1.
There are three bidders in an auction for a single object. Each individual
knows his own value for the object, but nobody knows anyone’s value but
their own. The other bidders and the seller think of any bidder’s value as
a random variable which is equally likely to be any amount between 0 and
$100. As it turns out, the actual values are as follows. Bidder 1 values the
object at $90, bidder 2 values it at $75 and bidder 3 values it at $45.
A) If the object is sold in an English auction and bidders all behave rationally,
approximately what price will the seller get for it?
$75
B) If the object is sold in a secondprice sealed bid auction and all bidders
bid rationally, approximately what price will the seller get for it?
$75
C) If the object is sold in a ﬁrstprice sealed bid auction, and each bidder
uses an equilibrium strategy, how much will each bidder bid? What price
does the seller receive for the object?
Each will bid 2/3 of his value. Seller will get $60.
2.
A) What is the ”revenue equivalence theorem?”
The revenue equivalence theorem is the result that in
a private values auction where a single object is sold and
where the probability distribution of individual values is
continuous and independent among individuals, the ex
pected revenue in a Nash equilibrium between bidders is
the same for any type of auction that has the following
characteristics. a) In equilibrium the object is sold to
the person with highest buyer value. b) The expected
winnings or losses of the person with lowest buyer value
is zero. The ﬁrst price sealed bid auction, the English
auction, the Dutch auction, and the second price sealed
bill auction all have these properties and hence yield the
same expected revenue under the above assumptions.
1
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 Fall '08
 Bergstrom,T
 Probability, Auction, revenue equivalence theorem, Sam Seatcover

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