DEFERRED ACCEPTANCE AUCTIONS AND RADIO SPECTRUM REALLOCATION

Transcription

1 DEFERRED ACCEPTANCE AUCTIONS AND RADIO SPECTRUM REALLOCATION PAUL MILGROM AND ILYA SEGAL STANFORD AND AUCTIONOMICS 1 June 26, 2014

2 INCENTIVE AUCTIONS R&O

3 September 28, 2012 F.C.C. BACKS PROPOSAL TO REALIGN AIRWAVES 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. June 26,

4 OVERALL STRUCTURE 1. Reverse auction buys TV broadcast rights. 2. Forward auction sells mobile broadband rights. 3. Clearing mechanism jointly determines how many channels to clear and licenses to sell in the two auctions. 4. Broadcasters that do not sell can be moved to new channels. June 26,

5 WHY SHOULD BE FCC BE INVOLVED? Why is any intermediary needed at all? Q: Why not just sell or give to broadcasters the full right to use their spectrum as they desire and allow trading to guide the allocation? A: Coordinated action of many parties is required to reorganize the spectrum in a way that respects complex, multilateral interference and other engineering requirements. Is a government agency the right intermediary? Q: Why not just sell or give an overlay license to a private party, who can then coordinate and organize the other parties? A: A successful intermediary must have and exercise power to specify standards for reasonable interference and to require broadcasters to relocate to different channels. June 26,

6 THE REPACKING CHALLENGE Broadcast interference constraints described by a graph. Stations as nodes in the graph Arcs for pairs of stations that cannot be assigned to the same channel. A set of bids is acceptable if the rejected bidders and nonparticipating stations can be repacked into the reduced set of available UHF channels. Testing the acceptability of a single set of bids is an NP-hard graph-coloring problem with about 130,000 constraints. We expect to test at least 1,000,000 combinations during the course of the reverse auction. June 26,

7 CO-CHANNEL INTERFERENCE CONSTRAINTS OET-69 Bulletin Coverage: 10 million cells (1km x 1km ) June 26, ,000 pairwise constraints 7

8 NON-UNIFORM GEOGRAPHY Midwest New York LA June 26,

9 A SPRING-LOADED MAP June 26,

10 SAT ENCODING FOR FEASIBILITY CHECKING Kevin Leyton-Brown s formulation Variable x s,c station s is assigned to channel c: one such variable for every station and channel. Station s must be on one of its allowable channels For every station s, a clause of the form (x s,c1) (x s,c2) Station s may broadcast on at most one channel For each pair of channels (x s,c1) (x s,c2) No harmful interference on pairs: (x s1,c1) (x s2,c2) Mostly 2-clauses, good for unit propagation. June 26,

11 ALGORITHM CONFIGURATION High-performance solvers for NP-complete problems like SAT are typically parameterized which branching heuristic, variable ordering, preprocessing strategy, clause learning technique, Address with algorithm configuration

12 SEQUENTIAL MODEL-BASED ALGORITHM CONFIGURATION (SMAC) [Hutter, Hoos & Leyton-Brown, 2011]

13 PERFORMANCE IMPROVEMENT!!

14 REVERSE AUCTION TASK Assume that each bidder owns a single stations and knows it value for its stations. Reasonable, because Encourage participation! Auction should be simple and obviously strategy-proof for single-station, single-minded bidders and, preferably, group-strategy-proof. In order to maximize clearing and raise net revenue, keep cost of acquiring stations low. June 26,

15 AN ADDITIONAL CHALLENGE For UHF station owners in the reverse auction, multiple options. 1. Cash for agreeing to go off-air 2. Less cash plus a LVHF license 3. Still less cash plus a HVHF license For VHF station owners, a chance to sell the LVHF or UVHF license. but don t add excessive complexity! June 26,

16 WHY NOT USE A VICKREY AUCTION? First, Vickrey prices are hard to compute. There are more than 2000 stations, so average station price is 0.05% of the total value. Vickrey price is the difference between the value of the total allocation with or without one station: the difference of two numbers on the order of 100% of total value. So, even tiny percentage errors in computing the maximum total value can lead to large percentage pricing errors. Second, Vickrey prices are hard to verify & explain. Third, Vickrey auctions are not group strategy-proof. 16

17 LITERATURE REVIEW COMPUTER SCIENCE AND ECONOMICS June 26,

18 HEURISTIC AUCTIONS The seminal paper is by Lehmann, O Callaghan, and Shoham (2002). Combinatorial auction for knapsack problem Single-minded bidders seek to buy space in the knapsack Goal: design a computationally simple, strategy-proof auction. LOS proposed a greedy acceptance auction Rank bids by price/unit Use a greedy algorithm to select acceptable bids Threshold prices: Each winning buyer pays the lowest price it could bid and still be winning. Related heuristic mechanisms. Cost-sharing: Moulin (1999), Mehta et al. (2007), Juarez (2007). Budget-constrained DA mechanism: Ensthaler and Giebe (2010,2011). June 26,

