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CS 286r. Computational Mechanism Design

CS 286r. Computational Mechanism Design. Issues in Scaling Up the 700 MHz Auction Design Second FCC Combinatorial Auction Conference October 27, 2001. Thanks for Karla Hoffman, GMU, for allowing use of these slides.

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CS 286r. Computational Mechanism Design

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  1. CS 286r. Computational Mechanism Design Issues in Scaling Up the 700 MHz Auction DesignSecond FCC Combinatorial Auction ConferenceOctober 27, 2001 Thanks for Karla Hoffman, GMU, for allowing use of these slides

  2. Issues in Scaling Up the 700 MHz Auction DesignWye River Conference IIOctober 27, 2001 Melissa Dunford Martin Durbin Karla Hoffman Dinesh Menon Rudy Sultana Thomas Wilson

  3. Introduction • Auction #31 rules were designed specifically for that auction • No computational problems for a 12 license auction • Seeking a design for large combinatorial auctions • Start with Auction #31, identify the problem areas in regards to scalability and then discuss alternatives

  4. Outline • Review of Auction #31 rules impacting scalability and possible alternatives • Determining maximum revenue each round • Choosing among ties • Minimum Acceptable Bid (MAB) • Mechanism for testing scalability

  5. Auction #31 Formulation • Choose among a set of bids such that: • Revenue to the FCC is maximized • Each license is awarded exactly once • No bidder has bids in a provisionally winning bid set from more than one round subject to: Ax = 1(each license awarded exactly once) Mutually Exclusive Bid Constraints

  6. Bid 1 2 3 4 5 6 7 8 Bid amt. $22e6 $12e6 $30e6 $16e6 $8e6 $11e6 $10e6 $7e6 Package ABD ABC AD C BC A D B x1 x1 1 1 0 1 0 1 0 0 1 1 + x3 0 1 + x4 0 1 0 + x4 1 0 0 1 0 0 1 1 0 0 + x7 0 1 0 + x8 0 1 0 0 = 1 A x1 + x2 + x3 + x6 = 1 B = 1 C = 1 D x3 + x5 + x6 Ax =1: Each license awarded once • This is called a set-partitioning problem. These types of problems have a very nice mathematical structure. Example: 4 licenses, 8 bids

  7. Create two “use-round” variables for Bidder A: Form the following constraints: Example: Mutually Exclusive Bid Constraints Considered Bids for Bidder A Set of bids made in Round 1: Set of bids made in Round 2:

  8. = 1 A = 1 B = 1 C = 1 D x1 + x2 + x3 + x4 + x5 - 5u(A,1) <= 0 R(A,1) x1 x1 + x3 + x4 + x4 + x7 + x8 - 3u(A,2) + x6 + x7 + x8 <= 0 R(A,2) + u(A,2) <= 1 u(A,1) UA 1 0 1 1 0 0 1 0 1 0 x1 + x2 + x3 + x6 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 -5 0 x3 + x5 + x6 0 0 0 0 0 1 1 1 0 -3 0 0 0 0 0 0 0 0 1 1 Solving Challenges: Mutually Exclusive Bid Rule • Mutually exclusive bid constraints alter the nice structure of set-partitioning (Ax =1) and make the problem harder to solve

  9. Solvability of Integer Optimizations • The size and complexity of “solvable” integer problems is growing at an extraordinary rate • In our testing most problems solve very quickly, but...

  10. Breaking Ties • Solvers work at getting an optimal solution quickly, no solver automatically provides tie information • Breaking ties randomly important for legal reasons • Cannot use a tie-breaking method that requires the generation of all tied sets (there could be millions of tied sets even for an auction with few licenses)

  11. Tie-Breaking Procedure • Ties will be broken randomly using a two-step process • First step selects a bid set that achieves the maximum revenue • Second step selects a bid set at random from all optimal bid sets

  12. subject to: Ax = 1 (each license awarded exactly once) Mutually Exclusive Bid Constraints = Maximum Revenue Choosing Randomly Among Winning Sets • Each considered bid is assigned a selection number • A bid’s selection number is the sum of n pseudo-random numbers where n is the number of licenses comprising the bid's package

