Post on 09-Aug-2015
Activity rule requires consistent bidding throughout 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
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Activity rule in clock rounds: Revealed preference or 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
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Activity rule in clock rounds: Example 1
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RoundInitial
R1Package Q1
Eligibility 100
Package size 100
Package price $140
RP constraints
R2Q2
100
100
$200
R3Q3
100
80
$180
R4Q4
80
80
$220
R5Q5
80
90
$200
RP3:5
R6Q6
80
85
$200
RP3:6
R7Q7
80
50
$190
R8Q8
50
50
$230
FinalR9Q9
50
55
$280
RP3:9RP7:9
Eligibility reducing round Eligibility reducing round
RP3:5 (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)
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
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Round 1: price difference < $500k; (2, 0) best
Product Price Bid Eligibility pointsUpper $1000k 2 100
Lower $600k 0 0
Total package $2000k (2, 0) 100
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Round 2: price difference > $500k; (1, 1) best
Product Price Bid Eligibility pointsUpper $1200k 1 50
Lower $650k 1 30
Total package $1850k (1, 1) 80
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Round 3: price difference < $500k; (2, 0) bestMust satisfy revealed preference w.r.t. round 2
Product Price Bid Eligibility pointsUpper $1250k 2 100
Lower $800k 0 0
Total package $2500k (2, 0) 100 (Eligibility is 80)
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(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
Round 4: price difference < $500k; (2, 0) bestBudget of $2800k reached
Product Price Bid Eligibility pointsUpper $1400k 2 100
Lower $1000k 0 0
Total package $2800k (2, 0) 100(Eligibility is 80)
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(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
Round 5: price difference < $500k; (1, 0) bestCan only afford 1 lot
Product Price Bid Eligibility pointsUpper $1650k 1 50
Lower $1200k 0 0
Total package $1650k (1, 0) 50
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Round 6: price difference > $500k; (0, 1) bestCan only afford 1 lot
Product Price Bid Eligibility pointsUpper $1800k 0 0
Lower $1250k 1 30
Total package $1250k (0,1) 30
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Round 7: price difference < $500k; (1, 0) bestRevealed preference with respect to round 2Product Price Bid Eligibility pointsUpper $1850k 1 50
Lower $1400k 0 0
Total package $1850k (1,0) 50(Eligibility is 30)
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Product PriceRound 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850kLower $600k $650k $800k $1000k $1200k $1250k $1400k
Eligibility 100 80 80 80 50 30 30Package (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
Round 7: price difference < $500k; (1, 0) bestRevealed preference with respect to round 5Product Price Bid Eligibility pointsUpper $1850k 1 50
Lower $1400k 0 0
Total package $1850k (1,0) 50(Eligibility is 30)
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(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
Product PriceRound 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850kLower $600k $650k $800k $1000k $1200k $1250k $1400k
Eligibility 100 80 80 80 50 30 30Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)
Round 7: price difference < $500k; (1, 0) bestRevealed preference with respect to round 6Product Price Bid Eligibility pointsUpper $1850k 1 50
Lower $1400k 0 0
Total package $1850k (1,0) 50(Eligibility is 30)
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(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
Product PriceRound 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850kLower $600k $650k $800k $1000k $1200k $1250k $1400k
Eligibility 100 80 80 80 50 30 30Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)
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
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Activity rule in supplementary round: Example 1
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RoundInitial
R1Package Q1
Eligibility 100
Package size 100
R2Q2
100
100
R3Q3
100
80
R4Q4
80
80
R5Q5
80
90
R6Q6
80
85
R7Q7
80
50
R8Q8
50
50
FinalR9Q9
50
55
Eligibility reducing round Eligibility reducing round
Small package S, size 45:RP 9: Sup. bid on S ≤ Sup. bid on Q9 + Price of S in R9 – Price of Q9 in R9
Medium 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 R9
Large 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
Activity rule in supplementary round: Example 2How large a bid can be placed on (1, 1)?
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(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
Product PriceRound 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
Upper $1000k $1200k $1250k $1400k $1650k $1800k $1850kLower $600k $650k $800k $1000k $1200k $1250k $1400k
Eligibility 100 80 80 80 50 30 30Package (2, 0) (1, 1) (2, 0) (2, 0) (1, 0) (0, 1) (1, 0)
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
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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
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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
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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