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Chapter Seventeen
Auctions
Who Uses Auctions?
Owners of art, cars, stamps,
machines, mineral rights etc.
Q: Why auction?
Who Uses Auctions?
Owners of art, cars, stamps,
machines, mineral rights etc.
Q: Why auction?
A: Because many markets are
imperfect and it is hard to discover
potential buyers’ true valuations of
your asset. Auctions help to
discover this information.
Types of Auctions
English auction:
– bids are public announcements
– bid price rises until there are no
further bids
– highest bid wins
– winner pays his bid.
Types of Auctions
Sealed-bid first-price auction:
– bids are private information
– bids are made simultaneously
– highest bid wins
– winner pays his bid.
Types of Auctions
Sealed-bid second-price auction:
– bids are private information
– bids are made simultaneously
– highest bidder wins
– winner pays second-highest bid
– also known as a Vickrey auction.
Types of Auctions
Dutch auction:
– auctioneer announces a high bid
and then gradually lowers the bid
– first buyer to accept wins and pays
that price.
Reserve Price
A seller specified bid level below
which no sale is made.
Economists’ Classification of
Auctions
Private-value auctions:
– every potential buyer knows for
sure her own valuation of the item
for sale
– all these individual valuations are
independent of each other.
Economists’ Classification of
Auctions
Common-value auctions:
– item for sale has the same value to
every potential buyer
– potential buyers differ in their own
estimates of this common value.
Auction Design
Goals:
– Pareto efficiency
– maximization of the seller’s profit.
Auction Design
Pareto efficiency:
– the item must sell to the buyer with
the highest valuation of the item.
Which auctions are Pareto efficient?
Auctions and Efficiency
English auction with no reserve price
must be efficient since, if a buyer
with a low valuation was about to
buy, the highest valuation buyer
would bid higher.
Auctions and Efficiency
English auction with a reserve price
need not be efficient since if the
reserve price is set above the
(unknown to the seller) highest buyer
valuation, then there will be no sale
and so no gains-to-trade.
Auctions and Efficiency
Dutch auction need not be efficient.
No buyer knows other buyers’
valuations, so the highest valuation
buyer may delay too long and lose to
another bidder.
Auctions and Efficiency
Sealed-bid first-price auction need
not be efficient. No buyer knows
other buyers’ valuations, so the
highest valuation buyer may bid too
low and lose to another bidder.
Auctions and Efficiency
Sealed-bid second-price auction is
Pareto efficient even though no
buyer knows the other buyers’
valuations (more on this later).
Why Use a Reserve Price?
Suppose there are 2 buyers.
The seller believes each buyer’s
valuation is $20 with chance 1/2 and
$50 with chance 1/2.
I.e. with chance 1/4 each, the seller
believes she faces buyer valuations
($20,$20), ($20,$50), ($50,$20) and
($50,$50).
Why Use a Reserve Price?
I.e. with chance 1/4 each, the seller
believes she faces buyer valuations
($20,$20), ($20,$50), ($50,$20) and
($50,$50).
Use an English auction.
Bids must be raised by at least $1.
With chance 1/4 each, winning bids
will be $20, $21, $21 and $50 if there
is no reserve price.
Why Use a Reserve Price?
With chance 1/4 each, winning bids
will be $20, $21, $21 and $50 if there
is no reserve price.
Seller’s expected revenue is
($20 + $21 + $21 + $50)/4 = $28
with no reserve price.
Why Use a Reserve Price?
With chance 1/4 each, the seller
believes she faces buyer valuations
($20,$20), ($20,$50), ($50,$20) and
($50,$50).
Set a reserve price of $50.
With chance 1/4 there will be no sale.
With chance 3/4 the winning bid will
be $50.
Why Use a Reserve Price?
Set a reserve price of $50.
With chance 1/4 there will be no sale.
With chance 3/4 the winning bid will
be $50.
Seller’s expected revenue is
3
1
× $50 + × $0 = $37 ⋅ 50 > $28.
4
4
Reserve Price and Efficiency
The reserve price causes an
efficiency loss since, with chance
1/4, there is no trade.
