U.S. Spectrum

Reallocation and

Heuristic Auctions Paul Milgrom and Ilya Segal

December 2012

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F.C.C. Backs Proposal to Realign

Airwaves 2 September 28, 2012 By EDWARD WYATT

WASHINGTON — The government took a big step on Friday to aid the

creation of new high-speed wireless Internet networks that could fuel the

development of the next generation of smartphones and tablets, and devices

that haven’t even been thought of yet.

The five-member Federal Communications Commission unanimously

approved a sweeping, though preliminary, proposal to reclaim public airwaves

now used for broadcast television and auction them off for use in wireless

broadband networks, with a portion of the proceeds paid to the broadcasters.

The initiative, which the F.C.C. said would be the first in which any

government would pay to reclaim public airwaves with the intention of selling

them, would help satisfy what many industry experts say is booming demand

for wireless Internet capacity.

Mobile broadband traffic will increase more than thirtyfold by 2015, the

commission estimates. Without additional airwaves to handle the traffic,

officials say, consumers will face more dropped calls, connection delays and

slower downloads of data.

The “Incentive Auction” Plan

“Reverse Auction”: buy TV broadcast licenses, providing

an “incentive” for broadcasters to participate.

Repack the remaining broadcasters into a smaller spectrum band.

CBO: $15 billion cost

“Forward Auction”: sell 4G wireless broadband

licenses.

Must first reorganize the cleared spectrum to create usable

licenses.

CBO: $40 billion revenue.

“Clearing Rule”: combine bids in the two auctions to

determine the amount of spectrum to be cleared and

the auctions’ “winners”.

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Background

“A Proposal for a Rapid Transition to Market

Allocation of Spectrum,” Evan Kwerel and

John Williams, OPP Working Paper 38, 2002.

National Broadband Plan, 2010 (pp. 84-85)

Middle Class Tax Relief and Job Creation Act,

February 16, 2012, Sec. 6101-6703

“Straw man” Appendix to FCC’s Notice for

Proposed Rule Making, Ausubel, Levin, Milgrom

(team leader), Segal, September 2012

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What kind of “Commodity” is Radio

Spectrum? 5

TV broadcast licenses 6

Each channel uses 6MHz of spectrum in one of three bands

Repurposed

in DTV

transition

Each of ≈2,500 TV licenses includes 7

Channel, location, and power restrictions

Protection from interference in current service area

From same channel or adjacent-channel stations

“Must-carry” rights on cable and satellite TV

Statute lets FCC retune non-participating station within home bands (compensating retuning costs)

Mandates “all reasonable efforts” to preserve interference-free population coverage

Stations can bid

to go off-air

to move to a lower band (preserving must-carry rights)

Interference Constraints 8

OET-69 Bulletin Coverage:

≈ 2 million cells (2km x 2km )

Pairwise constraints (0.5% threshold):

≈130,000 edges

Broadband (mobile) licenses 9

Must be separated in frequency from TV

Optimal license design depends on technology Frequency Division Duplexing: Separated Paired

Uplink & Downlink: Multiples of 2x5MHz; max speeds use 2x20MHz

Time Division Duplexing: Typically 10 MHz unpaired

Geographic coverage: National licenses, regional licenses, or a mix?

Overlap many TV stations’ license areas

FCC’s role in spectrum reallocation? 10

1. Allocate by administrative authority?

2. “Coasian” approach: sell to broadcasters the

property rights to use “their spectrum” as they

desire and allow trading?

Coordinated action of many parties is needed to

repurpose spectrum respecting engineering requirements.

3. “Market Design” approach:

Define spectrum and interference rights (e.g. FCC’s right

to retune) to minimize holdout, promote competition

Market mechanism for spectrum allocation with simple

participation and minimal scope for gaming

“New Paradigm for Spectrum Policy” 11

•FCC’s previous

auctions:

• Incentive

Auction: (Commissioner Robert McDowell)

“Reverse Auction”: Buying TV Licenses 12

Seek a mechanism to buy spectrum rights sufficient for a given goal, repacking remaining broadcasters

E.g. 120 MHz: clear channels 32-51

Goal may depend on the forward auction revenues

Assume:

Each station is separately owned

Each station is a “single-minded bidder”: bids on just one option (going off-air or to a lower band)

Assignment rule: which bids “win” (accepted) and “lose” (=rejected= assigned to home band)

Optimization-Based Reverse Auction?

