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
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
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, https://www.rtr.at/en/pr/PI28102013TK). 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 7.9.
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
3
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
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
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
2012.
8
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
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 𝑛𝑛𝑘𝑘 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� ≡ (𝑛𝑛1 , … , 𝑛𝑛𝐾𝐾 ). We use 𝜋𝜋 to denote generic packages, 𝜋𝜋 =
(𝜋𝜋1 , … , 𝜋𝜋𝐾𝐾 ) ∈ 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 𝑣𝑣𝑖𝑖𝛼𝛼 , 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
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 𝑝𝑝𝑡𝑡 = (𝑝𝑝1𝑡𝑡 , … , 𝑝𝑝𝐾𝐾𝑡𝑡 ), one price for each band. The (reserve) prices 𝑝𝑝1 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
𝑡𝑡
𝑡𝑡
𝑡𝑡
is denoted by 𝑑𝑑𝑖𝑖,𝑘𝑘
, and the whole package demand is denoted by 𝑑𝑑𝑖𝑖𝑡𝑡 = �𝑑𝑑𝑖𝑖,1
, … , 𝑑𝑑𝑖𝑖,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 𝛿𝛿𝑘𝑘 > 0: 𝑝𝑝𝑘𝑘𝑡𝑡+1 = 𝑝𝑝𝑘𝑘𝑡𝑡 + 𝛿𝛿𝑘𝑘 . Clock
prices for bands k in which there is no excess demand in round 𝑡𝑡 will not increase: 𝑝𝑝𝑘𝑘𝑡𝑡+1 = 𝑝𝑝𝑘𝑘𝑡𝑡 .
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 𝑒𝑒𝑘𝑘 eligibility points, 𝑒𝑒 = (𝑒𝑒1 , … , 𝑒𝑒𝐾𝐾 ), then the total eligibility of player
𝑡𝑡
𝑖𝑖 in round 𝑡𝑡 is given by 𝐸𝐸𝑖𝑖𝑡𝑡 = 𝑒𝑒 ∙ 𝑑𝑑𝑖𝑖𝑡𝑡 = ∑𝐾𝐾
𝑘𝑘=1�𝑒𝑒𝑘𝑘 𝑑𝑑𝑖𝑖,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., 𝐸𝐸𝑖𝑖𝑡𝑡 ≤ 𝐸𝐸𝑖𝑖𝑡𝑡−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
𝐸𝐸𝑖𝑖𝑡𝑡 < 𝐸𝐸𝑖𝑖𝑡𝑡−1, then round 𝑡𝑡 is called an anchor round, and bid 𝑑𝑑𝑖𝑖𝑡𝑡 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., http://stakeholders.ofcom.org.uk/binaries/spectrum/spectrumawards/awards-in-progress/notices/4g-final-results.pdf), 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
Package
Package eligibility
Package value
(1,2)
21
50
(2,1)
18
46.5
(1,1)
13
40
Table 1. Values and eligibility points for the three packages.
Bidder 𝑖𝑖 bids truthfully (or sincerely) in round 𝑡𝑡 if he demands a package 𝑑𝑑𝑖𝑖𝑡𝑡 that is (i)
available according to the current level of eligibility points, and (ii) maximizes bidder’s surplus
at current auction prices, i.e.,
𝑑𝑑𝑖𝑖𝑡𝑡 ∈ arg max (𝑣𝑣𝑖𝑖𝜋𝜋 − 𝑝𝑝𝑡𝑡 ∙ 𝜋𝜋) subject to 𝑒𝑒 ∙ 𝜋𝜋 ≤ 𝐸𝐸𝑖𝑖𝑡𝑡−1 .
𝜋𝜋∈Π
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 𝑝𝑝1 = (1,4) and the bid increments are
𝛿𝛿𝑘𝑘 = 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 1. 13 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
Round
no.
Prices
Package costs, values, and surplus
Package
(1,2)
1
(1,4)
(2,1)
(1,1)
(1,2)
2
(1,5)
(2,1)
(1,1)
(2,1)
(2,5)
3
(1,1)
(2,1)
4
(3,5)
(1,1)
(2,1)
5
(4,5)
(1,1)
(2,1)
6
(5,5)
(1,1)
(2,1)
7
(6,5)
(1,1)
(2,1)
8
(7,5)
(1,1)
Table 2. Clock phase development.
Cost
9
6
5
11
7
6
9
7
11
8
13
9
15
10
17
11
19
12
Value
50
46.5
40
50
46.5
40
46.5
40
46.5
40
46.5
40
46.5
40
46.5
40
46.5
40
Surplus
41
40.5
35
39
39.5
34
37.5
33
35.5
32
33.5
31
31.5
30
29.5
29
27.5
28
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 ≡ 𝑑𝑑𝑖𝑖𝑇𝑇 , denoted by 𝑏𝑏 𝑓𝑓 , as long as 𝑏𝑏 𝑓𝑓 ≥ 𝑝𝑝𝑇𝑇 ∙ 𝑑𝑑𝑖𝑖𝑇𝑇 ,
where 𝑝𝑝𝑇𝑇 = (𝑝𝑝1𝑇𝑇 , … , 𝑝𝑝𝐾𝐾𝑇𝑇 ) 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
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 = (𝜋𝜋1𝛼𝛼 , … , 𝜋𝜋𝐾𝐾𝛼𝛼 ) is
𝑓𝑓
𝛼𝛼
𝑇𝑇
𝐵𝐵 𝛼𝛼 = 𝑏𝑏 𝑓𝑓 + 𝑝𝑝𝑇𝑇 ∙ (𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝑓𝑓 ) = 𝑏𝑏 𝑓𝑓 + ∑𝐾𝐾
𝑘𝑘=1�𝑝𝑝𝑘𝑘 �𝜋𝜋𝑘𝑘 − 𝜋𝜋𝑘𝑘 ��.
In order to compute the cap for a package 𝜋𝜋 𝛽𝛽 , which requires a number of eligibility points
𝐸𝐸𝛽𝛽 = 𝑒𝑒 ∙ 𝜋𝜋 𝛽𝛽 > 𝐸𝐸 𝑓𝑓 , one first has to track the last round 𝑟𝑟 𝛽𝛽 in the clock phase when bidder 𝑖𝑖 had
𝛽𝛽
enough eligibility points to bid for package 𝜋𝜋 𝛽𝛽 . This 𝑟𝑟 𝛽𝛽 is uniquely determined by 𝐸𝐸 𝑟𝑟 < 𝐸𝐸𝛽𝛽 ≤
𝐸𝐸 𝑟𝑟
𝛽𝛽 −1
𝛽𝛽
. Round 𝑟𝑟 𝛽𝛽 is the anchor round for package 𝜋𝜋 𝛽𝛽 , and the package 𝜋𝜋�𝑟𝑟 𝛽𝛽 � = 𝑑𝑑𝑖𝑖𝑟𝑟 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
𝛽𝛽
𝛽𝛽
𝛽𝛽
𝑟𝑟
𝛽𝛽
𝐵𝐵 𝛽𝛽 = 𝑏𝑏�𝑟𝑟 𝛽𝛽 � + 𝑝𝑝𝑟𝑟 ∙ �𝜋𝜋 𝛽𝛽 − 𝜋𝜋�𝑟𝑟 𝛽𝛽 �� = 𝑏𝑏�𝑟𝑟 𝛽𝛽 � + ∑𝐾𝐾
𝑘𝑘=1 �𝑝𝑝𝑘𝑘 �𝜋𝜋𝑘𝑘 − 𝜋𝜋𝑘𝑘 �𝑟𝑟 ���.
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
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
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 3. 15 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
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).
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.
17
14
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 KO𝑓𝑓
bid 𝑏𝑏�𝑖𝑖 on the last clock round package 𝜋𝜋𝑖𝑖 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)
c)
If bidder 𝑖𝑖 bids 𝑏𝑏 𝑓𝑓 < 𝑏𝑏�𝑖𝑖 , then he may not win any package.
𝑓𝑓
𝑓𝑓
If 𝑁𝑁 = 2 or 𝐾𝐾 = 1, then 𝑏𝑏�𝑖𝑖 ≤ 𝑝𝑝𝑇𝑇 ∙ �𝜋𝜋� − ∑𝑗𝑗≠𝑖𝑖 𝜋𝜋𝑗𝑗 �, where 𝜋𝜋𝑗𝑗 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
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 𝑝𝑝𝑇𝑇 .
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.
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.
19
16
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 +
23. 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 Ψ𝑖𝑖 ⊂ Π be a set of packages that bidder 𝑖𝑖 bids on in either the clock
phase or in the supplementary round. Accordingly, let Φ𝑖𝑖 = {(𝑏𝑏𝑖𝑖𝛼𝛼 , 𝜋𝜋 𝛼𝛼 ): 𝜋𝜋 𝛼𝛼 ∈ Ψ𝑖𝑖 } be the set of
bids bidder 𝑖𝑖: (𝑏𝑏𝑖𝑖𝛼𝛼 , 𝜋𝜋 𝛼𝛼 ) reflects the monetary amount 𝑏𝑏𝑖𝑖𝛼𝛼 that bidder 𝑖𝑖 bids on package 𝜋𝜋 𝛼𝛼 . In
17
case a bidder makes several bids on the same package, only the highest bid is included in Φ𝑖𝑖 .
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 = 𝑑𝑑𝑖𝑖𝑡𝑡 in round 𝑡𝑡 ≥ 1 if either
a)
b)
Bidder 𝑖𝑖 maintains his eligibility points, i.e., 𝐸𝐸𝑖𝑖𝑡𝑡 = 𝐸𝐸𝑖𝑖𝑡𝑡−1, or
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 < 𝐸𝐸𝑖𝑖𝑡𝑡 < 𝐸𝐸𝑖𝑖𝑡𝑡−1 then (𝑣𝑣 𝛼𝛼 − 𝑝𝑝𝑡𝑡 ∙ 𝜋𝜋 𝛼𝛼 ) ≤ �𝑣𝑣 𝛽𝛽 − 𝑝𝑝𝑡𝑡 ∙ 𝜋𝜋 𝛽𝛽 � for
∀𝜋𝜋 𝛼𝛼 ∈ Π for which 𝐸𝐸𝑖𝑖𝑡𝑡 < 𝑒𝑒 ∙ 𝜋𝜋 𝛼𝛼 ≤ 𝐸𝐸𝑖𝑖𝑡𝑡−1, or
18
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.
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.