Second-Price, Sealed-Bid
Auction
– bids are private information
– bids are made simultaneously
– highest bidder wins
– winner pays second-highest bid
– also known as a Vickrey auction.
Second-Price, Sealed-Bid
Auction
No bidder knows any other bidder’s
true valuation of the item for sale.
Yet, it is individually rational for each
bidder to state truthfully his own
valuation. Why?
E.g. two bidders with true valuations
v1 and v2.
Second-Price, Sealed-Bid
Auction
E.g. two bidders with true valuations
v1 and v2.
Bids are b1 and b2.
Expected gain to bidder 1 is
(v1 − b2 ) Pr( win) + 0 × Pr(lose)
= (v1 − b2 ) Pr(b1 ≥ b2 ).
Second-Price, Sealed-Bid
Auction
Expected gain to bidder 1 is
(v1 − b2 ) Pr(b1 ≥ b2 ).
How is this maximized?
If v1 > b2, then maximize the prob. of
winning; i.e. set b1 = v1.
If v1 < b2, then minimize the prob. of
winning; i.e. set b1 = v1.
Either way, telling the truth is best!
Second-Price, Sealed-Bid
Auction
Since truth-telling is best for every
bidder, the highest valuation bidder
will win.
Hence the second-price, sealed-bid
auction is Pareto-efficient.
Common-Value Auctions
The item for sale has the same value
to every potential buyer.
Potential buyers differ in their own
estimates of this common value.
Bidder i’s estimate is vi = v + ε i
where v is the common value and ε i
is bidder i’s estimation error.
Common-Value Auctions
Bidder i’s estimate is vi = v + ε i
where v is the common value and ε i
is bidder i’s estimation error.
If every bid is truthful, the winner is
the bidder with the largest estimation
error ε i
so a truthful winner on average pays
more than the true value v -- the
winner’s curse.
Common-Value Auctions
If every bid is truthful, the winner is
the bidder with the largest estimation
max
error ε i -- call it ε i
max
> 0 then a truthful winner on
If ε i
average pays more than the true
value v -- the winner’s curse.
So bids should on average be less
than v in a common-value auction.

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Ch17

  • 2. Who Uses Auctions? Owners of art, cars, stamps, machines, mineral rights etc. Q: Why auction?
  • 3. Who Uses Auctions? Owners of art, cars, stamps, machines, mineral rights etc. Q: Why auction? A: Because many markets are imperfect and it is hard to discover potential buyers’ true valuations of your asset. Auctions help to discover this information.
  • 4. Types of Auctions English auction: – bids are public announcements – bid price rises until there are no further bids – highest bid wins – winner pays his bid.
  • 5. Types of Auctions Sealed-bid first-price auction: – bids are private information – bids are made simultaneously – highest bid wins – winner pays his bid.
  • 6. Types of Auctions Sealed-bid second-price auction: – bids are private information – bids are made simultaneously – highest bidder wins – winner pays second-highest bid – also known as a Vickrey auction.
  • 7. Types of Auctions Dutch auction: – auctioneer announces a high bid and then gradually lowers the bid – first buyer to accept wins and pays that price.
  • 8. Reserve Price A seller specified bid level below which no sale is made.
  • 9. Economists’ Classification of Auctions Private-value auctions: – every potential buyer knows for sure her own valuation of the item for sale – all these individual valuations are independent of each other.
  • 10. Economists’ Classification of Auctions Common-value auctions: – item for sale has the same value to every potential buyer – potential buyers differ in their own estimates of this common value.
  • 11. Auction Design Goals: – Pareto efficiency – maximization of the seller’s profit.
  • 12. Auction Design Pareto efficiency: – the item must sell to the buyer with the highest valuation of the item. Which auctions are Pareto efficient?
  • 13. Auctions and Efficiency English auction with no reserve price must be efficient since, if a buyer with a low valuation was about to buy, the highest valuation buyer would bid higher.
  • 14. Auctions and Efficiency English auction with a reserve price need not be efficient since if the reserve price is set above the (unknown to the seller) highest buyer valuation, then there will be no sale and so no gains-to-trade.
