Spiteful Bidding and Gaming. in Combinatorial Clock Auctions

Transcription

1 Spiteful Bidding and Gaming in Combinatorial Clock Auctions Maarten Janssen and Vladimir Karamychev Abstract. Combinatorial Clock Auctions (CCAs) have recently been used around the world to allocate mobile telecom spectrum. CCAs are claimed to significantly reduce the scope for strategic bidding. This paper shows, however, that bidding truthfully does not constitute an equilibrium if bidders also have an incentive to engage in spiteful bidding to raise rivals cost. The restrictions on further bids imposed by the clock phase of a CCA give certainty to bidders that certain bids above value cannot be winning bids, assisting bidders to engage in spiteful bidding. Key Words: Combinatorial auctions, Telecom markets, Spiteful biding, Raising rivals cost. JEL Classification: D440, L960. We have benefited from comments by Martin Bichler, Eric van Damme, Jacob Goeree, Bernhard Kasberger, Paul Klemperer, Jon Levin, Emiel Maasland, Tuomas Sandholm, David Salant, Andrzej Skrzypacz, and Achim Wambach on earlier versions of this paper, and seminar participants in Vienna, Moscow, Rotterdam, London, Cologne, and Zurich. The authors have advised bidders in recent auctions in Europe. University of Vienna and Higher School of Economics (Moscow) Erasmus University Rotterdam and Tinbergen Institute

2 1. Introduction Auction designs for allocating mobile telecom licenses have become increasingly complex. Complementarities between frequency bands with different properties in terms of (indoor) coverage and capacity require different bands to be allocated in one auction. Modern combinatorial clock auctions (CCAs) allow bidders to combine different frequencies into packages and express bids on these packages. Until recently, the more traditional simultaneous ascending auction (SAA) was the predominant auction form in telecom auctions, but it is wellknown that this auction format may induce bidders to manipulate the auction outcome by bidding strategically (see, e.g., Grim et al., 2003). The CCA is presented to national regulators as an alternative auction model that, because of its second-price rule, eliminates the scope for strategic bidding or gaming (see, e.g., Cramton, 2012). Despite the complexity of the auction model, bidders are advised to simply bid their valuations. 1 The CCA has a first clock phase where bidders express demand in each round at given prices, and prices increase between rounds until there is no excess demand. In a second supplementary phase, bidders can express many bids in a sealed-bid round. Bidding behavior in the clock phase imposes restrictions on supplementary round bids. Despite its use in highstake real-world auctions, not much is known about the theoretical properties of CCA. What is known is that if the VCG prices are in the core (on which more later) and if bidders only value the spectrum they win themselves and the price they pay for that spectrum, bidding truthfully constitutes one out of many possible equilibria (see, e.g., Ausubel, Cramton, and Milgrom, 2006, and Levin and Skrzypacz, 2014). In the clock round truthful bidding implies bidders decrease demand when price is larger than marginal value. In the supplementary round, truthful bidding implies that bidders bid their value on every package. This paper addresses two issues. First, it adds to our understanding of CCAs by providing a characterization of what types of bidding behaviors rational bidders may adopt, apart from 1 For example, in the abstract of his paper, Cramton (2012) says about CCA, the pricing rule and information policy are carefully tailored to mitigate gaming behavior. The Irish regulator Regcom (2012, p. 70) states that their consultancy firm DotEcon notes, the second price rule is utilized to disincentivize gaming behavior and encourage straightforward bidding. 2

3 truthful bidding. We do so, without restricting ourselves to specific examples such as considering two bidders or one frequency band only. Second, we show that the truthful bidding equilibrium is not robust to small changes in the bidders objective function. In particular, in part of our analysis we consider bidders having a spite motive and, ceteris paribus, preferring outcomes where rivals pay more for their winning allocation. That bidders in real-world auctions are likely to be interested in other bidders paying more for their licenses is confirmed in recent policy discussions. First, in the consultation phase in the United Kingdom, there was an active discussion on this point. 2 Second, after raising a very high revenue of 2 billion euro s for a country of 8 million inhabitants, the Austrian regulator RTR felt compelled to reveal an overview of the bidding behavior of the three participating bidders (see, 3 The three bidders actually submitted a total of more than 4,000 supplementary bids. More than 65% of these supplementary bids were submitted for the largest permissible combinations of frequency blocks, with a share of some 50% of available frequencies. In addition, the bidders utilized almost to the full the price limits that had applied to these large packages during the sealed-bid stage. These supplementary bids submitted on large frequency packages had a significant effect on the prices offered by the other bidders. At the same time, such bids generally only have a marginal likelihood of winning out in the end. If these bids for very large numbers of frequencies had been ignored when determining the winners and prices, the revenue from the auction would have settled at a level of about EUR 1 billion. Thus, the Austrian regulator argues that in the supplementary round bidders have made many bids on very large packages the bidders knew they were unlikely to win, and that these bids were effective in raising the prices other bidders had to pay. 2 See, for instance, Ofcom, 2012, page 122, point Because of legal procedures that are still running in different countries, other regulators have been reluctant to provide much information on the bidding behavior during the auction. 3