  13. = 1 A = 1 B = 1 C = 1 D x1 x1 + x3 + x4 + x4 + x7 + x8 x1 + x2 + x3 + x4 + x5 - 5u(A,1) <= 0 R(A,1) - 3u(A,2) + x6 + x7 + x8 <= 0 R(A,2) 1 0 1 1 0 0 1 0 1 0 + u(A,2) <= 1 x1 + x2 + x3 + x6 u(A,1) UA 1 1 1 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 -5 0 x3 + x5 + x6 22e6 0 12e6 0 30e6 0 16e6 0 8e6 0 11e6 1 10e6 1 7e6 1 0 0 0 -3 = 37e6 22e6x1 + 12e6x2 + 30e6x3 + 16e6x4 + 8e6x5 + 11e6x6 + 10e6x7 + 7e6x8 = 37e6 0 0 0 0 0 0 0 0 1 1 22 12 30 16 8 11 10 7 0 0 Solving Challenges: Tie-Breaking • The optimization required to break ties is a harder problem to solve than the maximum revenue problem due to a machine arithmetic problem Rev

  14. Tie-Breaking: Evaluation • Current method of random tie-breaking works • Due to time considerations in large auctions, tie-breaking procedure may be postponed until later rounds • Concept of stages

  15. Minimum Acceptable Bids A minimum acceptable bid is the greater of: • i. The minimum opening bid • ii. The bidder’s previous high bid on a license/package plus X% • iii. The bidder’s previous high bid on a license/package plus a share of the revenue needed to tie the provisional winners • (Pekec, Rothkopf; Weber; Milgrom)

  16. Shortfall Formulation • Determining Shortfall Revenue for Bid i subject to: Ax = 1(each license awarded exactly once) Mutually Exclusive Bid Constraints Bid i’s Shortfall = Maximum Revenue - Shortfall Revenuei

  17. W = {winning bids} subject to: Ax = 1 (each license awarded exactly once) Mutually Exclusive Bid Constraints = Shortfall Revenuei Distributing Shortfall • Pick the second-best allocation, with bid i, with the most provisional winning bidding units, then allocate a weighted-fraction to bid i.there can be more

  18. Deficit Calculations • Determining a deficit for Bid i Part 3 MABi = BidAmti + Deficiti

  19. Scaling-up Auction #31 • Largest auction run with 10 min round delay: • 50 licenses, 25 bidders, 25 packages • Deficit calculations are the biggest bottleneck • Reason: • A deficit is calculated for every license and every constructed package of every bidder • Up to (#Licenses + #Packages) * #Bidders, calculations per round • Each deficit calculation requires solving two integer programs • On average, 99.5% of the runtime was spent calculating deficits

  20. Alternatives to Deficit Calculations • Many authors suggest removing rule iii • Auction takes too long to close at 10% increment • Raising increment percentage may create a threshold problem • Other authors suggest using shadow prices to calculate MABs. • Rassenti, Smith, Bulfin (1982) • DeMartini, Kwasnisca, Ledyard, Porter (1999) • Milgrom (2001) • Shadow prices are the dual prices of the linear programming approximation to the integer problem • Shadow prices estimate the current value of each license

  21. Alternative Rule iii Shadow Prices A minimum acceptable bid is the greater of: i. The minimum opening bid ii. The bidder’s previous high bid on a package plus X% iii. The estimated value of a package b plus Y% Note: The estimated value of a package is the sum of the shadow prices of the licenses that make up the package

  22. Bid 1 2 3 4 5 Bid amt. $22 $24 $20 $16 $8 License C A B Package AB AC AB C BC Bid 1 2 3 4 5 Maximize + $22 b1 + $24 b2 $20 b3 $16 b4 + $8 b5 + Count 1 b3 License A b1 0 <= b4 0 1 0 License B b1 b2 b4 0 <= + + + + + + + + + + + + b3 License C 0 b2 1 0 b5 <= 0 b3 Variables 1 b1 0 b2 0 b4 1 b5 = 0 or 1 , , , , Auction Formulation Objective Value: $30 Note: This formulation of the maximum revenue problem ignores the mutual exclusive bid constraints.