Assignment rule maximizes the total value s.t.

interference constraints

a given clearing goal (e.g. clear channels 32-51).

Variation: incorporate revenue goal by maximizing

Myerson’s total “virtual value” conditioning on

stations’ characteristics

Computational challenge: Optimization is NP-hard –

can only be approximated

Associated payment rules:

Paid as bid? Induces overbidding

Ensure truthful bidding using Vickrey prices?

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Paid-as-bid?

Broadcaster’s optimal bid depends on its estimates of

bids of neighboring stations

algorithm used for computing the assignment

interference constraints used in the algorithm

bids in the forward auction, which help determine how much

spectrum is repurposed

post-auction value of licenses (common-value element)

⟹ Difficult, expensive for broadcasters to bid well!

Reduces participation in the auction.

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Vickrey Payments 15

Let S be a set of bids that can be feasibly rejected

(assigned to home bands into channels 2-31); let X be

the collection of all such sets.

Each station s submits a bid bs for its bidding option.

Set of bids to reject:

Stations in S* receive no payment.

Other bids are accepted, and paid:

S* Îargmax

SÎXb

ssÎSå

" ¢s ¢S *( )p

¢s= b

ss¢S*¢ - max{S¢X | ¢s ¢S }

bss¢S-{ ¢s }¢

Vickrey: Computational Problems 16

Vickrey price = difference between two amounts much

larger than the price itself ⟹ small % errors in

optimization can lead to large % errors in prices

Example (hypothetical):

True Vickrey price = 100 – 99 = 1

Approximate Vickrey price = 100 – 96 = 4

3% error in “second optimization” ⟹ 300% overpayment

Underpayment can also happen when “second optimization” is

more precise than overall optimization

These errors destroy incentives for truthful bidding and

thus ruin the auction’s supposed efficiency

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Greedy Heuristic Auctions

A “Greedy” Heuristic Algorithm

1. A (possibly imperfect) method to check whether a set

of bids can be feasibly “rejected” – assigned to their

home bands (with repacking).

2. A scoring function to prioritize bids.

Each bidder’s score is increasing its bid (e.g. score = bid/”volume”)

May be fixed or “adaptive” - depend on the current assignment, and

on bids already rejected

“Tie-breaking” is fixed as part of the scoring

Start with all bids active (provisionally accepted)

In each round, irreversibly reject the highest-scoring

still-active feasible bid

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Strategy-Proof Auctions 19

An auction is a deterministic assignment

rule coupled with a payment rule in which

only accepted bids receive payments.

An auction is strategy-proof if each

bidder i, regardless of other bids, cannot

gain by bidding an amount different from

its true value for its bidding option.

Assume each bidder is single-minded

Threshold Prices 20

An assignment rule is monotonic if for any

bidder j, increasing his bid bj never causes it to

win, regardless of the other bids b-j .

For any monotonic assignment rule and

any bidder j and competing bids b-j,

bidder j’s threshold price is the unique

amount pj = pj(b-j) such that j loses if bj > pj

and wins if bj < pj.

Characterization of

Strategy-Proof Auctions 21

A threshold auction collects bids and then applies

a monotonic station assignment rule

the corresponding threshold pricing rule, which

Pays each accepted bidder its threshold price

Pays zero to each rejected bidder

Theorem 1. An auction is strategy-proof if and

only if it is a threshold auction.

Greedy Threshold Auction 22

A greedy algorithm is monotonic.

Definition. A greedy threshold auction is a threshold auction whose assignment rule is computed by some greedy algorithm.

It is easy(!) to compute the exact threshold prices for accepted bids:

In each round n, for each still active bidder j, let pjn = his highest bid that would not be rejected in that round.

