E
XTRACT FROM
:
E
XPERT
A
DVICE
ON THE
A
UCTION
R
ULES
(P
URSUANT TO
SOW
C
LAUSE
2.3.1)
P
REPARED FOR
THE
A
USTRALIAN
C
OMMUNICATIONS
AND
M
EDIA
A
UTHORITY
(ACMA)
B
Y
P
OWER
A
UCTIONS
LLC
Revised
7 December
2011
CONFIDENTIAL
Expert Advice on the Auction Rules
Power Auctions LLC
Table of Contents
1.
Appropriateness of the Original Draft Rules
................................
................................
......
1
1.1
Improvements to the activity rules
................................
................................
.......................
1
1.1.1
Proposed activity rule for the clock rounds
................................
................................
..
1
1.1.2
Proposed activity rule for the supplementary round
................................
...................
2
1.1.3
Advantages of the proposed activity rule
................................
................................
.....
2
1.1.4
Directions of other spectrum regulators
................................
................................
......
3
1.2
Further explanatory text on the activity rules
................................
................................
.......
4
1.2.1
Activity rule for the clock rounds
................................
................................
..................
6
1.2.2
Activity rule for the supplementary round
................................
................................
...
7
1.2.3
Advantages and disadvantages of the propos
ed activity rule
................................
......
8
1.2.4
Experience to date
................................
................................
................................
........
9
1.3
Algorithms for solving allocation, pricing and assignment
................................
....................
9
1.3.1
Handling reserve prices
................................
................................
................................
9
1.3.2
Tie

breaking in winner determination
................................
................................
........
10
1.3.3
Weighting bidders of different sizes
................................
................................
...........
10
1.3.4
Explanation of pricing
................................
................................
................................
.
11
1.3.5
Tie

breaking in pricing
................................
................................
................................
17
1.3.6
Assignment stage bidding and optimization
................................
...............................
18
1.3.7
Verifying that the algorithms produce outcomes that are correct
............................
18
Appendix A: Example of bidding under the revised rules
................................
.......................
20
Appendix B: Example of how bidding in the supplementary round can change the allocation
................................
................................
................................
................................
..................
26
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1.
Appropriateness of the
Original Draft Rules
1.1
Improvements to the activity rules
Power Auctions has long believed that the greatest room for potential improvement in the
Combinatorial Clock Auction (CCA) lies in the activity rules. For example, the draft UK rules
proposed by Ofcom
in March 2011 in many ways reflect the state of the art in thinking
about the CCA, but the activity rules in the March 2011 rules have at least two significant
deficiencies:
1.
The activity rule for the clock stage prevents a bidder from placing bids on her
most
preferred package whenever the most preferred package exceeds her eligibility. For
example, the bidder may reduce her eligibility early in the auction but then need to
expand her eligibility when the price in a category she is bidding on increases muc
h
more than the price of a substitute category requiring more eligibility points. This
prevents the bidder from expressing her true preferences until the supplementary
round.
2.
The activity rule for the supplementary round (“Relative Cap”) fails to satisfy
a
desirable property that guarantees that the final clock package is unchanged as a
result of the supplementary round when there are no unallocated items in the final
clock round. Also, it is difficult for bidders to determine how to bid to guarantee
winni
ng the final clock package when items are unallocated in the final clock round.
Given these two significant deficiencies, Professors Larry Ausubel and Peter Cramton have
recently undertaken a substantive review of CCA activity rules, with the intention of
proposing improvements. Their resulting academic paper, “Activity Rules for the
Combinatorial Clock Auction,” can be found on the authors’ academic website; see:
www.ausubel.com/auction

papers/ausubel

cramton

activity

rules

for

cca.pdf
Power Auctions recommends that each of their two recommendations be adopted for
Australia.
In the next two sections, we provide informal descriptions of the recommended changes
,
first for the clock rounds and then for the supplementary round. Technically precise and
complete descriptions are provided in the academic paper.
1.1.1
Proposed activity rule for the clock rounds
Short name: RP/Eligibility

Point Hybrid
Long name: Revealed
Preference with an Eligibility

Point Safe Harbor
Informal description: In any round, the bidder can bid on a larger package than would be
permitted by the bidder’s current eligibility, provided that the package satisfies “revealed
preference” with respect
to each prior round’s bid in which eligibility was reduced.
(However, bidding on a larger package does not increase the bidder’s eligibility in
CONFIDENTIAL
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subsequent rounds.) At the same time, the bidder can always place a bid for any package
that is within the bidde
r’s current eligibility.
There are two ways for a bidder to think about this rule.
Consider a bidder who likes the simplicity of the
eligibility

point
monotonicity
. Under our
recommendation, such a bidder can continue to bid just as she would bid under the current
eligibility

point
rule. The only difference is that the bidder is given some extra flexibility to
bid on a larger package, provided that the larger packag
e satisfies revealed preference;
i.e.
the bid involves a switch to a package that has become relatively less
expensive. Note
that, in a pure
revealed

preference rule, a bidder may find it difficult to figure out how to
correct a bid submission that violate
s a
revealed

preference
constraint. However, under the
RP/
Eligibility

Point Hybrid rule, i
f the bidder is prevented from placing a larger package due
to a violation of revealed preference, the bidder knows at least one straightforward way to
correct the vi
olation: she can reduce the size of the package until it satisfies
eligibility

point
monotonicity.
Consider a bidder who has a consistent model of her values for all packages and adopts the
strategy of always bidding on her most preferred (i.e. most profit
able) package in every
round of the clock stage. Such a bidder will never be constrained by the activity rule.
Moreover, if for some reason the bidder’s values change during the auction, then it is
possible that a
revealed

preference
constraint will now bi
nd, but the bidder knows that she
can always fall back to a package whose eligibility points are consistent with her eligibility.
1.1.2
Proposed activity rule for the supplementary round
Name: Revealed

Preference Cap
Informal description:
All supplementary bids
must satisfy revealed preference with respect
to the bidder’s final clock package. In addition, supplementary bids for packages that are
larger than the final clock package must satisfy revealed preference with respect to each
clock round that resulted in
a reduction of eligibility, beginning with the last round in which
the bidder had sufficient eligibility to bid on the package.
Effectively, our proposed activity rule for the supplementary round strengthens the so

called Relative Cap by applying a reveal
ed

preference constraint relative to more rounds.
Under the Relative Cap,
supplementary bids for packages that are larger than the final clock
package must satisfy revealed preference with respect to the last round in which the bidder
had sufficient eligib
ility to bid on the package. Under the Revealed

Preference Cap,
supplementary bids for packages that are larger than the final clock package must also
satisfy revealed preference with respect to each eligibility

reducing clock round, beginning
with the las
t round in which the bidder had sufficient eligibility to bid on the package, as
well as with respect to the final clock round.
1.1.3
Advantages of the proposed activity rule
The proposed rule has many advantages, as shown in the academic paper.
The rule enable
s the bidder to bid on her most preferred package throughout the clock
stage, thereby improving price discovery (more revelation of relevant marginal value
CONFIDENTIAL
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information) and making the final clock allocation a better predictor of the auction
outcome.
The r
ule guarantees that the final clock allocation will not change if there are no unallocated
items
—
each winner is guaranteed to win her final clock package without making any
supplementary bids. And if there are unallocated items, then each winner can guaran
tee
winning at least her final clock package with a supplementary bid that increases the dollar
amount of her final clock package by the final clock price of the unallocated items.
The rule prevents a competitor from placing supplementary bids that have n
o chance of
winning that would increase the payments of rivals.
The emphasis on revealed preference with respect to the final clock package motivates the
bidder to bid on her most preferred package in the final clock round to improve her chances
of winning
her most preferred package. Since the bidder does not know which round will be
the final clock round, there is a persistent motivation to bid on the most preferred package
throughout the clock stage. This behavior is exactly what reveals the bidders’ trad
eoffs
among relevant packages and promotes efficient outcomes.
Revealed

preference
constraints that are not needed to prevent undesirable behavior are
not included. This simplifies the activity rule and gives the bidders greater flexibility
throughout the
auction. Supplementary bids are only constrained by revealed preference
with respect to the final round and relevant rounds in which the bidder reduced eligibility.
In the clock stage, the bidder is always able to place bids that are consistent with
eligib
ility

point
monotonicity. This provides one easy way for the bidder to see how a package can be
modified to satisfy the activity rule. The eligibility

