Incomplete Information and
Bayes-Nash Equilibrium
Why some fights start
Weak or tough?
• Two players are in dispute over a resource.
• Player 1 knows his own strength, but does not know
whether Player 2 is weak or tough and doesn’t know if
Player 1 will fight or yield.
• If Player 2 is weak, Player 1 could beat him in a fight,
but if Player 2 is tough, Player 1 would lose.
• Player 2 knows his own strength and also knows Player
1’s strength, but doesn’t know what Player 1 will do.
• Each player can either yield or fight. The extensive
form of the game is shown on the next slide.
Tough or Weak?
Nature
p
1-p
2 is strong
Fight
2 is weak
Player 2
Player 2
Fight
Yield
Yield
Player 1
Fight
100
-100
Yield
Fight
Fight
Yield
Fight
Yield
Yield
0
100
0
0
-100
100
100
0
0
100
Top number is payoff to Player 2. Bottom is payoff to Player 1
0
0
What do we need to specify for a
Bayes-Nash equilibrium
• Strategy for Player 1: Yield or Fight
• Strategy for Player 2: What he will do if strong.
What he will do if weak.
Is there a Nash Equilbrium where
Player 1 yields?
If Player 1 yields, then Fight is the best response
for Player 2, whether he is strong or weak.
If Player 2 always plays Fight, is Yield the best
response for Player 1?
Expected payoff for Player 1:
• From Yield is 0.
• From Fight is -100p+100(1-p)=100(1-2p)
• Payoff from Yield is larger than from Fight if
p>1/2
Bayes-Nash equilibrium Player 1
believes that the probability that 2 is
tough is p>1/2
Player 1 Yields.
Player 2 Fights if strong and Fights if weak.
Expected payoff to Player 1 is 0.
When is there an equilibrium where a
fight could happen?
If Player 1 fights, best response for Player 2 is to
Fight if he is strong and Yield if he is weak.
If Player 2 fights when he is strong and yields
when he is weak, expected payoff to Player 1
from Fight is -100p+100(1-p)=100(1-2p).
Expected payoff to Player 1 from Yield is 0.
Fight is best response if p<1/2.
Bayes-Nash Equilibrium if p<1/2
• Player 1 chooses Fight
• Player 2 chooses Fight if he is Strong and Yield
if he is weak.
• Expected payoff to Player 1 in this equilibrium
is 100(1-2p).
Hiring a Spy
• How much is it worth to Player 1 to find out
whether Player 2 is strong or weak?
• If Player 1 knows whether 2 is strong or weak,
Player 1 will Yield when 2 is strong and Fight
when Player 2 is weak.
In this case, Player 1 will get a payoff of 0 when
2 is strong and 100 when Player 2 is weak.
Expected payoff is then 0p+100(1-p)=100(1-p).
What is a spy worth?
• If p>1/2, knowing Player 2’s type increases
profits from 0 to 100(1-p), so Player 1 would be
willing to pay a spy up to 100(1-p).
• If p<1/2, knowing Player 2’s type would increase
Player 1’s profits from 100-2p to 100-p, so Player
1 would be willing to pay a spy up to
100(1-p)- 100(1-2p)=100p.
Value of spy’s information as a
function of p
Value of information
0
1/2
1
p
Private Information and Auctions
Auction Situations
• Private Value
– Everybody knows their own value for the object
– Nobody knows other people’s values.
• Common Value
– The object has some ``true value’’ that it would be
worth to anybody
– Nobody is quite sure what it is worth. Different
bidders get independent hints.
First Price Sealed Bid Auction
with private values
• Suppose that everyone knows their own
value V for an object, but all you know is that
each other bidder has a value that is equally
likely to be any number between 1 and 100.
• A strategy is an instruction for what you will
do with each possible value.
• Let’s look for a symmetric Nash equilibrium.
Case of two bidders.
• Let’s see if there is an equilibrium where both
bidders bid some fraction a of their values.
• Let’s see what that fraction would be.
• Suppose that you believe that if the other
guy’s value is X, he will bid aX.
• If you bid B, the probability that you will be
the high bidder is the probability that B>aX.
• The probability that B>aX is the probability
that X<B/a.
Two bidder case
• We have assumed that the probability
distribution of the other guy’s value is uniform on
the interval [0,100].
• For number X between 0 and 100, the
probability that his value is less than X is just
X/100.
• The probability that X<B/a is therefore equal to
B/(100 a).
• This is the probability that you win the object if
you bid B.
So what’s the best bid?
• If you bid B, you win with probability B/(100a).
• Your profit is V-B if you win and 0 if you lose.
• So your expected profit if you bid B is
(V-B) times B/(100a)=(1/100a)(VB-B2).
To maximize expected profit, set derivative with
respect to B equal to zero. We have V-2B=0 or
B=V/2.
