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U.S. Spectrum Reallocation and Heuristic Auctions Paul Milgrom and Ilya Segal December 2012 1.

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

2 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.

3 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”. 3

4 Background  “ A Proposal for a Rapid Transition to Market Allocation of Spectrum, ” Evan Kwerel and John Williams, OPP Working Paper 38,  National Broadband Plan, 2010 (pp )  Middle Class Tax Relief and Job Creation Act, February 16, 2012, Sec  “Straw man” Appendix to FCC’s Notice for Proposed Rule Making, Ausubel, Levin, Milgrom (team leader), Segal, September

5 What kind of “Commodity” is Radio Spectrum? 5

6 TV broadcast licenses 6  Each channel uses 6MHz of spectrum in one of three bands Repurposed in DTV transition

7 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)

8 Interference Constraints 8 OET-69 Bulletin Coverage: ≈ 2 million cells (2km x 2km ) Pairwise constraints (0.5% threshold): ≈130,000 edges

9 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

10 FCC’s role in spectrum reallocation? 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

11 “New Paradigm for Spectrum Policy” 11 FCC’s previous auctions: Incentive Auction: (Commissioner Robert McDowell)

12 “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  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)

13 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? 13

14 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. 14

15 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 b s for its bidding option.  Set of bids to reject:  Stations in S* receive no payment.  Other bids are accepted, and paid:

16 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

17 17 Greedy Heuristic Auctions

18 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 18

19 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

20 Threshold Prices 20  An assignment rule is monotonic if for any bidder j, increasing his bid b j 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 p j = p j (b -j ) such that j loses if b j > p j and wins if b j < p j.

21 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.

22 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 p jn = his highest bid that would not be rejected in that round.  When the algorithm terminates, for each accepted bid j, the threshold price is p j = min n p jn

23 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 23

24 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 24

25 Greedy Threshold Auctions 25 Descending Clock Auctions (assuming finite bid space)

26 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 26

27  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 27

28  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 28

29 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) 29

30 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) 30

31 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 31

32 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. 32

33 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) 33

34 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 34

35 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 35

36 Proof of “Only If”  Start by eliminating dominated bids below values. Then winning is strictly preferred to losing.  Bid b i is dominated by b i ' > b i  the change never causes bidder i to lose  it never affects his winning (by monotonicity)  it never affects outcome (by non-bossiness)  Bid b i is dominated by b i ' < b i b i never wins  all bids above b i 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 36

37  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 37

38 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 38

39 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 p A = C - B, p B = C - A  NOT group strategy-proof: A,B maximize each other’s prices by bidding 0  Pays “too much”: p A + p B = 2C-A-B > C = cost of “truthful” full-info Nash equilibrium of paid-as-bid optimizing auction (Bernheim-Whinston 1986)

40 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 40

41 Extensions  Post-Auction Resale  Multi-minded bidders  Clearing Rule 41

42 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” 42 price

43 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) 43

44 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 44

45 Clearing Rule: Efficiency-Revenue Trade-Off 45  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

46 An “Interleaved” Double Auction (Uniform-Product Illustration) 46 Net Revenue Target Reverse Price Forward Price Quantity Traded LOSS TV spectrum supply Broadband spectrum demand

47 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? 47

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