point safe harbor also provides additional
flexibility in the event that a bidder’s values change during
the clock stage as a result of
price discovery.
We are confident that the new rule can be implemented in a way that is easy for bidders.
We discuss this in the last section of this document. Indeed, one motivation for the new rule
is to further reduce comp
lex strategic behavior that can stand in the way of efficient
outcomes.
1.1.4
Directions of other spectrum regulators
As we understand it, the original draft auction rules were prepared by DotEcon, based on
the approach used for the CCA in some European countrie
s. For this reason, we think it is
important to call to the ACMA’s attention that European regulators appear to be taking very
seriously the same concerns about the activity rules as we have expressed above. The
clearest sign of this is that, on 24 October
2011, ComReg (the Ireland regulator) released
“Multi

band Spectrum Release: Draft Information Memorandum” (ComReg 11/75), available
at the following URL:
http://www.comreg.ie/_f
ileupload/publications/ComReg1175.pdf
We believe that t
h
is Draft Information Memorandum is
indicative of the direction that
ComReg
is heading.
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The Draft Information Memorandum for Ireland proposes substantial changes both to the
activity rule in the clock
rounds (see Clauses 4.69
–
4.94) and to the activity rule in the
supplementary round (see Clauses 4.144
–
4.152). Also see Annex 7.
Our preliminary assessment of these changes is as follows:
(1)
The approach being proposed with respect to the activity rule in
the supplementary
round in Europe is conceptually similar to what we are proposing, although we
believe that our proposal is more internally consistent. In any event, it appears that
the “Relative Cap” is being abandoned.
(2)
The approach being proposed with r
espect to the activity rule in clock rounds in
Europe appears to be much more complex than what we propose. In particular, it
involves defining “Relaxed Primary Bids” and “Binding Supplementary Bids”
—
in
effect, the European approach seems to allow bidders
to place clock bids that would
otherwise not be permitted by eligibility points, but at the cost of retroactively
increasing their earlier clock bids. The approach seems overly complex and possibly
unworkable; and it is unlikely to meet with acceptance by
bidders.
1.2
Further
explanatory text on
the activity rules
Activity rules are intended to promote truthful bidding throughout the auction. With more
truthful bidding, price discovery in the clock stage is improved. Truthful bidding provides
meaningful price and aggregate demand information to bidders, enabling the
m to focus
their valuation efforts on the set of packages most relevant to them. Activity rules
discourage a bidder from withholding true demands, since doing so will limit the bidder’s
ability to bid on what she really wants later in the auction.
In SMR
auctions, activity rules typically are based on eligibility points. A bidder’s eligibility in
the current round is based on her activity in the prior round. The activity rule requires that
eligibility can never increase as prices rise.
In a
CCA
, it is desi
rable to weaken this
eligibility

point
rule to assure that a bidder can bid on
her most preferred package throughout the clock stage. It is also necessary to introduce an
activity rule in the supplementary round that appropriately limits the supplementary
bids to
be consistent with the preferences expressed in the clock stage.
The activity rules in the
CCA
are based on both eligibility points and the more fundamental
notion of revealed preference.
The eligibility

points component considers the “size” of th
e package the bidder is bidding
on, where “size” is the sum of the eligibility points for each lot in the package.
Eligibility

point
monotonicity
requires bidders to bid on packages the same size or smaller as prices
rise. When a bidder switches to a packa
ge that is smaller than any package she has bid on
before, her eligibility is reduced to the size of this smaller package. A round in which a
bidder’s eligibility is reduced is called an eligibility

reducing round. These rounds will play a
special role in
the activity rules.
As an example, a bidder may start with eligibility of 100, but in round 3 the bidder switches
to a package of size 80. The bidder’s eligibility then is reduced to 80 and round 3 is an
eligibility

reducing round. The bidder’s eligibili
ty remains at 80 until she bids on a package
smaller than 80 in a future round. The beauty of the
eligibility

point
rule is its simplicity. The
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bidder just needs to keep track of a single number and this number cannot increase in
subsequent rounds. It is t
rivial to understand why a bid violates the
eligibility

point
rule (the
package is too big) and trivial to remedy the violation (make the package smaller). The
problem with this rule is that it may prevent desirable substitution between packages of
differe
nt sizes.
Revealed preference
in
contrast does not consider package size, but rather considers how
prices have changed and allows the bidder to shift toward packages that have become
relatively less expensive. The motivation for revealed preference is that
a bidder seeking to
maximize profits will shift to packages that result in a higher profit as prices change; that is
shift to packages that have gone up less in price if the revenue associated with the packages
is similar.
For example, suppose a bidder d
esires either a smaller package, X, or a larger package, Y,
but not both. At the current prices, X is preferred, but in subsequent rounds, the prices for
the lots in X go up much faster than the prices for the lots in Y. As a result, at the new prices
the
bidder prefers Y to X. Revealed preference allows the switch from X to Y, because Y is
now the better value.
By way of
contrast,
eligibility

point
monotonicity would not allow the
switch since Y is larger than X. This example illustrates the problem with t
he
eligibility

point
rule and the advantage of revealed preference.
More generally, in the clock stage, bidding on package Y in round t satisfies
revealed
preference with respect to round s
, if the price of package Y in round t has increased less
than the price of the package X that the bidder bid on in round s. That is, package Y has
become relativity less expensive than package X, so bidding on Y in round t is consistent
with the bidder’s
expressed preference for X in round s.
In the supplementary round, revealed preference puts a cap on the amount a bidder can bid
for a particular package. For example,
revealed preference with respect to the final clock
package
says that the bid for the p
ackage must be less than the bid for the final clock
package plus the price difference between the package and the final clock package
evaluated at the final clock prices. That is, the supplementary bid amount for the specified
package and the final clock
package needs to be consistent with the fact that the bidder
preferred the final clock package to the specified package at the final clock prices.
As an example, suppose that in the final round of the clock stage the bidder bid on the
package F at $200 at
the final clock prices. In the supplementary round, the bidder wants to
place a bid on the package S, which had a price of $250 at the final clock prices. Revealed
preference
(
with respect to the final clock package
)
puts a cap on the supplementary bid for
S based on the bid for F:
Supplementary bid for S < Supplementary bid for F + $50.
The $50 is just the difference in the price of S and F at the final clock prices ($250
–
200). If
the bidder thought that S was worth more than $50
greater
than F, then the
bidder should
have bid on S, not F, in the final clock round, since the
price
difference
between
S and F was
$50 in the final clock round. Thus, if the bidder wants to bid $349 for S, then a
supplementary bid of $300 or more is required on F.
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With this ba
ckground
,
we can now state the proposed activity rules for both the clock
rounds and the supplementary round.
A detailed example illustrating these rules is
provided in Appendix B.
1.2.1
Activity rule for the clock rounds
The following activity rule is required
in each clock round:
Revealed Preference with an Eligibility

Point Safe Harbor
: In any round, the bidder can bid
on a larger package than would be permitted by the bidder’s current eligibility, provided
that the package satisfies revealed preference with
respect to each prior eligibility

reducing
round. (However, bidding on a larger package does not increase the bidder’s eligibility in
subsequent rounds.) At the same time, the bidder can always place a bid for any package
that is within the bidder’s curren
t eligibility.
There are two ways for a bidder to think about this rule.
Consider a bidder who likes the simplicity of the
eligibility

point
monotonicity. Such a bidder
can continue to bid just as she would bid under the
eligibility

point
rule. The only
difference
is that the bidder is given some extra flexibility to bid on a larger package, provided that the
larger package satisfies revealed preference; i.e.
the bid involves a switch to a package that
has become relatively less expensive. Note that, in a
pure
revealed

preference
rule, a bidder
may find it difficult to figure out how to correct a bid submission that violates a
revealed

preference
constraint. However, under the Revealed Preference with an Eligibility