4 The above evidence on the Austrian auction and evidence from consultation phases in the UK suggest real-world bidders in telecom auctions are interested in raising rivals cost. We provide two arguments why this is so. First, after acquiring spectrum, winners have to invest large sums of money in developing or upgrading a network. Given imperfect capital markets, the more bidders pay for their licenses, the more expensive it is to externally finance future investments. 4 Raising what rivals pay for their spectrum holdings, effectively may delay or otherwise obstruct investments of competitors in increasing the quality of their network. This gives the spiteful bidder a competitive advantage in the market after the auction vis-à-vis its rivals. Second, the only way to evaluate after the auction the success of a firm s bidding strategy, is to compare the prices different firms have paid for the packages they have obtained. Due to governance issues between the bid team and senior management (or between senior management and shareholders), a bid strategy is considered to be unsuccessful if another firm paid much less for objectively similar spectrum. 5 As we want to see whether bidding truthfully remains an equilibrium strategy under slightly different objective functions, we model the spite motive in a lexicographic way, i.e., a bidder always prefers outcomes with a larger intrinsic surplus (the value of the winning package minus the payment). Rival payments only distinguish between outcomes with identical intrinsic surplus. 6 Thus, the lexicographic formulation of preferences acts as an elegant way to select among the many equilibria of the supplementary round resulting in the same spectrum allocation, but different payments. Inspired by the evidence from the Austrian regulator, most of our main results relate to the supplementary round. We first show that bidders that are active in the last clock round have a so-called knock-out bid, which is an amount they have to bid in the supplementary round to 4 There is a reasonably large literature on auctions with financial externalities, or auctions with a spite motive (see, e.g., Cooper and Fang, 2008; Morgan et al., 2003; Maasland and Onderstal, 2007; Shandra and Sandholm, 2010; and Lu, 2012). This literature typically deals with a single object auction. Our paper is different in that in a multi-object auction, bidders can be winners and raise rivals cost by placing bids on packages that are not winning themselves. This complicated gaming aspect is not present in single object auctions. 5 The Swiss auction outcome shows that payments for similar spectrum can be quite unequal creating quite a discussion at that time. See BAKOM (2010) for the Swiss auction design and BAKOM (2012) for the outcome. 6 As the value of spectrum also depends on how many winners the auction has, their identity (whether they are incumbents or entrants), and on the quality of the package of licenses the competitors get, bidders valuation may actually be endogenous to the auction outcome. The literature on this issue (see, e.g., Goeree, 2003; Jehiel and Moldovanu, 2000, 2003; Jehiel, Moldovanu, and Stacchetti, 1996; Janssen and Karamychev, 2007, 2010; Klemperer, 2002a, 2002b) does not consider CCAs. We do not consider this issue in much detail in this paper. 4

5 guarantee they are among the winners of the auction. A second result establishes that if all positive demands in the clock round satisfy a revealed preference condition (which is called anchor sincere bidding for reasons made clear later), then any combination of bids for different packages that includes a bid that is smaller than this knock-out bid on the last clock round package is weakly dominated. Our third result is that generally the supplementary phase has many undominated equilibria. All these equilibria are constrained efficient. Even if these equilibria result in the same allocation and prices, inefficiencies may arise if bidders do not coordinate on a specific equilibrium. Our final two results on the supplementary round use the lexicographic modeling of the spite motive to select among equilibria. Under special conditions, the supplementary round can be solved using iterative dominance bidders: bidders place their KO-bid on the last clock round package and raise bids on other packages to the maximum extent possible, given the behavior in the clock round. Bidders will not bid truthfully and may even bid on packages without intrinsic value. These special conditions include one spectrum band and market clearing in the last clock round. In this case (and also when there are only two bidders), the knock-out bid does not exceed the last clock round bid. In the clock phase, bidders engage in strategic demand reduction only if in this way they can finish the clock phase, reducing the possibilities that others raise their cost. If bidders believe the clock phase does not finish in the next round and others do not change their bids, they engage in strategic demand expansion thereby raising prices competitors pay. These results can explain aggressive bidding behavior in some recent European auctions. We also present an equilibrium analysis of the clock phase and the supplementary round of a one-band auction where bidders with lexicographic preferences to raise rivals cost engage in demand expansion. The example shows that the CCA provides bidders with the possibility to raise rivals cost if each of them has a reasonably accurate belief about the core demand of their competitor. There are different versions of the CCA rules. One element in which CCAs differ is the activity rule telling bidders what they can bid in future clock rounds and in the supplementary round. There is a so-called relative cap, a final cap and an intermediate cap, and other rules such as the weak or generalized axiom for revealed preferences (WARP or GARP); see, 5

6 e.g., Ausubel and Cramton (2011), Ausubel and Baranov (2014), and Ausubel and Baranov (2015). We discuss these different activity rules in more detail in the following Section. To focus our analysis, we concentrate on the relative cap rule as it has been often used in recent auction designs. The rule has been criticized recently by Ausubel and Baranov (2014) among others. Alternative rules have, however, also important side effects. This has led the UK regulator Ofcom to withdraw the final cap rule from its auction design and to revert to the relative cap rule. 7 It is therefore not clear which rule is more promising going forward. 8 Therefore, we mainly focus on the relative cap rule that has been frequently used. We discuss, however, how our results are sensitive to alternative activity rules and show that the equilibrium in the single-band auction that exhibits raising rivals cost is independent of the activity rule. Some recent papers discuss the CCA auction. Goeree and Lien (2012) show that the core selection principle introduced in the pricing rule used in CCAs implies that bidding valuation is no longer an optimal bidding strategy if VCG prices are not in the core. Beck and Ott (2011) show that this principle may imply that bidding both above and below valuations can be optimal in CCAs. Knapek and Wambach (2012) show in a series of examples that bidding in a CCA may be strategically complicated. Bichler et al. (2013) present experimental results on inefficient outcomes in CCAs. They attribute the inefficiency to the so-called missing bids problem. None of these papers provides a general analysis of how spiteful bidders can game the auction. In the Appendix, they also provide an example of spiteful bidding in a CCA. 9 A recent working paper by Levin and Skrzypacz (2014) is close in spirit to our paper. In a twobidder model with one divisible commodity, they focus on strategic bidding in the clock phase. Our main focus is on bidder behavior in the supplementary round with multiple units of different commodities. A more detailed discussion of the differences is postponed until Subsection 5. Ausubel and Baranov (2014) also provide some considerations on strategic 7 See, Ofcom (2012), Assessment of future mobile competition and award of 800 MHz and 2.6 GHz, 24 July Ausubel and Baranov (2014) argue The Final Cap appears to be the most natural implication of revealed preference theory that comes out of the clock rounds. Yet many regulators decide against the Final Cap for their CCAs relying on the Relative Cap only, viewing the Final Cap as giving excessive allocation stability between the clock stage and the supplementary round. 9 Salant (2013) provides an informed overview of some recently used auction design and a discussion on some outcomes of CCAs that have been held recently. 6