20
19
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
𝛽𝛽
with from 𝜋𝜋 𝛼𝛼 to 𝜋𝜋 𝛽𝛽 , then 𝑣𝑣𝑖𝑖𝛼𝛼 − 𝑝𝑝𝑡𝑡 ∙ 𝜋𝜋 𝛼𝛼 > 𝑣𝑣𝑖𝑖 − 𝑝𝑝𝑡𝑡 ∙ 𝜋𝜋 𝛽𝛽 so that the revealed preference
𝛽𝛽
constraint 𝐵𝐵 𝛽𝛽 = 𝑏𝑏 𝛼𝛼 + 𝑝𝑝𝑡𝑡 ∙ �𝜋𝜋 𝛽𝛽 − 𝜋𝜋 𝛼𝛼 � implies �𝐵𝐵 𝛽𝛽 − 𝑏𝑏 𝛼𝛼 � > �𝑣𝑣𝑖𝑖 − 𝑣𝑣𝑖𝑖𝛼𝛼 �. This means the bid
𝛽𝛽
𝑏𝑏 𝛽𝛽 for which �𝑏𝑏 𝛽𝛽 − 𝑏𝑏 𝛼𝛼 � = �𝑣𝑣𝑖𝑖 − 𝑣𝑣𝑖𝑖𝛼𝛼 � 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
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
clock-phase constrained efficient allocation and show that inefficiency of equilibria is not an
issue in the supplementary phase.
Definition 3. An allocation is clock-phase constrained efficient if it maximizes the total sum
of constrained values 𝑣𝑣�𝑖𝑖𝛼𝛼 .
Proposition 4. Any equilibrium of the supplementary phase that follows a clock phase where
bidders are known to have AS-bid is clock-phase constrained efficient.
Even if, for a given development of the clock phase, there is a unique clock-phase constrained
efficient allocation, there are many equilibrium strategies resulting in that allocation, but in
different prices. As the price bidders pay depends on what other bidders bid on packages they
do not win, bidders are indifferent between any of these strategies as long as they do not have
a preference for raising rivals’ cost. Proposition 4 basically says that bidders individually would
not want to raise the amount on non-winning packages if there is a risk that by doing so the
winning allocation is different from the clock-phase constraint efficient allocation.
Whether a bid of a bidder on a package that is not a part of a winning allocation will
become winning bid depends on all other bids that other bidders have made. The collection of
bids has to satisfy (only) one constraint, namely that they are just not winning. Many
combinations of bids satisfy this constraint. If bidders do not coordinate, however, the
allocation will be clock-phase constrained inefficient. Strategic uncertainty due to coordination
failures is an issue in the supplementary round.
3.1 Spiteful Bidding Behavior
We now consider which equilibria will be selected in the supplementary round if bidders have
a lexicographic motive to raise rivals’ cost. We formally model the spite motive as follows.
By the end of the supplementary round, any bidder bids a set of bids Φ = {(𝑏𝑏 𝛼𝛼 , 𝜋𝜋 𝛼𝛼 ): 𝜋𝜋 𝛼𝛼 ∈ Ψ}.
A bidder either wins one of these packages 𝜋𝜋 𝛼𝛼 and pays 𝑝𝑝𝛼𝛼 , or he does not win. His intrinsic
pay-off of bidding Φ thus equals (𝑣𝑣 𝛼𝛼 − 𝑝𝑝𝛼𝛼 ) or 0. If, for a fixed strategy profile of other bidders,
22
the intrinsic pay-off of two strategies is identical, bidders prefer the strategy that raises the sum
of rivals’ payments most. Thus, we can say a strategy Φ dominates another strategy Φ′ if:
1.
2.
For any bids of the other bidders, Φ never results in a lower intrinsic pay-off, or in the
same intrinsic pay-off and a lower sum of rivals’ payments than Φ′ ; and
For some of the others’ bids, either Φ results in a larger intrinsic pay-off, or it results in
the same intrinsic pay-off but raises the sum of rivals’ payments, as compared to Φ′ .
In what follows, we assume that the final clock round ends without excess supply. Without
assuming market clearing in the last clock round, there are too many possibilities to consider to
get analytic results. With market clearing, we first show that if there is one spectrum band only,
there exists a unique undominated equilibrium outcome.
Proposition 5. Let 𝐾𝐾 = 1, possible ties in the final allocation be resolved in favor of the last
clock round allocation, and let the clock phase end with market clearing.
Then, the
supplementary round is dominance solvable with bidders bidding maximum bids 𝐵𝐵𝑖𝑖𝛼𝛼 for all
𝑓𝑓
𝑓𝑓
packages 𝜋𝜋𝑖𝑖𝛼𝛼 ≠ 𝜋𝜋𝑖𝑖 , and winning their last clock round packages 𝜋𝜋𝑖𝑖 .
According to Proposition 5, with one band and market clearing, the only thing bidders can affect
in the supplementary round is the price competitors pay. Given the lexicographic nature of the
preferences, bidders bid their maximal feasible bids. This bidding to raise rivals’ cost is without
any risk in this case, as the final allocation is already determined by the bidding behavior in the
last clock round. Proposition 5 requires a tie-breaking rule because it may happen that raising
all supplementary round bids to the largest extent possible will create a tie with the final clock
round allocation. Without this tie-breaking rule, the supplementary phase will not possess an
equilibrium in pure strategies, similar to the argument why Bertrand competition with
differentiated costs needs a tie breaking rule to have an equilibrium in pure strategies. The
condition of AS-bidding is redundant with 𝐾𝐾 = 1 as with only one price (hence, without
changes in relative prices to take into account), bidders will anyway get their last clock round
package, irrespective of the supplementary round bids.
23
With more than one band, it is not necessarily the case that if the clock phase ends with
market clearing, players win their last clock round packages. The next example shows this in
the context of a simplified version of our example with two bidders.
Example. Not winning last clock round package.
Let us consider our example with two bidders and total supply being (2,2) instead of (3,3).
Suppose that at prices (𝑝𝑝𝑡𝑡1 , 𝑝𝑝𝑡𝑡2 ) in clock round 𝑡𝑡, bidder 1 was bidding for package (2,1) and
that bidder 2 continues biding for (1,1) from that clock round onwards. In clock round (𝑡𝑡 + 1),
when prices are (𝑝𝑝𝑡𝑡1 + 1, 𝑝𝑝𝑡𝑡2 ), bidder 1 switches to package (0,2), while in clock round (𝑡𝑡 + 2),
when prices are (𝑝𝑝𝑡𝑡1 + 1, 𝑝𝑝𝑡𝑡2 + 1), bidder 1 switches to package (1,1) and the clock phase
finishes.
(1,1)
Bidders are unrestricted on their last clock round packages and jointly bid 𝑏𝑏1
(1,1)
Bidder 1 can maximally bid 𝑏𝑏1
(1,1)
bid 𝑏𝑏1
(1,1)
+ 𝑏𝑏2
.
+ 𝑝𝑝𝑡𝑡1 + 2 on package (2,1), while bidder 2 can maximally
− 𝑝𝑝𝑡𝑡1 − 1 on package (0,1). Thus, on the combination of packages (2,1) and (0,1)
(2,1)
they can together bid 𝑏𝑏1
(0,1)
+ 𝑏𝑏2
(1,1)
> 𝑏𝑏1
(1,1)
+ 𝑏𝑏2
. Thus, if bidders bid their bid caps, they
do not win their last clock round packages, even though there is market clearing in all bands in
the last clock round.
///
The next result presents conditions under which the coordination problem associated with
multiple equilibria disappears even with more bands. We introduce the following terminology.
Given the development of the clock phase, it may happen that some feasible supplementary bid
on a certain package does not influence the winning allocation. If so, we call these bids
allocation irrelevant. When feasible bids do not influence the VCG pricing, we call them price
irrelevant. It is clear that if 𝑏𝑏𝑖𝑖𝛼𝛼 is allocation irrelevant and price irrelevant, then iterative
elimination of weakly dominated strategies does not put any restriction on these bids.
Proposition 6. Let it be common knowledge that before the supplementary round:
a)
All bidders have bid anchor-sincerely in the clock phase;
b)
In the final clock round, aggregate demand equals supply, i.e., ∑𝑖𝑖 𝜋𝜋𝑖𝑖 = 𝜋𝜋�;
𝑓𝑓
24
c)
d)
𝑓𝑓
𝑓𝑓
For each bidder 𝑖𝑖, the value 𝑣𝑣𝑖𝑖 exceeds the KO-bid 𝑏𝑏�𝑖𝑖 , i.e., 𝑣𝑣𝑖𝑖 > 𝑏𝑏�𝑖𝑖
𝑓𝑓
Each bidder 𝑖𝑖 bids only for the final clock round package 𝜋𝜋𝑖𝑖 and for packages that require
𝑓𝑓
𝑓𝑓
𝑓𝑓
strictly more eligibility points than 𝜋𝜋𝑖𝑖 , i.e., 𝑒𝑒 ∙ 𝜋𝜋𝑖𝑖𝛼𝛼 > 𝑒𝑒 ∙ 𝜋𝜋𝑖𝑖 for any 𝜋𝜋𝑖𝑖𝛼𝛼 ∈ Ψ𝑖𝑖 \�𝜋𝜋𝑖𝑖 �.
Then, the outcome is uniquely determined by iterative elimination of weakly dominated
𝑓𝑓
𝑓𝑓
strategies. Each bidder bids 𝑏𝑏𝑖𝑖 > 𝑏𝑏�𝑖𝑖 on package 𝜋𝜋𝑖𝑖 , and bids his entire caps 𝐵𝐵𝑖𝑖𝛼𝛼 on all price
𝑓𝑓
relevant packages 𝜋𝜋𝑖𝑖𝛼𝛼 . Each bidder wins 𝜋𝜋𝑖𝑖 , and pays not more than his KO-bid 𝑏𝑏�𝑖𝑖 .
The market clearing condition (b) in Proposition 6 guarantees that each bidder wins his final
𝑓𝑓
clock round package 𝜋𝜋𝑖𝑖 after the first round of elimination. When this condition fails, the first
elimination round does not pin down the auction allocation. In this case, bidding above true
values brings a risk of winning a package at a price above value, as in the VCG mechanism.