This means that if the other guy bids proportionally
to his value, you will too, and your proportion will
be a=1/2.
Bid-shading in this example
• In a first-price sealed bid private value
auction, where each believes that the other’s
value is uniformly distributed on the interval
[0,100], there is a Bayes-Nash equilibrium in
which each bidder bids half of her value.
Oil Lease Sales
Single bidder version
• A new oil field has come up for lease.
• The current owner has done no exploration of
the oilfield. All that the current owner knows is
that the field has two halves such that each half is
worth either $0 or $3 million dollars with
independent probabilities of ½ for each value for
each half.
• It is common knowledge that there is only one
possible buyer and that buyer has learned the
value of one half of the field but not the other.
Bayes-Nash equilibrium
• Strategy for Seller--Posts a price.
• Strategy for Buyer—Rule that specifies whether she
will buy or not at each possible price if the side she
knows is worth $0 and if it is worth $3 million.
• Buyer tries to maximize her expected profit.
• Buyer will buy if and only if the price is less than or
equal to her expected value, given her information.
• When seller posts his price, he doesn’t know whether
the side the buyer explored is worth 0 or $3 million.
Extensive Form of simplified game
Nature
Buyer sees 0
Buyer sees 3
Seller
P=1.5-e
P=4.5-e
Buyer
Buy
1.5-e
e
Don’t
0
0
P=1.5-e
Buyer
Buyer
Buy
4.5-e
e-1.5
Don’t
0
0
Buy
Don’t
P=4.5-e
Buyer
Buy
Don’t
0
0
0
1.5-e
4
0
3+e
.
Top number is Seller’s Profit. Bottom is Buyer’s Profit. 5e is a small number <1.
Trimming tree with subgame
perfection
Nature
Buyer sees 0
Buyer sees 3
Seller
P=1.5-e
Buyer
Buy
1.5-e
e
P=4.5-e
P=1.5-e
Buyer
Buyer
Don’t
0
0
P=4.5-e
Buy
1.5-e
3+e
Buyer
Buy
4
.
5
-
What is the probability
That the Buyer will buy the oil field if the price is
1.5-e?
That the Buyer will buy the oil field if the price is
4.5-e?
Clicker question
If the seller posts a price of just under $4.5
million, her expected profit is closest to.
A) $1 million
B) $1.5 million
C) $2 million
D) $2.25 million
E) $3 million
Clicker question
If the seller posts a price of just under $1.5
million, her expected profit is closest to
A) $.5 million
B) $.75 million
C) $1 million
D) $1.25 million
E) $1.5 million
In a Bayes-Nash Equilibrium
with a single buyer
A) Seller will post a price of (just under) $4.5
million. Buyer will buy if buyer’s side is worth $3
million and will reject if her side is worth $0.
B) Seller will post a price of (just under) $1.5
million. Buyer will buy whether her side is worth
$3 million or 0.
C) Seller will post a price of (just under) $1.5
million. Buyer will buy only if her side is worth $3
million.
In Bayes-Nash equilibrium for this game, the
expected revenue of the seller is
A) $1 million
B) $2 million
C) $2.25 million
D) $3 million
E) $4.5 million
What if there are two bidders?
• Each has explored a different half of the oil
field and knows the value of the half she
explored.
• The value of each side is either $3 million or 0,
which depended on the flip of a fair coin.
• Total value of field is the sum of the two sides
• Each bidder knows what her side is worth,
but not the other bidder’s side.
Posted Price
• The seller posts a price.
• If one buyer accepts and the other declines,
oilfield is sold to that bidder at posted price.
• If both buyers accept, the seller tosses a fair
coin to decide which buyer to sell to.
• If neither buyer accepts, oilfield remains
undeveloped and seller and both buyers
receive 0 payoff.
Would you buy?
• Suppose there are two buyers as described
above and suppose that the seller posts a
price of $4.1 million. If the half of the oil
field that you examined is worth $3 million,
would you buy it?
A) Yes
B) No
Expected values?
• Suppose there are two bidders and both would
offer to buy at a posted price of $4.1 million if
they saw $3 million and not if they saw 0 (Why
not? We found that your expected value of the
oilfield is $4.5 if you saw $3 million.)
• But wait... If the oilfield is worth 0, you get it for
sure. But if it is worth $6 million, you get it with
probability ½ because this means the other guy
also bid $4.5.
A winner’s curse?
• So if you offer to buy at $4.1 million, then
even though the oilfield is equally likely to be
worth $6 million or $3 million, you are more
likely to get it if its worth 3 than if its worth 6.
• We want to find the expected value of the
oilfield conditional on your getting it if you
offer to buy.
