightforward to see that in any efficient equilibrium it cannot be that for type θi∈[θθ]we have τ(θiθ)=p<u(0θ).Proposition 2 tells us that in an efficient equilibrium bidders do not reduce demand at this price. If p<u(0θ)≤u(1θ), the truthful demand of types just below θat pequals the full supply, while the truthful demand of the low type is strictly positive. Hence, the clock cannot end at τ(θiθ)=p<u(0θ). We now turn our attention to types “just below” θ,asitcouldbethatforevery>0 there is a type θi∈(θ− θ)with τ(θiθ)<u(0θ). From the previous case it follows that τ(θiθ)must be increasing in θi, because otherwise there would be types just below θwhose truthful demand is the full supply and who reduce demand, contradicting Proposition 2. We will show that for types just below θ,theclockmustendatu(0θ)if they meet the lowest type. It cannot end earlier due to an argument that relies on bidders having spiteful lexicographic preferences and which is akin to the proof of Proposition 1. For clock prices just below u(0θ), the truthful demand of types just below θis almost the full supply, while low types’ truthful demand is slightly above 0. The function τ(θiθ) increases in θiif demand is decreased continuously. Efficiency of equilibrium and the final cap require bidders to lower demand truthfully. Hence, the lowest type is active for all p<u(0θ). The marginal supplementary bids when the clock ends at u(0θ)must satisfy the relative cap s(xθ)≤u(xθ), which follows from truthful bidding at prices just below clock price u(0θ). Thus, the lowest type bids true utility on shares (slightly) 1302 Janssen and Kasberger Theoretical Economics 14 (2019) above 0. This follows from the relative cap being necessarily binding in an efficient and information revealing equilibrium. As a result, types just below θhave no incentive to lower demand truthfully at prices just below u(0θ), because they can expand demand until u(0θ)and still get the efficient share at the same CCA price if they meet a very low type.  The first claim of the lemma follows directly from the previous claim. For types just below θ, the clock does not end before u(0θ)in an efficient equilibrium. Thus, these types will expand demand for clock prices p<u(0θ)=pin equilibrium. The demand expansion weakens the (necessarily) binding constraint of the relative cap. The second claim of the lemma is that there is a δ>0such that types in [θθ]∪ [θ− θ]demand truthfully for p∈[p p+δ]. Bidders cannot reduce demand due to Proposition 2. The proof of Claim 2 rules out θj∈[θθ]demanding x>xjfor p<u j(0) and xj(uj(0)) =0.Claim 1 then shows that θj∈[θθ]bids truthfully for clock price p∈[p p+δ].Typeθi∈[θ− θ]also has to bid truthfully, as the low types bid truthfully. Demand expansion of a type whose truthful demand is arbitrarily close to the full supply at p∈[p p+δ]goes along with the possibility of market clearing. As the equilibrium is efficient, bidders have to demand truthfully at prices at which the clock can end with market clearing.  We prove the proposition by showing that types just below θcan make the CCA price dependent on the final clock price and this incentivizes some weak bidders to pool with lower types by reducing demand. Consider type θijust below θthat expands demand for prices just below p.Claim 3 of the proof of the lemma shows that the lowest equilibrium final clock price of θiis p.Letθjdenote the highest type for which the clock may end at pwhen meeting θiunder truthful bidding, that is, ui(˜ xi)=uj(˜ xj). It is clear that θj>θ. When the clock ends at p, efficiency requires the lowest type to bid true utility in a neighborhood of 0 in the supplementary phase. This follows from Sp(0)=0,thenecessity of bidding true marginal values on possible efficient shares, and the clock ending at pfor all types in (θ− θ]. Hence, when the clock stops at p, bidder i’s supplementary bid on the full supply must be Sp i(1)=minSp i(˜ xi)+p(1−˜ xi)Sp i(xi)+U(1−xi) Note that ˜ xi=x∗(θiθj)<xi=x∗(θiθ)for types just below θ. Since the clock ends at p when bidder imeets a type below θj, bidder imust bid true marginal values on [˜ xixi] in the supplementary phase when the clock ends at pin an efficient equilibrium, i.e., Sp i(xi)=Sp i(˜ xi)+Ui(xi)−Ui(˜ xi). Now we show that the relative cap is slack for θiin the supplementary phase if the clock ends at p,thatis,Sp i(1)=Sp i(xi)+U(1−xi)<S p i(˜ xi)+p(1−˜ xi). Suppose the inequality was false. Inserting the expression for Sp i(xi)and simplifying yields p(1−˜ xi+xi−xi)≤Ui(xi)−Ui(˜ xi)+U(1−xi) ⇒p(1−xi)+p(xi−˜ xi)<u i(˜ xi)(xi−˜ xi)+p(1−xi) Theoretical Economics 14 (2019) Combinatorial clock auction 1303 The implication uses decreasing marginal values (e.g., u(0θ)·x>U(x))andp=u(0θ). Hence, the last inequality is false and the relative cap must be slack in the supplementary phase that follows p. The presence of types close to θtherefore limits the types just below θfrom fully raising the supplementary bid on the entire supply after the clock ends at p. As the clock continues after p, types just below θlearn that the competitor’s type is at least θj>θ.Letp∗∈(pp+δ), where the neighborhood is given by Lemma 1. As demand is lowered truthfully, the clock can end with market clearing at clock price p∗> p in equilibrium. In the subsequent supplementary phase, bidder iwill raise the bid on the full supply so that the relative cap is binding. For shares in [xi(p∗) ˜ xi],the relative cap requires si(x) ≤ui(x) due to truthful bidding in the clock phase. Bidder i will also take these constraints as binding and will bid Si(˜ xi)=Si(x∗ i)+Ui(˜ xi)−Ui(x∗ i). As a result, the CCA price for types around θjjumps from Ui(xi)+U(1−xi)−Ui(x∗ i) to Ui(˜ xi)+p(1−˜ xi)−Ui(x∗ i). This discrete increase in the CCA price incentivizes types marginally larger than θjto inefficiently reduce demand at pto avoid the jump in the CCA price. Thus, there is no efficient equilibrium. Proposition 4. In the VCG mechanism with standard preferences, any strategy profile that survives any process of IEDS implements the efficient allocation. The VCG payments depend, however, on the order in which weakly dominated strategies are eliminated and onthechoiceofstrategyprofilethatsurvivesIEDS. Proof. First we show that with standard preferences, any strategy profile that survives any process of IEDS implements the efficient allocation. This proof has three parts. First, we show that bidding truthfully is always an optimal strategy. Therefore, it survives any process of iteratively eliminating weakly dominated strategies. Second, we argue that any bidder must be indifferent between any strategy that survived the IEDS and truthful bidding. In the third and final step, we show that only the efficient allocation can be implemented by strategies that survive IEDS. We use the following notation. The set Si is the set of strategies that survived IEDS for a bidder with type θi. The set of iteratively undominated strategy profiles is denoted as S=S1×S2. First, bidding truthfully is always an optimal strategy, i.e., it is a best response against any strategy profile of the other bidder Sj. To see this, let the other bidder use Sjand let ˆ xdenote the allocation implemented by the profile (UiSj),thatis,Ui(ˆ xi)+Sj(ˆ xj)≥ Ui(xi)+Sj(xj)for all other feasible allocations x. This inequality also says that the surplus of bidder iis at least as large under the allocation ˆ xthan under any other allocation, because one can simply subtract the constant maxySj(y) on both sides. Truthful bidding is always optimal and, therefore, Ui∈Si. Second, bidder imust be indifferent between all Si∈Siand Ui.ForallSi∈Si,itholds that for all other bidding functions Tiof bidder iand all bidding functions Sj∈Sj,the surplus of Siis at least as large as for Tior strictly higher than for Tifor at least one Sj. Recall that the surplus from Uiis at least as large as from Si. As a result, the strategy Siis iteratively not dominated if and only if for all Sj∈Sj, the surplus of Siand Uiis the same for all Sj∈Sj. 1304 Janssen and Kasberger Theoretical Economics 14 (2019) We now prove that any profile S∈Sstrictly implements the efficient allocation, i.e., Si(x∗ i)>Si(xi)for all feasible allocations x=x∗. First, note that the only share implemented by (UiSj)is the efficient share, that is, Uix∗ i+Sjx∗ j>U i(xi)+Sj(xj)for all x=x∗(6) To see this, suppose there is an allocation y=x∗with Ui(yi)+Sj(yj)≥Ui(x∗ i)+Sj(x∗ j).As bidder jis indifferent between Ujand Sj,wehavethatUj(yj)+Ui(yi)=Uj(x∗ j)+Ui(x∗ i). Strict concavity of Uimplies that there is a unique efficient allocation, implying that y= x∗, a contradiction. Hence, (UiSj)implements only the efficient share. The next step uses this property to show that (SiSj)also implements the efficient allocation. Again by