U.S. Spectrum Reallocation and Heuristic Auctions Paul Milgrom and Ilya Segal

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U.S. Spectrum Reallocation and Heuristic
AuctionsPaul Milgrom and Ilya Segal

• December 2012

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F.C.C. Backs Proposal to Realign Airwaves

• 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 havent 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.

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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 FCCs Notice for Proposed
  Rule Making, Ausubel, Levin, Milgrom (team
  leader), Segal, September 2012

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What kind of Commodity is Radio Spectrum?
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TV broadcast licenses

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

Repurposed in DTV transition
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Each of 2,500 TV licenses includes

• 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)

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Interference Constraints




















Pairwise constraints (0.5 threshold) 130,000
edges
OET-69 Bulletin Coverage 2 million cells (2km
x 2km )
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Broadband (mobile) licenses

• 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

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FCCs role in spectrum reallocation?

• Allocate by administrative authority?
• 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.
• Market Design approach
• Define spectrum and interference rights (e.g.
  FCCs right to retune) to minimize holdout,
  promote competition
• Market mechanism for spectrum allocation with
  simple participation and minimal scope for gaming

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New Paradigm for Spectrum Policy

• FCCs previous auctions
• Incentive Auction
• (Commissioner Robert McDowell)

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Reverse Auction Buying TV Licenses

• 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)

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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
  Myersons 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?

• Broadcasters 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

• 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

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Vickrey Computational Problems

• 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 auctions supposed
  efficiency

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Greedy Heuristic Auctions
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A Greedy Heuristic Algorithm

• A (possibly imperfect) method to check whether a
  set of bids can be feasibly rejected assigned
  to their home bands (with repacking).
• A scoring function to prioritize bids.
• Each bidders 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

• 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

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Threshold Prices

• 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 js threshold
  price is the unique amount pj pj(b-j) such that
  j loses if bj gt pj and wins if bj lt pj.

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Characterization of Strategy-Proof Auctions

• 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.

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Greedy Threshold Auction

• 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

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Nice Properties of Greedy Threshold Auctions

• Computationally Simpler
• Strategy-Proof
• Equivalent to Descending Clock Auctions
• (Weakly) Group Strategy-Proof
• Outcome-equivalent to full-info Nash equilibrium
  of paid-as-bid auction with same assignment rule
• i.e. threshold pricing may not cost us
• 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, OCallaghan, 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
Descending Clock Auctions
(assuming finite bid space)
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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 bidders 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 bidders 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 lt his value
• ? his final threshold price cant 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 gt 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 lt
  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 doesnt matter
  (Marx-Swinkels)
• Theorem 5. Consider a paid-as-bid non-bossy
  monotonic auction with finite bid spaces.
• The auction is dominance-solvable if and only if
  it can be implemented via a greedy heuristic.
• In this case, the outcome in (1) is also a unique
  Nash equilibrium outcome in undominated
  strategies.
• In one bid profile consistent with both iterated
  dominance and undominated Nash, losers bid
  value and winners bid threshold prices

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Proof of If

• Start by eliminating dominated bids below values.
  Then winning is strictly preferred to losing.
• As clock prices descend, quitting to lose is
  undominated/in support of NE only for sure
  losers, who might as well bid value (by
  non-bossiness)
• When clock stops, this leaves winners, for whom
  bidding below the final clock price ( threshold
  price) is dominated

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Proof of Only If

• Start by eliminating dominated bids below values.
  Then winning is strictly preferred to losing.
• Bid bi is dominated by bi' gt bi
• ? the change never causes bidder i to lose
• ? it never affects his winning (by monotonicity)
• ? it never affects outcome (by non-bossiness)
• Bid bi is dominated by bi' lt bi ? bi never wins
• ? all bids above bi never win (by monotonicity)
• Until unique outcome is established, we can find
  one bidder whose highest remaining bid never wins
  ? can decrement clock price to this bidder in
  this round

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• Corollary. A paid-as-bid surplus-maximizing
  auction is dominance-solvable if and only if it
  has the substitutes property.
• Proof
• Surplus maximization ? non-bossiness,
  monotonicity
• For surplus maximization, implementation via a
  greedy heuristic is equivalent to the substitutes
  property

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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
C
A
B

• One channel available ? can assign either AB or
  C
• Optimization CANNOT be achieved via greedy
  heuristic or a clock auction
• AB lt C ? C assigned, Vickrey prices pA C - B,
  pB C - A
• NOT group strategy-proof A,B maximize each
  others prices by bidding 0
• Pays too much pA pB 2C-A-B gt C cost of
  truthful full-info Nash equilibrium of
  paid-as-bid optimizing auction (Bernheim-Whinston
  1986)

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Simulations

• Complementarities are present
• However, greedy heuristic outcome with good
  feasibility checking looks close to Vickrey in
  efficiency and cost
• Cost may be even lt 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 (k1)st lowest
  value
• Price the highest post-auction equilibrium
  market price of stations
• Truthful bidding in the auction is resale-proof

price
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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
  channelpop ( 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

• 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. gt 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

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An Interleaved Double Auction(Uniform-Product
Illustration)
TV spectrum supply
LOSS
Broadband spectrum demand
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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?