1. 1 Multiunit Auctions Part II Thanks to Larry Ausubel and especially to Peter Cramton for sharing their notes.

2. 2 The Vickrey Auction: Lovely but Lonely? Potential Problems: If bidder valuations are not substitutes, then we can have…. –Potential equilibria with no or zero revenue –Seller’s revenue non-monotone in set of bidders and amounts bid –Vulnerable to collusion by a coalition of losing bidders –Vulnerable to use of multiple bidding identities by a single bidder –Vickrey payoffs not in the core General problems –Loses incentive properties with budget constraints As do other mechanisms-but then tradeoffs are important

3. 3 Zero Revenue Possibility ObjectsBidder 1Bidder 2Bidder 3 A alone022 B alone022 A and B222 Vickrey revenue: 0 Ascending uniform-price auction for package: 2

4. 4 Non-Monotonicity When Removing Bidder 3 ObjectsBidder 1Bidder 2Bidder 3 A alone022 B alone022 A and B222 Vickrey revenue: 2 Bidders can exploit these features, undermining both revenue and efficiency…

5. 5 Incentives for Collusion ObjectsBidder 1Bidder 2Bidder 3 A alone0.5 B alone0.5 A and B2.5 Vickrey revenue: 1 Bidders 2 and 3 can collude and each report 2 instead of.5, and win and pay price 0

6. 6 Incentives for Shill Bidding ObjectsBidder 1Bidder 2Bidder 3 A alone0.5 B alone0.5 A and B21.5 Vickrey revenue: 1 Bidder 2 can create a copy of himself, bid 2 for a single unit in each case, and win with price 0

7. 7 Distorted Merger Incentives ObjectsBidder 1Bidder 2Bidder 3Bidder 2 and 3 merged A alone0222 B alone0222 A and B2225 Merger is efficient, but raises bidders’ payments from 0 to 2. So bidders don’t go forward with merger.

8. 8 The “Core”: A Cooperative Game Theory Concept Definition: Set of feasible allocations that cannot be improved upon by a subset (a coalition) of the economy's consumers. –A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle. Captures the idea of stability

9. 9 Vickrey Payoffs and the Core Theorem. If the Vickrey payoff vector π is in the core, then it is the bidder-dominant point in the core. If the Vickrey payoff vector π is not in the core, then there is no bidder-dominant point in the core and the seller’s Vickrey payoff is strictly less than the smallest of the seller’s core payoffs. Bidder preferences satisfy substitutes property implies Vickrey is in core.

10. 10 5 bidder example with bids on {A,B} b 1 {A} = 28 b 2 {B} = 20 b 3 {AB} = 32 b 4 {A} = 14 b 5 {B} = 12 Winners Vickrey prices: p 1 = 14 p 2 = 12

11. 11 The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment 14 12 3228 20 The Core Efficient outcome

12. 12 The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment Vickrey prices 14 12 3228 20 Vickrey prices: How much can each winner’s bid be reduced holding others fixed? Problem: Bidder 3 can offer seller more (32 > 26)!

13. 13 The Core b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment Vickrey prices 14 12 3228 20 Bidder-optimal core prices: Jointly reduce winning bids as much as possible Bidder-optimal core Problem: bidder- optimal core prices are not unique!

14. 14 Unique core prices b 4 {A} = 14 b 3 {AB} = 32 b 5 {B} = 12 b 1 {A} = 28 b 2 {B} = 20 Bidder 2 Payment Bidder 1 Payment Vickrey prices 14 12 3228 20 Core point closest to Vickrey prices 17 15 Minimize incentive to distort bid!

