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

Seller prefers N “Value of market power bulow and klemperer 1996 • Assume symmetric values • Given the choice of: Which should the seller prefer? Theorem (Bulow and Klemperer 1996) 1 bidders and second-price auction with no reserve. Value of thickness” Seller should increase participation 1 • N + 1 bidders, but English auction with no reserve • N bidders, but revenue-maximizing mechanism

“Value of market power bulow and klemperer 1996 • Assume symmetric values • Given the choice of: Which should the seller prefer? Theorem (Bulow and Klemperer 1996) reserve. Value of thickness” Seller should increase participation 1 • N + 1 bidders, but English auction with no reserve • N bidders, but revenue-maximizing mechanism Seller prefers N + 1 bidders and second-price auction with no

bulow and klemperer 1996 • Assume symmetric values • Given the choice of: Which should the seller prefer? Theorem (Bulow and Klemperer 1996) reserve. Seller should increase participation 1 • N + 1 bidders, but English auction with no reserve • N bidders, but revenue-maximizing mechanism Seller prefers N + 1 bidders and second-price auction with no ⇒ “Value of market power ≤ Value of thickness” ⇝

• Consider the intermediate mechanism: Run optimal proof sketch (kirkegaard 2006) • So constrained optimal mechanism must be weakly better 1 auction on N English Intermediate mechanism Optimal on N • (Myerson 1981) 1 bidders is constrained optimal • English auction on N • Allocates object with probability 1 • A mechanism is constrained optimal if it is revenue • Same revenue as the optimal mechanism on N bidders. not sold. 1 st bidder if mechanism on N bidders. Give object to N the optimal mechanism on N bidders 1 bidders beats • A constrained optimal mechanism on N probability 1. maximizing, constrained on the object being sold with 2

• Consider the intermediate mechanism: Run optimal proof sketch (kirkegaard 2006) • English auction on N 1 auction on N English Intermediate mechanism Optimal on N • (Myerson 1981) 1 bidders is constrained optimal • Allocates object with probability 1 • So constrained optimal mechanism must be weakly better • A mechanism is constrained optimal if it is revenue • Same revenue as the optimal mechanism on N bidders. not sold. 1 st bidder if mechanism on N bidders. Give object to N the optimal mechanism on N bidders probability 1. maximizing, constrained on the object being sold with 2 • A constrained optimal mechanism on N + 1 bidders beats

proof sketch (kirkegaard 2006) • English auction on N 1 auction on N English Intermediate mechanism Optimal on N • (Myerson 1981) 1 bidders is constrained optimal • So constrained optimal mechanism must be weakly better • A mechanism is constrained optimal if it is revenue • Allocates object with probability 1 • Same revenue as the optimal mechanism on N bidders. not sold. the optimal mechanism on N bidders probability 1. maximizing, constrained on the object being sold with 2 • A constrained optimal mechanism on N + 1 bidders beats • Consider the intermediate mechanism: Run optimal mechanism on N bidders. Give object to ( N + 1 ) st bidder if

proof sketch (kirkegaard 2006) • English auction on N 1 auction on N English Intermediate mechanism Optimal on N • (Myerson 1981) 1 bidders is constrained optimal • So constrained optimal mechanism must be weakly better • A mechanism is constrained optimal if it is revenue • Allocates object with probability 1 • Same revenue as the optimal mechanism on N bidders. not sold. the optimal mechanism on N bidders probability 1. maximizing, constrained on the object being sold with 2 • A constrained optimal mechanism on N + 1 bidders beats • Consider the intermediate mechanism: Run optimal mechanism on N bidders. Give object to ( N + 1 ) st bidder if

proof sketch (kirkegaard 2006) • English auction on N 1 auction on N English Intermediate mechanism Optimal on N • (Myerson 1981) 1 bidders is constrained optimal • So constrained optimal mechanism must be weakly better • A mechanism is constrained optimal if it is revenue • Allocates object with probability 1 • Same revenue as the optimal mechanism on N bidders. not sold. the optimal mechanism on N bidders probability 1. maximizing, constrained on the object being sold with 2 • A constrained optimal mechanism on N + 1 bidders beats • Consider the intermediate mechanism: Run optimal mechanism on N bidders. Give object to ( N + 1 ) st bidder if

