Bidding on an Antique
An antique auction
• An antique is sold in a sealed-bid auction. It
will go to the high bidder at the price the high
bidder bids.
• You do not know if it is a fake or not. You
know that 20% of all antiques that look like
this one are fakes.
• If it is a fake, it will be worthless to you. If it is
real, you would be willing to pay $1,000 for it.
What is your expected value of this
antique?
•
•
•
•
•
A) $0
B) $100
C) $800
D) $1000
E) $1200
Another bidder
• You know that an expert dealer has also looked at
it. This dealer can always tell a fake and so would
never bid on a fake.
• You know that if the table is genuine, the dealer
will either bid $800 for it or $500 for it, depending
on whether he has similar items in stock.
• You believe that if the table is genuine, the dealer
is equally likely to bid $500 or to bid $800.
Diagram of Events
Event A
Event C
Event B
P(A)=.2
P(C0=.4
P(B)=.4
Event A : Antique is a Fake and Dealer doesn’t bid
Event B: Dealer bids $500
Event C: Dealier bids $800
If you bid B where 0<B<$500 and you
win the auction, what is the probability
the antique is a fake?
A)
B)
C)
D)
1
½
1/10
2/10
If you bid $100 for the antique and
you get it, what is the expected value
of the antique?
A)
B)
C)
D)
E)
0
$100
$200
$400
$800
If you bid $501 for the antique,
what is the probability you get it?
A)
B)
C)
D)
E)
0
½
.6
.8
1
When do you get it?
• You get the antique with a bid of $501 if it is a
fake, and also if it is real and the dealer bids only
$500.
• Let A be the event that the antique is a fake.
Then P(A)=.2
• Let B be the event Dealer bids $500. Then P(B)=.4
• You will get it with a bid of $501. either if Event A
occurs or Event B occurs This happens with
probability
P(A or B)=P(A)+P(B)= .2+.4=.6.
If you bid $501 for the antique and you
get it, what is the probability that it is
a fake?
A)
B)
C)
D)
E)
1/10
¼
1/3
½
1
Diagram of Events
Event A
Event C
Event B
P=.2
P=.4
P=.4
A bid of $501 gets the object if Event A or Event B occurs, but
not if Event C occurs. Given that Event A or B occurs, the conditional
Probability that A occurs is P(A)/((P(A)+P(B))=.2/(.2+.4)=1/3
The conditional probability that
you get a fake if you bid $501
• If you bid $501 the probability that you get it
is P(A or B)=.6.
• The probability that it is a fake, given that
you get it is
P(A|AorB)= P(A)/ P(AorB)=.2/.6=1/3
Your expected profit if you bid $501
• If you bid $501, and get the antique, the
probability that it is good is 2/3
• Your expected net gain if you get the antique
is (1000)2/3-501 =165.66.
• A bid of $501 will get the object with
probability .6.
• So your expected profit from a bid of $501 is
.6 times $165.66=$99.396.
What is the probability that the antique
is fake if you bid $801 and you get it.
A)
B)
C)
D)
E)
0
1/5
2/5
½
1
What is your expected profit if you bid
$801?
A)
B)
C)
D)
-$100
-$1
$100
$199
Bidding strategy
•If you bid
Where does Bayes’ Nash come in?
• Where does Bayes’ Nash come in? You don’t
know the dealer’s type. That is, you don’t know
whether he knows the antique is a fake or what
he will bid if it is real.
• You have some a prior probability distribution of
the dealer’s type and you assume that the dealer
does what is in his best interest, given his type.
• Notice that your expected value of the antique
conditional on winning the auction is not the
same as your expected value of the antique.
Some lessons
If you are bidding against experts for an object
of unknown quality, a low bid is a poor idea,
since you are likely to get stuck with the item if
it is bad.
If you are bidding against experts, it doesn’t
pay to bid at all, unless you believe the object
is worth “enough” more to you than to the
experts.
Oil Lease Auctions and the
Winners’ Curse
Geologists’ estimates of value differ
widely
Company that makes highest estimate bids the
highest. Often loses money.
Oil companies became aware of this in the
1960’s and now take it into account.
From industry experts
An article in the Journal of Petroleum Technology,
1971 by three employees of Atlantic Richfield Oil
Company reports that many oil companies found, in
retrospect that they bid too much for offshore oil
leases, particularly those in the Gulf of Mexico.
`` a lease winner tends to be the bidder who most
overestimates reserves potential… ``successful”
bidders may not be so successful after all.”
“Competitive Bidding in High-Risk Situations”,
By Capen, Clapp, and Campbell 1971
Our simplified set-up
• You own an oil company. A new field has come
up for lease.
• There are two bidders. You and another firm.
• Each of you has explored half of the oil field and
knows the value of the half they explored.
• The value of each side is either $3 million or 0,
which nature determined by the flip of a fair coin.
• Total value of field is the sum of the two sides
• You know what your side is worth, but not the
other company’s side.