When the algorithm terminates, for each accepted bid j, the threshold price is pj = minn pjn

Nice Properties

of Greedy Threshold Auctions

1. Computationally Simpler

2. Strategy-Proof

3. Equivalent to Descending Clock Auctions

4. (Weakly) Group Strategy-Proof

5. Outcome-equivalent to full-info Nash equilibrium of

paid-as-bid auction with same assignment rule

i.e. threshold pricing “may not cost us”

6. Can implement any assignment rule in which

bidders are substitutes (if computationally feasible)

Vickrey fails (3)-(5) when bidders are not substitutes

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Earlier Heuristic Auctions

Lehmann, O’Callaghan, Shoham (2002), Babaioff-Blumrosen

(2008): Greedy heuristic auction for selling, trivial feasibility

checking

Our auction irreversibly rejects bids (deferred acceptance),

theirs irreversibly accept bids ⟹ NOT equivalent to a clock

auction (price computation requires more info)

Moulin (1999), Mehta et al. (2007), Juarez (2007):

Cost-Sharing Mechanisms that are (W)GSP

Special cases of clock auction: losers cannot affect others’

assignments or payments

Ensthaler-Giebe (2009,2010): Heuristic sealed-bid and clock

auctions for budget-constrained knapsack problem

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Greedy

Threshold

Auctions

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Descending

Clock Auctions (assuming finite bid space)

Descending Clock Auctions

Definition: A descending clock auction is a dynamic

mechanism in which bidder-specific prices are

initialized at reserves and descend over time. In

every round, the auction:

Selects a still-active bidder who can feasibly “quit” – be

assigned to its home band

Decrements the selected bidder’s price and gives him the

option to quit

When no more bidder can feasibly quit, auction ends,

accepting all still-active bids at their current prices

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Theorem 2(a): Any greedy threshold auction is

equivalent to a descending clock auction.

Proof: The equivalent clock auction selects for

price reduction the highest-scoring bidder

among those who could be feasibly rejected

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Theorem 2(b): Any descending clock auction is

equivalent to a greedy threshold auction.

Proof: An equivalent greedy auction gives

each active bidder a “score” equal to inverse

of the number of clock rounds, starting from

current threshold prices, in which he would quit

by bidding truthfully if no other bidder quits

before him

This score is increasing in the bidder’s value

The highest-scoring active bidder is the next to quit

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Advantages of descending clock

auctions

Optimality of truthful bidding for single-

minded bidders is obvious (also in experiments)

Winners need not reveal/know their exact

values

With common values, permit information

feedback to help aggregation (Milgrom-

Weber 1982)

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Group Strategy-Proofness

“Broadcasters Considering FCC Incentive Auctions

Launch Coalition” (National Journal, Nov 13, 2012)

Definition: An auction is Weakly Group Strategy-

Proof if no coalition has a strict Pareto improving

deviation from truthtelling, for any bids of others

Side payments not allowed

Weak Pareto improvements not considered

Theorem 3: Any greedy heuristic auction is Weakly

Group Strategy-Proof.

Generalizes Mehta, Roughgarden, Sundararajan (2007)

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Proof of WGSP

No assigned (“losing”) bidder can be in the deviating coalition

⟹ Deviation cannot affect payments to winners (determined

by losers’ bids) unless it changes the assignment

Consider the first round of the heuristic affected by deviation

Losers are truthful ⟹ bidder supposed to be assigned in

this round must have underbid to remain unassigned

⟹ his current threshold price < his value

⟹ his final threshold price can’t be any higher

⟹ he does not gain from the deviation

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Paid-as-Bid vs. Threshold Auction:

Full-Information Equivalence

Theorem 4. A paid-as-bid auction whose assignment rule is

computed by a greedy algorithm, for any vector of values, has

a full-information Nash equilibrium in which losers bid their

values and winners bid their threshold prices.

The equilibrium assignment and payments are the same as in

the corresponding threshold auction.

Proof:

A winner has no profitable deviation: its threshold price >

value, and is the highest payment it could get.

A loser has no profitable deviation: to win it would have to

bid at most its threshold price < value.

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Ruling out other NE outcomes

Definition: An assignment rule is non-bossy if a bidder

cannot affect assignment without changing his own.

Prevents losers (who are indifferent) from affecting

allocation

Winners are always non-bossy in a greedy heuristic

Examples:

Surplus-maximizing assignment

“Stationary” greedy algorithms:

bidders’ scores are fixed (e.g., score = bid/population)

feasibility checking is “static” (feasibility of a set S is history-

independent) and “monotone” (S is feasible ⟹ so is any subset of S)

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Dominance-Solvability of Paid-as-Bid

Auctions

An auction is dominance-solvable if, under full information, iterated

deletion of dominated strategies yields a unique outcome (allocation and

winning bids).