Nevertheless, if a bidder knows that a package 𝜋𝜋 𝑈𝑈 remains unsold in the last clock round, he
can safely bid his entire caps for those packages that exceed (in terms of eligibility points) his
𝑓𝑓
final package plus the unsold package �𝜋𝜋 𝑈𝑈 + 𝜋𝜋𝑖𝑖 �, because these packages cannot be winning,
but are likely to determine the prices other bidders pay. In that case, bidders can still engage in
spiteful bidding without risking to get packages at prices above value. 21
In addition to the market clearing condition (b), there are two important conditions in
Proposition 6, conditions (c) and (d). These conditions are irrelevant in case 𝐾𝐾 = 1 as in that
case all the relevant constraints on supplementary round bidding are clear to all bidders, and
final prices are only determined by what others bid on packages that are larger than their last
clock round package. When condition (d) fails and bidders also bid on packages that require
fewer eligibility points than the final clock round package, the supplementary round is not
dominance solvable, and multiple equilibria re-appear again. The danger of bidding maximum
caps is that such bids may become winning bids in combination with bids of others on smaller
packages. Our previous example with two bidders clearly illustrates this point. In that case,
21
For example, if there are four units to be auctioned with three bidders, and the clock phase ends with all bidders
demanding one unit (excess supply of one unit), then (depending on further details) each bidder may bid their KObid on one unit, bid in addition the marginal value difference on two units, and bid an amount above value on three
units. This last bid is likely to determine prices others pay, but is unlikely to win, given that others bid their KObid on one unit. The bid on two units is constructed such that the bidder would be happy to acquire the second
unit as the price he has to pay is not more than the value difference.
25
bidders may be reluctant to express high bids on larger packages with the sole purpose of raising
rivals’ cost.
The next Example illustrates Proposition 6. In addition, it shows that if bidders know that
others have low valuations for small packages (even if they do not know the precise values for
sure) they may be able to figure out that price relevant bids on these small packages are
dominated, and thus that others do not make such bids. In that case, the supplementary round
is dominance solvable even if condition (d) does not hold.
Example. Iterative elimination of bidding strategies.
We have seen that if bidders were bidding truthfully in the clock phase in our example, then
conditions (a) and (b) of Proposition 6 are fulfilled, and from the previous part of the example,
we know that condition (c) is also fulfilled. Suppose, first, that it is common knowledge that
bidders only bid for packages (1,1), (2,1), (1,2), (3,0), (0,3). Subsequently, we argue that
bidders can figure out that no one will bid for other packages if it is common knowledge their
intrinsic value is low.
Above, we have shown that bidders have a KO-bid of 29. Thus, following Proposition 6,
any bidding behavior in the supplementary round where bidders bid 𝑏𝑏 < 29 on (1,1) is weakly
dominated. In the second elimination round, the spite motive for raising rivals’ cost ranks all
the undominated strategies with respect to the price that others pay. Thus, the only strategies
surviving the second round of elimination are such that bidders bid their maximum bid caps.
Thus, only strategies of the form (𝑏𝑏, 𝑏𝑏 + 7, 𝑏𝑏 + 11, 𝑏𝑏 + 9, 𝑏𝑏 + 14) with 𝑏𝑏 ≥ 29 survive
iterative elimination of weakly dominated strategies. In the resulting equilibrium, each bidder
gets package (1,1) and pays an auction price of 23 (as the particular bid history is such that no
combination of bidders can make others pay more than 23).
Now consider the possibility that bidders make supplementary bids on smaller packages.
If every bidder bids maximally on all nine possible packages, then the winning allocation will
be (0,3), (1,0), and (2,0) as the total bid on this combination is (3𝑏𝑏 + 11) and prices will be
26, 7 and 14 correspondingly. Bidding high on packages that are larger than the final clock
round package may seem to be risky then. We will argue, however, that if bidders have
sufficiently low intrinsic value for packages other than the three main packages we consider
26
here, any combination of bids that survives iterative elimination of dominated strategies, does
not have bids on packages (1,0) and (2,0) that are either pay-off or allocation relevant.
To see this first consider whether bids on packages (1,0) and (2,0) that are allocation
irrelevant could potentially be price relevant, and whether bidders would like to make such bids
in the supplementary round. If this were so, bidders could safely bid on these packages without
winning them. The answer to this question, however, is negative. Indeed, these bids can only
be price relevant if they form the highest total value in conjunction with a bid of one other
player. Given that players can only bid for three licenses at most (and that there are three units
of each license), it must be that the best combination of bids is with the other bidder’s bid for
(0,3). As the other bidder can bid at most 𝐵𝐵 (0,3) = 𝑏𝑏 (1,2) + 3, the total bid on the two packages
together cannot be larger than 𝑏𝑏 (1,2) + 3 + (𝑏𝑏 − 5) if in combination with the bid for (1,0), or
𝑏𝑏 (1,2) + 3 + (𝑏𝑏 + 2 ) if in combination with the bid for (2,0).
However, instead of (or, in addition to) making a price relevant bid on either (1,0) or (2,0)
the bidder can also make a bid of (𝑏𝑏 + 6 ) on package (2,1). 22 Such a bid can be combined
with the rival bid on (1,2) to make a possible joint bid of 𝑏𝑏 (1,2) + (𝑏𝑏 + 6 ). As this is larger
than both expressions given above, a combination of bids that includes price relevant bids on
(1,0) and (2,0) are dominated by other bids. 23 Moreover, it is clear that if the intrinsic
valuations are low enough, bids on packages (1,0) and (2,0) that are allocation relevant are
weakly dominated as the minimum price for each unit is the (positive) reserve price. A similar
argument holds true for packages (0,1) and (0,2).
///
It is difficult to have a general result saying how much auction outcomes are influenced by a
lexicographic spite motive. The example shows, however, that even a lexicographic preference
can have a very large, increasing effect on the price bidders pay given truthful bidding in the
clock phase. Instead of paying the VCG prices of 16.5 for package (1,1), which would have
resulted had they bid values on all packages, they are now ending up with the same winning
allocation, but having to pay 23!
As the bidder bids less than his marginal valuation on (2,1) compared to the bid on (1,1), he will not regret if
he wins that package instead of (1,1).
23
Note that bids that are allocation irrelevant or price irrelevant remain so if a smaller set of possible strategies
by other bidders is considered.
22
27
One may wonder whether it is risky to bid above value on packages to raise the price others
pay for the licenses they win. In the example, bidders have to pay much more to protect
themselves against other bidders bidding high on packages that do not have an intrinsic value,
such as packages (3,0) and (0,3). The answer depends on whether bidders are reasonably
certain about the competitors’ values and their rationality. Rational bidders protect themselves
against not winning, by calculating the KO-bid and bidding this amount (Proposition 3). If
bidders expect others to be rational, they can predict this behavior and there is no risk to bid
above value on packages to raise the price others pay. If one of the bidders does not understand
the iterative dominance logic, however, there may be bidders winning a package for a price
above value.
It is interesting to note the difference with the VCG mechanism. In the VCG mechanism,
bidding above value is risky as it may result in negative surplus (when the bid is winning), but
it may also result in positive surplus with competitors paying more. In a CCA, the restrictions
on supplementary bids imposed by the clock phase give certainty that certain bids above value
cannot be winning bids, assisting bidders to engage in spiteful bidding.
4. The Clock Phase
We now consider whether it is optimal to bid truthfully in the clock round. The next Proposition
provides some support for bidders engaging in demand expansion in the clock phase in order
for them to be able to optimally increase competitors’ prices. Consider the final clock round in
which one bidder 𝑖𝑖 has switched from package 𝜋𝜋 𝛼𝛼 = 𝑑𝑑𝑖𝑖𝑇𝑇−1 to his final package 𝜋𝜋 𝑓𝑓 = 𝑑𝑑𝑖𝑖𝑇𝑇
thereby ending the auction. The Proposition states that such behavior is strictly suboptimal,
and extending the clock phase is preferred if all KO-bids remain below values. This behavior
is consistent with AS-bidding in the clock phase.
Proposition 7. Suppose conditions (a) – (d) of Proposition 6 hold, and in the final clock round
𝑓𝑓
𝑓𝑓
𝑇𝑇, bidder 𝑖𝑖 has switched from package 𝜋𝜋𝑖𝑖𝛼𝛼 = 𝑑𝑑𝑖𝑖𝑇𝑇−1 to his final package 𝜋𝜋𝑖𝑖 = 𝑑𝑑𝑖𝑖𝑇𝑇 , 𝜋𝜋𝑖𝑖 ≠ 𝜋𝜋𝑖𝑖𝛼𝛼 .
This behavior is dominated by continuing bidding for package 𝜋𝜋𝑖𝑖𝛼𝛼 in round 𝑇𝑇 and switching to
28
𝑓𝑓
𝜋𝜋𝑖𝑖 in round 𝑇𝑇 + 1 ending the clock phase if all other bidders do not change their demands, i.e.,
𝑓𝑓
𝑑𝑑𝑗𝑗𝑇𝑇+1 = 𝑑𝑑𝑗𝑗𝑇𝑇 = 𝜋𝜋𝑗𝑗 for all 𝑗𝑗 ≠ 𝑖𝑖.
According to Proposition 7, if other bidders do not change their bids when the clock phase lasts
one more round, then the bidder under consideration prefers to prolong the clock phase by one
round. In doing so, he relaxes the caps on bid increments on his own non-winning bids, without
affecting the final allocation of packages increasing the payment of others. At the same time,
the caps of the other bidders for packages that are larger than the last clock round packages
(determining his KO-bid) remain intact, and so does the price he pays. The reason we use the
Proposition 6 conditions, is that in this case we know that the supplementary round has a unique
equilibrium, but the logic applies more generally.
If all bidders expand demand this way, final payments may be considerably larger than
when all bidders would bid truthfully. The clock phase has a Prisoners’ Dilemma aspect: it is
individually rational to extend the clock phase when the others have already reduced demands
to levels they do not want to lower even further. If everyone acts this way, the bidders
collectively get a worse outcome. Our example shows an extreme case where bidders extend
the clock phase so much that in the supplementary round they dissipate their surplus.
Example. Demand expansion in the clock phase.
We illustrate the demand expansion strategy of Proposition 7 and show it may even hold if
others’ bidding behavior is not kept constant. Suppose that from 𝑡𝑡 = 2 onwards, bidders
continue bidding on package (2,1) until the prices reach the amount (13,5) and then reduce
demand to (1,1). The example shows that the clock phase cannot end with all bidders bidding
for package (1,1) until the moment the KO-bid is larger than the value of 40 for that package.
In particular, if others are bidding (1,1) in round 𝑡𝑡, each individual bidder has an incentive to
continue bidding at least one more round for (2,1) before reducing to (1,1).
If the clock phase ends with (1,1) in round 𝑡𝑡, then in the supplementary phase bidders can
bid 𝑏𝑏 (3,0) ≤ 𝑏𝑏 (1,1) + (2𝑡𝑡 − 7) on package (3,0), while they can bid 𝑏𝑏 (0,3) ≤ 𝑏𝑏 (1,1) + (𝑡𝑡 + 6) on
(0,3). Thus, the KO-bid on (1,1) equals (3𝑡𝑡 − 1). Suppose bidder 1 deviates in the way
described above. If the final allocation remains the same, then as the non-deviating bidders will
29
certainly bid their KO-bids, provided they are below value, the deviating bidder is able to raise
rivals’ cost as he has larger bidding caps. As the final allocation and the KO-bid of the deviating
bidder are not affected, the bidder 1 is certainly better off.