7 bidding due to the second-price principle adopted in a CCA and argue that it may be desirable to move towards a first-price version of a CCA. The rest of the paper is organized as follows. Section 2 provides a more detailed description of the different stages of the CCA. Section 3.1 discusses optimal bidding in the supplementary round with and without the spite motive. Section 4 discusses our results on the clock phase. To make the material more accessible, we use one example throughout these three sections to illustrate how a CCA works and what drives our results. Section 5 discusses in more detail the differences with Levin and Skrzypacz (2014) and presents the equilibrium analysis of an example of a single-band auction where bidders have a lexicographic preference for raising rivals cost. Section 6 concludes with a discussion. Proofs are in Appendix I, and other derivations are in Appendix II. 2. Combinatorial Clock Auctions and Truthful Bidding Most recent spectrum auctions allocate different frequency bands. Let there be KK different frequency bands with nn kk blocks in band kk = 1,, KK. Spectrum in different frequency bands has different properties in terms of geographic (or indoor) coverage and capacity. Mobile telecom companies, therefore, want to acquire a combination of different units in different frequency bands. The set of all feasible combinations (packages) is denoted by Π, and the aggregate supply is denoted by ππ, ππ (nn 1,, nn KK ). We use ππ to denote generic packages, ππ = (ππ 1,, ππ KK ) Π, and use the Greek letter superscripts to refer to specific packages, e.g., ππ αα. The intrinsic valuation of bidder ii for any package ππ αα is denoted by vv αα ii, and when no confusion is possible we drop subscript ii. We use the language of spectrum auctions as these are the CCA auctions that have recently attracted most attention. It is clear, however, that the format applies to any allocation problem with multiple units of different commodities. 7

8 A combinatorial clock auction allocates the available spectrum using two integrated phases. 10 The first clock phase is divided into several rounds. In each round tt, the auctioneer announces prices pp tt = (pp 1 tt,, pp KK tt ), one price for each band. The (reserve) prices pp 1 are set in advance of the auction. At these given prices, bidders express in each band how many blocks they would like to acquire. The number of blocks in band kk demanded by player ii in round tt is denoted by dd tt ii,kk, and the whole package demand is denoted by dd tt ii = dd tt tt ii,1,, dd ii,kk Π. At the end of a round, bidders are typically only informed about the total demand in each band. 11 If, in round tt, there is excess demand in band kk, clock prices for that band will increase in clock round (tt + 1) by a predetermined price increment δδ kk > 0: pp kk tt+1 = pp kk tt + δδ kk. Clock prices for bands k in which there is no excess demand in round tt will not increase: pp kk tt+1 = pp kk tt. It is possible though that in later rounds prices for these bands start increasing again when bidders switch their demand from other bands. The clock phase of the auction stops when there is no excess demand in any band. The intention of the clock phase is to imitate the price discovery offered by an ascending auction (Ausubel, Cramton and Milgrom, 2006). At the end of the clock phase, no allocation of spectrum takes place yet. Each block in a spectrum band is given a certain number of so-called eligibility points. These points are used to provide a one-dimensional measurement of a player s demand. If a block in band kk requires ee kk eligibility points, ee = (ee 1,, ee KK ), then the total eligibility of player ii in round tt is given by EE ii tt = ee dd ii tt = KK kk=1 tt ee kk dd ii,kk. In any clock round, a bidder cannot express a demand requiring a larger number of eligibility points than the bid expressed in the previous round, i.e., EE ii tt EE ii tt 1. If bidder ii expresses a demand for a package in round tt that requires a strictly smaller number of eligibility points than the bid in the previous round, i.e., if EE ii tt < EE ii tt 1, then round tt is called an anchor round, and bid dd ii tt is called an anchor bid. Anchor rounds, the band prices in these anchor rounds, and anchor bids play an important role in determining which bids a bidder can express in the supplementary round. 10 Spectrum auctions typically have a third (assignment) phase where specific locations in a band are allocated. As this phase typically does not raise many concerns and the revenues of this phase are low compared to the revenues of the supplementary phase (see, e.g., we do not discuss this assignment phase. 11 In the 2010 Austrian auction, even that information was not available. Bidders could only infer there was excess demand in a certain band from the fact that the clock prices in these bands increased in a clock round. 8

