Week 7
Auctions
Ch 9.6-9.8
Setup
• Single object up for auction
• ⫫ private values viaFlu Ftuffu
◦ Each player has a valuationvie0,1
◦ Players valuation does not depend upon his opponents valuation or information
• Each bidder is risk neutral, ie.
Vi Vi wonthe
auction p
p ayment
expected
Second price sealed bid auction
Theorem: in a second price sealed bid auction with ⫫ private values it is weakly dominant strategy
to bid
Proof:
• Player i bids his valuation and wins
◦ Bidding above v doesn’t change the payo as price remains the same and i still wins
◦ Bidding below v lowers the probability of winning, so decreases payo
• Player i bids his valuation and loses
◦ Bidding below v doesn’t change anything
◦ Bidding above v increases the probability of winning and having to pay strictly more than v,
leading to a negative payo
secondprice iwinningPr iw ins
BR ti P.tv II
bi bi 0 or
b vzbabebeer
wingin ni an so
First price sealed bid
Let VinV10.1
Biddingfunction Bi vi At
IU v b Pr i wins v b Pr bi BVj Hj i v b
IIPrbi BluD
SymmetricBNEinlinearstrategies B vi Kui
Prbi Blvd Pr biskv Pryet Pr v e Prvets Pr une k Prvie
Vi Ivi bi
Auctions
Ch 9.6-9.8
Setup
• Single object up for auction
• ⫫ private values viaFlu Ftuffu
◦ Each player has a valuationvie0,1
◦ Players valuation does not depend upon his opponents valuation or information
• Each bidder is risk neutral, ie.
Vi Vi wonthe
auction p
p ayment
expected
Second price sealed bid auction
Theorem: in a second price sealed bid auction with ⫫ private values it is weakly dominant strategy
to bid
Proof:
• Player i bids his valuation and wins
◦ Bidding above v doesn’t change the payo as price remains the same and i still wins
◦ Bidding below v lowers the probability of winning, so decreases payo
• Player i bids his valuation and loses
◦ Bidding below v doesn’t change anything
◦ Bidding above v increases the probability of winning and having to pay strictly more than v,
leading to a negative payo
secondprice iwinningPr iw ins
BR ti P.tv II
bi bi 0 or
b vzbabebeer
wingin ni an so
First price sealed bid
Let VinV10.1
Biddingfunction Bi vi At
IU v b Pr i wins v b Pr bi BVj Hj i v b
IIPrbi BluD
SymmetricBNEinlinearstrategies B vi Kui
Prbi Blvd Pr biskv Pryet Pr v e Prvets Pr une k Prvie
Vi Ivi bi