Suppose then, that another bidder reacts to the deviation in period 𝑡𝑡 and the final allocation
is different.
As in equilibrium, no player can win a package he does not value, the
(1,1)
supplementary bids have then to satisfy 𝑏𝑏1
(1,2)
𝑏𝑏1
(1,1)
= 𝑏𝑏1
(1,1)
+ 𝑏𝑏2
(1,1)
+ 𝑏𝑏3
(1,2)
≤ 𝑏𝑏1
(2,1)
+ 𝑏𝑏2
. By bidding
, bidder 1 can ensure, however, that he either wins (1,2) at the price of (1,1)
before the deviation, or all bidders win (1,1). In the former case, bidder 1 gets a higher surplus,
while in the latter case, the above argument applies and bidders 2 and 3 pay more.
Thus, the clock phase can only end with all bidders bidding for (1,1) until the KO-bid is
larger than value, which happens when 3𝑡𝑡 − 1 ≥ 40, or 𝑡𝑡 = 14. In that case, the symmetric
equilibrium bids in the supplementary round are 𝑏𝑏 (1,1) = 40, 𝑏𝑏 (2,1) = 53, 𝑏𝑏 (1,2) = 57, 𝑏𝑏 (3,0) ∈
[60,61], and 𝑏𝑏 (0,3) = �120 − 𝑏𝑏 (3,0) �, and all surplus is dissipated. There is a continuum of
these equilibria where the latter constraint illustrates observations we have made after
Proposition 4 that there is a coordination element in raising rivals’ cost.
///
The above example suggests that spiteful bidders engage in demand expansion in the clock
phase. The example does not provide a full equilibrium analysis of the clock phase. Even in
case of our simple three-player two-band example, a full equilibrium analysis is difficult to
provide. The next Section discusses whether to expect demand expansion or demand reduction
in the clock phase.
5. Demand Expansion or Demand Reduction in the Clock Phase?
So far, we have emphasized that bidders are able to exercise their spite motive in the
supplementary phase by expanding demand given the clock phase bids. Bidders do so by
bidding on packages that cannot be winning, but that do raise competitors’ prices. The
lexicographic spite motive solves the indeterminacy that otherwise exists in favor of bids that
raise rivals’ cost. In Section 4, we have shown that under certain conditions this motive may
30
lead to bidders relaxing the constraints in the clock phase by expanding their demand in the
clock phase as well.
These results may seem to be inconsistent with results by Levin and Skrzypacz (2014) –
hereafter LS – who indicate in one of their models (model B) that in anticipation of the
competitor trying to raise rivals’ cost in the supplementary phase, one bidder engages in demand
reduction in the clock phase. LS analyze a simplified version of the CCA with two bidders and
one band. Importantly, they restrict their analysis to linear proxy strategies. LS show that
anticipating the other bidder engages in spiteful bidding, the linear strategy one bidder chooses
effectively has lower demand at any given clock price than the demand implied by truthful
bidding.
It is important to understand that neither our paper, nor LS, provides a full equilibrium
analysis of both phases of the CCA under lexicographic preferences to raise rivals’s cost.
Without lexicographic preferences, LS show that there are many equilibria involving both
demand expansion and demand reduction in the clock phase. Their analysis with lexicographic
preferences restricts one bidder, however, to linear proxy strategies in the clock phase, but this
class of strategies is clearly not optimal if the bidder wants to raise rivals’ cost.
It is difficult to fully characterize the perfect Bayesian equilibria of both phases of the CCA
under lexicographic preferences because of the many possible rounds, the simultaneous nature
of bidding behavior within a round, and the possibility of multiple equilibria in the
supplementary round. The next example shows, however, that one can provide equilibria for
the full game in some specific example. This equilibrium has bidders bidding anchor sincerely
and expanding demand in the clock phase.
Example. An Equilibrium with Demand Expansion and Anchor Sincere Bidding
Let there be two bidders and two identical objects to be auctioned in a CCA. Bidder 𝑖𝑖’s
valuation for one object is 𝑣𝑣𝑖𝑖 (1) = 11, and for two objects is 𝑣𝑣𝑖𝑖 (2) = 11 + 𝜀𝜀𝑖𝑖 , where 𝜀𝜀𝑖𝑖 is
uniformly distributed over [0, 10] and is private information to bidder 𝑖𝑖. In the clock phase of
the CCA, the price increases continuously from 𝑝𝑝 = 0 upwards. The bidders are only informed
about whether there is excess demand at the current price level (to simplify the description of
the strategies for the moment). If after the supplementary round there are multiple combinations
31
of winning bids, then the following tie-breaking rule is used: if the last-clock round allocation
is among the value-maximizing allocations it is chosen as the final auction allocation, otherwise
the allocation with the largest number of winners is chosen.
We propose the following equilibrium strategy of the clock phase:
a)
b)
c)
if 𝑝𝑝 < 10, bidder 𝑖𝑖 demands 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 2 units;
if 𝑝𝑝 = 10, bidder 𝑖𝑖 demands 𝑑𝑑𝑖𝑖 (10, 𝜀𝜀𝑖𝑖 ) = 1 unit;
if 𝑝𝑝 > 10, bidder 𝑖𝑖 demands 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 0 units.
In the proposed equilibrium, both bidders switch to demanding one unit at the 𝑝𝑝 = 10, and the
clock phase stops at that moment. Both bidders submit supplementary round bids 𝑏𝑏𝑖𝑖 (1) = 10
and 𝑏𝑏𝑖𝑖 (2) = 20, and win 1 object each at the auction price 𝑝𝑝(1) = 10.
If a bidder 𝑖𝑖 deviates and demands 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 1 units at prices 𝑝𝑝 ≥ 𝑝𝑝1 , where 0 ≤ 𝑝𝑝1 <
10, then the supplementary round bids of this bidder 𝑖𝑖 are capped by 𝑏𝑏𝑖𝑖 (2) ≤ 𝑏𝑏𝑖𝑖 (1) + 𝑝𝑝1 so
that the other bidder 𝑗𝑗 ≠ 𝑖𝑖 wins his unit at a price of 𝑝𝑝1 < 10, whereas the surplus of bidder 𝑖𝑖
remains the same (and equal to 1). Thus, this deviation does not affect the final allocation, the
price that bidder 𝑖𝑖 pays, but only lowers the price that bidder 𝑗𝑗 ≠ 𝑖𝑖 pays. Thus, this deviation
is not profitable.
If bidder 𝑖𝑖 deviates and demands 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 1 units at prices 𝑝𝑝 ∈ [𝑝𝑝1 , 𝑝𝑝0 ), where 0 ≤ 𝑝𝑝1 ≤
𝑝𝑝2 < 10, and bids for 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 0 units at prices 𝑝𝑝 ≥ 𝑝𝑝0, then the supplementary round bids
of this bidder 𝑖𝑖 are capped by 𝑏𝑏𝑖𝑖 (1) ≤ 𝑝𝑝0 𝑏𝑏𝑖𝑖 (2) ≤ 𝑝𝑝0 + 𝑝𝑝1 so that the other bidder 𝑗𝑗 ≠ 𝑖𝑖 wins
both units, and bidder 𝑖𝑖 gets zero surplus. Thus, this deviation is also not profitable.
If bidder 𝑖𝑖 deviates and demands 𝑑𝑑𝑖𝑖 (𝑝𝑝, 𝜀𝜀𝑖𝑖 ) = 2 units at all prices 𝑝𝑝 ∈ [0, 𝑝𝑝1 ), where 𝑝𝑝1 >
10, then this bidder wins both units at price 𝑝𝑝𝑖𝑖 (2) = 20, and his surplus is 𝑣𝑣𝑖𝑖 (2) − 𝑝𝑝𝑖𝑖 (2) =
𝜀𝜀𝑖𝑖 − 9. Thus, for all types 𝜀𝜀𝑖𝑖 ∈ [0,10], this deviation is also not profitable. 24
Note that the bidding behavior is anchor-sincere, according to Definition 1. Thus, this
example shows that anchor sincere bidding and then maximally raising rivals’ cost is an
equilibrium of the full game.
///
24
If we allow the bidders to observe the aggregate demand (which is equivalent in this example to let them observe
the demand of the competitor), the equilibrium outcome remains intact. For example, if the other bidder 𝑗𝑗 ≠ 𝑖𝑖
switches to 𝑑𝑑𝑗𝑗 = 1 at price 𝑝𝑝 ∈ (0, 10), bidder 𝑖𝑖 keeps bidding for 2 units.
32
The crucial point in the example is that the given the equilibrium strategy of the other bidder(s),
the only way a bidder can prevent prices to further increase in the clock phase is by dropping
demand to zero. This implies, however, he does not win anything. He therefore prefers to
continue bidding for the maximal number of bands to raise rivals’ cost. Note that this conclusion
is independent of the activity rule used. In particular, since there is only one anchor round - the
last one, the same equilibrium construction applies for the intermediate cap rule. Moreover, the
equilibrium also continues to hold under a GARP-based activity rule combined with the final
cap rule. A GARP-based rule does not have an implications as long as a bidder does not change
his demand from the initial demand in the first clock round. In the example, the bidders only
change their demand in the last clock round, and the final cap rule allows them to bid up to 10
more for the second unit in the supplementary.
In general, the demand reduction strategy adopted by LS requires that bidders can predict
the final prices they have to pay if they stop the clock phase. With more than two bidders and
more than one band, Proposition 1 and the example following it show that this is impossible.
First, the KO-bid can be (much) larger than the final clock round bid. Second, the final price
will depend on the specific bidding history of the different individual bidders, while bidders
have only information about total demand in each clock round (at best). Third, with three or
more bidders, individually a bidder may not be able to stop the clock phase (as in our example
when all bidders are bidding for (2,1)). Complementarities create further complications if
demand reduction in the clock phase implies bidding on packages with a relatively low value.
With complementarities between bands and/or units, it may well happen that for all packages
that are still feasible to bid on in a clock round the value is smaller than the bid a bidder can
place, so that reducing demand may imply winning a package at a price above value.
6. Discussion and Conclusion
This paper has extended our knowledge of combinatorial clock auctions (CCAs) in several
directions. First, most of the analysis on the supplementary round of the CCA is not restricted
to specific cases where only two bidders are considered or only one spectrum band. We show
that a so-called KO-bid plays a key role in analyzing the optimal bidding strategy in the
33
supplementary round. This KO-bid is the lowest bid on the final clock round package that is
such that the bidder is guaranteed to be among the winners of the auction. This KO-bid can be
(much) larger than the final clock round bid itself. A supplementary round bid that involves
bidding below this KO-bid on the final clock round package is dominated.
