Bidding Strategy and Auction
Design
Josh Ruffin, Dennis Langer, Kevin
Hyland and Emmet Ferriter
Auctions
Auctions entail the transfer of a particular object from a
seller to a buyer for a certain price. It is a useful way
to sell commodities of undetermined quality. Auctions
can be used for single items such as a work of art
and for multiple units of a homogeneous item such
as gold or Treasury securities. It is the purest of
markets: a seller wishes to obtain as much money as
possible, and a buyer wants to pay as little as
necessary.
Why do we care?
Auctions are a multi-billion dollar business
The US Treasury uses auctions to determine
mortgage rates
The FCC uses competitive bidding to allocate
licenses for broadcasting on the
electromagnetic spectrum
Stores such as Filene’s Basement use a
pricing strategy that reduces the price on
items remaining on the racks for longer than
a certain time
Auction Formats
First Price Auction – The individual who submits the highest bid wins the
auction, and pays that submitted highest bid; can be applied to any
type of auction
Second Price Auctions – The individual who submits the highest bid wins
the auction, and pays the amount submitted by the second highest
bidder; can be applied to any type of auction
Sealed Bid Auctions – Bids are submitted privately and each bid is
evaluated simultaneously; the highest bidder becomes the winner; can
be applied to any type of auction
Open-outcry Auctions – Bids are submitted openly and publicly for rival
bidders to evaluate; the highest bidder becomes the winner; can be
applied to any type of auction
English Auction – The auctioneer announces a low price and invites
ascending bids until no one is willing to go above the last bid made;
the last bidder wins
Dutch Auction – The auctioneer announces a high price and then
announces successively lower bids; the bidder who calls a halt to such
announcements first wins the auction
Common-value Auctions – The value of the object is the same for
all bidders; each bidder’s estimated valuation vary slightly; also
known as “Objective-value”
Private-value Auctions – Each bidder places his/her own and
unique valuation to the good; for example, valuations can be
influenced by sentimental issues and/or imprecise monetary
estimates; also known as “Subjective-value”
Winner’s Curse – When the winner, and highest bidder, are forced
to bid and pay a higher price for the good than its true value (to
win)
Risk-averse Bidders – A bidder is more concerned about the losses
caused by underbidding than the costs associated with bidding
at or close to their true valuations; risk-averse bidders want to
win without ever overbidding
Which format is the best?
The answer depends upon many variables.
1.Seller’s perspective: - tries to reach the highest selling price
- decrease incentives to cheat
-affiancy (for perisible items)
Winner’s Curse
Vickrey’s Truth Serum
In open outcry auctions, the winning bidder essentially pays the
valuation of the second highest bidder, since the winner will not
bid more than the minimum on the final bid.
In sealed envelope auctions, the winner pays his bid, regardless of
the distance between it and the second highest bid.
As a result, a strategic bidder shades his bid lower than his true
valuation in order to retain some profit.
William Vickrey devised a method to ensure that bidders would bid
their true valuation: the highest bidder wins, but only pays the
second highest bid.
So, the second-price sealed-bid auction is called “Vickrey’s Truth
Serum.”
The Seller’s Choice
Before showing why the second-price auction works, we
will consider the position of the seller.
It’s clear that, by selling the item for the second highest
price, the seller is making less profit. In essence, he
is buying information (the true valuation of each
bidder).
However, in the first-price auction, the seller is also
making less profit when the bidders shade their bids.
The seller must decide which form of auction will
reduce his profit less.
Vickrey’s Claim
For any sealed-bid auction, the bidder has three
strategies:
A) bid their true valuation
B) bid under their valuation
C) bid over their valuation
For a first-price sealed-bid auction, the best strategy is
for the bidder to bid under his valuation.
Vickrey says that in the second-price auction, the best
strategy for every bidder is to bid their true valuation.
Why Vickrey’s Claim is True
Consider a second-price sealed-bid private-value auction
for some item, and let
v = your true valuation of the item
b = your bid
r = the highest bid besides yours
All bids less than r are irrelevant, since they have no
effect on whether you win or lose.