contradiction, let ˆ x=x∗be the allocation implemented by (SiSj). Bidder iis indifferent between Siand Ui,soUi(ˆ xi)+Sj(ˆ xj)=Ui(x∗ i)+Sj(x∗ j), contradicting inequality (6). As aresult,(SiSj)must implement the efficient allocation. The proof that the VCG prices depend on the process of IEDS is constructive. We construct a sequence of eliminations that ends with a set of undominated strategies. Strategies in the set will have the desired properties. To show that a strategy is dominated, one needs to find an alternative strategy that yields weakly higher utility against all admissible strategy profiles of the other bidders and a strictly higher utility against some admissible strategy profiles. Above, we have seen that bidding Uiis always optimal. In the subsequent three steps of iterative elimination, we have to find only a strategy Sjto show that the alternative of truthful bidding is strictly preferred. Let Bbe the set of all bidding functions, i.e., the set of all S:[θ θ]×[01]→R+.Note that the optimality of a function depends on the type θi. Step 1. Strategies Sifor which there exists ˜ x<1such that Si(˜ x) > Ui(˜ x) are dominated. Bidder juses the bidding function Sj(x) =⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ max ySi(y) +Si(˜ x) for x=1 max ySi(y) for x=1−˜ x 0else The bidding profile Simplements the allocation in which bidder iwins ˜ xand bidder jwins 1−˜ x, since ties are broken in favor of interior allocations. Bidder i’s surplus is Ui(˜ x) −Si(˜ x) < 0, whereas the surplus from bidding truthfully is nonnegative. Remove these dominated strategies to obtain S1⊂B. Step 2. Strategies are dominated that satisfy Si(1)>V i(θ). Note that for low types, Vi(θ)>U i(1). For high types it can be the case that xi=1,soUi(1)=Vi(θ). Bidder jbids Sj(x) =0for x<1and Sj(1)=Si(1)−ε,withε∈(0Si(1)−Vi(θ)). Bidder iwins the full supply at a price higher than utility. Truthful bidding is, therefore, strictly better against this strategy profile of other bidders. Remove these dominated strategies to get S2⊂S1. Step 3. Strategies are dominated for which there exists ˜ x∈[xi xi]with Ui(˜ xi)>S i(˜ x). Let x∈argmaxySi(y).Letε∈(0Ui(˜ x) −Si(˜ x)).Inthecaseof˜ x<1, suppose bidder j uses the bidding function Sj(1)=Si(x)+ε,Sj(1−˜ x) =Si(x)−Si(˜ x),andSi(x) =0for all other x. Under this bidding function, bidder iwins nothing and gets zero surplus. If Theoretical Economics 14 (2019) Combinatorial clock auction 1305 the bid on ˜ xis raised to Ui(˜ x), then bidder iwins ˜ xand gets strictly positive surplus. For ˜ x=1,letSj(1)=Si(1)+εand 0otherwise. Bidder iwins nothing if the bid is below true utility level and wins the full supply if the bid equals utility. The set S⊂S2is obtained by eliminating these dominated strategies. After the three steps of elimination, all remaining strategies implement the efficient allocation. To see this, let bidders use the admissible strategy profile S∈S. The value jointly expressed for the efficient allocation is higher than the value jointly expressed for any other feasible allocation x.Letxi<1for all i.Then Si(xi)+Sj(xj)≤Ui(xi)+Uj(xj)≤Uix∗ i+Ujx∗ j=Six∗ i+Sjx∗ j where the first inequality follows from Step 1, the second inequality follows from the definition of efficiency, and the last equality follows from Steps 1, 2, and 3. For an allocation such that there is an iwith xi=1,wehave Si(xi)+Sj(x) =Si(1)≤Vi(θ)≤Uix∗ i+Ujx∗ j=Six∗ i+Sjx∗ j where the first equality follows from Step 1, the first inequality follows from Step 2, the second inequality follows from the definition of efficiency, and the last equality ffollows rom Steps 1, 2, and 3. All strategy profiles in Simplement the efficient allocation. There are no further dominated strategies, as any strategy that survives IEDS yields the same expected utility as bidding truthfully. To see that the VCG prices depend on the chosen strategy profile, consider a bidder with θisufficiently small so that Vi(θ)<U i(1)and the other player has the lowest possible type θ. The value of the efficient allocation is V(θ iθ). Suppose bidder ichooses Si∈Siwith Si(x) =Ui(x) for x<1and Si(1)=Vi(θ), and the other bidder plays Sj=Uj. Hence, the VCG price for bidder jis Vi(θ)−Ui(x∗ i). If the strategy profile was such that Si=Ui, then the VCG price would be strictly less than that and equal to Ui(1)−Ui(x∗ i). Note that the strict inequality and continuity imply that the