We introduce a spite motive in a lexicographic way and show that this spite motive selects
among the many equilibria of the supplementary round. We have provided some evidence that
a spite motive is important in markets where CCAs are used. With the spite motive, bidders
place their KO-bid on the final clock round package and place high bids on other packages that
cannot be winning.
CCAs give rise to many gaming opportunities. By placing high bids on non-winning
packages, bidders have the ability to raise rivals’ cost. Information provided in the clock phase,
together with the rules capping the bids in the supplementary round, allows bidders to calculate
the “knock-out” price. Given that rational bidders bid at least this “knock-out” amount, it is
without risk to bid in the supplementary round above value on other packages to raise rivals’
cost.
We have identified conditions under which bidders will raise their bids on non-winning
packages to the maximal extent possible. If these conditions are satisfied, this behavior
generates the unique outcome that survives iterative elimination of dominated strategies. If the
conditions are not satisfied, there are multiple equilibria with possibly large inefficiencies
resulting if bidders do not coordinate on the same equilibrium. We have provided an example
where in equilibrium in the clock phase, bidders engage in demand expansion to soften the
constraints on supplementary phase bids.
In the context of CCAs, many issues remain unknown, leading to different possible areas
for future research.
In terms of bidding behavior, it is important to better understand
equilibrium behavior in the clock phase. In terms of design, we identify the following three
issues. A first issue is how much information bidders should get in the clock phase. We have
pointed at the fact that bidders can use the information provided in the clock rounds (either
about individual or total demand) to calculate a KO-bid allowing them to raise rivals’ cost by
bidding high on non-winning packages. This raises the issue whether the clock round should
provide less information (such as in the 2010 Austrian auction where bidders were only
34
informed about clock round prices) or whether there should be a clock phase at all. Without
the clock phase, a CCA reduces to a VCG mechanism where bidding above value is risky. The
clock phase is meant to allow for price discovery, but if bidders will manipulate the price
discovery process the question is how useful the clock phase information is. A second design
issue relates to the pricing rule. If bidders also bid strategically under the opportunity cost
pricing rule of the CCA, the main advantage of using a second price rule is lost and a first price
rule (like in Romania in 2012) may be preferred.
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Appendix I: Proofs
Proof of Proposition 1.
We first assume that the development of the clock phase is public information. Then, we relax
this assumption. At the end of the proof, we consider cases 𝑁𝑁 = 2 and 𝐾𝐾 = 1.
37
𝑓𝑓
𝑓𝑓
Suppose bidder 𝑖𝑖 only bids 𝑏𝑏𝑖𝑖 on 𝜋𝜋𝑖𝑖 . We also assume for a moment that the clock round
𝑓𝑓
bids of bidder 𝑖𝑖 are not taking part in computing the optimal allocation, i.e., Ψ𝑖𝑖 = �𝜋𝜋𝑖𝑖 � and
𝑓𝑓
𝑓𝑓
𝑓𝑓
Φ𝑖𝑖 = ��𝑏𝑏𝑖𝑖 , 𝜋𝜋𝑖𝑖 ��. We first show that there is an amount 𝑏𝑏�𝑖𝑖 such that if 𝑏𝑏𝑖𝑖 > 𝑏𝑏�𝑖𝑖 , bidder 𝑖𝑖
𝑓𝑓
necessarily wins 𝜋𝜋𝑖𝑖 and pays at most 𝑏𝑏�𝑖𝑖 . Then, after taking into account his clock phase bids
and allowing him to bid on all packages in the supplementary round, he remains a winner, which
ends this argument.
We use the following notation. Let bidder 𝑖𝑖 win package 𝜋𝜋𝑖𝑖 . Let the maximum of the
bids’ sum of all other bidders for an allocation �𝜋𝜋𝑗𝑗𝛼𝛼 � of the remaining (𝜋𝜋� − 𝜋𝜋𝑖𝑖 ) blocks be:
𝑉𝑉−𝑖𝑖 (𝜋𝜋𝑖𝑖 ) ≡
max
�𝜋𝜋𝑗𝑗𝛼𝛼 �:∑𝑗𝑗≠𝑖𝑖 𝜋𝜋𝑗𝑗𝛼𝛼 ≤(𝜋𝜋
�−𝜋𝜋𝑖𝑖 )
∑𝑗𝑗≠𝑖𝑖 𝑏𝑏𝑗𝑗𝛼𝛼 .
If bidder 𝑖𝑖 does not win, we formally set 𝜋𝜋𝑖𝑖 = 0 in the above definition. When bidder 𝑖𝑖 has
𝑓𝑓
𝑓𝑓
𝑓𝑓
only one bid 𝑏𝑏𝑖𝑖 on package 𝜋𝜋𝑖𝑖 , he either wins 𝜋𝜋𝑖𝑖 and pays the VCG price
𝑓𝑓
𝑓𝑓
𝑝𝑝𝑖𝑖 = 𝑉𝑉−𝑖𝑖 (0) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 �,
𝑓𝑓
𝑓𝑓
(1)
𝑓𝑓
𝑓𝑓
𝑓𝑓
or he does not win any package. He wins 𝜋𝜋𝑖𝑖 when 𝑏𝑏𝑖𝑖 + 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 � ≥ 𝑉𝑉−𝑖𝑖 (0), i.e., 𝑏𝑏𝑖𝑖 ≥ 𝑝𝑝𝑖𝑖 . We
𝑓𝑓
show that 𝑝𝑝𝑖𝑖 has an upper-bound which we denote by 𝑏𝑏�𝑖𝑖 :
𝑏𝑏�𝑖𝑖 =
sup
�𝑏𝑏𝑗𝑗𝛼𝛼 �:𝜋𝜋𝑗𝑗𝛼𝛼 ∈Π,𝑗𝑗≠𝑖𝑖
𝑓𝑓
𝑝𝑝𝑖𝑖 =
sup
�𝑏𝑏𝑗𝑗𝛼𝛼 �:𝜋𝜋𝑗𝑗𝛼𝛼 ∈Π,𝑗𝑗≠𝑖𝑖
𝑓𝑓
�𝑉𝑉−𝑖𝑖 (0) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 ��,
where the maximum is taken over all feasible (satisfying the relative cap rule) bids 𝑏𝑏𝑖𝑖𝛼𝛼 for all
feasible packages 𝜋𝜋𝑖𝑖𝛼𝛼 of all other bidders 𝑗𝑗 ≠ 𝑖𝑖. We show now that 𝑏𝑏�𝑖𝑖 is finite.
The bid for any package 𝜋𝜋𝑗𝑗𝛼𝛼 that bidder 𝑗𝑗 can make in the supplementary round is capped
𝛽𝛽
by a certain amount relative to the bid for package 𝜋𝜋𝑗𝑗 , where 𝛽𝛽 is the package that bidder 𝑗𝑗 bid
𝛽𝛽
for in the anchor (for package 𝜋𝜋𝑗𝑗𝛼𝛼 ) round, i.e., 𝑏𝑏𝑗𝑗𝛼𝛼 ≤ 𝑏𝑏𝑗𝑗 + 𝑐𝑐𝑗𝑗𝛼𝛼 . Continue this process recursively,
𝛽𝛽
𝛽𝛽
𝛾𝛾
𝑓𝑓
we write 𝑏𝑏𝑗𝑗 ≤ 𝑏𝑏𝑗𝑗 + 𝑐𝑐𝑗𝑗 and so on, until we reach the final clock round package 𝜋𝜋𝑗𝑗 . Thus, the
𝑓𝑓
relative cap rule implies 𝑏𝑏𝑗𝑗𝛼𝛼 ≤ 𝑏𝑏𝑗𝑗 + 𝑐𝑐̅𝑗𝑗𝛼𝛼 , where the caps 𝑐𝑐̅𝑗𝑗𝛼𝛼 only depend on the clock phase
development. Now we can evaluate 𝑏𝑏�𝑖𝑖 .
𝑓𝑓
𝑓𝑓
𝑓𝑓
𝑓𝑓
First, as ∑𝑗𝑗 𝜋𝜋𝑗𝑗 ≤ �𝜋𝜋� − 𝜋𝜋𝑖𝑖 �, it follows that 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 � ≥ ∑𝑗𝑗≠𝑖𝑖 𝑏𝑏𝑗𝑗 . Then:
𝑉𝑉−𝑖𝑖 (0) ≤
max
�𝜋𝜋𝑗𝑗𝛼𝛼 �:∑𝑗𝑗 𝜋𝜋𝑗𝑗𝛼𝛼 ≤𝜋𝜋
�
∑𝑗𝑗�𝑏𝑏𝑗𝑗𝑓𝑓 + 𝑐𝑐̅𝑗𝑗𝛼𝛼 � ≤ 𝑉𝑉−𝑖𝑖 (0) ≤ 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖𝑓𝑓 � +
38
max
�𝜋𝜋𝑗𝑗𝛼𝛼 �:∑𝑗𝑗 𝜋𝜋𝑗𝑗𝛼𝛼 ≤𝜋𝜋
�
∑𝑗𝑗�𝑐𝑐̅𝑗𝑗𝛼𝛼 �.
Hence, we can write:
𝑏𝑏�𝑖𝑖 =
sup
�𝑏𝑏𝑗𝑗𝛼𝛼 �:𝜋𝜋𝑗𝑗𝛼𝛼 ∈Π,𝑗𝑗≠𝑖𝑖
𝑓𝑓
�𝑉𝑉−𝑖𝑖 (0) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 �� ≤
max
�𝜋𝜋𝑗𝑗𝛼𝛼 �:∑𝑗𝑗 𝜋𝜋𝑗𝑗𝛼𝛼 ≤𝜋𝜋
�
∑𝑗𝑗�𝑐𝑐̅𝑗𝑗𝛼𝛼 �.
This implies that 𝑏𝑏�𝑖𝑖 is finite and, by construction, is achievable.
So far, we made use of the assumption that bidders knew the demand of individual bidders
in each clock round. If only total demands in each clock round are known, a KO-bid can be
calculated for any combination of individual demands that is consistent with the information
concerning total demand and own bidding behavior. One can then take the maximal KO-bid of
all possible KO-bids by considering all (finitely many) possibilities.
In order to proof part (c) of the proposition, we use the following lemma.
Lemma 1. Let bidder 𝑖𝑖 bid 𝑏𝑏 𝑓𝑓 = 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋 𝑓𝑓 in round 𝑇𝑇. Then, his relative cap for any package
𝜋𝜋 𝛼𝛼 does not exceed the value of the aggregate supply 𝜋𝜋� at the prices 𝑝𝑝𝑇𝑇 , i.e., 𝐵𝐵 𝛼𝛼 ≤ 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋�.
Proof.