Conditional Probability
P(A|B) =P(A and B)/P(B)) (Bayes’ rule)
Let Event A be other guy saw $3 million.
Let Event B be: Your offer is accepted.
P(A and B) = ¼.
P(B)=1/4+1/2=3/4.
P(A|B)=1/4÷3/4=1/3.
If you get the object, the probability that the other
guy saw $3 million is only 1/3.
Probability he saw 0 is 2/3.
Conditional expected value and
winner’s curse
Suppose that you saw $3 million on your side.
Although the probability that the other guys saw
$3 million is ½ on his side, the probability that
he saw $3 million, given that you get the object,
is only 1/3.
The conditional expected value of the object
given that you get it is therefore only
$6 million x 1/3 + $3 million x 2/3= $4 million.
Winner’s curse in common value
auctions
If you win something in an auction or other
bidding contest, makes it more likely that the
object is of low value to other bidders.
If low value to other bidders implies lower value
to you, then it is a mistake to bid as much as
your expected value given only your own
information.
Possible equilibrium?
• Suppose the seller sets price at (just under) $4
million.
• Buyers will offer to buy if they saw $3 million.
• Buyers will not offer to buy if they saw $0.
You will get a sale if at least one of the buyers
saw $3 million.
Bayes-Nash equilibrium with two
buyers
• There would be a Bayes-Nash equilibrium in
which seller sets price at $4 million and buyers
will offer to buy if their side is worth $3
million and will not offer to buy if their side is
worth 0.
• Expected revenue of seller would then be
(1/4)x0+(3/4)x4 million= $3 million
One more auction
First-bidder, sealed bid auction. Two bidders.
Object goes to high bidder. If there is a tie, coin
flip decides who gets it.
Each knows his own value, which is either 3, 5,
or 8. Each believes that other’s value is 3 with
probability .3, 5 with probability .3, and 8 with
probability .4.
Checking an Equilibrium Candidate
• Is there a symmetric Bayes-Nash equlibrium in
which each bidder bids 2 if his value is 3, 3 if
his value is 4, and 4 if his value is 8?
• To check this out, suppose that the other
bidder is using this strategy. See if this is then
the best strategy for you.
• Recall that each believes that other’s value is
3 with probability .3, 5 with probability .3, and
8 with probability .4.
What is probability my bid wins?
If I bid less than 2, I never win.
If I bid 2, I win only if other bidder has value 3 and I win
coin toss. So probability of winning if I bid 3 is always
(1/2)x.3=.15
If I bid 3, I win if other bidder bids 2 or if other bidder bids
3 and I win coin toss. This happens with probability
.3 +(1/2)x.3=.45.
If I bid 4, I win if other bids 2 or 3 or if he bids 4 and I win
coin toss. This happens with probability .3+.3+(1/2)x.4=.8
If I bid more than 4, I get it for sure.
Table of winnings
My
Bid
Prob I get it Expected payoff:
V=3
Expected
payoff:V=5
Expected payoff: V=8
1
0
0
0
0
2
0.15
1x.15=0.15* 3x.15=0.45 6x.15=0.9
3
.45
0x.45=0
2x.45=0.9* 5x.45=2.25
4
0.8
-1x.8=-0.8
1x.8=0.8
4x.8=3.20*
5
1
-2x1=-2
0x1=0
3x1=3.0
Checking it out
• Look at table on previous slide.
• Note that if other guy is bidding 2 when his value
is 3, 3 when his value is 4, and 4 when his value is
8, your best choices are:
– 2 if 3
– 3 if 5
– 4 if 8
• So there is a symmetric Bayes Nash equilibrium
where each bidder uses this strategy.
Another Bayes Nash equilibrium?
• Suppose each bids 2 if 3, 3 if 5 and 5 if 8.
What are your probabilities of winning?
Same as before with bids of 1, 2, or 3, but not
the same with bids of 4 or 5.
If other bids 2 if 3, 3 if 4, and 5 if 8
My Bid
Prob I get Expected payoff:
it
V=3
Expected
payoff:V=5
Expected payoff: V=8
1
0
0
0
0
2
0.15
1x.1=0.15 *
3x.15=0.45
6x.15=.9
3
.45
0x.45=0
2x.45=0.90* 5x.45=2.25
4
.6
-1x..6= -.6
1x.6=.6
4x.6 =2.4*
5
0.8
-2x1= -2
0x.45=0
3x.8=2.4*
6
1
-3x1= -3
-1x1=-1
2x1=2
Equilibrium
• So we see from the table that if the other bids
2 when value is 3, 3 when value is 5 and 5
when value is 8, that it is a best response for
you to bid 2 when your value is 3, 3 when
your value is 5 and 5 when your value is 8.
• Each is doing a best response to the other’s
action. So this is also a symmetric Nash
equilibrium.
Going, Going,…