9 Package Package eligibility Package value (1,2) (2,1) (1,1) Table 1. Values and eligibility points for the three packages. Bidder ii bids truthfully (or sincerely) in round tt if he demands a package dd ii tt that is (i) available according to the current level of eligibility points, and (ii) maximizes bidder s surplus at current auction prices, i.e., dd ii tt arg max ππ Π (vv ii ππ pp tt ππ) subject to ee ππ EE ii tt 1. The clock phase imposes restrictions on what bidders can bid in the supplementary round. Before we explain these restrictions, we introduce the basic ingredients of the example we use throughout this paper to illustrate the working of the CCA and the results we obtain. Example. Set up and truthful bidding in clock phase. Our leading example has three (ex ante identical) bidders, two bands, and three blocks that are available in each of the bands. The reserve prices are pp 1 = (1,4) and the bid increments are δδ kk = 1 for both bands. 12 We assume the auction rules restrict bids to no more than three licenses in total. Intrinsic values and eligibility points are given in Table Bidders are intrinsically interested in three packages (xx, yy), where xx and yy stand for the number of blocks in bands 1 and 2. Blocks in band 1 and 2 cost 5 and 8 eligibility points, respectively. Truthful bidding in the clock phase of the example is represented in Table 2. Bidders start bidding on package (1,2) as this is the most profitable package. As there is excess demand in band 2, the band 2 price increases. In round 2, bidders have enough eligibility points for all three packages and truthful bidding implies they choose the package that they prefer the most at these prices, which is (2,1). By doing so, bidders reduce their eligibility points and cannot 12 We set the reserve price for band 2 blocks at 4 only in order to make the clock phase shorter. For our results in Section 3.1, it is important that the clock price ratio is at some round different from the ratio of eligibility points. To get this difference quickly in a simple example, we assume some difference right from the start. 13 The example captures important elements of telecom auctions. The number of bidders is limited (usually, only incumbents bid, but sometimes there is a limited number of entrants as well). As incumbents have been in the market together for some time, they have a similar understanding of the market and imitate market strategies that work well. Thus, their values are not too dissimilar and they expect others to have similar considerations as they do themselves. Finally, the set of spectrum bands available is also limited. 9

10 Round Package costs, values, and surplus Prices no. Package Cost Value Surplus (1,2) (1,4) (2,1) (1,1) (1,2) (1,5) (2,1) (1,1) (2,5) (2,1) (1,1) (3,5) (2,1) (1,1) (4,5) (2,1) (1,1) (5,5) (2,1) (1,1) (6,5) (2,1) (1,1) (7,5) (2,1) (1,1) Table 2. Clock phase development. Optimal package Eligibility (1,2) 21 (2,1) 18 (2,1) 18 (2,1) 18 (2,1) 18 (2,1) 18 (2,1) 18 (1,1) 13 bid in the clock phase for package (1,2) anymore. From round 3 onwards, prices increase for block 1 only. When prices reach (7,5) in round 8, bidders prefer package (1,1) over package (2,1) and truthful bidding implies that out of the two packages for which the bidders have enough eligibility points in that round, they choose to bid for (1,1). As in that round there is no excess demand in any band, the clock phase of the CCA is over. /// The supplementary round is a simultaneous bid round, where in one round bidders can express bids for all possible packages, subject to constraints. Bidders cannot bid less than the bids they expressed in the clock phase. In addition, the most commonly used constraint is the so-called relative cap, which works as follows. 14 In the supplementary round, bidder ii is unconstrained on his bid on the final clock round TT package ππ ff dd TT ii, denoted by bb ff, as long as bb ff pp TT dd TT ii, where pp TT = (pp TT 1,, pp TT KK ) are the final clock round prices. On all other packages, a bidder is 14 This constraint was used in auctions in Austria (2010, 2013), Switzerland (2012), the Netherlands (2012), and the United Kingdom (2013), among others. 10

11 constrained, and the constraints are calculated relative to bb ff as follows. For all packages ππ αα that require a number of eligibility points EE αα = ee ππ αα smaller than or equal to the number of points EE ff = EE TT in the final clock round TT, round TT is the anchor round. The maximum bid BB αα in the supplementary round that can be expressed on package ππ αα = (ππ αα 1,, ππ αα KK ) is BB αα = bb ff + pp TT (ππ αα ππ ff ) = bb ff + pp TT kk ππ αα kk ππ ff kk=1 kk. KK In order to compute the cap for a package ππ ββ, which requires a number of eligibility points EE ββ = ee ππ ββ > EE ff, one first has to track the last round rr ββ in the clock phase when bidder ii had enough eligibility points to bid for package ππ ββ. This rr ββ is uniquely determined by EE rrββ < EE ββ EE rrββ 1. Round rr ββ is the anchor round for package ππ ββ, and the package ππ rr ββ rr = dd ββ ii that bidder ii has bid on in round rr ββ is the anchor package for ππ ββ. The maximum price bidder ii can place on package ππ ββ can be computed iteratively as follows, starting from the final clock round rr ββ = TT. Let bb rr ββ be the highest bid in the clock or supplementary round for anchor package ππ rr ββ. Then, the maximum bid BB ββ that a bidder can express on package ππ ββ is BB ββ = bb rr ββ + pp rrββ ππ ββ ππ rr ββ = bb rr ββ + rr pp ββ kk ππ ββ kk=1 kk ππ kk rr ββ. The relative cap expresses upper bounds on what bidders can bid for packages relative to their bid for the final clock round package. These bounds depend on clock prices and individual bids and thus rely only on an individual bidder s own information. The relative cap has a clear economic interpretation in terms of revealed preference (see, e.g., Ausubel, Cramton, and Milgrom, 2006). To see this, note that when in a certain clock round a bidder reduces its eligibility points, it will give up the possibility to bid on certain packages. According to revealed preference, the package he was bidding on at these clock prices is preferred to any alternative package that was still feasible. The iterative procedure is necessary to keep track of which packages were still feasible in which clock round. Alternative activity rules are the so-called intermediate and final cap rule (see, e.g., Ausubel and Baranov, 2015). The difference with the relative cap rule lies in the rounds to which the revealed-preference constraint should be applied. The relative cap rule we defined above requires that a bid for package x satisfies the revealed-preference constraint with respect to the last clock round where the bidder had enough eligibility points to bid for package xx. The intermediate cap rule requires that the bid for package x satisfies the revealed-preference 11 KK