The Auction
• The lease for the entire field is up for auction. A bid
must be an integer number (possibly 0) of million $.
• There are two bidders, you and the company that
explored the other side.
• You know what your side is worth.
• Entire field will be leased to the higher bidder in a
sealed bid auction. If there are tie bids, winner is
chosen by coin flip.
• If you win the auction, your profit or loss is the value
of the total field minus your bid.
A strategy
• A strategy states the amount you will bid if
your side is worth $0 and the amount you
will bid if your side is worth $3 million.
Finding a symmetric Bayes-Nash
equilibrium
• A symmetric Bayes-Nash equilibria is one in which
all players use the same strategy.
• Note that a strategy tells what you do for each
type you could be.
• This means that two players using the same
strategy might take different actions (because
they turn out to be of different types.
• In this game, two players using the same strategy
might bid differently because one saw $3 million
on his half and the other saw $0.
Lets find your strategy: What would
you bid if your side is worth $0?
A) $0
B) $1 million
C) $2 million
D) $3 million
E) $4 million
What would you bid if your side is
worth $3 million?
A) $1 million
B) $2 million
C) $3 million
D) $4 million
E) $5 million
Some things to think about
• What would be your expected profit if the
company you bid against uses the same
strategy that you do?
• If your side is worth $0 and you win the
auction, what do you expect the total oilfield
to be worth?
• What does this tell us about symmetric
equilibrium strategies?
Is (0,4) a symmetric Bayes-Nash
equilibrium?
• Suppose other guy bids 0 when he sees 0 and
$4 million when he sees $3 million on his own
side.
• If (0,4) is a symmetric Bayes-Nash equilibrium,
your best response has to be (0,4)?
• Lets see if it is
– What if you see 0?
– What if you see $3 million?
What is your best response if you see
$3 million?
• If the other guy bids 0 when he sees 0 and $4m
when he sees $3m on his own side.
• If I bid $4m when I see $3m
– If other guy sees 0, he bids 0. The object is worth $3
and I pay $4. So I would lose $1m.
– If other guy sees $3m, he bids $4. Bids are a tie. If I
win coin toss, I make a profit of $6m-4m=$2m.
•
With probability ½ I lose $1m. With probability ¼
I make $2m.
• My expected winnings if I see 3 are 1/2x1+1/4x2=0.
Suppose I bid only $1m when I see
$3m
• If the other guy bids 0 when he sees 0 and
$4m when he sees $3m on his own side.
• If I bid $1m when I see $3m
– If other guy sees 0, he bids 0. The object is worth
$3m. I pay $1m. I get a profit of $2m.
– If other guy sees $3m, he bids $4. I don’t get
field. My profit is 0.
• My expected winnings are ½ 2 +1/2 0=1.
• This is a better payoff than I get if I bid $4m
when I see $3m.
What if I bid just $1m when I see $3m?
• If the other guy bids 0 when he sees 0 and $4
million when he see $3m on his own side.
• If I see $3 million, then I am sure to get the object
if he sees 0.
• If instead I played (0,1), I would get the object
only when I see $3 million and the other guy sees
0. This happens with probability ¼ and then my
profit would 2. So my expected profit would be
1/2.
• Therefore (0,4) is not a best response to (0,4). So
(0,4) is not a symmetric Bayes-Nash equilibrium.
Is (0,3) a symmetric Bayes-N.E?
• If other guy is playing (0,3) and I play (0,3) , I will get the object if I saw 3
and he saw 0 OR if we both saw 3 and the coin flip came out my way.
• I make 0 profit if I saw 3 and he saw 0. I make a profit of 3 if we both saw
3 and I won the coin flip, which happens with probability 1/8. So my
expected payoff would be 3/8.
• If I play (0,1), I get the object if I saw 3 and he saw 0. This happens with
probability ¼ and in this case my profit is 3-1=2. So my expected profit if I
play (0,1) is 2x(1/4)=1/2.
• So (0,3) is not a best response to (0,3). Hence (0,3) is not a symmetric NE
strategy.
What about (0,2).
• Suppose other guy is playing (0,2).
If I play (0,2) my expected payoff is
½x0+½(½x1+½x½x4)=¾.
If I play (0,1) my expected payoff is ½
If I play (0,3) my expected payoff is ¾.
You can check out that (0,x) is worse for any
other x and so (0,2) for both players is a
symmetric Bayes-Nash equilibrium.
Signaling Games
Econ 171
General form
• Two players– a sender and receiver.
• Sender knows his type. Receiver does not. It is
not necessarily in the sender’s interest to tell
the truth about his type.
• Sender chooses an action that receiver
observes
• Receiver observes senders action, which may
influence his belief about receiver’s type.
• Receiver takes action
Perfect Bayes Nash equilibrium for
signaling game
• Sender’s strategy specifies an action for each type
that she could be. Her action maximizes her
expected payoff for that type, given the way the
receiver will respond.
• For each action of the sender, receiver’s strategy
specifies an action that maximizes his expected
payoff.
