Activity 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

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Activity rule - Peter Cramton - Power Auctions

  1. 1. Activity rule Peter Cramton
  2. 2. Activity rule requires consistent biddingthroughout auction • Eliminates bid snipping  Promotes price and assignment discovery • Single item  If “In” at high price, must be “In” at lower prices (can’t enter late) • Single product  Maintain or reduce demand as price rises (can’t increase demand) • Many products  Eligibility point rule  Move toward smaller packages as prices rise  Revealed preference rule  Move toward packages that have become better values 2
  3. 3. Activity rule in clock rounds: Revealed preferenceor eligibility point monotonicity • Eligibility for next round is minimum of current eligibility and size of current package • Eligibility-reducing round is any round with eligibility > size of package bid in round • Can always bid on packages that are the same size or smaller than your eligibility (eligibility point monotonicity) • Can bid on larger packages that satisfy revealed preference with respect to each prior eligibility reducing round  Revealed preference with respect to a prior round is satisfied if the current package has gone up less than the prior package since the prior round; current package is now relatively less expensive 3
  4. 4. Activity rule in clock rounds: Example 1 Eligibility reducing round Eligibility reducing round Initial Final Round R1 R2 R3 R4 R5 R6 R7 R8 R9 Package Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Eligibility 100 100 100 80 80 80 80 50 50 Package size 100 100 80 80 90 85 50 50 55 Package price $140 $200 $180 $220 $200 $200 $190 $230 $280 RP constraints RP3:5 RP3:6 RP3:9 RP7:9RP3:5 (Price of Q5 in R5) – (Price of Q5 in R3) ≤ (Price of Q3 in R5) – (Price of Q3 in R3)RP3:6 (Price of Q6 in R6) – (Price of Q6 in R3) ≤ (Price of Q3 in R6) – (Price of Q3 in R3)RP3:9 (Price of Q9 in R9) – (Price of Q9 in R3) ≤ (Price of Q3 in R9) – (Price of Q3 in R3)RP7:9 (Price of Q9 in R9) – (Price of Q9 in R7) ≤ (Price of Q7 in R9) – (Price of Q7 in R7) 4
  5. 5. Activity rule in clock rounds: Example 2 • Bidder desires 1 or 2 lots of 700 MHz • Upper lot has 50 eligibility points • Lower lot has 30 eligibility points • Bidder has budget of $2800k • Package = (# Upper, # Lower) • (2,0) better than (1,1) if price difference < $500k • (1,0) better than (0,1) if price difference < $500k 5
  6. 6. Round 1: price difference < $500k; (2, 0) bestProduct Price Bid Eligibility pointsUpper $1000k 2 100Lower $600k 0 0Total package $2000k (2, 0) 100 6
  7. 7. Round 2: price difference > $500k; (1, 1) bestProduct Price Bid Eligibility pointsUpper $1200k 1 50Lower $650k 1 30Total package $1850k (1, 1) 80 7
  8. 8. Round 3: price difference < $500k; (2, 0) best Must satisfy revealed preference w.r.t. round 2 Product Price Bid Eligibility points Upper $1250k 2 100 Lower $800k 0 0 Total package $2500k (2, 0) 100 (Eligibility is 80)(2, 0) must have gone up less than (1, 1) from round 2 to round 3:(Price of (2, 0) in R3) – (Price of (2, 0) in R2) ≤ (Price of (1, 1) in R3) – (Price of (1, 1) in R2)($2500k – $2400k) ≤ ($2050k – $1850k)$100k ≤ $200k 8
  9. 9. Round 4: price difference < $500k; (2, 0) best Budget of $2800k reached Product Price Bid Eligibility points Upper $1400k 2 100 Lower $1000k 0 0 Total package $2800k (2, 0) 100 (Eligibility is 80)(2, 0) must have gone up less than (1, 1) from round 2 to round 4:(Price of (2, 0) in R4) - (Price of (2, 0) in R2) ≤ (Price of (1, 1) in R4) – (Price of (1, 1) in R2)($2800k – $2400k) ≤ ($2400k – $1850k)$400k ≤ $550k 9
  10. 10. Round 5: price difference < $500k; (1, 0) bestCan only afford 1 lotProduct Price Bid Eligibility pointsUpper $1650k 1 50Lower $1200k 0 0Total package $1650k (1, 0) 50 10
  11. 11. Round 6: price difference > $500k; (0, 1) bestCan only afford 1 lotProduct Price Bid Eligibility pointsUpper $1800k 0 0Lower $1250k 1 30Total package $1250k (0,1) 30 11
  12. 12. Round 7: price difference < $500k; (1, 0) best Revealed preference with respect to round 2 Product Price Bid Eligibility points Upper $1850k 1 50 Lower $1400k 0 0 Total package $1850k (1,0) 50 (Eligibility is 30) Price Product Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850k Lower $600k $650k $800k $1000k $1200k $1250k $1400k Eligibility 100 80 80 80 50 30 30 Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)(Price of (1, 0) in R7) – (Price of (1, 0) in R2) ≤ (Price of (1, 1) in R7) – (Price of (1, 1) in R2) ($1850k – $1200k) ≤ ($3250k – $1850k) $650k ≤ $1400k 12