12 constraint with respect to all eligibility-reducing clock rounds after the last clock round where the bidder had enough eligibility points to bid for package x. The final cap rule is different in that it is usually not combined with an eligibility point-based activity rule, but instead with revealed preference rule like WARP or GARP (see, e.g., Ausubel and Cramton, 2011, and Ausubel and Baranov, 2014). The rule itself states that the bid for package x should satisfy revealed-preference constraint with respect to the final clock round. When not explicitly stated otherwise, we assume the relative cap rule is relevant. Example. Caps in the supplementary round under truthful bidding in clock phase. In our example, the caps on supplementary bids implied by the relative cap rule can be determined as follows. For package (1,1) bidders are unconstrained. Suppose they bid bb (1,1) = bb for (1,1). For their supplementary bid on package (2,1), the final round is the anchor round, and at these prices (7,5), package (2,1) costs 7 more than package (1,1). The fact that a bidder was bidding on (1,1) reveals that he was not willing to pay 7 more for package (2,1) than for package (1,1). Thus, for package (2,1) bidders can bid up to BB (2,1) = bb + 7 in the supplementary round. For package (1,2), round 2 is the anchor round, and at the prevailing prices (1,5) package (1,2) costs 4 more than package (2,1). Therefore, revealed preference implies that bidders can bid for package (1,2) up to BB (1,2) = bb (2,1) + 4, where bb (2,1) BB (2,1) is the actual highest bid for package (2,1) in either round. If a bidder bids truthfully in the clock phase, he can bid value on all packages in the supplementary round. Since bidders are unconstrained in their supplementary round bid for package (1,1), they can bid their value of 40. As they can bid for package (2,1) maximally 7 more than what they bid for package (1,1), they can bid their value 46.5 if they bid 40 for (1,1). Finally, they can bid for package (1,2) maximally 4 more than what they bid for package (2,1). This means they can bid 50 if they bid 46.5 for (2,1). In addition, in the supplementary round bidders can make bids on packages they have not bid for in the clock phase, such as (3,0) and (0,3), or on packages that are smaller than (1,1). Bidders do not have an intrinsic value for these packages, but as we will see, spiteful bidders may nevertheless want to bid on these packages. For package (3,0), and all packages requiring less eligibility points than (1,1), the anchor round is the final clock round, and at these prices 12

13 package (3,0) costs 9 more than package (1,1). Therefore, bidders can bid for package (3,0) up to bb + 9. For other, smaller packages, bidders can maximally make the following bids: bb (1,0) (bb 5), bb (0,1) (bb 7), bb (0,2) (bb 2), and bb (2,0) (bb + 2). For package (0,3), the anchor round is the first clock round, and at these prices it costs 3 more than package (1,2). Therefore, bidders can bid for package (0,3) up to BB (0,3) = bb (1,2) + 3, where bb (1,2) BB (1,2) is the actual bid for package (1,2) in any round. Under the intermediate cap rule, the constraints on packages for which the last clock round is the anchor round are identical to what we have calculated above for the relative cap rule. For packages (1,2) and (0,3) the constraints are different, however. It is easy to see that bidders can bid up to BB (1,2) = bb + 5 on package (1,2) and up to BB (0,3) = bb + 3 on package (0,3) in the supplementary round. This implies that under the intermediate cap rule, the bid history may be such that bidders cannot bid truthfully in the supplementary round. The problem is that once a bidder has reduced demand to (2,1) and 18 eligibility points, he cannot bid for (1,2) anymore even though this would be his most preferred package at the clock prices in round This is not a problem for the weaker relative cap as it allows the bidder to bid the value difference in the supplementary round. Given the same bid history, the final cap rule has the same implications as the intermediate cap. However, as we noted above, the final cap rule is usually implemented together with a strict activity rule as WARP or GARP, instead of an eligibilitypoint-based rule and under GARP a bidder would be able to bid for (1,2) in round 3 even though he has bid for (2,1) in round 2. /// Finally, we explain how the winners of the auction and final auction prices are determined. The winners are determined in the same way as in the Vickrey-Clark-Groves (VCG) mechanism. Of all feasible combinations of bids, one per bidder, a combination is selected that maximizes the total sum of the bids. Bids that are part of this combination, are winning bids, and bidders who have submitted these bids win their winning bids packages. The CCA prices are equal to 15 The fact that for certain clock phase bid histories, bidders cannot bid truthfully in the supplementary round under the intermediate cap rule is not so surprising if one realizes that by reducing demand early in the clock phase bidders cannot bid on certain packages in later clock rounds. The intermediate cap rule still imposes, however, constraints on the bids on these packages that are relevant as if the bidders could have bid on these packages in later clock rounds. 13