difference in VCG prices holds for an open set of types. Similarly, if Step 1 was such that we also eliminate Si(1)>U i(1), then the first VCG price would not be possible. References Ausubel, Lawrence M. (2004), “An efficient ascending-bid auction for multiple objects.” American Economic Review, 94, 1452–1475. [1274] Ausubel, Lawrence M. and Oleg V. Baranov (2014), “Market design and the evolution of the combinatorial clock auction.” American Economic Review: Papers & Proceedings, 104, 446–451. [1276,1297] Ausubel, Lawrence M. and Oleg V. Baranov (2019), “Core-selecting auctions with incomplete information.” Unpublished paper, Forthcoming International Journal of Game Theory. [1278] Ausubel, Lawrence M., Peter Cramton, and Paul Milgrom (2006), “The clock-proxy auction: A practical combinatorial auction design.” In Combinatorial Auctions (P. Cramton, Y. Shoham, and R. Steinberg, eds.). MIT Press. [1272,1276,1284] 1306 Janssen and Kasberger Theoretical Economics 14 (2019) Bichler, Martin, Pasha Shabalin, and Jürgen Wolf (2013), “Do core-selecting combinatorial clock auctions always lead to high efficiency? An experimental analysis of spectrum auction designs.” Experimental Economics, 16, 511–545. [1276] BT (2015), “Response to: Public sector spectrum release (PSSR) award of the 2.3 GHz and 3.4 GHz bands.” http://goo.gl/xaoDKY, accessed on January 8, 2019. [1275] Cramton, Peter (2013), “Spectrum auction design.” Review of Industrial Organization, 42, 161–190. [1272,1276,1277] Cramton, Peter and Axel Ockenfels (2017), “The German 4G spectrum auction: Design and behaviour.” Economic Journal, 127, F305–F324. [1274] Day, Robert and Peter Cramton (2012), “Quadratic core-selecting payment rules for combinatorial auctions.” Operations Research, 60, 588–603. [1278] Day, Robert and Paul Milgrom (2008), “Core-selecting package auctions.” International Journal of Game Theory, 36, 393–407. [1278] Erdil, Aytek and Paul Klemperer (2010), “A new payment rule for core-selecting package auctions.” Journal of the European Economic Association, 8, 537–547. [1278] Goeree, Jacob K. and Yuanchuan Lien (2016), “On the impossibility of core-selecting auctions.” Theoretical Economics, 11, 41–52. [1278] Gretschko, Vitali, Stephan Knapek, and Achim Wambach (2017), “Bidding complexities in combinatorial clock auctions.” In Handbook of Spectrum Auction Design (Martin Bichler and Jacob Goeree, eds.). Cambridge University Press. [1276] Grimm, Veronika, Frank Riedel, and Elmar Wolfstetter (2003), “Low price equilibrium in multi-unit auctions: The GSM spectrum auction in Germany.” International Journal of Industrial Organization, 21, 1557–1569. [1271] Janssen, Maarten and Vladimir A. Karamychev (2016), “Spiteful bidding and gaming in combinatorial clock auctions.” Games and Economic Behavior, 100, 186–207. [1276] Kroemer, Christian, Martin Bichler, and Andor Goetzendorf (2016), “(Un)expected bidder behavior in spectrum auctions: About inconsistent bidding and its impact on efficiency in the combinatorial clock auction.” Group Decision and Negotiation, 25, 31–63. [1291] Laffont, Jean-Jacques and Jean Tirole (1988), “The dynamics of incentive contracts.” Econometrica, 56, 1153–1175. [1273,1284] Levin, Jonathan and Andrzej Skrzypacz (2016), “Properties of the combinatorial clock auction.” American Economic Review, 106, 2528–2551. [1272,1275,1276,1278,1279, 1281,1285,1291] Milgrom, Paul (2004), Putting Auction Theory to Work. Cambridge University Press. [1274] Ofcom (2012), “Assessment of future mobile competition and award of 800 MHz and 2.6 GHz.” http://goo.gl/KfQBsX, accessed on January 8, 2019. [1275] Theoretical Economics 14 (2019) Combinatorial clock auction 1307 Ofcom (2014), “Public sector spectrum release (PSSR) award of the 2.3 GHz and 3.4 GHz bands.” http://goo.gl/L4FjnM, accessed on January 8, 2019. [1275,1278] Ofcom (2015), “Annex 8—Recent European awards.” http://goo.gl/CYwDbM.[1291] Power Auctions LLC (2015), “Auction design considerations for the public sector spectrum release, prepared for Hutchison 3G UK.” http://goo.gl/rwacrm, accessed on January 8, 2019. [1278] Telekom Austria (2013), “Results of the Austrian spectrum auction.” http://goo.gl/ ZZV8eB, accessed on January 8, 2019. [1297] Co-editor Simon Board handled this manuscript. Manuscript received 12 February, 2018; final version accepted 2 May, 2019; available online 14 May, 2019.