Point Safe
Harbor, if the bidder is prev
ented from bidding on a larger package due to a violation of
revealed preference, the bidder knows at least one straightforward way to correct the
violation: she can reduce the size of the package until it satisfies
eligibility

point
monotonicity.
Consider
a bidder who has a consistent model of her values for all packages and adopts the
strategy of always bidding on her most preferred (i.e. most profitable) package in every
round of the clock stage. Such a bidder will never be constrained by the activity ru
le.
Moreover, if for some reason the bidder’s values change during the auction, then it is
possible that a
revealed

preference
constraint will now bind, but the bidder knows that she
can always fall back to a package whose eligibility points are consistent
with her eligibility.
Below is an example of the activity rule in the clock stage of an auction with nine clock
rounds. The bidder reduced eligibility in two rounds, round 3 and round 7 (these “eligibility

reducing rounds” are highlighted in yellow). In t
hree rounds (rounds 5, 6, and 9), the bidder
bid on a package that was larger than the bidder’s eligibility.
Initial
Final
Round
R1
R2
R3
R4
R5
R6
R7
R8
R9
Package
Q1
Q2
Q3
Q4
Q5
Q6
Q7
Q8
Q9
Eligibility
100
100
100
80
80
80
80
50
50
Package
size
100
100
80
80
90
85
50
50
55
Package price
$140
$200
$180
$220
$250
$260
$190
$230
$280
RP constraints
RP3:5
RP3:6
RP3:9
RP7:9
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This
behavio
u
r results in four
revealed

preference
constraints as follows:
RP3:5
(Price of Q5 in
R5)
–
(Price of Q5 in R3) ≤ (Price of Q3 in R5)
–
(Price of Q3 in R3)
;
RP3:6
(Price of Q6 in R6)
–
(Price of Q6 in R3) ≤ (Price of Q3 in R6)
–
(Price of Q3 in R3)
;
RP3:9
(Price of Q9 in R9)
–
(Price of Q9 in R3) ≤ (Price of Q3 in R9)
–
(Price of Q3 in R3)
;
RP7:9
(Price of Q9 in R9)
–
(Price of Q9 in R7) ≤ (Price of Q7 in R9)
–
(Price of Q7 in R7)
.
1.2.2
Activity rule for the supplementary round
Each bid in the supplementary round must satisfy the following activity rule:
Revealed

Preference Cap
: For
packages tha
t consist of the final clock package plus one or
more lots that remain unallocated after the final clock round, the supplementary bid must
satisfy revealed preference with respect to the final clock round.
All other packages must
satisfy revealed preferenc
e with respect to the final clock round, as well as
with respect to
each eligibility

reducing round, beginning with the last round in which the bidder had
sufficient eligibility to bid on the package.
The activity rule for the supplementary round sometime
s strengthens the so

called Relative
Cap (as proposed by Ofcom) by applying a revealed

preference constraint with respect to
additional rounds. Under the Relative Cap,
supplementary bids for packages that are larger
than the final clock package must satisf
y revealed preference with respect to the last round
in which the bidder had sufficient eligibility to bid on the package. Under the Revealed

Preference Cap, supplementary bids for packages that are larger than the final clock
package must also satisfy rev
ealed preference with respect to each eligibility

reducing clock
round, beginning with the last round in which the bidder had sufficient eligibility to bid on
the package, as well as with respect to the final clock round. However, if the package
consists o
f the final clock package plus one or more lots that are unallocated at the end of
the final clock round, then the bid is only constrained by revealed preference with respect
to the final clock package. This potentially weakens the Relative Cap, so as to g
ive bidders
greater flexibility to expand their final clock packages to include unallocated lots.
Continuing the example above, suppose that the bidder submits three supplementary bids,
one of size 45 (S), one of size 60 (M) that includes the final clock package plus some
unallocated lots at the end of the clock stage, and one of size 90 (L) that does
not consist
entirely of the bidder’s final clock package plus lots that are unallocated in the final clock
round. The package S of size 45 has to satisfy a single
revealed

preference
constraint:
Sup. bid on S < Sup. bid on Q9 + Price of S in R9
–
Price of
Q9 in R9.
Likewise, the package M of size 60 has to satisfy a single
revealed

preference
constraint,
since it consists entirely of the bidder’s final clock package plus lots that are unallocated in
the final clock round:
Sup. bid on M < Sup. bid on Q9 + Pr
ice of M in R9
–
Price of Q9 in R9.
The package L of size 90 has to satisfy three
revealed

preference
constraints (since L does
not consist entirely of the bidder’s final clock package plus lots that are unallocated in the
final clock round, the bid must a
lso satisfy revealed preference with respect to the two
eligibility

reducing rounds beginning with R3, the last round that the bidder could bid on L):
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Sup. bid on L < Sup. bid on Q9 + Price of L in R9
–
Price of Q9 in R9
;
Sup. bid on L < Sup. bid on Q7 + P
rice of L in R7
–
Price of Q7 in R7
;
Sup. bid on L < Sup. bid on Q3 + Price of L in R3
–
Price of Q3 in R3.
1.2.3
Advantages
and disadvantages
of the proposed activity rule
The proposed rule has many advantages.
The rule enables the bidder to bid on her most
preferred (i.e. most profitable) package
throughout the clock stage, thereby improving price discovery (more revelation of relevant
marginal value information) and making the final clock allocation a better predictor of the
auction outcome. Otherwise, the
re is an incentive for a bidder to only choose large
packages when prices are low to avoid losing the opportunity to bid on these packages as
prices increase.
The rule guarantees that the final clock allocation will not change if there are no unallocated
i
tems
—
each winner is guaranteed to win her final clock package without making any
supplementary bids.
This is illustrated in Appendix C.
And if there are unallocated items,
then each winner can guarantee winning at least her final clock package with a
suppl
ementary bid that increases the dollar amount of her final clock package by the final
clock price of the unallocated items.
The emphasis on revealed preference with respect to the final clock package motivates the
bidder to bid on her most preferred packa
ge in the final clock round to improve her chances
of winning her most preferred package. Since the bidder does not know which round will be
the final clock round, there is a persistent motivation to bid on the most preferred package
throughout the clock s
tage. This behavior is exactly what reveals the bidders’ trade

offs
among relevant packages and promotes efficient outcomes.
Revealed

preference
constraints that are not needed to prevent undesirable behavior are
not included. This simplifies the activity
rule and gives the bidders greater flexibility
throughout the auction. Supplementary bids are only constrained by revealed preference
with respect to the final round and certain rounds in which the bidder reduced eligibility.
In the clock stage, the bidder
is always able to place bids that are consistent with
eligibility

point
monotonicity. This provides one easy way for the bidder to see how a package can be
modified to satisfy the activity rule. The eligibility

point safe harbor also provides additional
f
lexibility in the event that a bidder’s values change during the clock stage as a result of
price discovery.
The
main disadvantage that could be argued against the
proposed
activity
rule
is that
it
is
more
complicated
to state
than a
“
simple
”
eligibility

o
nly activity rule
. However,
there is an
important distinction between brevity of the auction rules and the level of strategic
complexity that they may require. T
he less
sophisticated
bidder can always just ignore the
additional complexity and follow an eli
gibility

only activity rule,
without
concern
ing itself
about the
seeming
additional complexity.
Meanwhile,
the
more
sophisticated
bidder
will
appreciate
(and utilise)
the additional flexibility that the
proposed
rule provides.
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In conclusion, a
lthough the activity rules may
initially
seem complex, the motivation for the
activity rules is to further reduce complex strategic behavio
u
r that can stand in the way of
efficient outcomes.
1.2.4
Experience to date
To date there is limited experience with CCA a
ctivity rules: two auctions in the UK, followed
by auctions in the Netherlands, Denmark, and Austria.
The UK auctions illustrate well the problems with an activity rule based only on eligibility
points
.
(
See P.
Cramton
,
2009, “Spectrum Auction Design
,
”
fo
r
details
.
)
A pure eligibility

point
rule sharply distorts bidding in the clock stage. Bidders are motivated to bid on the
largest package that remains profitable, rather than
to
bid on the most profitable package.
This drives clock prices well above compe
titive equilibrium levels, as bidders face strong
incentives not to reduce demands
consistent with marginal values as prices rise. The end
result is less effective price discovery and a large spread between the final clock prices and
the base prices paid b
y winners.
The UK and other countries (the Netherlands, Denmark, and Austria)
initially
switched to the
use of
the “Relative Cap” (a version of a revealed

preference constraint) in the
supplementary round, while maintaining a pure
eligibility

point
rule
in
the clock stage. This
approach
performs much better, although it is now well understood that there are
remaining conceptual problems with th
ose
rules. All countries using the C
C
A for future
auctions appear to be moving
in the direction of
a rule along the
lines we present here.
1.3
Algorithms for solving allocation, pricing and assignment
We now turn to the specific approaches that we would recommend for handling reserve
prices, tie