Non-bossiness ⟹ order of deletion doesn’t matter (Marx-Swinkels)

Theorem 5. Consider a paid-as-bid non-bossy monotonic auction with finite

bid spaces.

1. The auction is dominance-solvable if and only if it can be implemented

via a greedy heuristic.

2. In this case, the outcome in (1) is also a unique Nash equilibrium outcome

in undominated strategies.

3. In one bid profile consistent with both iterated dominance and

undominated Nash, losers bid “value+” and winners bid threshold prices

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What about the Vickrey auction?

It is strategy-proof ⟹ a threshold mechanism

Definition: Bidders are substitutes in the assignment

rule if raising one bid cannot cause another to lose.

Theorem: Any monotonic assignment rule in which

bidders are substitutes can be implemented with a

clock auction (⟺ greedy threshold auction).

Proof: decrement price to a bidder who would lose

given the current prices and already-rejected bids

Substitutes ⟹ Since other active bids can only go

down, this bidder could never win at his current bid

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Vickrey with Complementarity 39

A C B

One channel available ⟹ can assign either A+B or C

Optimization CANNOT be achieved via greedy heuristic or a clock auction

A+B < C ⟹ C assigned, Vickrey prices pA= C - B, pB = C - A

NOT group strategy-proof: A,B maximize each other’s prices by bidding 0

Pays “too much”: pA + pB = 2C-A-B > C = cost of “truthful” full-info Nash

equilibrium of paid-as-bid optimizing auction (Bernheim-Whinston 1986)

Simulations

Complementarities are present

However, greedy heuristic outcome with good

feasibility checking looks “close” to Vickrey in

efficiency and cost

Cost may be even < Vickrey cost if scoring is used to

curb stations’ inforents

Conjecture: large number of channels (e.g. at least 16

for UHF) creates substitutability that outweighs

complementarities

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Extensions

Post-Auction Resale

Multi-minded bidders

Clearing Rule

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Post-Auction Resale

Consider an isolated region with n identical stations, of which

we must clear exactly k

Greedy heuristic (= Vickrey) clears the k lowest-value stations

at the (k+1)st –lowest value

Price = the highest post-auction equilibrium market price of stations

Truthful bidding in the auction is “resale-proof”

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price

Resale of Heterogeneous stations

“Resale-proofness” generally not achievable nor desirable

Resale can raise efficiency by moving programming across “sticks”

Example: liquid post-auction resale market will value “sticks”

proportionally to their coverage “pops”

⟹ efficiency means maximizing total on-air channel*pop

(= average # of channels per resident)

Under full information, scoring almost entirely by value/pop eliminates

all inforents, and can get close to efficiency

Dispersed common-value information can be aggregated via a clock

auction with information feedback (as in Milgrom-Weber)

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Multi-minded Bidders

A clock auction quoting bidder-specific

prices for different bidding options may

permit bidders to switch bidding option

as prices fall

Strategy-proofness is lost for such bidders

But incentives to manipulate may be small in

large markets

Similarly for owners of multiple stations

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Clearing Rule: Efficiency-Revenue Trade-Off

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Theorem (Segal-Whinston 2012, generalizing Myerson-Satterthwaite):

Independent Private Values

Each agent has an “opt-out type,” whose non-participation is efficient regardless

of the others’ types

The core is nonempty with prob. 1, multivalued with prob. > 0

Then any efficient voluntary mechanism runs an expected

deficit.

Proof idea:

To ensure incentives and voluntary participation, each agent must get at least his

expected marginal contribution to the total surplus

Multivalued core ⟹ marginal contributions add up to more than the total surplus

To yield revenues, must reduce trade

E.g. McAfee (1992): prohibit one least valuable trade

An “Interleaved” Double Auction

(Uniform-Product Illustration) 46

Net Revenue Target

Reverse Price

Forward Price

Quantity

Traded

LOSS

TV

spectrum

supply

Broadband

spectrum

demand

Conclusion

A heuristic, interleaved clock double auction

approach to spectrum repurposing

Things to do:

A good “feasibility checker” for TV channel repacking:

to reduce cost/maximize clearing s.t. net revenue target

Allow other types of bids: accept interference, channel-

share

Lose exact strategy-proofness

Allow non-uniform regional clearing to sidestep

“holdout” stations in scarce-spectrum areas?

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