proof sketch (kirkegaard 2006) not sold. (Myerson 1981) • So constrained optimal mechanism must be weakly better • Allocates object with probability 1 • A mechanism is constrained optimal if it is revenue • Same revenue as the optimal mechanism on N bidders. the optimal mechanism on N bidders probability 1. maximizing, constrained on the object being sold with 2 • A constrained optimal mechanism on N + 1 bidders beats • Consider the intermediate mechanism: Run optimal mechanism on N bidders. Give object to ( N + 1 ) st bidder if • English auction on N + 1 bidders is constrained optimal • = ⇒ Optimal on N = Intermediate mechanism < English auction on N + 1

a puzzle Figure 1: “CHARITY AUCTIONS: 12+ INSANELY USEFUL TIPS FOR YOUR AUCTION,” OneCause Fundraising Solutions. https://www.onecause.com/charity-auction/ 3

model with entry • Submit bids What should the seller choose? • N potential bidders 0 1 potential bidders but • N New question: Given the choice of • Bidders see value after entering Two-stage mechanism • Auction • Potential bidders i • Entry decisions 4 • Entering costs a random c i (Drive to the auction site) • Seller can choose to institute fee φ • Cost of entrance is c i + φ for potential bidder i

model with entry Two-stage mechanism • Entry decisions • Potential bidders i • Auction • Bidders see value after entering • Submit bids New question: Given the choice of • N potential bidders What should the seller choose? 4 • Entering costs a random c i (Drive to the auction site) • Seller can choose to institute fee φ • Cost of entrance is c i + φ for potential bidder i • N + 1 potential bidders but φ = 0

N potential bidders. Cost c i are i.i.d. 0.05 0.00 0.05 0.10 0.15 Revenue advantage of admission fee Theoretical value of admission fee advantage Plot the revenue advantage of 95% -confidence interval 0.20 2 4 6 8 10 12 14 N Figure 2: is better for large N our result Theorem as a function of N 1 bidders Revenue with N Revenue with : 1 . N 1 Set (not necessarily optimal) fee Exponential with mean N . Uniform 0 1 . Values are i.i.d. Proof by simulation. the extra potential bidder. 5 Under certain conditions, the seller prefers φ > 0 rather than

0.05 0.00 0.05 0.10 0.15 Revenue advantage of admission fee Theoretical value of admission fee advantage 95% -confidence interval 0.20 2 4 6 8 10 12 14 N Figure 2: is better for large N our result Theorem as a function of N 1 Set (not necessarily optimal) fee Exponential with mean N . Proof by simulation. the extra potential bidder. 5 Under certain conditions, the seller prefers φ > 0 rather than N potential bidders. Cost c i are i.i.d. Uniform [ 0 , 1 ] . Values are i.i.d. φ = N − 1 . Plot the revenue advantage of φ : ( Revenue with φ ) − ( Revenue with N + 1 bidders )

our result Exponential with mean N . Theorem as a function of N 1 Set (not necessarily optimal) fee 5 the extra potential bidder. Proof by simulation. Under certain conditions, the seller prefers φ > 0 rather than N potential bidders. Cost c i are i.i.d. 0.05 Uniform [ 0 , 1 ] . Values are i.i.d. 0.00 0.05 φ = 0.10 N − 1 . 0.15 Revenue advantage of admission fee Plot the revenue advantage of φ : Theoretical value of admission fee advantage 95% -confidence interval 0.20 2 4 6 8 10 12 14 ( Revenue with φ ) N − ( Revenue with N + 1 bidders ) Figure 2: φ is better for large N

• Single bidder, Uniform 0 1 value, willingness to pay is 1 • If bidder knew his value, optimal reserve is 1 • Seller collects 1 optimal revenue is 1 2 half the time • Bidder gains 1 4 “information edge” by observing his value. 4 . intuitions 2 . before knowing the value. 2 • Charging bidders up front puts a price on information • Seller is a monopolist selling seats to the auction • Extra potential bidder may not enter 6 Reasons for seller to prefer φ > 0:

• Single bidder, Uniform 0 1 value, willingness to pay is 1 • If bidder knew his value, optimal reserve is 1 • Seller collects 1 optimal revenue is 1 2 half the time • Bidder gains 1 4 “information edge” by observing his value. 4 . intuitions 2 . before knowing the value. 2 • Charging bidders up front puts a price on information • Seller is a monopolist selling seats to the auction • Extra potential bidder may not enter 6 Reasons for seller to prefer φ > 0: