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Winner determination and pricing rule - Peter Cramton - Power Auctions

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Presentation by Professor Peter Cramton of Power Auctions to the January 2012 digital dividend stakeholder workshops

Presentation by Professor Peter Cramton of Power Auctions to the January 2012 digital dividend stakeholder workshops

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  • 1. Winner determination and pricing Peter Cramton
  • 2. Outline of CCA • Auction begins with a clock stage for price discovery  Auctioneer names prices; bidders name packages  Price increased if there is excess demand  Process repeated until no excess demand • Supplementary bids collected in a final sealed-bid round  Improve clock bids  Bid on other relevant packages • Optimization to determine allocation and prices based on all the clock bids and the supplementary bids • Assignment stage to determine specific assignment 2
  • 3. Winner determination • Select allocation that maximizes total value (as bid)  Accept at most one package bid from each bidder  Standard “set packing” problem  Solved with commercial optimizers (CPLEX and GLPK)  Unique solution found from tie-breaking rule  If tie, favor final clock allocation  If still tie, favor fewer unsold lots  If still tie, then randomize 3
  • 4. Motivation for pricing rule• Second pricing lets bidders bid full value  Price is set by competitive bids of others• Single item with second pricing  In ascending auction stay in until your value is reached  Bidder with the highest value wins and pays second-highest bid  In sealed-bid auction bid full value  Bidder with the highest value wins and pays second-highest bid• Second pricing generalizes to auctions with many items  Vickrey pricing or a variant called bidder-optimal core pricing  Price is set by competitive bids of others 4
  • 5. Bidder-optimal core pricing • Core = allocation and payments such that  Efficient: Value-maximizing allocation  Unblocked: No collection of bidders offered seller a better deal • Bidder-optimal core pricing finds the smallest payments that are consistent with the core  Examples  For the auction of a single item, bidder-optimal core pricing is identical to second pricing: the high bidder wins and pays the second-highest price  For auctioning many items that are substitutes, bidder-optimal core pricing is identical to Vickrey pricing: the winners form the value- maximizing assignment and each pays the opportunity cost of its winnings  For auctioning many related items with both substitutes and complements, then bidder-optimal core prices may be higher than Vickrey prices in order to respect competition constraints 5
  • 6. Five-bidder example with bids on {A,B}• Strong bidder on A: b1{A} = 28 Winners• Strong bidder on B: b2{B} = 20• Bidder needing both: b3{AB} = 32 Vickrey prices: p1= 14• Weak bidder on A: b4{A} = 14• Weak bidder on B: b5{B} = 12 p2= 12 The sum of bidder 1 and 2’s values exceeds the value to bidder 3, as well as the sum of values of bidder 4 and 5. Simple second pricing results in prices of 14 for A and 12 for B. Simple second pricing fails to account for the impact of bidder 3’s package bid. 6
  • 7. The Core: Efficient outcome and payments thatrespect competitive constraints b4{A} = 14 b1{A} = 28 Bidder 2 Payment b3{AB} = 32 Efficient outcome 20 b2{B} = 20 The Core 12 b5{B} = 12 Bidder 1 Payment 14 28 32 7
  • 8. Vickrey prices: How much can each winner’s bidbe reduced (while holding others fixed)? b4{A} = 14 b1{A} = 28 Bidder 2 Payment b3{AB} = 32 b2{B} = 20 20 The Core 12 Vickrey b5{B} = 12 prices Problem: Bidder 3 offered to pay more (32 > 26)! Bidder 1 Payment 14 28 32 8
  • 9. Bidder-optimal core prices: Jointly reduce winningbids as much as possible (while remaining in core) b4{A} = 14 b1{A} = 28 Bidder 2 Payment b3{AB} = 32 b2{B} = 20 20 The Core 12 Vickrey b5{B} = 12 prices Problem: bidder- optimal core prices are not unique! Bidder 1 Payment 14 28 32 9
  • 10. Core point closest to Vickrey prices: a simple wayto determine unique bidder-optimal core prices b4{A} = 14 b1{A} = 28 Bidder 2 b3{AB} = 32 Payment Reserve price A = Reserve price B b2{B} = 20 20 Unique prices determined 15 12 Vickrey b5{B} = 12 prices Each pays proportionate share above Vickrey Bidder 1 Payment 14 17 28 32 10
  • 11. Extra above Vickrey is proportionate to packagesize (nearest Vickrey weighted by size) b4{A} = 14 b1{A} = 28 Bidder 2 b3{AB} = 32 Payment Reserve price A = 2 x Reserve price B b2{B} = 20 20 Unique prices determined 14 12 Vickrey b5{B} = 12 prices A winner pays $4 extra B winner pays $2 extra Bidder 1 Payment 14 18 28 32 11
  • 12. Use of optimization in CCA • Optimization methods are required in package auctions  To determine the value-maximizing assignment  To price the winning packages • These methods are now commonly used in government auctions, especially in spectrum auctions • The algorithms result in a unique determination of both assignment and prices • Solutions are well-tested and verified • Thus, a high level of transparency is still maintained • And activity rule assures a high level of price and allocation discovery in the clock stage 12