14 the VCG prices, if the VCG prices are in the core. If the VCG prices are not in the core, CCA typically uses some adjustment such that the CCA prices are in the core. 16 As we do not want our arguments to depend on the specific core selection principle that is used (as Goeree and Lien (2012) have already shown that core-selecting pricing rules imply that it is not optimal to bid straightforwardly), we assume that CCA bids are such that the CCA prices are in the core with respect to bids. In this case, CCA prices coincide with VCG prices. We have made sure that this assumption holds true in all our examples. The VCG price a bidder pays for the package he wins equals the opportunity cost he imposes on others. Thus, bidder ii pays the additional price others jointly have expressed to be willing to pay for acquiring the spectrum bidder ii won in addition to the spectrum they won. Example. Winners and Auction Prices under Truthful Bidding. If bidders bid truthfully in both the clock and supplementary phase, then every bidder wins package (1,1). This is because this allocation generates a total value of 120, which is larger than, e.g., the total value of 96.5 generated by allocating packages (2,1) and (1,2) to two bidders and nothing to the third bidder. Each bidder imposes an opportunity cost of 16.5 on the two competitors. This is also the VCG price for package (1,1). As opportunity costs may be considered natural market prices, 16.5 may be considered the market price. 17 /// The final and intermediate cap rules imply that if the clock phase ends with market clearing, then the final allocation equals the allocation reached in the final clock round and the supplementary round only determined the prices bidders pay. This is the reason why Ofcom decided not to use the final cap rule in its 2013 auction (see, Ofcom, 2012). 16 See, e.g., Day and Raghavan (2007), Day and Milgrom (2008), and Erdil and Klemperer (2010). 17 It is easy to see that in this example, VCG prices are in the core so that the core selection requirements that CCA imposes on prices are not binding. 14

15 3. The Supplementary Phase We first analyze bidding behavior in the supplementary round without recourse to the spite motive. The results we present provide a characterization of the different possible bidding behaviors in the supplementary round. We then select among these possible bidding behaviors by introducing the spite motive in a lexicographical way. Throughout the analysis, values are private information and that information about total, but not individual, demand is revealed during the clock phase. Our first result on bidding in the supplementary round relates to the existence of a certain bid on the final clock round package ππ ff that guarantees that a bidder who was still active in the final clock round is among the winners of the auction. The proposition below states that for any clock phase development, each bidder can calculate this so-called knock out bid bb, KO-bid hereinafter. Proposition 1. For any outcome of the clock phase and for every bidder ii, there exist a KObid bb ii on the last clock round package ππ ii ff with the following properties: a) If bidder ii bids bb ff > bb ii in the supplementary round on package ππ ff, he surely wins some package. Moreover, if he wins package ππ ff, he pays not more than bb ii. b) If bidder ii bids bb ff < bb ii, then he may not win any package. c) If NN = 2 or KK = 1, then bb ii pp TT ff ππ jj ii ππ jj, where ππ ff jj is the last clock round package of bidder jj. In this case, the KO-bid is independent of the particular bid history. Under the simpler final cap rule, it is known that a knock-out bid exist (see, e.g., Bichler et al., 2013, and independent work by Ausubel and Cramton, 2011). For the same bid history, the actual value of the knock-out bid depends, however, on the activity rule. As explained in Section 2, bidding behavior of the clock phase imposes restrictions on the feasible bids in the supplementary round. If a bidder knew the bidding history of each individual bidder in the clock phase, he could compute the maximal amount each other bidder can bid on any package in addition to the unrestricted supplementary round bid on the final clock round package. This is important information as it imposes an upper bound on what other 15

16 bidders can jointly bid more on a combination of packages resulting in the bidder not being part of the winning combination. If a bidder knew the full bidding history, this upper bound would be the bidder s knock-out bid. However, bidders are usually not informed about the full bidding history and are only informed about total demand in each bidding round, or not even that. In these cases, bidders have to consider all possible individual bidding histories that are consistent with the information they received (which at least include the clock prices and when the clock phase ended). The knock out bid bb ii is then essentially the maximum bid difference the other bidders can generate to eliminate the bidder from winning, given all possible bid histories that are consistent with the information bidders have concerning the clock phase. 18 Obviously, with many commodities and many bidders, depending on the actual bid history and the amount of information bidders received about the clock phase behavior, this knock out bid bb ii may be quite high. 19 The dependence of the KO-bids on the constraints imposed by the clock phase is, in general, very complex. The next example shows that the KO-bid is typically much larger than the final clock round bids and dependent on the information bidders have about the clock phase. Only with one band, or two bidders the KO-bid is independent of whether bidders know the individual bid history of all bidders, or only the total demand in every round. In these cases, the knock out bid bb does not exceed the cost of the last clock round package ππ ff extended with all ( unsold ) blocks in excess supply at the last clock round prices pp TT. Whether or not the bidder wins his final clock round package ππ ff depends on all bids that are made. A bidder can make sure he wins ππ ff by bidding high on this package. Proposition 1 then guarantees that the price pp ff he has to pay for ππ ff is at most bb. Example. Calculation of KO-bid given truthful bidding in clock phase. In the supplementary round, players bids under the relative cap rule can be expressed as follows: 18 Note that the VCG mechanism does not have a knock-out bid as bids are unrestricted. 19 Independent of the context, by definition of the KO-bid, it is the case that this is the lowest bid a bidder has to place on his final clock round package to ensure he wins some package in the auction. By making this bid, the bidder is exposed to the full sum as (given the available information), there is at least one bid history in which the bidder may have to pay the KO-bid. Whether the bidder wins his last clock round package depends, among other things, on whether he also bids high on other packages. 16