19 RELATED ECONOMICS Ausubel, Lawrence (2006). An Efficient Dynamic Auction for Heterogeneous Commodities. American Economic Review 96(3): Bikhchandani, Sushil, Sven de Vries, James Schummer and Rakesh Vohra (2011), An Ascending Vickrey Auction for Selling Bases of a Matroid, Operations Research 59(2): Hatfield, John and Paul Milgrom (2005), Matching with Contracts, American Economic Review 95(4): Kelso, Alexander and Vincent Crawford (1982), Job Matching, Coalition Formation and Gross Substitutes, Econometrica 50(6): June 26, 2014 Auctionomics 19

20 DEFERRED ACCEPTANCE ALGORITHMS THEORY FOR THE SINGLE-OPTION CASE June 26,

21 BIDS AND THRESHOLD PRICES Each bidder has a finite set of possible bids (positive real numbers). Each bidder s possible values lie in a compact interval [0,B]. Given a bid profile b, the set of winning bids is α(b). Bidder j s threshold price is its highest bid that, given the other bids, would be winning in this procurement auction. June 26,

22 STRATEGY-PROOF AUCTIONS Definition: An auction is strategy-proof if, for almost all values, reporting true value rounded up to the next higher bid is a weakly dominant strategy. Theorem: The auction (α(. ),p(. )) is strategy-proof if and only if α is monotonic and p is its threshold pricing function. June 26, 2014 Auctionomics 22

23 DEFERRED ACCEPTANCE ALGORITHMS Notation Finite set of bidders N with typical element i. Product set of bid profiles B, a finite subset of R N + Possible bidder values [0,v i ] A = set of still-active bidders during the algorithm Scoring function s ia : B i xb N\A à R + (and realized scores) Any constraints are coded into in the scoring function. Different DA algorithms distinguished by their scoring functions. Deferred acceptance algorithm 1. Initialize round number t=1 and A 1 = N 2. If for all i in A t, s A i 0, stop and accept bidders in A t. 3. Set A t+1 = A t argmax i s A i. ( Reject a bid that is too high. ) 4. Increment t and return to step 2. June 26,

24 HOW TO USE A SCORING FUNCTION As examples, a bidder s score might 1. be its bid (becoming zero when the bidder becomes infeasible), replicating a greedy rejection algorithm. 2. be its bid divided by its degree in the interference graph, for better heuristic packing efficiency. 3. depend on the set of remaining items, to adapt to a limited budget. 4. depend on previously rejected bids, to allow benchmark pricing in areas with little supply competition. 5. depend on the population covered, for example to proxy the cost of accepting impairments. June 26,

25 INFORMAL SUMMARY: MAIN RESULTS 1. Every DA threshold auction is strategically equivalent to some descending clock auction with cut-off strategies, and vice versa. 2. Every descending clock auction (with or without the cut-off restriction) are weakly group strategy-proof. 3. The price and allocation from any DA threshold auction is the same as the unique full-information undominated Nash equilibrium outcome of its corresponding paid-asbid auction. ( Revenue equivalence ) 4. Non-bossy paid-as-bid auctions are generically dominance solvable if and only if they are DA auctions. June 26,

26 NON-BOSSY MECHANISMS Definition: An assignment α:r +N à {0,1} N is bossy if for some bid profiles b and c and some bidder j, α j (b)=α j (b -j,c j ) and α(b) α(b -j,c j ). non-bossy otherwise. Examples Non-bossy rules Surplus-maximization. Simplest DA rule Bossy rules (using B N\A ) Budget-constrained scoring. Yardstick competition. June 26, 2014 Auctionomics 26

27 DESCENDING CLOCK AUCTIONS Notation Active bidders at round t: A t Cumulative history: A t =(A 1,,A t ) Set of possible histories: H. Pricing function: p:hà R N such that p(a t ) p(a t-1 ). E t is the set of active bidders who exit at round t. Clock algorithm Set prices p(a t ) for round t. If p(a t )=p(a t-1 ), the auction ends and assignments are made. Otherwise, A t+1 = A t E t, increment t, and iterate. Information: bidders know own price only. (Other possibilities.) June 26,