Now, we must consider two cases:
a) where b < v
b) where b > v
Shading Up
Suppose you bid higher than your true
valuation (b > v)
Then if
I) (r < v) i.e., the next highest bid is less
than your valuation, so you win the
item and turn a profit. However, if you
had bid (b = v), you would’ve still won,
and would’ve turned the same profit.
Shading Up
Else if
II) (v < r < b) i.e., the next highest bid is
between your valuation and your bid,
then you win the item, but you must
purchase it for more than your
valuation. So, you should’ve bid (b = v);
although you would’ve lost the item, you
wouldn’t have sustained a loss.
Shading Up
Else if
III) (b < r) i.e., you do not have the
highest bid. If you had bid (b = v), you
still would’ve lost.
1st Summary
In cases I and III, bidding (b > v) has the
same result as bidding (b = v).
In case II, bidding (b > v) is worse than
bidding (b = v).
So, there is no reason to bid (b > v)
instead of (b = v) since 1/3 of the time,
the result is worse, and 2/3 of the time,
the result is equal.
Shading Down
Suppose you bid lower than your true
valuation (b < v)
Then if
I) (r < b) i.e., the next highest bid is less
than your bid, so you win the item and
turn the profit. However, if you had bid
(b = v), you would’ve still won, and
would’ve turned the same profit.
Shading Down
Else if
II) (b < r < v) i.e., you do not have the
highest bid, which is less than your
valuation. So, you should’ve bid (b = v);
you would’ve won the item and turned a
profit.
Shading Down
Else if
III) (v < r) i.e., you do not have the
highest bid, which is above your
valuation. If you had bid (b = v), you
would’ve still lost.
2nd Summary
In cases I and III, bidding (b < v) has the
same result as bidding (b = v).
In case II, bidding (b < v) is worse than
bidding (b = v).
Again, there is no reason to bid (b < v)
instead of (b = v) since 1/3 of the time,
the result is worse, and 2/3 of the time,
the result is equal.
Overview
So, we have shown that bidding your true
valuation is better than both bidding
under your valuation and bidding over
your valuation.
Therefore, it is clear that, in a secondprice auction, the best strategy for each
bidder is to bid their true valuation.
So Vickrey’s Truth Serum works
Jack v. Jill
Let’s consider a specific example. Suppose Jack and Jill
are bidding on a painting. Assume the following:
a) Jill values the painting at $100.
b) Jill considers it equal possible that Jack could
value the painting at $100 or at $80.
c) In the event that Jack and Jill make the same
bid, the winner is decided by a coin toss.
d) Jack and Jill can only make bids of $100 and
$80.
So, this example is more limited than the last.
Jill’s Point of View
Jill know that she values the painting at $100.
She considers the following possibilities:
a) Jack values the painting at $80 with
probability ½. In this case, she wins the
painting with probability 1.
b) Jack values the painting at $100 with
probability ½ also. In this case, she wins the
painting with probability of ½, according to a
coin toss.
So she calculates her odds at winning at:
( 1 )( ½ ) + ( ½ )( ½ ) = ( ¾ ).
Jill’s Payoff
So, Jill’s expected gain is
( ¾ )(100) + ( ¼ )(0) = 75.
However, this equation ignores the fact that Jill
must pay the seller. Since this is a secondprice auction, she pays $80 with probability
½ and she pays $100 with probability
( ½ )( ½ ) = ( ¼ ).
So, Jill’s net gain is
( ¾ )(100) – (80)( ½ ) – (100)( ¼ ) = 10.
Can Jill Increase Her Payoff?
Can Jill increase her payoff by bidding
$80?
To answer this, Jill must consider two
payoff matrices:
a) if Jack values the painting at $100
b) if Jack values the painting at $80.
If Jack’s Value is $100
If Jack values the painting at $100, Jill
considers this matrix:
100
80
100
(0, 0) (20, 0)
80
(0, 20) (10, 0)
So, a strategy of bidding $100 is
dominant for Jill.
If Jack’s Value is $80
If Jack values the painting at $80, Jill
considers this matrix:
100
80
100
(0, 10) (20, 0)
80
(0, 0) (10, 0)
So, again, a strategy of bidding $100 is
dominant for Jill.
Overview
So, again we can see from this specific,
yet more limited example that truthful
bidding is the dominant strategy is the
second-price auction.
Again, Vickrey’s Truth Serum proves to be
effective.