breaking in the winner determination, tie

breaking in the pricing algorithm,
assignment stage bidding and optimisation, and verifying that the algorithms produce
correct results.
1.3.1
Handling reserve prices
We recommend handling reserve prices with “reserve bidders.” With this approach
,
a
reserve
bid at the
opening
price is included for each generic lot on behalf of
the
ACMA in
the allocation phase, effectively enforcing a lower bound on the marginal price paid for any
lot in the base price computation. If the marginal value of a lot is less
than
the
opening
price,
then the lot would go unsold
in the auction
. The interpretation is that
the
ACMA
values the unsold lots at the
opening
price
s
and has set the reserve prices accordingly.
Note
that this means that the
reserve
bids impact the core in the exactly the same wa
y as all
other bids (i.e. the core is defined by all relevant bids, including
reserve
bids).
The use of
reserve
bids at the reserve value has been tested extensively by Power Auctions
and has
been used
in the peer

reviewed academic literature.
(See L.
Ausubel and P.
Cramton, “Vickrey Auctions with Reserve Prices,”
Economic Theory
, 2004.)
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1.3.2
Tie

breaking in winner determination
We recommend
the following
tie

breaking
procedure for
winner determination
: first,
select
the allocation of packages in the final
clock round, if this is one of the ways of maximising
the aggregate value of the selected bids; second,
select the value

maximi
s
ing solution that
maximizes the
eligibility
points
allocated to bidders; and,
finally
, g
iven multiple maximum

eligibility winner
determination solutions,
implement
a pseudo

random selection procedure
that
select
s
exactly one maximum

bid

value, maximum

eligibility

points allocation.
Efficient winner determination (i.e. selecting an allocation based on the maximisation of
accepted bi
d amounts) is the only selection criterion for combinatorial auctions that is given
any serious consideration in the economics literature. In general, ties in the winner
determination problem are unlikely, as in most instances a tie requires multiple bidde
rs to
name identical pric
es independently.
The likeliest scenario for a tie is if value maximisation includes a bid by a given bidder for a
given product at the opening price. (This scenario becomes reasonably likely when there is
never any excess demand f
or the given product in the clock rounds, and so the clock price
never rises from the opening price.) In that event, there may be a second value

maximising
solution in which the given bidder’s bid is replaced by a dummy bid at the opening price. In
that in
stance, we would want to favour the solution in which the given product is allocated
to the given bidder
—
the reason is that, in all likelihood, the given bidder values the product
at more than the opening price
;
but, under the auction rules, the bidder was never required
to bid any
higher,
due to the absence of excess demand. Observe that any protocol in which
the primary tie

breaking factor is the maximisation of the eligibility points allocated to
bidders favou
rs the desired outcome.
We recommend
preservation of the allocation of packages in the final clock round as a
n
initial
tie

breaking factor even before maximisation of the eligibility points allocated to
bidders, as this maximises the predictability between
the clock rounds and the
supplementary round. Tie

breaking based on maximum eligibility has been used in other
European spectrum auctions and serves to favour the allocation of more licenses and the
service of a greater population. Rules invoking pseudo

r
andomization have also been used
as a fail

safe when all other decision criteria are met
—
and some form of a final fail

safe
condition is required to assure uniqueness.
Other
additional
selection criteria could be applied before pseud
o

random selection, if
desired. An example of another possible tie

breaking factor is the maximisation of the
number of winners. In the interest of simplicity, we recommend against inclusion of any
other factors
, particularly in the initial consultation phase
.
The likelihood of
a relevant tie in
the winner determination problem is fairly remote, and specifying other tie

breaking factors
would only serve as a distraction. By contrast, a multiplicity in minimum

revenue core prices
(as described in section 1.3.3)
is much more likely
, but uniqueness in the pricing problem is
accomplished by minimizing a (weighted) distance from
specified
reference prices.
1.3.3
Weighting
bidders of different sizes
Before discussing alternative mechanisms for tie

breaking in pricing, let us make an
observati
on about “weighting”. In all combinatorial clock auctions to date that have used a
nearest

Vickrey rule, the auction system has selected the minimum

revenue core outcome
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that minimises the
conventional
E
uclidean distance
from the Vickrey
outcome. This means
that the auctioneer is implicitly weighting
any
two bidders equally and requiring them to
divide equally the burden of exceeding the bids of another coalition, even if one bidder is
much larger than the other. As a substantive change i
n the auction rules, we propose
instead to weight the bidders according to some measure of their size and to apply a
weighted
, rather than an unweighted, Euclidean distance.
The use of a weighted Euclidean distance rather than an unweighted Euclidean dista
nce
provides an improvement to enhance
the
fairness of pricing, beyond that of the unweighted
distances used in Europe recently. For a simple example, if Abe wins two generic lots and Bill
wins one generic lot, while Carl has a competing (losing) bid of 6
for all three lots, an
unweighted rule has Abe and Bill each paying 3, while a rule weighted by the reserve price
of each bundle would have Abe paying 4 and Bill paying 2. That is, a reserve

weighted
Euclidean distance has any required payments above the r
eference price in proportion to
their winnings, with Abe paying twice as much for two lots as Bill pays for one lot. Using the
Vickrey prices as the measure of winner size has similar benefits.
Power Auctions recommends modifying the auction rules so as to
weight the Euclidean
distance by winner size. Weighting the distance in this way
represents a new enhancement
of existing rules, but
it is
one so
natural and
appealing that we are confident in
recommending it on its own merit.
Its implementation involves
no risks.
1.3.4
Explanation of pricing
Prices are determined at two points in the auction, after the clock stage, including the
supplementary bids, to determine the base prices for the winners in the value

maximizing
generic assignment, and after the assignment
stage to determine the additional payments
for specific assignments.
The pricing rule plays a major role in fostering incentives for truthful bidding. Pay

as

bid
pricing in a clock auction or a simultaneous ascending auction creates incentives for demand
reduction (Ausubel and Cramton 2002). Large bidders shade their bids, recognizing their
impact on price. This bid shading both complicates bidding strategies and also leads to
inefficiency.
In contrast, Vickrey pricing provides ideal incentives for truthfu
l bidding. Each winner pays
the social opportunity cost of its winnings, and therefore receives 100 percent of the
incremental value created by its bids. This aligns the maximization of social value with the
maximization of individual value for every bidde
r. Thus, with private values, it is a dominant
strategy to bid truthfully. See Ausubel (2004, 2006) for an analysis in a clock auction.
Unfortunately, as a result of complements, it may be that the Vickrey prices are too low in
the sense that one or more b
idders would be upset with the assignment and prices paid,
claiming that they had offered the seller more. For example, suppose there are two items, A
and B, and three bidders. Bidder 1 bids $4 for A, bidder 2 bids $4 for B, and bidder 3 bids $4
for A and
B. The Vickrey outcome is for 1 to win A, 2 to win B, and each winner pays $0.
Bidder 3 in this case has a legitimate complaint, “Why are you giving the goods to bidder 1
and 2, when I am offering $4 for the pair?” The basic problem is that with complement
s, the
Vickrey outcome may not be in the core. Some coalition of bidders may have offered the
seller more than the sum of the Vickrey prices. (The core is defined as a set of payments
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that support the efficient assignment in the sense that there does not e
xist an alternative
collation of bidders that has collectively offered the seller more.) This point has been
emphasized in Ausubel and Milgrom (2002).
The solution is to increase one or more prices to assure that the prices are in the core. In
order to pro
vide the best incentives consistent with core pricing, the auctioneer finds the
lowest payments that are in the core; that is, such that no alternative coalition of bidders
has offered the seller more than the winning coalition is paying.
If we are auction
ing a single item, then this is the second

price auction. Suppose the highest
bidder bids $100 and the second

highest bidder bids $90. The item is awarded to the
highest bidder, who pays the second

highest price of $90―the social opportunity cost of
awardi
ng the good to the highest bidder. Alternatively, we can think of assigning the item to
maximize value, so we assign it to the highest bidder, and then we find the smallest
payment that satisfies the core constraints. In this case, the second

highest bidde
r would be
upset if the highest bidder paid less than $90, so $90 is the bidder

optimal core price. When
the items are substitutes, then the bidder

optimal core point is unique and identical to the
Vickrey prices.
The payment minimizing core prices, or bid
der

optimal core prices, typically are not unique
when the Vickrey prices are outside the core. Thus, it will be important to have a method of
selecting a unique bidder

optimal core point when there are many. One sensible approach
adopted in each of the re
cent Ofcom auctions for both the base prices and the assignment
prices is to select the payment minimizing core prices that are closest to the Vickrey prices.
This is what
we
call nearest