17 bb (1,1) = bb, bb (2,1) = bb + xx, bb (1,2) = bb + xx + yy, bb (3,0) = bb + zz, bb (0,3) = bb + xx + yy + tt. As argued above, the relative cap implies that given the behavior in the clock phase reported in Table 2, xx 7, yy 4, zz 9, and tt 3. If clock phase bids of individual bidders are public information, this implies that the maximal sum of bids of a two-winner combination is 2bb + 23, which is a combination of bb (3,0) and bb (0,3) : bb (3,0) + bb (0,3) = 2bb + zz + xx + yy + tt 2bb To guarantee to be among the winners of the auction, a bidder should bid at least bb = 23, the KO-bid for this public information case, on the final clock round package (1,1). As by doing so, all three bidders win, it must be the case that they all win package (1,1). If, as in most real-world auctions, bidders are only informed about total demands in each clock round, they have to consider any possible development of the clock phase that is consistent with their own bidding and the information on total demand. The maximum of all such bids is the true KO-bid guaranteeing the bidder wins irrespective of the precise development of the clock phase. If bidders do not bid for more than 3 licenses, there are 14 generically different scenarios that generate the same joint demand of bidders 2 and 3. Computing KO-bids for each scenario yields a maximum KO-bid of 29. To get this, let us consider bidder 1. He knows that the other two bidders have expressed a total demand of (2,4) in round 1, of (4,2) in rounds 2-7, and a total demand of (2,2) in round 8. He may consider this demand comes from the following individual demands: both other bidders demanding (1,2) in round 2, one bidder demanding (3,0) while the other demanding (1,2) in rounds 2-7 and finally, one bidder demanding (1,0) and the other (1,2) in the last clock round. With this detailed bid history, bidders 2 and 3 can generate bids totaling 29 more on the full spectrum supply (3,3) than their highest total bid on package (2,2). Details are given in Appendix II. Under the final and intermediate cap rules, the knock-out bid is easy to calculate. Given market clearing, the final clock round bid is the knock-out bid as others are constrained by the final clock round prices. Thus, the knock-out bid equals 12. /// The KO-bid has an important role in determining which strategies in the supplementary round are weakly dominated. Let Ψ ii Π be a set of packages that bidder ii bids on in either the clock phase or in the supplementary round. Accordingly, let Φ ii = {(bb αα ii, ππ αα ): ππ αα Ψ ii } be the set of bids bidder ii: (bb αα ii, ππ αα ) reflects the monetary amount bb αα ii that bidder ii bids on package ππ αα. In 17

18 case a bidder makes several bids on the same package, only the highest bid is included in Φ ii. In what follows, we drop the subscript ii, whenever convenient and not confusing. Unlike, the VCG mechanism, a CCA does not have a dominant strategy. First, truthful bidding, which is dominant in CCA, might not be feasible given the caps. Second, in a CCA bidding the KO-bid guarantees winning. Using the KO-bid, the following proposition shows that quite some supplementary round bids are nevertheless weakly dominated in a CCA. Note that this Proposition is independent of the activity rule that is used. Proposition 2. If Φ is such that bb ff < bb and bb ββ < vv ββ for all ππ ββ Ψ, then bid Φ is weakly dominated. There are two conditions that ensure a strategy Φ is weakly dominated. First, the bid bb ff on the last clock round package ππ ff must be smaller than the KO-bid bb. Second, all bids bb ββ, including bb ff, must be below values vv ββ. When these two conditions hold, raising all bids in Φ by a small amount εε > 0 weakly dominates Φ. If the first condition fails, i.e., bb ff bb, then Φ is already winning so that raising all bids bb ββ by the same amount does not change the allocation nor the price bidder ii pays. If the second condition fails, i.e., bb ββ vv ββ, raising all bids might result in winning ππ ββ at a price above vv ββ. In the previous section, we have argued that if bidders bid truthfully in the clock phase, they can also bid truthfully in the supplementary phase. This property holds not only for truthful bidding, but also for a weaker condition, which we call anchor-sincere bidding, AS-bidding hereinafter. This is important as we show in Section 4 and 5 that bidding truthfully is not an equilibrium strategy in the clock phase, although bidding anchor sincerely may well be. Definition 1. Bidder ii bids anchor-sincerely for ππ ββ = dd tt ii in round tt 1 if either a) Bidder ii maintains his eligibility points, i.e., EE tt ii = EE tt 1 ii, or b) Bidder ii switches to a non-empty package with fewer eligibility points that is more profitable at round tt prices than any other package ππ αα which is feasible in round (tt 1) and non-feasible in round tt, i.e., if 0 < EE ii tt < EE ii tt 1 then (vv αα pp tt ππ αα ) vv ββ pp tt ππ ββ for ππ αα Π for which EE ii tt < ee ππ αα EE ii tt 1, or 18

