Activity rule

Peter Cramton

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

2

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

3

Activity rule in clock rounds: Example 1

4

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

5

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

6

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

7

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)

9

(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

10

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)

13

(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)

14

(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

19

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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

Conclusion

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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