28 EQUIVALENCE MAPPING Implement the DA algorithm as follows. 1. At round t of the procedure, determine which bidder j would be rejected next if all bidders were to make their maximum still-available bids: call that bid p jt. 2. If bidder j s bid amount is p jt, reject it. Otherwise, eliminate bid p jt from j s bid set. 3. Iterate. Consider a clock algorithm that mimicks this procedure by reducing the price to bidder j at round t to p jt, leaving other prices unchanged, and checking whether b jt =p jt. This mapping is invertible to find a DA auction for any clock auction. June 26,

29 EQUIVALENCE THEOREM Cutoff strategy: bidder j plans to exit at any price below b j (using only own current price information). Theorem 2. Every deferred acceptance threshold auction is strategically equivalent to some descending clock auction with cutoff strategies, and Every descending clock auction with cutoff strategies is strategically equivalent to some deferred acceptance threshold auction. June 26,

30 CLOCK AUCTION ADVANTAGES Strategy-proofness is obvious Fewer bidding mistakes: experimental evidence. Clock auction variations can reveal more information to bidders. Improves bidder confidence. Bidders with low values may not need to compute their values exactly to bid optimally. Windfall profits are never observed June 26,

31 QUANTITY ADJUSTMENT AND CLEARING (UNIFORM-PRODUCT ILLUSTRATION) Reverse Price TV spectrum supply LOSS Net Revenue > Target Forward Price Broadband spectrum demand Quantity Traded June 26,

32 GROUPS OF BIDDERS Definition: An auction is group strategy-proof if there is no value profile v and coalition S such that S has a strict Pareto-improving deviation from truth-telling. Side payments among S not allowed Weak Pareto improvements not considered Theorem: Every clock auction is group strategyproof. Generalizes Mehta, Roughgarden, Sundararajan (2007) 32

33 PROOF OF GROUP STRATEGY-PROOFNESS Losing bidders determine the prices. So, a coalition S can gain only if some loser in S becomes winning. Consider the first clock round altered by the deviation. No currently assigned ( losing ) bidder can be in S a bidder j who would otherwise have been assigned at this step is in S Bidder j bid less than his value and would have been assigned if truthful the current clock price is equal to his value his final threshold price must be lower he is either assigned or paid less than his value. 33

34 PAID-AS-BID COMPARISONS OUTCOME EQUIVALENCE June 26,

35 EXPECTED COSTS Literature: Bernheim-Whinston menu auction analysis finds paid-asbid auction equilibrium outcomes are in the core. Ausubel-Milgrom analysis finds that bidders Vickrey payoffs are always weakly higher and seller s payoff weakly lower than any core payoffs. So, with this equilibrium selection and complete information, Vickrey cannot perform better and can perform worse. Are incentives costly for deferred acceptance auctions as well? June 26,

36 DOMINANCE-SOLVABILITY OF PAID-AS-BID AUCTIONS An auction is (full-information) dominance-solvable if iterated deletion of weakly dominated strategies in any order yields a unique allocation and unique prices. Non-bossiness order of deletion doesn t matter (Marx- Swinkels TDI condition) Theorem 5. Assume discrete bids and finite reserve prices. One bid profile consistent with both iterated elimination and undominated Nash has each station bid the greater of its value+ and its threshold price. A paid-as-bid non-bossy monotonic auction is dominance-solvable if and only if it is a deferred acceptance auction has a a unique undominated Nash equilibrium outcome 36

37 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 37

38 PROOF OF ONLY IF Start by eliminating dominated bids below values. Then winning is strictly preferred to losing. Bid bi is dominated by b i ' > b i the change never affects bidder i s winning (by monotonicity) it never affects others winning (by non-bossiness) Bid bi is dominated by b i ' < b i bi never wins all bids above bi never win either Hence until unique outcome is established, we can find one bidder who never wins unless his highest possible bid is reduced can decrement clock price to this bidder in this round 38

39 OPTIMAL AUCTIONS USING DEFERRED ACCEPTANCE ADAPTING THE MYERSON ANALYSIS June 26,

40 OPTIMAL AUCTIONS FORMULATION Formulate as follows min x,t E TotalPayment(x,t) = min x,t E J t j (c)x j (c) j=1 subject to (x,t) satisfies Incentive Constraints (for all c j ) Participation Constraints (for all c j ) Feasibility Constraints (x(c) S for all c) Observations (IC) identical to Myerson s model (PC) reversed but standard Analysis begins identically to Myerson s model June 26,

41 OPTIMAL & HEURISTIC SOLUTIONS Optimal auction assignments: regular case In every realization, maximize the total virtual value of stations assigned to continue broadcasting, subject to the clearing target and assignment feasibility. DA threshold auctions: regular case Apply the deferred acceptance heuristic to find a nearoptimal solution to the virtual value maximization problem. June 26,

42 END June 26,