1. Market Design and the Evolution of the Combinatorial ClockAuction
Lawrence M. Ausubel, University of Maryland
Oleg V. Baranov, University of Colorado
5 January 2014

2. Combinatorial Clock Auction (current generation)
Has been used for digital dividend auctions of spectrum in Australia, Austria, Canada, Denmark, Ireland, the Netherlands, Slovakia, Switzerland and the UK
Has been adopted for a second (upcoming) spectrum auction in Canada
In the past few years, CCAs have been used for more major spectrum auctions worldwide than any other auction format
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3. Combinatorial Clock Auction (current generation)
A three-phase auction with package bidding, consisting of a clock auction phase, followed by a supplementary round, and concluding with an assignment round
Clock auction phase: Auctioneer announces prices and bidders respond with quantities; bidders are constrained by eligibility points and/or a WARP-based activity rule
Supplementary round: Bidders can supplement their bids from the clock rounds with additional package bids, constrained by aWARP-based activity rule
Assignment stage: generic spectrum mapped to frequencies
Allocation and pricing are given by a core-selecting mechanism
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4. Combinatorial Clock Auction process
Bidders select quantities to bid at current prices
Round opens
Bidders’ choices are constrained by activity rule
Auctioneer announces aggregate demand and next round’s prices
Round closes
Calculate excess demand and determine if clock rounds should continue
open
Clock Auction phase ends
Conduct Supplementary Round
Conduct Assignment Stage
close
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5. Evolution of the Combinatorial Clock Auction
The CCA has been in constant evolution since it was proposed:
Ausubel, Cramton & Milgrom, “The Clock-Proxy Auction: A Practical Combinatorial Auction Design,” academic article presented at Wye River Conf. in 2003 and published in 2006:
Two stages: clock rounds and supplementary round
Activity rules incorporate revealed preference
Price determination is done using the ascending proxy auction
Trinidad and Tobago Spectrum Auction (TATT and MDI/Power Auctions, June 2005):
Introduction of assignment stage
In allocation phase, bidders bid for generic spectrum
In assignment stage, bidders bid for specific frequencies
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6. Evolution of the Combinatorial Clock Auction
Proposal to Ofcom (MDI/Power Auctions, June 2006):
Ascending proxy auction is replaced by core-selecting mechanism
UK 10 – 40 GHz and L-Band auctions (Ofcom/DotEcon, February and May 2008):
First implementations of a three-stage combinatorial clock auction
Eligibility-point-based activity rule, rather than revealed pref, in clock
Absolute cap, rather than relative cap, used in supplementary round
Various recent auctions (Power Auctions and DotEcon):
Relative cap activity rule introduced for supplementary bids
Hybrid revealed-preference/eligibility-point activity rule for clock rounds
First implementations of CCA with regional licenses (Australia, Canada)
Shift to use of a “reserve bidder” approach (UK, Australia, Canada)
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7. Critiques of Current Versions of the CCA
With regional licenses, bidders may have difficulty expressing their preferences using mutually-exclusive bids
Current versions of the CCA use point-based activity rules or limited introductions of revealed-preference considerations
Most dynamic auction formats are “iterative first-price”—they are literally first-price (while effectively second-price)
For example, in the English auction and in other formats commonly used for spectrum auctions, bids are submitted which, if they turn out to win, specify the amounts that will actually be paid
However, the CCA as used for spectrum auctions is closer to being “iterative second-price”
This exacerbates some basic tensions within the CCA
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8. Evolution of the Combinatorial Clock Auction
This paper discusses three evolutionary enhancements to the CCA which would address the previous critiques:
Introducing OR bidding in the supplementary round
A GARP-based activity rule
Converting the CCA from an iterative second-price auction to an iterative first-price auction
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9. Part I: OR bidding in the supplementary round

10. XOR versus OR bids
In the current version of the CCA, all bids by a given bidder are treated as mutually exclusive (“XOR” bids)
There is a good reason for this: otherwise, a bidder cannot express “I would like A or B, but not both”
Or, to put it differently, an XOR bidding language is fully general
However, with N regional licenses, there are 2N possible packages
Thus, it may become difficult for a bidder to express valuations for all relevant packages
It may also contribute to the “missing bid problem”
A modest enhancement would be to permit bidders, on a voluntary basis, to include supplementary bids that may be accepted on an and/or basis (“OR” bids)
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11. An Example of a Collection of OR bids
Suppose that a bidder’s Final Clock Package is {A, B, C}, while the licenses D, E and F are unsold
A bidder might submit a bid for a base package of {A, B, C} together with OR bids for D, E and F
Submitting this is the same as enumerating XOR bids for:
{A, B, C}
{A, B, C, D}
{A, B, C, E}
{A, B, C, F}
{A, B, C, D, E}, etc.