  13. 13. Round 7: price difference < $500k; (1, 0) best Revealed preference with respect to round 5 Product Price Bid Eligibility points Upper $1850k 1 50 Lower $1400k 0 0 Total package $1850k (1,0) 50 (Eligibility is 30) Price Product Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850k Lower $600k $650k $800k $1000k $1200k $1250k $1400k Eligibility 100 80 80 80 50 30 30 Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)(Price of (1, 0) in R7) – (Price of (1, 0) in R5) ≤ (Price of (1, 0) in R7) – (Price of (1, 0) in R5) ($1850k – $1650k) ≤ ($1850k – $1650k) $200k ≤ $200k 13
  14. 14. Round 7: price difference < $500k; (1, 0) best Revealed preference with respect to round 6 Product Price Bid Eligibility points Upper $1850k 1 50 Lower $1400k 0 0 Total package $1850k (1,0) 50 (Eligibility is 30) Price Product Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850k Lower $600k $650k $800k $1000k $1200k $1250k $1400k Eligibility 100 80 80 80 50 30 30 Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)(Price of (1, 0) in R7) – (Price of (1, 0) in R6) ≤ (Price of (0, 1) in R7) – (Price of (0, 1) in R6) ($1850k – $1800k) ≤ ($1400k – $1250k) $50k ≤ $150k 14
  15. 15. Activity rule in supplementary round:Revealed preference cap• For packages = final clock package + any unallocated lots,  supplementary bid satisfies RP with respect to final clock round• For all other packages,  supplementary bid satisfies RP with respect to final clock round  supplementary bid satisfies RP with respect to each eligibility reducing round beginning with the last round in which the bidder had eligibility to bid on the package 15
  16. 16. Activity rule in supplementary round: Example 1 Eligibility reducing round Eligibility reducing round Initial Final Round R1 R2 R3 R4 R5 R6 R7 R8 R9 Package Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Eligibility 100 100 100 80 80 80 80 50 50 Package size 100 100 80 80 90 85 50 50 55Small package S, size 45:RP 9: Sup. bid on S ≤ Sup. bid on Q9 + Price of S in R9 – Price of Q9 in R9Medium package M, size 60, final clock package + some unallocated lots:RP 9: Sup. bid on M ≤ Sup. bid on Q9 + Price of M in R9 – Price of Q9 in R9Large package L, size 90:RP 9: Sup. bid on L ≤ Sup. bid on Q9 + Price of L in R9 – Price of Q9 in R9RP 7: Sup. bid on L ≤ Sup. bid on Q7 + Price of L in R7 – Price of Q7 in R7RP 3: Sup. bid on L ≤ Sup. bid on Q3 + Price of L in R3 – Price of Q3 in R3 16
  17. 17. Activity rule in supplementary round: Example 2 How large a bid can be placed on (1, 1)? Price Product Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850k Lower $600k $650k $800k $1000k $1200k $1250k $1400k Eligibility 100 80 80 80 50 30 30 Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)(Sup Bid on (1,1)) ≤ (Highest Bid on (1,0)) + (Price of (1,1) in R7) – (Price of (1,0) in R7) (Sup Bid on (1,1)) ≤ $1850k + $3250k – $1850k (Sup Bid on (1,1)) ≤ $3250k(Sup Bid on (1,1)) ≤ (Highest Bid on (0,1)) + (Price of (1,1) in R6) – (Price of (0,1) in R6) (Sup Bid on (1,1)) ≤ $1250k + $3050k – $1250k (Sup Bid on (1,1)) ≤ $3050k(Sup Bid on (1,1)) ≤ (Highest Bid on (1,0)) + (Price of (1,1) in R5) – (Price of (1,0) in R5) (Sup Bid on (1,1)) ≤ $1850k + $2850k – $1650k (Sup Bid on (1,1)) ≤ $3050k 17
  18. 18. Motivation for activity rule • Can bid on most preferred package if bid consistently • Strong incentive to bid on most preferred package in final clock round • Strong incentive to bid on most preferred package in each eligibility reducing round 18
  19. 19. Price and allocation discovery in clock stage • If clock stage ends with no unallocated lots, then the final allocation is the allocation at the end of the clock stage  Supplementary round cannot change allocation  Clock stage determines the final allocation 19
  20. 20. Properties with substitutes • Bidding on the most profitable package is best • Clock yields competitive equilibrium with efficient allocation and supporting prices • Final allocation = allocation at end of clock stage  The clock stage does all the work 20
  21. 21. Properties in general • Supplementary bids needed if the clock stage ends with unallocated lots as a result of complementarities • Nonetheless, even in this case, the bidder can guarantee winning its final clock package by raising its bid by final clock price of unallocated lots 21
  22. 22. Conclusion
  23. 23. Combinatorial clock auction improves on SMRA • Eliminates exposure by allowing package bids • Minimizes gaming as a result of package bids and the use of an effective pricing rule and activity rule • Enhances substitution by using generic lots whenever possible • Simplifies bidding 23

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