Vickrey core pricing. Since the set of core prices is convex―a
polyt
ope formed from the intersection of half

spaces―and the Vickrey prices are always
unique, there is a unique vector of core prices that is closest in Euclidean distance to the
Vickrey prices. Not only are the prices unique, but since they are bidder

optimal

core prices,
they maximize the incentive for truthful bidding among all prices that satisfy core
constraints (Day and Milgrom 2007).
The approach then is to take all the bids from the clock stage and the supplementary bids,
determine the value maximizing
assignment, and then determine the payment minimizing
core prices that are closest to the Vickrey prices. Prices are as small as possible subject to
the competitive constraints.
Calculating the winning assignments and prices involves solving a sequence of
standard
optimization problems. The basic problem is the winner determination problem, which is a
well

understood set

packing problem. The main winner determination problem is to find
the value maximizing assignment. To guarantee uniqueness, there is a seq
uence of
lexicographic objectives, such as: 1) maximize total value, 2) minimize concentration, 3)
maximize quantity sold, and 4) random. First the auctioneer maximizes total value. Then a
constraint that the value equals this maximum value is added and co
ncentration is
minimized. Then another constraint that concentration equals this minimum level is added
and the quantity sold is maximized. Finally, the constraint that the quantity sold equals this
maximum quantity is added and an objective based on rando
m values for each bid is
maximized. This guarantees uniqueness.
Calculating the prices is a bit more involved. First, we determine the Vickrey prices by
solving a sequence of winner determination problems, essentially removing one winner at a
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time to deter
mine each winner’s social opportunity cost of winning its package. Then we
determine the bidder

optimal core prices using a clever constraint generation method
proposed in Day and Raghavan (2007). Having found the Vickrey prices, we solve another
optimizat
ion to find the most violated core constraint. If there is none, then we are done,
since the Vickrey prices are in the core. Otherwise, we add this most

violated constraint and
resolve the optimization, again finding the most violated core constraint. We a
dd it to the
optimization and re

solve. We keep doing this until there is no violated core constraint, and
then we are done.
The reason that that Day

Raghavan approach is a highly efficient method of solution is
because in practice there are typically only
a handful of violated core constraints; thus, the
procedure stops after just a few steps. In contrast the number of core constraints grows
exponentially with the number of bidders and that makes including all the core constraints
explicitly an inefficient
method of solving the problem, both in time and memory.
As mentioned, the tie

breaking rule for prices is important, since typically ties will arise
along the southwest face of the core polytope. Finding the prices that are closest to the
Vickrey
prices involves solving a simple quadratic optimization. This gives us a unique set of
prices. Uniqueness is important. It means that there is no discretion in identifying the
outcome, either in the assignment or the prices.
An example will help illustrat
e all of these concepts. Suppose there are five bidders, 1, 2, 3,
4, 5, bidding for two lots, A and B. The following bids are submitted:
b
1
{A} = 28
b
2
{B} = 20
b
3
{AB} = 32
b
4
{A} = 14
b
5
{B} = 12
Bidders 1 and 4 are interested in A, bidders 2 and 5 are
interested in B, and bidder 3 is
interested in the package A and B.
Determining the value maximizing assignment is easy in this example. Bidder 1 gets A and
bidder 2 gets B, generating 48 in total value. No other assignment yields as much. Vickrey
prices a
re also easy to calculate. If we remove bidder 1, then the best assignment gives A to
bidder 4 and B to bidder 2, resulting in 34, which is better than the alternative of awarding
both A and B to bidder 3, which yields 32. Thus, the social opportunity cost
of bidder 1
winning A is 34
–
20 = 14 (the value lost from bidder 4 in this case). Similarly, if we remove
bidder 2, then the efficient assignment is for bidder 1 to get A and bidder 5 to get B,
resulting in 40. Then the social opportunity cost of bidder
2 winning B is 40
–
28 = 12 (the
value lost from bidder 5). Hence, the Vickrey outcome is for bidder 1 to pay 14 for A and for
bidder 2 to pay 12 for B. Total revenues are 14 + 12 = 26. Notice that bidder 3 has cause for
complaint, since bidder 3 offered 3
2 for both A and B.
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Now consider the core for this example. The core is represented in the payment space of
the winning bidders―in this case the payments of bidders 1 and 2. Each bid defines a half

space of the payment space:
Bidder 1’s bid of 28 for A imp
lies
bidder
1 cannot pay more than 28 for A.
Bidder 2’s bid of 20 for B implies
bidder
2 cannot pay more than 20 for B.
Bidder 3’s bid of 32 for AB implies that the sum of the payments for A and B must be at
least 32.
Bidder 4’s bid of 14 for A implies tha
t bidder 1 must pay at least 14 for A.
Bidder5’s bid of 12 for B implies that bidder 2 must pay at least 12 for B.
The core is the intersection of these half

spaces as shown in Figure 6.
Figure 6. The Core
This example is quite general. First, unlike in
some economic settings, in an auction, the core
is always nonempty. The reason is that the core always includes the efficient outcome. The
reason is that all the constraints are southwest of the efficient point, since the efficient
point maximizes total va
lue. Second, the core is always a convex polytope, since it is the
intersection of numerous half

spaces. Third, complementarities, like bidder 3’s bid for AB,
are the source of the constraints that are neither vertical nor horizontal. These are the
constra
ints that can put the Vickrey prices outside the core. Without complementarities, all
the constraints will be vertical and horizontal lines, and there will be a unique extreme point
to the southwest: the Vickrey prices.
The Core
b
4
{A} = 14
b
3
{AB} = 32
b
5
{B} = 12
b
1
{A} = 28
b
2
{B} = 20
Bidder 2
Payment
Bidder 1
Payment
14
12
32
28
20
Efficient outcome
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Figure 7. Vickrey prices: how much c
an each winner’s bid be reduced holding others fixed?
The graphical representation of the core is also a useful way to see the Vickrey prices.
Vickrey is asking how much can each winner unilaterally reduce its bids and still remain a
winner. As shown in
Figure 7, bidder 1 can reduce its bid to 14 before bidder 1 is displaced
by bidder 4 as a winner. Similarly, bidder 2 can reduce its bid to 12 before being displaced
by bidder 5. Thus, the Vickrey prices are 14 and 12. The problem is that these payments su
m
to 26, which violates the core constraint coming from bidder 3’s bid of 32 for AB.
Figure 8. Bidder

optimal core prices: jointly reduce winning bids as much as possible
The Core
b
4
{A} = 14
b
3
{AB} = 32
b
5
{B} = 12
b
1
{A} = 28
b
2
{B} = 20
Bidder 2
Payment
Bidder 1
Payment
Vickrey
prices
14
12
32
28
20
Problem: Bidder 3
can offer seller
more (32 > 26)!
The Core
b
4
{A} = 14
b
3
{AB} = 32
b
5
{B} = 12
b
1
{A} = 28
b
2
{B} = 20
Bidder 2
Payment
Bidder 1
Payment
Vickrey
prices
14
12
32
28
20
Problem: bidder

optimal core prices
are not unique!
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Bidder

optimal core prices can also be thought of as maximal reductions in the bids
of
winners, but rather than reducing the bids of each winner one at a time, we jointly reduce
all the winning bids, as shown in Figure 8, until the southwest face of the core is reached. As
can be seen, this does not result in a unique core point, since t
he particular point on the
southwest face depends on the rate at which each winner’s bids are reduced. The bidder

optimal core points consist of the entire southwest face of the core. If the southwest face is
a unique point, then it is the Vickrey prices;
if the southwest face is not unique then the face
is a core constraint involving complementarities, and the Vickrey prices lie outside the core.
Figure 9. Core point closest to Vickrey prices
Nonetheless, there is always a unique bidder

optimal core poin
t that is closest to the
Vickrey prices. This is seen in Figure 9, as the bidder

optimal core point that forms a 90
degree angle with the line that passes through the Vickrey prices. This point minimizes the
Euclidean distance from the Vickrey prices.
Whe
n the
bidders
win packages of different size, where size is measured using the reserve
prices, then it makes sense to use the weighted Euclidean distance from the Vickrey prices,
where the weights are the reciprocal of the reserve price of
the
winning pack
age.
For
example, if two winners collectively must increase payments by $3, but one bidder wins a
package that is twice as big as the other, then the added payment is split $2 for the larger
winner and $1 for the smaller winner. In the example in Figure 9,
it is assumed that A and B
have the same reserve price; thus, the $6 additional payment required by bidders 1 and 2 is
split equally ($3, $3).
Nearest

Vickrey core pricing was adopted in each of the UK
spectrum auctions, both the two
that have already been held as well as the proposed auctions for the 2.6 GHz spectrum and
the digital dividend spectrum. Nearest

Vickrey core pricing was also used in the other CCA’s
to date. Erdil and Klemperer (2009) argu
e that marginal incentives for truthful bidding may
Unique
core prices
b
4
{A} = 14
b
3
{AB} = 32
b
5
{B} = 12
b
1
{A} = 28
b
2
{B} = 20
Bidder 2
Payment
Bidder 1
Payment
Vickrey
prices
14
12
32
28
20
17
15
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be improved by using a reference point other than the Vickrey prices for selecting among
bidder

optimal core prices. In particular, they recommend a reference point that is
independent of the winners’ bid
s. A natural alternative reference point is the reserve prices.
The approach is the same as described above, except reserve prices are substituted for
Vickrey prices.
We discuss alternative tie

breaking rules in the next section.
Bidder

optimal core pricin
g has several advantages. First, it minimizes the bidders’ incentive
to distort bids in a Pareto sense: there is no other pricing rule that provides strictly better
incentives for truthful bidding. Bidder

optimal core pricing implies Vickrey pricing, whene
ver
Vickrey is in the core. For example, when lots are substitutes, Vickrey is in the core, and the
bidders have an incentive to bid truthfully. Since the prices are in the core, it avoids the
problem of Vickrey prices being too low as a result of compleme
nts.
1.3.5
Tie

breaking in pricing
Given multiple minimum

revenue

core pricing possibilities, payments are determined to
minimize a weighted Euclidean
distance
from a set of reference prices. In words, we select
the payments to minimize the weighted sum (summed
over all winners) of the squared
deviations between the payments and the reference prices. We recommend using either
Vickrey price (opportunity cost) or the reserve price of each winner’s package as the
reference price for each winner. We recommend the us
e of
the
reserve prices
of
winning
packages as weights in the weighted Euclidean
distance
to enforce a fair allocation of core
coalitional payment requirements.
To reiterate, s
uch a rule essentially splits the extra
payments that are required in proportion to the size of the bidder’s winnings, where the size
of the winning package is measured
using
reserve prices. Thus, larger bidders contribute
proportionately more than small b
idders.
Two alternatives seem most compelling for the tie

breaking rule for payments:
1.
Nearest

reserve with reserve weights, or
2.
Nearest

Vickrey with
reserve
weights.
Recent research suggests that the nearest

reserve rule may do a better job of mitigating
in
centives for bidders to shade their bids below value.
See, for example, “Core

Selecting
Auctions with Incomplete Information,” by Professors Larry Ausubel and Oleg Baranov,
available at the following URL:
http://www.ausubel.com/auction

papers/ausubel

baranov

core

selecting

auctions

with

incomplete

info.pdf
The intuition
favouring the nearest

reserve rule is that it utilises reference price
s that are
exogenous, rather than endogenous. With an endogenous reference price such as the
Vickrey prices, one effect of a bidder raising its own bid is to improve its opponents
reference prices, effectively costing the bidder money and providing a furth
er incentive to
shade its bid.
However, the
nearest

reserve
rule
also has a potentially significant drawback. It
increases
the extent that the
bidders’
payments depend on the relative reserve prices, as set by
the
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regulator
.
By contrast, with
the nearest

Vickrey rule
, the regulator’s choice of reserve prices
appears to have a substantially smaller impact on
the
bidders’
payments.
Our initial assessment is that t
he best choice
between a nearest

reserve rule and a nearest

Vickrey rule
depends on
the trade

of
f between these two effects.
Either of the
se
proposed payment rules (nearest

Vickrey or nearest

reserve) maintains all of
the benefits of minimum

revenue

core pricing as described in the academic literature, most
notably
in
providing optimal incentives am
ong all core

selecting payment rules and
in
mitigati
ng
certain forms of collusion. While the nearest

Vickrey version of minimum

revenue

core pricing has been implemented in several European spectrum auctions, the
nearest

reserve rule may further reduce the
already weakened incentives for bid shading
(i.e., bidding less than true value), in particular negating
any
benefit of shading small
amounts in many situations. However, as mentioned the nearest

reserve rule makes the
payments more dependent on the relat
ive magnitudes of reserve prices, which may be
undesirable in situations where the regulator does not have a good sense of relative prices.
At the end of the day, we do not currently have strong views between nearest

Vickrey or
nearest

reserve.
1.3.6
Assignment
stage bidding and optimization
In the event
that
there are multiple regions, the draft rules currently propose that the
assignments are done independently, region by region. The advantage of this approach is
that it is simple. The
difficulty
with this app
roach is that it will potentially lead to many
instances where a bidder winning blocks in adjacent regions wins different blocks in each
region. This can cause additional interference issues at the border and also make handoffs
more difficult.
1.3.7
Verifying th
at the algorithms produce outcomes that are correct
In order to verify and validate the results of the winner determination problem and
calculation of the prices during both the allocation stage and assignment stage, we propose
to use an independent implem
entation of the optimization algorithms performed by two
separate validation teams. Such an approach is common among regulators across the globe
because of its relative simplicity and high reliability.
For solving the optimizations
in a production environm
ent
, Power Auctions
typically
uses
the IBM ILOG CPLEX Optimizer, the leading commercial high

performance mathematical
programming solver for linear programming, mixed integer programming, and quadratic
programming.
However, Power Auctions also has a second
development team
that
implements optimi
s
ations using the GNU Linear Programming Kit (GLPK)
, a completely
distinct optimization platform. The two optimization platforms (CPLEX and GLPK) are coded
inde
pendently and serve as verifications of each other’s acc
uracy.
CPLEX
is the industry standard commercial optimizer for
winner determination problems.
According to IBM, i
t is used by
more than 1,300 commercial customers,
including one

third
of the Global 500
.
The optimizer not only is used extensively in
industry, but is commonly
used in relevant academic research testing the computational complexity combinatorial
auction problems. See for example
: K. Leyton

Brown and Y. Shoham, “A Test Suite for
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Combinatorial Auctions,” chapter 18 of
Combinatorial Auction
s
(P.
Cramton,
Y. Shoham
and
R.
Steinberg,
eds.), MIT Press, 2006; K. Leyton

Brown, E. Nudelman and Y. Shoham,
“Empirical Hardness Models for Combinatorial Auctions,” chapter 19 of
Combinatorial
Auctions
(P.
Cramton,
Y. Shoham
and
R.
Steinberg,
eds.), MIT
Press, 2006; R.
Day and
S.
R
aghavan,
“
Fair Payments for Efficient Allocations in Public Sector Combinatorial Auctions,
”
Management Science
, 2007;
and
R. Day and P. Cramton, “Quadratic Core

Selecting Payment
Rules for Combinatorial Auctions,”
Operations
Research
, forthcoming, 2012
.
Further information from IBM is also provided in the following link:
http://www.ibm.com/common/ssi/fcgi

bin/ssialias?infotype=PM&subtype=SP&appname=SWGE_WS_WS_USEN&ht
m
lfid=WSD14044USEN&attachment=WSD14044USEN.PDF
A number of case studies are provided here:
http://www

01.ibm.com/software/websphere/products/optimization/case_studi
es/
CPLEX solves the winner determination and pricing problems using state of the art
optimization techniques. Our approach using CPLEX finds a unique solution to the
optimization problem. CPLEX does not stop until the solution is found. Given that the
opt
imization problem is NP hard, we cannot be certain that the solution will be found in a
reasonable amount of time. However,
our software testing phase will demonstrate the size
and type of problems that can be solved. The experience to date with CCAs is th
at the actual
auctions are solved in a matter of seconds. The auction rules will include a limit on the
number of supplementary bids. This limit is set so that the probability of finding a solution in
a realistic amount of time is close to 1.
The mathemati
cal details of the algorithms used for winner determination and pricing are
presented in
R.
Day and
P.
Cramton,
“Quadratic Core

Selecting Payment Rules for
Combinatorial Auctions,”
Operations Research
,
forthcoming,
2012
.
During a software
testing phase, each software implementation undergoes a series of
accuracy tests that include carefully constructed examples and pseudo

randomly generated
data sets, based on the expected size and complexity of the actual setting. Given the
theoretical pro
perties of the problem at hand and a fixed set of rules for tie

breaking, this
testing verifies that both implementations always result in the same outcomes in each
phase of optimi
sation. Both optimis
ation platforms use a combination of branch

and

bound
an
d cutting plane algorithms, but the use of distinct platforms with specialized internal
heuristics enhances the reliability of the results and the ability to identify numerical or
computational anomalies before implementation. Exhaustive testing on problem
s of similar
size and complexity provides assurance of algorithmic accuracy and expediency before the
critical application.
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Appendix
A
: Example of bidding under the revised rules
Assumptions
For the purposes of this example, we will consider only the 700 M
H
z band
, which is
assumed
to be offered on a
nationwide
basis
. Suppose that each Upper lot has an eligibility of 50
points, while
the
Lower lot has an eligibility of 30 points.
We will consider a single Bidder, who we will call Bidder A. Bidder A would
like to obtain
two lots of the Upper Product. However, if the price difference between the Lower and the
Upper Product becomes large enough, Bidder A would prefer one lot of the Upper Product
and one lot of the Lower Product. The threshold for this chang
e in preference is a price
difference of $500k.
Eventually, if the price becomes too high, Bidder A will be unable to afford two lots and
must reduce its demand to one lot. In this case, Bidder A again prefers the Upper Product,
but will switch to the Low
er Product if the price difference becomes greater than $500k.
Bidder A’s total budget is $2800k. If the price of obtaining two blocks becomes greater than
this, Bidder A must reduce its demand to one lot.
Round 1
In Round 1, the opening prices of $1000k
for the Upper Product and $600k for the Lower
Product are announced. As Bidder A prefers two lots of the Upper Product unless the price
difference is greater than $500k, bidder A will bid for two lots of the Upper Product:
Product
Price
Bid
Eligibility P
oints
Upper
$1000k
2
100
Lower
$600k
0
0
Total Package
$2000k
(2, 0)
100
Round 2
In Round 1, several other Bidders shared Bidder A’s preference for the Upper Product, while
few Bidders bid on the Lower Product. As a result, the prices in Round 2 are
$1200k for the
Upper Product and $650k for the Lower Product, for a price difference of $550k. So, Bidder
A now prefers one lot of each Product.
Product
Price
Bid
Eligibility Points
Upper
$1200k
1
50
Lower
$650k
1
30
Total Package
$1850k
(1, 1)
80
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Round 3
In Round 2, the low price of the Lower Product caused many bidders to switch demand to
that product. As a result, the price of the Lower Product has increased at a faster rate than
the price of the Upper Product. The Round 3 prices are $1250k for
the Upper Product and
$800k for the Lower Product. This price difference is only $450k, so Bidder A would prefer
to switch back to bidding on two Lots of the Upper Product.
Using only an Eligibility

Points monotonicity rule, switching back at this point
would be
impossible. This limitation would have the effect of creating a disincentive for Bidder A to
bid on its most favorable package in Round 2. Bidder A would be forced to bid on a less
profitable package in order to maintain its eligibility for as m
any rounds of the auction as
possible.
With a rule of Revealed Preference with an Eligibility

Point Safe Harbor, however, Bidder A
is free to switch back:
Product
Price
Bid
Eligibility Points
Upper
$1250k
2
100
Lower
$800k
0
0
Total Package
$2500k
(2, 0
)
100 (Eligibility is 80)
In order to place a bid with eligibility points greater than its current Eligibility, Bidder A must
meet the revealed

preference constraint with respect to each prior eligibility

reducing
round. In this case, the only
eligibility

reducing round is Round 2. In words, the
requirement to switch from (1,1) to (2,0) is that the (2,0) package has become relatively
cheaper than the (1,1) package. Mathematically, the revealed

preference constraint is
stated as:
(Price of (2, 0
) in R3)

(Price of (2, 0) in R2) ≤ (Price of (1, 1) in R3)
–
(Price of (1, 1) in R2)
($2500k
–
$2400k) ≤ ($2050k
–
$1850k)
$100k ≤ $200k
This constraint is satisfied, so Bidder A is permitted to place the bid.
Round 4
In Round 4, the price on the Upper
Product increases to $1400k, while the price on the
Lower Product increases to $1000k. Bidder A bids on the same package as in Round 3:
Product
Price
Bid
Eligibility Points
Upper
$1400k
2
100
Lower
$1000k
0
0
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Total Package
$2800k
(2, 0)
100
(Eligibility is 80)
Bidder A’s Eligibility is still only equal to 80, however, so it must meet the revealed

preference constraint in order to place this bid. As before, the requirement is that the (2,0)
package needs to be relatively cheaper than the (1,
1) package (as compared to Round 2):
(Price of (2, 0) in R4)

(Price of (2, 0) in R2) ≤ (Price of (1, 1) in R4)
–
(Price of (1, 1) in R2)
($2800k
–
$2400k) ≤ ($2400k
–
$1850k)
$400k ≤ $550k
This constraint continues to be satisfied, so Bidder A is permitt
ed to place this bid.
Round 5
In Round 5, the price continues to increase on both products, with the Upper Product at
$1650k and the Lower Product at $1200k. As a result, both two

lot combinations now
exceed Bidder A’s budget of $2800k. Bidder A must
decrease its demand to one lot. Since
the price of the Upper Product is $450k greater than the price of the Lower Product which is
less than $500k, Bidder A places a bid for 1 Lot of the Upper Product. This bid reduces
Bidder A’s eligibility further to 5
0 points. Bidder A is within its Eligibility of 80 points, so
there are no revealed

preference constraints on this bid.
Product
Price
Bid
Eligibility Points
Upper
$1650k
1
50
Lower
$1200k
0
0
Total Package
$1650k
(1, 0)
50
Round 6
In Round 6, the
price on the Upper Product increases at a faster rate, increasing the price
difference to $550k, which is greater than the $500k threshold. Bidder A thus switches its
bid to one Lot of the Lower Product, reducing its Eligibility further to 30 points:
Prod
uct
Price
Bid
Eligibility Points
Upper
$1800k
0
0
Lower
$1250k
1
30
Total Package
$1250k
(0,1)
30
Round 7
In Round 7, the price on the Lower Product increases at a faster rate, causing Bidder A to
again desire to switch:
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Product
Price
Bid
Eligibility
Points
Upper
$1850k
1
50
Lower
$1400k
0
0
Total Package
$1850k
(1,0)
50 (Eligibility is 30)
In order to place this bid, Bidder A must satisfy the revealed preference rule relative to each
round in which it has reduced its eligibility. It is helpful
to summarize the prices up to this
point:
Product
Price
Round
1
Round 2
Round
3
Round
4
Round 5
Round 6
Round 7
Upper
$1000k
$1200k
$1250k
$1400k
$1650k
$1800k
$1850k
Lower
$600k
$650k
$800k
$1000k
$1200k
$1250k
$1400k
The constraints are as follows:
(Price of (1, 0) in R7)

(Price of (1, 0) in R2) ≤ (Price of (1, 1) in R7)
–
(Price of (1, 1) in R2)
($1850k
–
$1200k) ≤ ($3250k
–
$1850k)
$650k ≤ $1400k
(Price of (1, 0) in R7)

(Price of (1, 0) in R5) ≤ (Price of (1, 0) in R7)
–
(Price of (1, 0) in
R5)
($1850k
–
$1650k) ≤ ($1850k
–
$1650k)
$200k ≤ $200k
(Price of (1, 0) in R7)

(Price of (1, 0) in R6) ≤ (Price of (0, 1) in R7)
–
(Price of (0, 1) in R6)
($1850k
–
$1800k) ≤ ($1400k
–
$1250k)
$50k ≤ $150k
All three constraints are satisfied, so Bi
dder A is permitted to place this bid.
Supplementary round
In the bidding of Round 7, the aggregate demand drops sufficiently that the clock stage
concludes. Bidder A is in the position of having a Final Clock Package of one Lot of the
Upper Product. No
te that if there had only been an Eligibility

Points monotonicity rule in
the clock stage, Bidder A would likely have, instead, a Final Clock Package of one Lot of the
Lower Product, a less desirable package.
With the Revealed

