admission fees in auctions Jiafeng (Kevin) Chen and Scott Duke - - PowerPoint PPT Presentation

admission fees in auctions

Jiafeng (Kevin) Chen and Scott Duke Kominers October 14, 2018

Ec 2099 Harvard University

bulow and klemperer 1996

• Assume symmetric values
• Given the choice of:
• N + 1 bidders, but English auction with no reserve
• N bidders, but revenue-maximizing mechanism

Which should the seller prefer? Theorem (Bulow and Klemperer 1996) Seller prefers N 1 bidders and second-price auction with no reserve. “Value of market power Value of thickness” Seller should increase participation

1

bulow and klemperer 1996

• Assume symmetric values
• Given the choice of:
• N + 1 bidders, but English auction with no reserve
• N bidders, but revenue-maximizing mechanism

Which should the seller prefer? Theorem (Bulow and Klemperer 1996) Seller prefers N + 1 bidders and second-price auction with no reserve. “Value of market power Value of thickness” Seller should increase participation

1

bulow and klemperer 1996

• Assume symmetric values
• Given the choice of:
• N + 1 bidders, but English auction with no reserve
• N bidders, but revenue-maximizing mechanism

Which should the seller prefer? Theorem (Bulow and Klemperer 1996) Seller prefers N + 1 bidders and second-price auction with no reserve. ⇒ “Value of market power ≤ Value of thickness” ⇝ Seller should increase participation

1

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N

1 bidders beats the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to N 1 st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N

1 bidders is constrained optimal (Myerson 1981)

• Optimal on N

Intermediate mechanism English auction on N 1

2

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N + 1 bidders beats

the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to N 1 st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N

1 bidders is constrained optimal (Myerson 1981)

• Optimal on N

Intermediate mechanism English auction on N 1

2

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N + 1 bidders beats

the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to (N + 1)st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N

1 bidders is constrained optimal (Myerson 1981)

• Optimal on N

Intermediate mechanism English auction on N 1

2

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N + 1 bidders beats

the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to (N + 1)st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N

1 bidders is constrained optimal (Myerson 1981)

• Optimal on N

Intermediate mechanism English auction on N 1

2

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N + 1 bidders beats

the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to (N + 1)st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N

1 bidders is constrained optimal (Myerson 1981)

• Optimal on N

Intermediate mechanism English auction on N 1

2

proof sketch (kirkegaard 2006)

• A mechanism is constrained optimal if it is revenue

maximizing, constrained on the object being sold with probability 1.

• A constrained optimal mechanism on N + 1 bidders beats

the optimal mechanism on N bidders

• Consider the intermediate mechanism: Run optimal

mechanism on N bidders. Give object to (N + 1)st bidder if not sold.

• Same revenue as the optimal mechanism on N bidders.
• Allocates object with probability 1
• So constrained optimal mechanism must be weakly better
• English auction on N + 1 bidders is constrained optimal

(Myerson 1981)

• =

⇒ Optimal on N = Intermediate mechanism < English auction on N + 1

2

a puzzle

Figure 1: “CHARITY AUCTIONS: 12+ INSANELY USEFUL TIPS FOR YOUR AUCTION,” OneCause Fundraising Solutions. https://www.onecause.com/charity-auction/

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model with entry

Two-stage mechanism

• Entry decisions
• Potential bidders i
• Entering costs a random ci (Drive to the auction site)
• Seller can choose to institute fee φ
• Cost of entrance is ci + φ for potential bidder i
• Auction
• Bidders see value after entering
• Submit bids

New question: Given the choice of

• N

1 potential bidders but

• N potential bidders

What should the seller choose?

4

model with entry

Two-stage mechanism

• Entry decisions
• Potential bidders i
• Entering costs a random ci (Drive to the auction site)
• Seller can choose to institute fee φ
• Cost of entrance is ci + φ for potential bidder i
• Auction
• Bidders see value after entering
• Submit bids

New question: Given the choice of

• N + 1 potential bidders but φ = 0
• N potential bidders

What should the seller choose?

4

• ur result

Theorem Under certain conditions, the seller prefers φ > 0 rather than the extra potential bidder. Proof by simulation.

N potential bidders. Cost ci are i.i.d. Uniform 0 1 . Values are i.i.d. Exponential with mean N. Set (not necessarily optimal) fee

1 N 1.

Plot the revenue advantage of : Revenue with Revenue with N 1 bidders as a function of N

2 4 6 8 10 12 14 N 0.20 0.15 0.10 0.05 0.00 0.05 Revenue advantage of admission fee Theoretical value of admission fee advantage 95%-confidence interval

Figure 2: is better for large N

5

• ur result

Theorem Under certain conditions, the seller prefers φ > 0 rather than the extra potential bidder. Proof by simulation.

N potential bidders. Cost ci are i.i.d. Uniform[0, 1]. Values are i.i.d. Exponential with mean N. Set (not necessarily optimal) fee φ =

1 N−1.

Plot the revenue advantage of φ: (Revenue with φ) − (Revenue with N + 1 bidders) as a function of N

2 4 6 8 10 12 14 N 0.20 0.15 0.10 0.05 0.00 0.05 Revenue advantage of admission fee Theoretical value of admission fee advantage 95%-confidence interval

Figure 2: is better for large N

5

• ur result

Theorem Under certain conditions, the seller prefers φ > 0 rather than the extra potential bidder. Proof by simulation.

N potential bidders. Cost ci are i.i.d. Uniform[0, 1]. Values are i.i.d. Exponential with mean N. Set (not necessarily optimal) fee φ =

1 N−1.

Plot the revenue advantage of φ: (Revenue with φ) − (Revenue with N + 1 bidders) as a function of N

2 4 6 8 10 12 14 N 0.20 0.15 0.10 0.05 0.00 0.05 Revenue advantage of admission fee Theoretical value of admission fee advantage 95%-confidence interval

Figure 2: φ is better for large N

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intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform 0 1 value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time

• ptimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform 0 1 value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time

• ptimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform[0, 1] value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time

• ptimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform[0, 1] value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time

• ptimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform[0, 1] value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time =

⇒ optimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

intuitions

Reasons for seller to prefer φ > 0:

• Extra potential bidder may not enter
• Seller is a monopolist selling seats to the auction
• Charging bidders up front puts a price on information
• Single bidder, Uniform[0, 1] value, willingness to pay is 1

2

before knowing the value.

• If bidder knew his value, optimal reserve is 1

2.

• Seller collects 1

2 half the time =

⇒ optimal revenue is 1

4.

• Bidder gains 1

4 “information edge” by observing his value.

6

adversarial results

If we loosen restrictions, we can derive some extreme results. Theorem Assume cost profile ci is nonstochastic.1 For any N, there exists a value distribution F and cost profile c such that

• 1. Every potential bidder enters the auction when there are

no fees

• 2. Under the optimal admission fee, only 1 bidder enters.

1The joint cost distribution is over permutations of fixed cost values.

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