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Winner determination and pricing           Peter Cramton
Outline of CCA • Auction begins with a clock stage for price discovery     Auctioneer names prices; bidders name packages...
Winner determination • Select allocation that maximizes total value (as bid)     Accept at most one package bid from each...
Motivation for pricing rule• Second pricing lets bidders bid full value    Price is set by competitive bids of others• Si...
Bidder-optimal core pricing • Core = allocation and payments such that     Efficient: Value-maximizing allocation     Un...
Five-bidder example with bids on {A,B}• Strong bidder on A: b1{A} = 28                                                    ...
The Core: Efficient outcome and payments thatrespect competitive constraints                           b4{A} = 14         ...
Vickrey prices: How much can each winner’s bidbe reduced (while holding others fixed)?                                 b4{...
Bidder-optimal core prices: Jointly reduce winningbids as much as possible (while remaining in core)                      ...
Core point closest to Vickrey prices: a simple wayto determine unique bidder-optimal core prices                          ...
Extra above Vickrey is proportionate to packagesize (nearest Vickrey weighted by size)                                  b4...
Use of optimization in CCA • Optimization methods are required in package auctions     To determine the value-maximizing ...
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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

Published in: Technology, Business
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Transcript of "Winner determination and pricing rule - Peter Cramton - Power Auctions"

  1. 1. Winner determination and pricing Peter Cramton
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
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