This imposes an additive structure on bidders’ valuations, which may be a reasonable approximation on the margin
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12. OR bids
OR bidding languages have been used successfully in sealed-bid combinatorial environments
A large number of OR bids tagged onto a base bid might cause a bidder to lose control over how much he wins, but control can be regained by adding control elements such as a size‑of‑package limit or a bid-amount limit defined by the bidder
One (perhaps the biggest) challenge with introducing OR bids into the CCA is integrating them with an effective activity rule
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13. Part II: Activity Rules

14. Current Activity Rules
Activity rules are intended to prevent “bid sniping”: a bidder must submit bids in early rounds of the clock stage in order to be allowed to continue to submit bids in later rounds (and in the supplementary round)
Spectrum auctions have traditionally required monotonicity in eligibility points
But this approach bumps up against the following result:
Theorem: For any choice of eligibility points, there exist bidder valuations and price histories such that the bidder is prevented from bidding its true valuations by an activity rule requiring monotonicity in eligibility points; and if bidders attempt to bid straightforwardly, the outcome will necessarily be inefficient
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15. Current Activity Rules
A few CCAs (Ireland, Australia, Canada) have incorporated revealed preference considerations:
WARP: (ps – pt)  (xs – xt)  0, for s  t
Verbally, bids should only involve switches to packages that have become relatively less expensive
“Hybrid” rule—abidder can bid on a package iff it is within the bidder’s current eligibility points or consistent with WARP
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16. Critiques of the current version of the CCA
The hybrid rule is an improvement upon using only eligibility points, but it is only a halfway measure
Only some of the revealed-preference constraints are imposed, in order to avoid “dead-ends”
This creates potential anomalies
Finally, it may be possible for a bidder to influence which revealed-preference constraints are imposed, leaving the bidder insufficiently constrained in the supplementary round
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17. Generalized Axiom of Revealed Preference (GARP)
Given price-quantity pairs (x1 , p1), … , (xt , pt), package xi is said to be revealed preferred to xm if there is a sequence j, k, …, l such that pi xi  pi xj , pj xj  pj xk , … , pl xl  pl xm .
The data (x1 , p1), … , (xt , pt) are said to satisfy GARP if:
xi is revealed preferred to xm  pm xm  pm xi .
Intuitively, this requires that there is “cyclical consistency”: ifpackage xi is revealed to be preferred to package xm through a sequence of decisions, package xm cannot be revealed to be strictly preferred to the original package xi .
Afriat’s Theorem (1967): A finite set of data is consistent with utility maximization if and only if it satisfies GARP.
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18. GARP-based activity rules
Definition: The GARP-based activity rule for the clock rounds is the requirement that, given bid history (x1 , p1), … , (xt–1 , pt–1), the bidder is permitted to bid xt in round t if and only if the data (x1 , p1), … , (xt , pt) is consistent with GARP.
This is a stronger activity rule than requiring WARP to hold in every period, which in turn is a stronger activity rule than is used in today’s CCAs—see the example on the next slide.