Preference Cap, Bidder A is n
ow guaranteed to win its Final Clock
Package if there was no undersell in the clock stage of the auction. If there were some
unallocated lots, Bidder A can still guarantee winning its Final Clock Package by submitting a
bid increasing the dollar amount by
the final clock price of those unallocated items.
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Now, suppose Bidder A wishes to increase its bid on the package (1,1) (i.e. one lot of the
Upper Product and one lot of the Lower Product) to its maximum budget of $2800k. This
package is larger than Bidd
er A’s Final Clock Package. Therefore, Bidder A must satisfy
revealed preference with respect to the final package, as well as with respect to each
eligibility

reducing round beginning with the last round in which Bidder A had sufficient
eligibility to bi
d on the (1,1) package, i.e. beginning with Round 5.
The revealed

preference constraints are as follows, starting with the Final Clock Package
constraint:
(Sup Bid on (1,1)) < (Highest Bid on (1,0)) + (Price of (1,1) in R7)
–
(Price of (1,0) in R7)
(Sup
Bid on (1,1)) < $1850k + $3250k
–
$1850k
(Sup Bid on (1,1)) < $3250k
(Sup Bid on (1,1)) < (Highest Bid on (0,1)) + (Price of (1,1) in R6)
–
(Price of (0,1) in R6)
(Sup Bid on (1,1)) < $1250k + $3050k
–
$1250k
(Sup Bid on (1,1)) < $3050k
(Sup Bid on (1,
1)) < (Highest Bid on (1,0)) + (Price of (1,1) in R5)
–
(Price of (1,0) in R5)
(Sup Bid on (1,1)) < $1850k + $2850k
–
$1650k
(Sup Bid on (1,1)) < $3050k
Thus, Bidder A’s Supplementary Bid on the (1,1) package is most constrained by the
revealed

preferenc
e constraints relative to Round 5 and Round 6. In this case, the
constraining amount is $3050k, allowing Bidder A to place the $2800k bid.
Increasing supplementary bid prices
If Bidder A wishes to increase its package bid for (1,1) beyond $3050k, it must
increase its
supplementary bids both for its Final Clock Package (the (1,0) package) and for the (0,1)
package.
Let us restate the revealed

preference constraints in terms of any supplementary bids that
Bidder A submits:
(Sup Bid on (1,1)) < (Sup Bid on (
1,0)) + (Price of (1,1) in R7)
–
(Price of (1,0) in R7)
(Sup Bid on (1,1)) < (Sup Bid on (1,0)) + ($3250k
–
$1850k)
(Sup Bid on (1,1)) < (Sup Bid on (1,0)) + $1400k
(Sup Bid on (1,1)) < (Sup Bid on (0,1)) + (Price of (1,1) in R6)
–
(Price of (0,1) in R6)
(Sup Bid on (1,1)) < (Sup Bid on (0,1)) + ($3050k
–
$1250k)
(Sup Bid on (1,1)) < (Sup Bid on (0,1)) + $1800k
(Sup Bid on (1,1)) < (Sup Bid on (1,0)) + (Price of (1,1) in R5)
–
(Price of (1,0) in R5)
(Sup Bid on (1,1)) < (Sup Bid on (1,0)) + ($2850k
–
$1650k)
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(Sup Bid on (1,1)) < (Sup Bid on (1,0)) + $1200k
Note that the first constraint is redundant with the third constraint. However, the second
constraint may also be binding. Even if Bidder A places a large supplementary bid on the
Final Clock Packa
ge, its bid for the (1,1) package is still capped at the relatively low value of
$3050k. In order to increase this bid, Bidder A must also place a supplementary bid on the
(0, 1) package. With both the (1,1) and (0,1) package caps increased, the Bidder w
ill be able
to increase its (1,1) package bid beyond $3050k. For example, Bidder A will be permitted to
place a supplementary bid of up to $3500k on the (1,1) package, provided that it places a
supplementary bid of $2300k on the (1,0) package
and
a supplem
entary bid of $1700k on
the (0,1) package.
Revealed

preference constraints for the (0,1) package
The (0, 1) package is also subject to revealed

preference constraints. These are based on the
packages in each eligibility

reducing round beginning with the la
st round in which Bidder A
had sufficient eligibility to bid on the (0,1) package, as well as in the Final Clock Round. In
this example, the last round in which Bidder A had sufficient eligibility to bid on the (0,1)
package was Round 7
—
which is also th
e Final Clock Round. So, the only constraint on the
bid for the (0,1) package is:
(Sup Bid on (0,1)) < (Sup Bid on (1,0)) + (Price of (0,1) in R7)
–
(Price of (1,0) in R7)
(Sup Bid on (0,1)) < (Sup Bid on (1,0)) + ($1400k
–
$1850k)
(Sup Bid on (0,1)) <
(Sup Bid on (1,0))
–
$450k
So, without any supplementary bids placed on the Final Clock Package, the highest
supplementary bid that Bidder A can place on the (0,1) package is $1350k. However, if for
example Bidder A places a supplementary bid on its Final
Clock Package of $2300k, then
Bidder A may also place a supplementary bid on the (0,1) package of up to $1850k.
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Appendix
B
: Example of
how bidding in the supplementary
round can
change the
allocation
In this appendix
,
we provide an
example in which
the
cl
ock stage
end
s
with no excess supply
and
the
supplementary bids
will be
required to satisfy
the Relative Cap
,
yet
the final
allocation
will not be
the same as the tentative allocation at the end of the
final
clock
round
.
We repeat the example under the
Re
vealed

Preference Cap (our
proposed
revision to the
activity rule for the supplementary round)
to illustrate that
our proposed revision
places
additional restrictions on supplementary bids to
help
ensure that
the
final allocation will be
the same as the te
ntative allocation.
An example
using the
Relative Cap
Suppose that three bidders are bidding in an auction with four categories
—
A, B, C and D
—
each comprising a single item. The tables below list the prices at each round, and the
highlighted cells indicate
the items demanded for by the given bidder in the given round.
Clock Stage
Bidder
1
Eligibility
points
10
30
20
20
Lot
A
B
C
D
Eligibility
Round 1
90
90
90
110
80
Round 2
100
100
100
120
60
Round 3
110
100
100
120
50
Bidder
2
Eligibility
points
10
30
20
20
Lot
A
B
C
D
Eligibility
Round 1
90
90
90
110
20
Round 2
100
100
100
120
20
Round 3
110
100
100
120
20
Bidder
3
Eligibility
points
10
30
20
20
Lot
A
B
C
D
Eligibility
Round 1
90
90
90
110
60
Round 2
100
100
100
120
10
Round 3
110
100
100
120
10
Suppose that Bidder 1 wants to submit supplementary bids for {A, B, C} and {B, C, D}.
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Observe that, in round 2, Bidder 1 revealed that it preferred paying 300 for {A, B, C} than
320 for {B, C, D}, so
. In round 3, Bidder 1 revealed that it
preferred paying 200 for {B, C} than 310 for {A, B, C}, so
. So any
bid that Bidder 1 submits for
{A, B, C}
in the supplementary bid rounds is constrained by
. If Bidder 1 w
ants to submit a bid for {B,
C, D}, then that bid is
constrained by
(as long as it also submits the maximum possible bid for
{A, B, C}).
So Bidder 1 is permitted by the Relative Cap to submit the following set of supplementary
bids:
{
B, C}: 200 (no additional supplementary bid)
{A, B, C}: 310
{B, C, D}: 330 .
We will suppose that Bidders 2 and 3 do not submit any supplementary bids.
In the tentative allocation at the end of the final clock round,
Bidder 1 wins {B, C}, bidder 2
wins
{D} and Bidder 3 wins {A}. This provides a combined bid value of 430.
However, observe that the alternative allocation in which Bidder 1 wins {B, C, D} and Bidder
3 wins {A} provides a combined bid value of 330 + 110 = 440. It can further be seen that thi
s
allocation maximizes the combined bid value over all feasible selections of supplementary
bids.
The clock stage ended with no excess supply and the supplementary bids were required to
satisfy the Relative Cap, yet the final allocation was not the same a
s the tentative allocation
at the end of the final clock round.
The same example under the
Revealed

Preference Cap
The Revealed

Preference Cap adds an additional inequality to the requirements of the
Relative Cap in this example: any supplementary bid for
{B
,
C, D}
must satisfy the
strict
revealed

preference inequality with respect to the final clock bid. This is writt
en:
.
Thus,
.
Consequently, if no supplementary bid is submitted for
{B
,
C}
above the highest clock bid of 200
, then the highest supplementary bid that can be
submitted for
{B
,
C, D}
is 319. (Similarly, since
we are imposing strict inequality, the highest
bid that can be submitted for
{
A,
B
,
C}
is 309.) Hence, Bidder 1’s set of supplementary bids
would be trimmed by the Revealed

Preference Cap to be no higher than:
{B, C}: 200 (no additional supplementary bid)
{A, B, C}: 309
{B, C, D}: 319 .
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Observe now that there does not exist any alternative allocation yielding a combined bid
value of 429. Indeed, if
the clock
stage
end without excess supply and if the supplementary
bids are required to satisfy the Revealed

Preference Cap, then the final allocation will be
guaranteed to be the same as the tentative allocation at the end of the final clock round.
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