  23. Bid 1 2 3 4 5 0.5 0.5 0.5 b3 0 0 Variables b1 b2 b4 b5 b3 Variables b1 b2 b4 b5 >= 0 = 0 or 1 , , , , A B C License Maximize + + + $22 b1 $24 b2 $20 b3 $16 b4 $8 b5 + + + + 1sA 1 sB 1sC sA sB Minimize Bid amt. + Count 0 sB + 0 $22 >= + sA 0 1 b3 License A b1 0 <= b4 0 + Bid 1 $24 sC >= + sA sB + 1 0 License B b1 b2 b4 0 <= $20 sC >= Bid 2 + 0 0 + + + + + + + + + + + + + $16 0 >= b3 License C 0 b2 1 0 b5 <= Bid 3 + $8 sC >= Bid 4 , , Bid 5 9 sA sB 13 0 sC 11 Variables >= Dual Formulation LP Formulation Objective Value: $33 , , , , Dual Problem Objective Value: $33

  24. Dual prices: Two Problems • Greater than primal • Not unique

  25. Solution: Pseudo-Duals • Rassenti, Smith and Bulfin (1982) as well as DeMartini, Kwasnica, Ledyard and Porter (1999) suggest solving for pseudo-dual prices in order to adjust the over-estimated shadow prices • Use the provisional winners as the “mark” for adjusting down the shadow prices • Result... • The shadow prices sum to equal the revenue obtained from the integer solution • The sum of the shadow prices of a winning package equals the bid amount of the package

  26. = >= + 2 2 >= + 3 3 >= + 4 4 >= = >=  Bid amt. $22 Bid 1 $24 $20 Bid 2 $16 Bid 3 9 13 8 $8 Bid 4 3 Variables 2 4 >= 0 , , 3 3 0 Bid 5 Adjusted Shadow Prices Pseudo-Dual Problem Objective Value: $30 A B C License + + 1sA 1 sB 1sC Minimize + + 0 sA sB + + 0 sB sC + + sA 0 sC + + sA sB 0 + + 0 0 sC 0 , , >= sA sB sC Variables

  27. License Shadow Price 5 9 A B 5 1 2 8 value (B) in $ $10 $8 $5 $1 $5 $10 $2 $9 value (A) in $ 1 Alternative Shadow Prices Bid 1 Bid amt. $10 Package AB Solution 1

  28. License Shadow Price Bid 1 2 3 A Bid amt. $10 $5 $2 B Package AB A B Solution 1 0 0 Dual Constraints A B 2 + value (B) in $ $5 Bid 2 >= + $2 Bid 3 >= $10 $2 3 $10 $5 value (A) in $ 1 Alternative Shadow Prices

  29. License Shadow Price Bid 1 2 3 6.5 5 A Bid amt. $10 $5 $2 B 5 3.5 Package AB A B Solution 1 0 0 8 2 value (B) in $ $10 $5 $3.5 $2 $6.5 $8 $5 $10 value (A) in $ 1 Alternative Shadow Prices Solution: Uses a centering algorithm to choose among alternative optima

  30. Scaling-up: Testing • Need a method to test the new auction design in larger environments

  31. BidBot Characteristics • Provides basic bidder functionality • Place new bids (with increments) • Renew previous bids • Dynamically create new packages • Utilize activity waivers • Reduce eligibility • Bidders are given budget constraints • Eligibility based on historic data • Many small bidders with some large bidders • Adjacency matrix for geographic synergies that determine the value of packages • Value and straightforward bidders with random increment bidding based on historical data

  32. Computational Staging For each of these alternatives, answer the question: How large an auction can be run? 1. Exact optimization calculations for all aspects of the auction Staging: 2. Early in the auction, use shadow prices instead of deficit calculations to determine minimum acceptable bid 3. Early in the auction, use shadow prices and do not choose among ties, i.e. take whatever solution obtained in Maximum Revenue calculation 4. Early in the auction, use shadow prices, do not break ties, and stop optimization after x minutes of calculation or y percent of optimality 5. Other changes to rules At later stage of auction -- Always do everything exactly

  33. Where do we go next? • More testing using both BidBots and mock auctions • Shadow price testing: • What is the best way to adjust shadow prices? • What happens when licenses have few or no bidders? • How can we obtain a bidder-specific MAB that uses shadow prices? • Click-box bidding versus greater flexibility in submission of bids: • Does providing more digits of accuracy help to eliminate ties? • Issues of transparency, speed of auction, collusion • When should computational staging occur?

  34. Package Bidding SystemFor Auction 31Wye River ConferenceOctober 27, 2001

  35. bidder5 *****

  36. 20 MHz nationwide

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