  • 15. Auctions and Efficiency Dutch auction need not be efficient. No buyer knows other buyers’ valuations, so the highest valuation buyer may delay too long and lose to another bidder.
  • 16. Auctions and Efficiency Sealed-bid first-price auction need not be efficient. No buyer knows other buyers’ valuations, so the highest valuation buyer may bid too low and lose to another bidder.
  • 17. Auctions and Efficiency Sealed-bid second-price auction is Pareto efficient even though no buyer knows the other buyers’ valuations (more on this later).
  • 18. Why Use a Reserve Price? Suppose there are 2 buyers. The seller believes each buyer’s valuation is $20 with chance 1/2 and $50 with chance 1/2. I.e. with chance 1/4 each, the seller believes she faces buyer valuations ($20,$20), ($20,$50), ($50,$20) and ($50,$50).
  • 19. Why Use a Reserve Price? I.e. with chance 1/4 each, the seller believes she faces buyer valuations ($20,$20), ($20,$50), ($50,$20) and ($50,$50). Use an English auction. Bids must be raised by at least $1. With chance 1/4 each, winning bids will be $20, $21, $21 and $50 if there is no reserve price.
  • 20. Why Use a Reserve Price? With chance 1/4 each, winning bids will be $20, $21, $21 and $50 if there is no reserve price. Seller’s expected revenue is ($20 + $21 + $21 + $50)/4 = $28 with no reserve price.
  • 21. Why Use a Reserve Price? With chance 1/4 each, the seller believes she faces buyer valuations ($20,$20), ($20,$50), ($50,$20) and ($50,$50). Set a reserve price of $50. With chance 1/4 there will be no sale. With chance 3/4 the winning bid will be $50.
  • 22. Why Use a Reserve Price? Set a reserve price of $50. With chance 1/4 there will be no sale. With chance 3/4 the winning bid will be $50. Seller’s expected revenue is 3 1 × $50 + × $0 = $37 ⋅ 50 > $28. 4 4
  • 23. Reserve Price and Efficiency The reserve price causes an efficiency loss since, with chance 1/4, there is no trade.
  • 24. Second-Price, Sealed-Bid Auction – bids are private information – bids are made simultaneously – highest bidder wins – winner pays second-highest bid – also known as a Vickrey auction.
  • 25. Second-Price, Sealed-Bid Auction No bidder knows any other bidder’s true valuation of the item for sale. Yet, it is individually rational for each bidder to state truthfully his own valuation. Why? E.g. two bidders with true valuations v1 and v2.
  • 26. Second-Price, Sealed-Bid Auction E.g. two bidders with true valuations v1 and v2. Bids are b1 and b2. Expected gain to bidder 1 is (v1 − b2 ) Pr( win) + 0 × Pr(lose) = (v1 − b2 ) Pr(b1 ≥ b2 ).
  • 27. Second-Price, Sealed-Bid Auction Expected gain to bidder 1 is (v1 − b2 ) Pr(b1 ≥ b2 ). How is this maximized? If v1 > b2, then maximize the prob. of winning; i.e. set b1 = v1. If v1 < b2, then minimize the prob. of winning; i.e. set b1 = v1. Either way, telling the truth is best!
  • 28. Second-Price, Sealed-Bid Auction Since truth-telling is best for every bidder, the highest valuation bidder will win. Hence the second-price, sealed-bid auction is Pareto-efficient.
  • 29. Common-Value Auctions The item for sale has the same value to every potential buyer. Potential buyers differ in their own estimates of this common value. Bidder i’s estimate is vi = v + ε i where v is the common value and ε i is bidder i’s estimation error.
  • 30. Common-Value Auctions Bidder i’s estimate is vi = v + ε i where v is the common value and ε i is bidder i’s estimation error. If every bid is truthful, the winner is the bidder with the largest estimation error ε i so a truthful winner on average pays more than the true value v -- the winner’s curse.
  • 31. Common-Value Auctions If every bid is truthful, the winner is the bidder with the largest estimation max error ε i -- call it ε i max > 0 then a truthful winner on If ε i average pays more than the true value v -- the winner’s curse. So bids should on average be less than v in a common-value auction.