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19. Dead-ends in bidding using WARP-based activity rules
Consider:
The bidding and WARP have produced a “dead-end”: the only allowable bid in Round 4 under WARP is (0, 0, 0).
However, the reason for the dead-end is that the bid in Round 3 already violated GARP.
Under a GARP activity rule, x3 is not allowed to be bid.
Round
Price vector
Bidder’s demand 1
p1 = (1, 1, 1)
x1 = (1, 0, 0) 2
p2 = (1, 1, 4)
x2 = (0, 1, 0) 3
p3 = (4, 1, 4)
x3 = (0, 0, 1) 4
p4 = (4, 3, 5)
x4 = ???
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20. GARP-based activity rules
Definition: The GARP-based activity rule for the clock rounds is the requirement that, given bid history (x1 , p1), … , (xt–1 , pt–1), the bidder is permitted to bid xt in round t if and only if the data (x1 , p1), … , (xt , pt) is consistent with GARP.
Notwithstanding the activity rule’s strength, it satisfies:
Straightforward bidding is always permitted by the GARP-based activity rule.
No dead-ends: At least one of the bidder’s priorbids, x1 , … , xt–1 , is always permitted in round t by the GARP-based activity rule.
Subsets: If bidding x in round t is allowed by the GARP-based activity rule, then bidding y  x is also allowed.
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21. GARP-based activity rules
Definition 1: The GARP-based activity rule for the clock rounds is the requirement that, given bid history (x1 , p1), … , (xt–1 , pt–1), the bidder is permitted to bid xt in round t if and only if the data (x1 , p1), … , (xt , pt) is consistent with GARP.
Notwithstanding the strength of the activity rule:
Proposition 1: Straightforward bidding is always permitted by the GARP-based activity rule.
Proposition 2 (No dead-ends): At least one of the bidder’s priorbids, x1 , … , xt–1 , is always permitted in round t by the GARP-based activity rule.
Proposition 3 (Smaller bids): If bidding x in round t is allowed by the GARP-based activity rule, then bidding y  x is also allowed.
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22. “Test” of GARP in the UK auction: H3G
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23. “Test” of GARP in the UK auction: EE
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24. “Test” of GARP in the UK auction: Vodafone
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25. Part III: Pricing

26. Tensions from second pricing in the current CCA
There is a tension between strict activity rules and the use of a core-selecting mechanism (“second pricing”):
With strict activity rules and without undersell, each bidder is guaranteed to win its package from the final clock round
So bidders may lack incentive to submit supplementary bids
There may also be minimal consequences to a bidder inflating its expressed opportunity cost
Since the price determination depends on supplementary bids, prices may be too low (the “missing bid” problem) or too high (“spiteful bidding”)
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27. Transforming the CCA from second- to first-pricing
One can ask the question: Given opponents’ bids, and given the activity rule, what is the maximum amount that a bidder could ever have to pay to win a package if this were the final clock round of the auction (“exposure calculation”)?
Using the nominal clock prices as the bid amount for a package, as is done in the current CCA, is an arbitrary convention; all that matters are relative prices
Instead, for example, the bid amount for a package could be discounted, based upon the bidder’s exposure calculation
Furthermore, one could utilize the discounted bid amounts to make the pricing pay-as-bid
With first-pricing of the final clock packages, it doesn’t matter if there is insufficient incentive for supplementary bids
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28. The CCA as an iterative first-price auction
Much of the rationale for the strong GARP-based activity rule is that it generates reasonable exposure calculations
Under second price, supplementary bids serve both an allocation and a pricing role; under first price, attention is focused on the allocation role
A strong activity rule assures a high degree of stability and predictability in going from the clock rounds to the supplementary round (by contrast, a weak activity rule allows bid-sniping)
The use of first-pricing would also tend to reduce the importance of budget constraints
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29. Conclusion
The CCA has been reasonably successful in its use for spectrum auctions
The CCA has been continually evolving since its inception
There are new directions in which the CCA can further evolve that will make it still more effective
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