• Receiver’s beliefs about sender’s type, conditional
on actions observed are consistent.
Types of equilibria.
• Separating equilibria. Different types of
senders take different actions.
• Pooling equilibria Different types of senders
take same actions.
Breakfast: Beer or quiche?
A Fable *
*The original Fabulists are game theorists, David Kreps and In-Koo Cho
Breakfast and the bully
• A new kid moves to town. Other kids don’t
know if he is tough or weak.
• Class bully likes to beat up weak kids, but
doesn’t like to fight tough kids.
• Bully gets to see what new kid eats for
breakfast.
• New kid can choose either beer or quiche.
Preferences
• Tough kids get utility of 1 from beer and 0 from
quiche.
• Weak kids get utility of 1 from quiche and 0 from
beer.
• Bully gets payoff of 1 from fighting a weak kid, -1
from fighting a tough kid, and 0 from not fighting.
• New kid’s total utility is his utility from breakfast
minus w if the bully fights him and he is weak and
utility from breakfast plus s if he is strong and
bully fights him.
Nature
Toug
h
Wea
k
New Kid
New Kid
Beer Quiche
Fight
1+s
-1
Beer
Quiche
Bully
Don’t
1
0
Fight
Don’t
Bully
B
Fight
s
-1
-w 0
0
1
Fight
Don’t
0
0
1-w
1
Don’t
1
0
How many possible strategies are there for the
bully?
A) 2
B) 4
C) 6
D) 8
What are the possible strategies for
bully?
Fight if quiche, Fight if beer
Fight if quiche, Don’t if beer
Fight if beer, Don’t if quiche
Don’t if beer, Don’t if quiche
What are possible strategies for New
Kid
•
•
•
•
Beer if tough, Beer if weak
Beer if tough, Quiche if weak
Quiche if tough, Beer if weak
Quiche if tough, Quiche if weak
Separating equilibrium?
• Is there an equilibrium where Bully uses the
strategy Fight if the New Kid has Quiche and
Don’t if the new kid has Beer.
• And the new kid has Quiche if he is weak and
Beer if he is strong.
• For what values of w could this be an
equilibrium?
Best responses?
• If bully will fight quiche eaters and not beer
drinkers:
• weak kid will get payoff of 0 if he has beer, and 1w if he has quiche.
– So weak kid will have quiche if w<1.
• Tough kid will get payoff of 1 if he has beer and s
if he has quiche.
– So tough kid will have beer if s<1
– Tough kid would have quiche if s>1. (explain)
Suppose w<1 and s<1
• We see that if Bully fights quiche eaters and not
beer drinkers, the best responses are for the new
kid to have quiche if he is weak and beer if he is
strong.
• If this is the new kid’s strategy, it is a best
response for Bully to fight quiche eaters and not
beer drinkers.
• So the outcome where Bully uses strategy “Fight if
quiche, Don’t if beer “ and where New Kid uses
strategy “Quiche if weak, Beer if tough” is a Nash
equilibrium.
Clicker question
• The equilibrium in which the new kid has
quiche if weak and beer if tough and where
the bully fights quiche-eaters but doesn’t fight
beer-drinkers is
A) A separating equilibrium
B) A pooling equilibrium
C) Neither of these
If w>1
Then if Bully uses strategy “Fight if quiche,
Don’t if beer”, what will New Kid have for
breakfast if he is weak?
Pooling equilibrium?
• If w>1, is there an equilibrium in which the
New Kid has beer for breakfast, whether or
not he is weak.
• If everybody has beer for breakfast, what will
the Bully do?
• Expected payoff from Fight if quiche, Don’t if
beer depends on his belief about the
probability that New Kid is tough or weak.
Payoff to Bully
• Let p be probability that new kid is tough.
• If new kid always drinks beer and bully
chooses Don’t Fight if Beer, Fight if Quiche,
Bullie’s payoff is 0.
• If Bully chooses a strategy that Fight if Beer,
(anything) if Quiche, Bullie’s expected payoff is
-1xp+1x(1-p)=1-2p. If p>1/2, Fight if Beer, Don’t
fight if Quiche is a best response for Bully.
Pooling equilibrium
• If p>1/2, there is a pooling equilibrium in
which the New Kid has beer even if he is weak
and prefers quiche, because that way he can
conceal the fact that he is weak from the
Bully.
• If p>1/2, a best response for Bully is to fight
the New Kid if he has quiche and not fight him
if he has beer.
What if p<1/2 and w>1?
There won’t be a pure strategy equilibrium.
There will be a mixed strategy equilibrium in
which a weak New Kid plays a mixed strategy
that makes the Bully willing to use a mixed
strategy when encountering a beer drinker.
What if s>1?
• Then tough New Kid would rather fight get in
a fight with the Bully than have his favorite
breakfast.
• It would no longer be Nash equilibrium for
Bully to fight quiche eaters and not beer
drinkers, because best response for tough
New Kid would be to eat quiche.