15. 15 Why core pricing? Truthful bidding nearly optimal –Simplifies bidding –Improves efficiency Same as Vickrey if Vickrey in core (substitutes) Avoids Vickrey problems with complements –Prices that are too low Revenue is monotonic in bids and bidders Minimizes incentive to distort bids

16. 16 Auctions for Heterogeneous Units with Complements Spectrum auctions Bus routes Online advertising

17. 17 Exercise Two bidders (L & R), two items (A & B) –L needs both (value = 2 times birth month) –R needs one (value = birth month) Three auction formats (bidding on individual items!) –Simultaneous first price –Simultaneous second price –Simultaneous ascending auction

18. 18 Exposure problem exercise: Optimal strategy in SAA

19. 19 Simultaneous Ascending Auction Auction rules Simultaneous –All lots at the same time Ascending –Can raise bid on any lot Stopping rule –All lots open until no bids on any lot Activity rule –Must be active to maintain eligibility

20. 20 Strategy in SAA Negotiate split of licenses Accommodate and retaliate –Stake a reasonable claim –Punish intruders

21. 21 Simultaneous ascending auction Strengths –Simple price discovery process –Allows arbitrage across substitutes –Piece together desirable packages –Reduces winner’s curse Weaknesses –Demand reduction –Tacit collusion –Parking –Exposure –Hold up –Limited substitution –Complex bidding strategies

22. 22 Limited substitution: US AWS 90 MHz, 161 rounds, $14 billion

23. 23 AWS price for 10 MHz by block Day 3 Day 4 Day 5 Stage 2 Final 40% discount 6 REAGs 176 EAs 734 CMAs

24. 24 Alternatives/Improvements to SAA: Needed Enhancements Anonymous bidding Generic lots Package bidding with clock –Porter-Rassenti-Roopnarine-Smith (2003) –Ausubel-Cramton (2004) –Ausubel-Cramton-Milgrom (2006) “Second” pricing Revealed preference activity rule

25. 25 Package clock auction Auctioneer names prices; bidder names package –Price adjusted according to excess demand –Process repeated until no excess demand Supplementary bids –Revised clock bids –Bids on other packages Optimization to determine assignment/prices No exposure problem (package auction) Second pricing to encourage truthful bidding Activity rule to promote price discovery

26. 26 UK Spectrum Auctions

27. 27 UK upcoming auctions 10-40 GHz: fixed wireless or backhaul L-Band: mobile broadcast 2.6 GHz: 4G mobile wireless Requirements Technology neutral Flexible spectrum usage rights Efficient assignment

28. 28 UK (Ofcom) process Find spectrum (rule making on likely use) Design auction (and rule making) Build system Test system –Full-scale experiments with expert PhD students and realistic scenarios – Two independent optimizers Conduct bidder seminar and mock auction Auction!

29. 29 UK 2.6 GHz auction proposal 190 MHz (38 lots of 5 MHz) How much paired vs. unpaired?

30. 30 Let auction determine band plan

31. 31 Key design choices Generic 5 MHz lots –Lots are perfect substitutes Package bids –No exposure problem Clock stage –How many paired? How many unpaired? Supply = 38 –Continue until no excess demand Supplementary bids –Improve clock bids; add other packages Principal stage –Find value maximizing generic assignment and base prices Assignment stage –Contiguous spectrum –Top-up bid to determine specific assignment Activity rule

32. 32 Pricing Rule

33. 33 Pricing rule In clock stage? In assignment stage? Pay-as-bid pricing –Incentives for demand reduction, bid shading Bidder-optimal core pricing –Maximize incentives for truthful bidding

34. 34 Bidder-optimal core pricing Minimize payments subject to core constraints Core = assignment and payments such that –Efficient: Value maximizing assignment –Unblocked: No subset of bidders prefers to offer seller a better deal

35. 35 Optimization Core point that minimizes payments readily calculated –Solve Winner Determination Problem –Find Vickrey prices –Constraint generation method (Day and Raghavan 2005) Find most violated core constraint and add it Continue until no violation Tie-breaking rule for prices is important –Minimize distance from Vickrey prices

36. 36 Package Clock Auction Summary Package clock auction –Eliminates exposure –Eliminates gaming –Enhances substitution –Allows auction to determine band plan –Readily customized to a variety of settings –Many other applications (airport slot auctions) Open questions: budget constraints