Let {𝑡𝑡𝑙𝑙 }, 𝑙𝑙 = 1, … , 𝐿𝐿, be the finite decreasing sequence of anchor rounds of bidder 𝑖𝑖, 𝑡𝑡1 = 𝑇𝑇,
𝑡𝑡𝑙𝑙 > 𝑡𝑡𝑙𝑙+1 . Using the relative cap rule definition, we write:
𝐵𝐵 𝛼𝛼 ≤ 𝑏𝑏 𝑓𝑓 + 𝑝𝑝𝑡𝑡1 ∙ (𝑑𝑑 𝑡𝑡2 − 𝑑𝑑 𝑓𝑓 ) + 𝑝𝑝𝑡𝑡2 ∙ (𝑑𝑑𝑡𝑡3 − 𝑑𝑑𝑡𝑡2 ) + ⋯ + 𝑝𝑝𝑡𝑡𝐿𝐿 ∙ (𝜋𝜋 𝛼𝛼 − 𝑑𝑑 𝑡𝑡𝐿𝐿 ),
(2)
𝐵𝐵 𝛼𝛼 ≤ 𝑏𝑏 𝑓𝑓 + (𝑝𝑝𝑇𝑇 − 𝑝𝑝𝑡𝑡2 ) ∙ 𝑑𝑑 𝑡𝑡2 + (𝑝𝑝𝑡𝑡2 − 𝑝𝑝𝑡𝑡3 ) ∙ 𝑑𝑑 𝑡𝑡3 + ⋯ + 𝑝𝑝𝑡𝑡𝐿𝐿 ∙ 𝜋𝜋 𝛼𝛼 − 𝑝𝑝𝑇𝑇 ∙ 𝑑𝑑 𝑓𝑓
(3)
𝛼𝛼
where 𝑡𝑡𝐿𝐿 = 𝑟𝑟 𝜋𝜋 is the anchor round for package 𝜋𝜋 𝛼𝛼 . By rearranging, (2) becomes:
≤ 𝑏𝑏 𝑓𝑓 + 𝑝𝑝𝑇𝑇 ∙ (𝜋𝜋� − 𝜋𝜋 𝑓𝑓 ) = 𝑝𝑝 𝑓𝑓 ∙ 𝜋𝜋�.
The last inequality is due to 𝑝𝑝𝑡𝑡𝑙𝑙 ≥ 𝑝𝑝𝑡𝑡𝑙𝑙+1 and 𝑑𝑑 𝑡𝑡𝑗𝑗 ≤ 𝜋𝜋�. This ends the proof.
■
Let 𝑁𝑁 = 2, 𝑗𝑗 ≠ 𝑖𝑖, and let bidder 𝑖𝑖 bid 𝑏𝑏 𝑓𝑓 = 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋 𝑓𝑓 in the last clock round. If he wins 𝜋𝜋 𝑓𝑓 , he
𝑓𝑓
𝑓𝑓
𝑓𝑓
𝑓𝑓
pays auction price (1). Then 𝑏𝑏𝑗𝑗 = 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋𝑗𝑗 implies 𝑉𝑉𝑗𝑗 �𝜋𝜋𝑖𝑖 � ≥ 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋𝑗𝑗 , and Lemma 1 implies
𝑓𝑓
𝑓𝑓
𝑓𝑓
𝑉𝑉𝑗𝑗 (0) ≤ 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋�. Hence, the auction price 𝑝𝑝𝑖𝑖 = 𝑉𝑉−𝑖𝑖 (0) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 � ≤ 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋� − 𝑝𝑝𝑇𝑇 ∙ 𝜋𝜋𝑗𝑗 , and the
𝑁𝑁 = 2 case of the proposition follows.
When 𝐾𝐾 = 1, prices and packages in (2) are all scalars, which simplifies (2) to 𝐵𝐵 𝛼𝛼 ≤ 𝑏𝑏 𝑓𝑓 +
𝑓𝑓
𝑝𝑝𝑇𝑇 (𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝑓𝑓 ). Since bids 𝑏𝑏𝑗𝑗 of bidders 𝑗𝑗 ≠ 𝑖𝑖 do not affect the maximal price bidder 𝑖𝑖 pays in
39
𝑓𝑓
case of winning, we can set 𝑏𝑏𝑗𝑗 = 𝑝𝑝𝑇𝑇 𝜋𝜋 𝑓𝑓 in computing (1), which yields 𝐵𝐵 𝛼𝛼 = 𝑝𝑝𝑇𝑇 𝜋𝜋 𝛼𝛼 , and,
𝑓𝑓
𝑓𝑓
therefore, 𝑉𝑉−𝑖𝑖 (0) ≤ 𝑝𝑝𝑇𝑇 𝜋𝜋�. At the same time, 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 � ≥ 𝑝𝑝𝑇𝑇 ∑𝑗𝑗≠𝑖𝑖 𝜋𝜋𝑗𝑗 . Hence,
𝑓𝑓
𝑓𝑓
𝑓𝑓
𝑓𝑓
𝑝𝑝𝑖𝑖 = 𝑉𝑉−𝑖𝑖 (0) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋𝑖𝑖 � ≤ 𝑝𝑝𝑇𝑇 𝜋𝜋� − 𝑝𝑝𝑇𝑇 ∑𝑗𝑗≠𝑖𝑖 𝜋𝜋𝑗𝑗 = 𝑝𝑝𝑇𝑇 �𝜋𝜋� − 𝜋𝜋−𝑖𝑖 �,
which ends the proof of the proposition.
■
Proof of Proposition 2.
Let us take an 𝜀𝜀 > 0 and consider a set of bids Φ′ = {(𝑏𝑏 𝛼𝛼 + 𝜀𝜀, 𝜋𝜋 𝛼𝛼 ): 𝜋𝜋 𝛼𝛼 ∈ Ψ}. Under the
conditions of the proposition, Φ′ is feasible in the supplementary round as all relative cap rules
are respected. By taking 𝜀𝜀 sufficiently small, we can guarantee that 𝑏𝑏 𝛼𝛼 + 𝜀𝜀 < 𝑣𝑣 𝛼𝛼 for all
packages 𝜋𝜋 𝛼𝛼 ∈ Ψ.
Then, bid Φ′ weakly dominates the original bid Φ, as in the VCG
mechanism: either both bids are winning with equal auction price for bidder 𝑖𝑖, or they are both
not winning and yield zero surplus to bidder 𝑖𝑖, or bid Φ′ is winning whereas bid Φ is not. In
the latter case, whatever package 𝜋𝜋 𝛼𝛼 bidder 𝑖𝑖 wins, its price is 𝑝𝑝𝛼𝛼 < 𝑏𝑏 𝛼𝛼 + 𝜀𝜀 < 𝑣𝑣 𝛼𝛼 and,
therefore, the surplus is positive. Finally, by construction of 𝑏𝑏�, there exists sets of bids of other
bidders so that Φ is not winning whereas bid Φ′ is, which ends the proof of the proposition. ■
Proof of Proposition 3.
Let bidder 𝑖𝑖 submit a set of bids Φ = {(𝑏𝑏 𝛼𝛼 , 𝜋𝜋 𝛼𝛼 ): 𝜋𝜋 𝛼𝛼 ∈ Ψ}. First, consider Φ such that 𝑏𝑏 𝑓𝑓 <
min�𝑏𝑏�, 𝑣𝑣 𝑓𝑓 �. At the end of the proof we consider Φ to be such that 𝑏𝑏 𝛽𝛽 < 𝑏𝑏 𝑓𝑓 + �𝑣𝑣�𝛽𝛽 − 𝑣𝑣 𝑓𝑓 �. We
define a marginal bid Δ𝑏𝑏 𝛼𝛼 and a marginal value Δ𝑣𝑣 𝛼𝛼 for package 𝜋𝜋 𝛼𝛼 as Δ𝑏𝑏 𝛼𝛼 ≡ (𝑏𝑏 𝛼𝛼 − 𝑏𝑏 𝑓𝑓 ) and
Δ𝑣𝑣 𝛼𝛼 ≡ (𝑣𝑣 𝛼𝛼 − 𝑣𝑣 𝑓𝑓 ). The relative bid for package 𝜋𝜋 𝛼𝛼 is defined as 𝜃𝜃 𝛼𝛼 ≡ (Δ𝑏𝑏 𝛼𝛼 − Δ𝑣𝑣 𝛼𝛼 ). We say
that a bidder bids below his value for package 𝜋𝜋 𝛼𝛼 if 𝜃𝜃 𝛼𝛼 < 0.
The relative cap rule, written as 𝑏𝑏 𝛼𝛼 ≤ 𝑏𝑏 𝑓𝑓 + 𝑐𝑐̅ 𝛼𝛼 in the proof of Proposition 1, can be written
as 𝜃𝜃 𝛼𝛼 ≤ 𝑐𝑐̅ 𝛼𝛼 − Δ𝑣𝑣 𝛼𝛼 ≡ 𝛾𝛾 𝛼𝛼 , where the caps 𝛾𝛾 𝛼𝛼 are recursively determined by actual bids in Φ,
i.e., 𝛾𝛾 𝛼𝛼 = 𝛾𝛾 𝛼𝛼 (Φ) depends on Φ. By construction, 𝜃𝜃 𝑓𝑓 ≡ 0 and 𝛾𝛾 𝑓𝑓 ≡ 0.
Case 1. 𝜃𝜃 𝛼𝛼 ≤ 0 for all 𝜋𝜋 𝛼𝛼 ∈ Ψ .
� = ��𝑏𝑏�𝛼𝛼 , 𝜋𝜋 𝛼𝛼 ��, where 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 + �min�𝑏𝑏�, 𝑣𝑣 𝑓𝑓 � − 𝑏𝑏 𝑓𝑓 � >
We consider a different set of bids Φ
� is feasible as it has the same relative bids: 𝜃𝜃�𝛼𝛼 = 𝜃𝜃 𝛼𝛼 ≤ 0. We will show that Φ
� weakly
𝑏𝑏 𝛼𝛼 . Φ
dominates Φ.
40
� yield the same allocation, or
Similar to the analysis of VCG mechanisms, either Φ and Φ
� is winning whereas set Φ is not. When Φ
� and Φ yield the same allocation, bidder 𝑖𝑖 pays the
Φ
�.
same price and gets the same surplus, whereas other bidders pay weakly higher prices under Φ
� is winning and Φ is not, it suffices to show that the surplus is nonnegative. Let Φ
� wins
When Φ
package 𝜋𝜋 𝛼𝛼 at price 𝑝𝑝�𝛼𝛼 . Since 𝑝𝑝�𝛼𝛼 ≤ 𝑏𝑏�𝛼𝛼 , the surplus 𝑠𝑠�𝛼𝛼 of the bidder is:
𝑠𝑠�𝛼𝛼 = 𝑣𝑣 𝛼𝛼 − 𝑝𝑝�𝛼𝛼 ≥ 𝑣𝑣 𝛼𝛼 − 𝑏𝑏 𝛼𝛼 − �min�𝑏𝑏�, 𝑣𝑣 𝑓𝑓 � − 𝑏𝑏 𝑓𝑓 � ≥ 𝑣𝑣 𝛼𝛼 − 𝑏𝑏 𝛼𝛼 − (𝑣𝑣 𝑓𝑓 − 𝑏𝑏 𝑓𝑓 ) = −𝜃𝜃 𝛼𝛼 ≥ 0.