19 c) Bidder ii drops out of the clock phase by reducing demand to zero in all bands. Anchor sincere bidding only requires that if a bidder reduces eligibility, then he should have a good reason (in the form of a kind of revealed preference argument) to do so as he constrains his future possibilities: he must prefer the package he bids on to all the packages that are not feasible anymore after that round. Anchor sincere bidding allows for demand expansion as well as for a severe form of demand reduction, namely reducing demand to zero in all bands. Bidding truthfully is a special case of AS-bidding, as a truthful bidder always bids for the most profitable, feasible package, i.e., condition (b) of Definition 1 is always fulfilled. If a bidder AS-bids in the clock phase, then all his anchor bids are sincere relative to all larger packages for which the bid is the anchor bid. Bid caps under the relative cap rule, but not necessarily under the intermediate cap rule (see our lead example in the previous Section), allow him to bid truthfully in the supplementary phase on all packages requiring more eligibility points than the last clock round package. For smaller packages, bid caps may restrict bidders from bidding truthfully. We define a bidder s clock-phase constrained values as follows. Definition 2. Given a development of the clock phase, 20 i.e., given the iterative definition of BB ββ in Section 2, for all ππ ββ Π, a bidder s clock-phase constrained values vv ββ are defined by: vv ββ = vv ff + min BB ββ bb ff, vv ββ vv ff = min BB ββ bb ff + vv ff, vv ββ, where the bid cap BB ββ is computed such that its anchor bid bb rr ββ equals its cap, i.e., bb rr ββ = BB αα, where ππ αα = ππ rr ββ is the package the bidder bids on in the anchor round. Given a bidder s bidding in the clock phase and the unconstrained bid on the final clock round package, the bidder may not be able to bid value on all packages in the supplementary round. The clock-phase constrained values are the closest a bidder can come to bidding value for a certain package. Using definitions 1 and 2, we strengthen Proposition 2 by showing which bids are weakly dominated if bidder ii has bid anchor-sincerely in the clock phase. 20 Note that even if a bidder did not bid for package α in the clock phase, the bidder is still constrained on bidding for this package in the supplementary phase by the development of the clock phase. 19

20 Without constraints imposed by clock phase bidding, bidding truthfully in the supplementary round is known to be a weakly dominant strategy. To establish this result, part of the argument runs as follows. Suppose a bidder bids on at least two packages ππ αα and ππ ββ with bids bb αα and bb ββ with vv αα bb αα > vv ββ bb ββ 0. A strategy that includes these bids is weakly dominated by an alternative strategy where the bid bb αα is slightly increased so that the bid difference bb αα bb ββ is more in line with the value difference vv αα vv ββ. This bid increase either has no effect on the final allocation and the winning price of the bidder, or results in winning ππ αα instead of ππ ββ with a higher surplus, or results in winning ππ αα with positive surplus instead of not winning at all. In a CCA, bidders are constrained by caps and, therefore, may not be able to bid truthfully in the supplementary round. However, if a bidder bids anchor-sincerely in an anchor round with from ππ αα to ππ ββ, then vv αα ii pp tt ππ αα > vv ββ ii pp tt ππ ββ so that the revealed preference constraint BB ββ = bb αα + pp tt ππ ββ ππ αα implies BB ββ bb αα > vv ββ ii vv αα ii. This means the bid bb ββ for which bb ββ bb αα = vv ββ ii vv αα ii is always feasible. The next Proposition uses this argument to link anchor-sincere bidding to bid strategies that are weakly dominated. Proposition 3. Let bidder ii bid anchor-sincerely in the clock phase. If Φ is such that bb ff < min bb, vv ff or bb ββ bb ff < vv ββ vv ff for some package ππ ββ Ψ, then Φ is weakly dominated. According to Proposition 2, bids that simultaneously satisfy two conditions are dominated. Proposition 3 differs from Proposition 2 in two respects. First, it uses the clock-phase constrained values vv ββ instead of true values vv ββ, and, second, it shows that a bid is dominated if either of the conditions holds. For this result, it is important that clock bids on all anchor packages satisfy revealed preference with respect to all packages that are not feasible anymore after that round, i.e., condition (b) of Definition 1 holds. If a bidding strategy includes a bid on the final clock round package that is smaller than min bb, vv ff and all other bids are below clock-phase constrained marginal values bb ff + vv ββ vv ff, then it is better to raise the bids on all packages in Ψ by the same amount as doing 20

21 so does not change the package he wins, nor does it affect the price paid. If bb vv ff, the bidder should bid his entire value vv ff on package ππ ff to increase the chance of winning. If vv ff > bb, by bidding bb ff = bb, the bidder guarantees he is winning; which package he wins depends on the relative bids in combination with rivals bids. Second, like in the VCG mechanism, a strategy is also dominated if a bidder does not express at least the marginal value differences on other packages if doing so is feasible. For packages that require more eligibility points than the last clock round package, AS-bidding implies it is always feasible, i.e., vv ββ = vv ββ for such packages; for other packages this may not be the case, and bidders will bid their maximum cap, realizing the clock-phase constrained marginal values. Without AS-bidding, or with the intermediate cap rule, truthful bidding in the supplementary round is not generally feasible. In such cases, it may happen that a bidder can only bid the clock-phase constrained value on one package if he bids above the true value on another package. As a result, some bids that are lower than clock-phase constrained values remain undominated. The next example demonstrates that Proposition 3 does not hold in this case. Example. Undominated bids that are below clock-phase constrained values. Suppose bidder ii has not bid anchor-sincerely and his bid caps are BB (2,1) = bb (1,1) + 8 and BB (1,2) = bb (2,1). Let ππ ff = (1,1) and bb ff = 40. Then, the clock-phase restricted values are vv (1,1) = 40, vv (2,1) = 46.5, and vv (1,2) = 48. The supplementary round set of bids bb (1,1) = 40, bb (2,1) = bb (1,2) = 47 is undominated, however, even though bb (1,2) < vv (1,2). The reason is that raising bb (1,2) requires raising bb (2,1), which is already above the (marginal, unrestricted) value. Thus, it may well be that by raising bb (1,2) and bb (2,1), the winning package will change from (1,1) to (2,1) and that the bidder has to pay more than the marginal value for obtaining (2,1) instead of (1,1), which is clearly not optimal. /// Equipped with Proposition 3, we conclude that if bidders AS-bid in the clock phase, in any undominated equilibrium of the supplementary phase, all bidders who are active in the last clock round are winners, provided their values are above their KO-bids. We next define a 21