� weakly dominates Φ.
Thus, Φ
Case 2. 𝜃𝜃 𝛼𝛼 > 0 for all 𝜋𝜋 𝛼𝛼 ∈ A and 𝜃𝜃𝛽𝛽 ≤ 0 for 𝜋𝜋 𝛽𝛽 ∈ Ψ \Α. Since Φ is feasible, it must be that
𝜃𝜃 𝛼𝛼 ≤ 𝛾𝛾 𝛼𝛼 (Φ). We denote 𝜃𝜃 𝐴𝐴 = min 𝜃𝜃 𝛼𝛼 > 0, and let
𝛼𝛼∈Α
𝑧𝑧 = min��min�𝑏𝑏�, 𝑣𝑣 𝑓𝑓 � − 𝑏𝑏 𝑓𝑓 �, 𝜃𝜃 𝐴𝐴 � > 0.
� = ��𝑏𝑏�𝛼𝛼 , 𝜋𝜋 𝛼𝛼 �� where
We consider a different a set of bids Φ
𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 for 𝜋𝜋 𝛼𝛼 ∈ Α, and 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 + 𝑧𝑧 for 𝜋𝜋 𝛽𝛽 ∈ Ψ𝑖𝑖 \Α.
� is feasible, i.e., 𝜃𝜃�𝛼𝛼 ≤ 𝛾𝛾 𝛼𝛼 �Φ
� �, and that Φ
� dominates Φ.
We show that Φ
First note that 𝑏𝑏�𝑓𝑓 = 𝑏𝑏 𝑓𝑓 + 𝑧𝑧. Let 𝑟𝑟(𝛼𝛼) be the anchor round for 𝜋𝜋 𝛼𝛼 ≠ 𝜋𝜋 𝑓𝑓 and let 𝜋𝜋 𝛽𝛽 ≡
𝑟𝑟(𝛼𝛼)
𝑑𝑑𝑖𝑖
be the corresponding anchor package. Let 𝑏𝑏 𝛽𝛽 be the actual bid in the supplementary round
expressed for package 𝜋𝜋 𝛽𝛽 , i.e., �𝑏𝑏 𝛽𝛽 , 𝜋𝜋 𝛽𝛽 � ∈ Φ. According to the relative cap rule, 𝑏𝑏 𝛼𝛼 ≤ 𝐵𝐵 𝛼𝛼 =
𝑏𝑏 𝛽𝛽 + 𝑝𝑝𝑟𝑟(𝛼𝛼) ∙ �𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝛽𝛽 �. Depending on the signs of 𝜃𝜃 𝛼𝛼 and 𝜃𝜃𝛽𝛽 , four cases are possible.
1. 𝜃𝜃 𝛼𝛼 ≤ 0 and 𝜃𝜃𝛽𝛽 ≤ 0. In this case, 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 + 𝑧𝑧 and 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 + 𝑧𝑧, so that
�𝛼𝛼 .
𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 + 𝑧𝑧 ≤ 𝐵𝐵 𝛼𝛼 + 𝑧𝑧 = 𝑏𝑏 𝛽𝛽 + z + 𝑝𝑝𝑟𝑟 ∙ �𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝛽𝛽 � = 𝑏𝑏�𝛽𝛽 + 𝑝𝑝𝑟𝑟 ∙ �𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝛽𝛽 � = 𝐵𝐵
2.
3.
4.
�𝛼𝛼 and is feasible.
Hence, bid 𝑏𝑏�𝛼𝛼 satisfies the relative cap rule, 𝑏𝑏�𝛼𝛼 < 𝐵𝐵
�𝛼𝛼 .
𝜃𝜃 𝛼𝛼 > 0 and 𝜃𝜃𝛽𝛽 ≤ 0. In this case, 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 and 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 + 𝑧𝑧, so that 𝑏𝑏�𝛼𝛼 < 𝐵𝐵
�𝛼𝛼 .
𝜃𝜃 𝛼𝛼 > 0 and 𝜃𝜃𝛽𝛽 > 0. In this case, 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 and 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 , so that again 𝑏𝑏�𝛼𝛼 ≤ 𝐵𝐵
𝜃𝜃 𝛼𝛼 ≤ 0 and 𝜃𝜃𝛽𝛽 > 0. In this case, 𝜃𝜃𝛽𝛽 ≥ 𝜃𝜃 𝐴𝐴 , 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 + 𝑧𝑧 and 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 .
Since 𝜃𝜃 𝑓𝑓 = 0, it follows that 𝜋𝜋 𝛽𝛽 ≠ 𝜋𝜋 𝑓𝑓 , and switching anchor-sincerely from 𝜋𝜋 𝛾𝛾 to 𝜋𝜋 𝛽𝛽 in
round 𝑟𝑟(𝛼𝛼) < 𝑇𝑇 at prices 𝑝𝑝𝑟𝑟(𝛼𝛼) implies that 𝑣𝑣 𝛼𝛼 − 𝑝𝑝𝑟𝑟(𝛼𝛼) ∙ 𝜋𝜋 𝛼𝛼 ≤ 𝑣𝑣 𝛽𝛽 − 𝑝𝑝𝑟𝑟(𝛼𝛼) ∙ 𝜋𝜋 𝛽𝛽 , and,
therefore, Δ𝑣𝑣 𝛼𝛼 − Δ𝑣𝑣 𝛽𝛽 = 𝑣𝑣 𝛼𝛼 − 𝑣𝑣 𝛽𝛽 ≤ 𝑝𝑝𝑟𝑟(𝛼𝛼) ∙ �𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝛽𝛽 �. Hence, the relative price cap
41
�𝛼𝛼 − 𝑏𝑏�𝛼𝛼 = 𝐵𝐵 𝛼𝛼 − 𝑏𝑏 𝛼𝛼 −
implies 𝐵𝐵 𝛼𝛼 = 𝑏𝑏 𝛽𝛽 + 𝑝𝑝𝑟𝑟(𝛼𝛼) ∙ �𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝛽𝛽 � ≥ 𝑏𝑏 𝛽𝛽 + Δ𝑣𝑣 𝛼𝛼 − Δ𝑣𝑣 𝛽𝛽 . Thus, 𝐵𝐵
𝑧𝑧 satisfies the following inequality:
�𝛼𝛼 − 𝑏𝑏�𝛼𝛼 ≥ 𝑏𝑏 𝛽𝛽 + Δ𝑣𝑣 𝛼𝛼 − Δ𝑣𝑣 𝛽𝛽 − 𝑏𝑏 𝛼𝛼 − 𝑧𝑧 = 𝜃𝜃𝛽𝛽 − 𝑧𝑧 − 𝜃𝜃 𝛼𝛼 ≥ 𝜃𝜃𝛽𝛽 − 𝜃𝜃 𝐴𝐴 ≥ 0,
𝐵𝐵
which implies that 𝑏𝑏�𝛼𝛼 is feasible.
� dominates Φ. Similar to the case where Α = ∅, if Φ
� and Φ yield the
It remains to show that Φ
� is winning whereas Φ is not, the bidder prefers Φ
� . The only difference
same allocation, or if Φ
� the bidder wins a package 𝜋𝜋 𝛽𝛽 ∈ Ψ\Α, whereas by bidding Φ he wins
here is that by bidding Φ
package 𝜋𝜋 𝛼𝛼 ∈ Α .
Using (1) we write the price bidder 𝑖𝑖 pays if he wins 𝜋𝜋 𝛼𝛼 as 𝑝𝑝𝛼𝛼 = 𝑉𝑉−𝑖𝑖 (∅) − 𝑉𝑉−𝑖𝑖 (𝜋𝜋 𝛼𝛼 ). His
surplus then is 𝑠𝑠 𝛼𝛼 = 𝑣𝑣 𝛼𝛼 − 𝑝𝑝𝛼𝛼 = 𝑣𝑣 𝛼𝛼 + 𝑉𝑉−𝑖𝑖 (𝜋𝜋 𝛼𝛼 ) − 𝑉𝑉−𝑖𝑖 (∅). Similarly, when he wins 𝜋𝜋 𝛽𝛽 , he pays
𝛽𝛽 = 𝑣𝑣 𝛽𝛽 + 𝑉𝑉 �𝜋𝜋 𝛽𝛽 � − 𝑉𝑉 (∅).
price 𝑝𝑝�𝛽𝛽 = 𝑉𝑉−𝑖𝑖 (∅) − 𝑉𝑉−𝑖𝑖 �𝜋𝜋 𝛽𝛽 �, and gets the surplus of 𝑠𝑠�
−𝑖𝑖
−𝑖𝑖
Moreover, it follows that 𝑏𝑏�𝛽𝛽 + 𝑉𝑉−𝑖𝑖 �𝜋𝜋 𝛽𝛽 � ≥ 𝑏𝑏�𝛼𝛼 + 𝑉𝑉−𝑖𝑖 (𝜋𝜋 𝛼𝛼 ), since 𝑏𝑏�𝛼𝛼 is not anymore winning.
𝛽𝛽 − 𝑠𝑠 𝛼𝛼 can be written as follows:
Using this inequality, the difference 𝐷𝐷 = 𝑠𝑠�
𝐷𝐷 = 𝑣𝑣 𝛽𝛽 + 𝑉𝑉−𝑖𝑖 �𝜋𝜋 𝛽𝛽 � − 𝑣𝑣 𝛼𝛼 − 𝑉𝑉−𝑖𝑖 (𝜋𝜋 𝛼𝛼 ) ≥ 𝑣𝑣 𝛽𝛽 − 𝑏𝑏�𝛽𝛽 + 𝑏𝑏�𝛼𝛼 − 𝑣𝑣 𝛼𝛼 .
� dominates Φ.
Finally, 𝑏𝑏�𝛽𝛽 = 𝑏𝑏 𝛽𝛽 + 𝑧𝑧 and 𝑏𝑏�𝛼𝛼 = 𝑏𝑏 𝛼𝛼 , imply 𝐷𝐷 ≥ −𝜃𝜃𝛽𝛽 ≥ 0. Thus, Φ
The argument for why Φ is dominated if it contains bids 𝑏𝑏 𝛽𝛽 < 𝑏𝑏 𝑓𝑓 + �𝑣𝑣 𝛽𝛽 − 𝑣𝑣 𝑓𝑓 � for some
other package 𝜋𝜋 𝛽𝛽 ∈ Ψ is similar to that from VCG. Therefore, we only provide a sketch of the
proof. In such a case, the relative bid 𝜃𝜃𝛽𝛽 < 0 and, therefore, the lowest relative bid is 𝜃𝜃 ≡
min 𝜃𝜃 𝛼𝛼 < 0. We define Α ≡ �𝜋𝜋 𝛼𝛼 ∈ Ψ: 𝜃𝜃 𝛼𝛼 = 𝜃𝜃� to be the set of packages with the lowest
𝜋𝜋 𝛼𝛼 ∈Ψ
relative bid 𝜃𝜃, and 𝑧𝑧 ≡ 𝛼𝛼min 𝜃𝜃 𝛼𝛼 − 𝜃𝜃 > 0 to be the amount by which we can raise all bids in
𝜋𝜋 ∈Ψ\Α
Φ for packages 𝜋𝜋 𝛼𝛼 ∈ Α such that the relative bid for these packages 𝜋𝜋 𝛼𝛼 ∈ Α are still the lowest
�.
(and are equal 𝜃𝜃 + 𝑧𝑧). We call the resulting set of bids Φ
� dominates Φ. This is so because if Φ
� and Φ win different
It can be easily shown that Φ
� wins a package 𝜋𝜋 𝛼𝛼 ∈ Α whereas Φ
packages (and this is the only relevant case to consider), Φ
wins package 𝜋𝜋 𝛾𝛾 ∈ Ψ\Α, and the relative bid for 𝜋𝜋 𝛼𝛼 is lower than for 𝜋𝜋 𝛾𝛾 , so that the surplus
from winning 𝜋𝜋 𝛼𝛼 is higher than from winning 𝜋𝜋 𝛾𝛾 .
42
The above argument is directly applicable to the packages 𝜋𝜋 𝛼𝛼 for which bidding relative
value, i.e., 𝜃𝜃 𝛼𝛼 = 0, is feasible. Anchor-sincere bidding may result in 𝑣𝑣�𝛽𝛽 < 𝑣𝑣 𝛽𝛽 for packages 𝜋𝜋 𝛽𝛽
for which the last round 𝑇𝑇 is the anchor round. Nevertheless, since 𝑏𝑏 𝛽𝛽 < 𝑏𝑏 𝑓𝑓 + �𝑣𝑣�𝛽𝛽 − 𝑣𝑣 𝑓𝑓 � and
𝐵𝐵 𝛽𝛽 = 𝑏𝑏 𝑓𝑓 + �𝑣𝑣�𝛽𝛽 − 𝑣𝑣 𝑓𝑓 � in this case, raising 𝑏𝑏 𝛽𝛽 all the way up to 𝐵𝐵 𝛽𝛽 is feasible. Thus, the above
argument is applicable to all packages, including the packages 𝜋𝜋 𝛽𝛽 for which the bidder is
■
capped below his relative value.
Proof of Proposition 4.
According to Proposition 3, all bidders bid KO-bids on 𝜋𝜋 𝑓𝑓 , and clock-phase constrained values
on all other packages 𝜋𝜋 𝛼𝛼 . The CCA allocation, as it is the case for the VCG mechanism,
maximizes the sum of the winning bids. Thus, it maximizes the sum of clock-phase constrained
values.
Proof of Proposition 5.
𝑓𝑓
According to Proposition 1, bidders do not need to increase their bids on 𝜋𝜋𝑖𝑖 as 𝑏𝑏�𝑖𝑖 ≤
𝑓𝑓
𝑓𝑓
𝑝𝑝𝑇𝑇 �𝜋𝜋� − ∑𝑗𝑗≠𝑖𝑖 𝜋𝜋𝑗𝑗 � = 𝑝𝑝𝑇𝑇 𝜋𝜋𝑖𝑖 . Then, the value of the last clock round allocation is 𝑝𝑝𝑇𝑇 𝜋𝜋�. Since
𝐵𝐵 𝛼𝛼 ≤ 𝑏𝑏 𝑓𝑓 + 𝑝𝑝𝑇𝑇 (𝜋𝜋 𝛼𝛼 − 𝜋𝜋 𝑓𝑓 ) = 𝑝𝑝𝑇𝑇 𝜋𝜋 𝛼𝛼 for all bidders, the value ∑𝑗𝑗 𝑏𝑏𝑗𝑗𝛼𝛼 of any other allocation {𝜋𝜋𝑖𝑖𝛼𝛼 },
∑𝑗𝑗 𝜋𝜋𝑖𝑖𝛼𝛼 ≤ 𝜋𝜋�, does not exceed
∑𝑗𝑗 𝑏𝑏𝑗𝑗𝛼𝛼 ≤ ∑𝑗𝑗 𝐵𝐵𝑗𝑗𝛼𝛼 ≤ ∑𝑗𝑗 𝑝𝑝𝑇𝑇 𝜋𝜋 𝛼𝛼 = 𝑝𝑝𝑇𝑇 ∑𝑗𝑗 𝜋𝜋 𝛼𝛼 ≤ 𝑝𝑝𝑇𝑇 𝜋𝜋�.
𝑓𝑓
Therefore, for the chosen tie-breaking rule, the allocation �𝜋𝜋𝑖𝑖 � is the final auction allocation.
𝑓𝑓
𝑓𝑓
Bids 𝑏𝑏𝑖𝑖𝛼𝛼 do not affect price 𝑝𝑝𝑖𝑖 bidder 𝑖𝑖 pays, but do affect prices 𝑝𝑝𝑗𝑗 other bidders pay.
Thus, bidding maximal caps weakly dominate all other strategies. As a result, each bidder wins
𝑓𝑓
■
𝜋𝜋𝑖𝑖 and pays the maximal price given the clock phase development.
Proof of Proposition 6.
𝑓𝑓
After all bids with 𝑏𝑏 𝑓𝑓 < 𝑏𝑏� are eliminated for all bidders, each bidder 𝑖𝑖 bids 𝑏𝑏𝑖𝑖 ≥ 𝑏𝑏�𝑖𝑖 and wins a
𝑓𝑓
package, let it be package 𝜋𝜋𝑖𝑖𝑊𝑊 . Under conditions (b) and (d), 𝜋𝜋𝑖𝑖𝑊𝑊 = 𝜋𝜋𝑖𝑖 . The prices that
competitors pay will necessarily depend on price-relevant bids that a bidder puts on other
packages. Therefore, all bids 𝑏𝑏𝑖𝑖𝛼𝛼 which are lower than the maximal caps 𝐵𝐵𝑖𝑖𝛼𝛼 and which are used
43
in computing prices for at least one other bidder are dominated by 𝑏𝑏𝑖𝑖𝛼𝛼 = 𝐵𝐵𝑖𝑖𝛼𝛼 . In the second
round of elimination, we eliminate low bids 𝑏𝑏𝑖𝑖𝛼𝛼 for all such packages.
■
Proof of Proposition 7.
Under the conditions of the proposition, packages 𝜋𝜋 𝑓𝑓 are the same independently of whether
bidder 𝑖𝑖 ends the clock phase in round 𝑇𝑇 or (𝑇𝑇 + 1). Because the bids of all other bidders 𝑗𝑗 ≠ 𝑖𝑖
are also assumed independent of that, the KO-bid 𝑏𝑏 𝑓𝑓 of bidder 𝑖𝑖 is also the same. Since
continuing bidding for package 𝜋𝜋𝑖𝑖𝛼𝛼 in round 𝑇𝑇 relaxes the cap restrictions for bidder 𝑖𝑖,
Proposition 6, which uses previous AS-bidding, still holds. According to Proposition 6, bidder
𝑖𝑖 wins 𝜋𝜋 𝑓𝑓 and pays 𝑝𝑝 𝑓𝑓 < 𝑏𝑏� in any case. What is different is the price the others pay.
If bidder 𝑖𝑖 ends the clock phase in (𝑇𝑇 + 1), his relative cap will allow him to submit a
higher bid on 𝜋𝜋 𝛼𝛼 and, consequently, on all other packages that require more eligibility points
than this package does. According to Proposition 6, this leads to higher prices other bidders
pay. Therefore, bidder 𝑖𝑖 strictly prefers not switching from 𝜋𝜋 𝛼𝛼 to 𝜋𝜋 𝑓𝑓 in round 𝑇𝑇.
■
Appendix II: Computing KO-bid
When only total demand is revealed to bidders, there are 14 different feasible scenarios (up to
the symmetry of bidders 2 and 3) of the clock phase development that generate the same
aggregate demand as Table 2 presents and that are consistent with the bidding behavior of
bidder 1. Table 3 presents all 14 different ways of individual bidding of bidders 2 and 3.
Scenario 14 is the actual clock phase development.
44
Scenario
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Round 1
Rounds 2-7
Round 8
Bidder 2
Bidder 3
Bidder 2
Bidder 3
Bidder 2
Bidder 3
(2,1)
(0,3)
(3,0)
(1,2)
(1,0)
(1,2)
(1,2)
(2,1)
(1,2)
(2,1)
(1,2)
(2,1)
(1,2)
(2,1)
(1,2)
(2,1)
(1,2)
(2,1)
(1,2)
(1,2)
(0,3)
(1,2)
(0,3)
(1,2)
(0,3)
(1,2)
(0,3)
(1,2)
(0,3)
(1,2)
(0,3)
(1,2)
Table 3. Clock phase scenarios.
(3,0)
(1,2)
(3,0)
(1,2)
(3,0)
(1,2)
(2,1)
(2,1)
(2,1)
(2,1)
(3,0)
(1,2)
(3,0)
(1,2)
(2,1)
(2,1)
(2,1)
(2,1)
(3,0)
(1,2)
(3,0)
(1,2)
(2,1)
(2,1)
(2,1)
(2,1)
(1,0)
(0,1)
(0,1)
(0,1)
(0,1)
(2,0)
(2,0)
(2,0)
(2,0)
(1,1)
(1,1)
(1,1)
(1,1)
(1,2)
(2,1)
(2,1)
(2,1)
(2,1)
(0,2)
(0,2)
(0,2)
(0,2)
(1,1)
(1,1)
(1,1)
(1,1)
KObid
28
29
28
28
24
23
28
29
24
23
28
29
24
23
For each scenario, we can compute the KO-bid according to Proposition 1. These bids are
shown in the last column. As one can see, the maximal KO-bid is 29, and it occurs in scenarios
2, 8, and 12. Thus, when the clock phase development is not public information, Proposition 1
remains valid, and the KO-bid must be calculated as a maximum KO-bid across all possible
scenarios.
45