Midterm Scores
Total of 200 points, 40 per question.
A--155-200
B—120-154
C—80-119
D—50-79
F <50
Median Score 125, Top Score 180
Bayes-Nash equilibrium with
Incomplete Information
What’s New here?
Incomplete information:
In previous examples we had “Imperfect Information”. Players
knew each others payoffs, but might not know other’s moves.
In these problems, players know their own payoffs, but may not
know other Players’ payoffs
•
Bayesian Game
• Nature determines each player’s type.
• One’s type determines one’s payoffs.
• Player learns own type, but only the
probability distribution of others’ types.
• A strategy specifies what a player would do
conditional on being of each possible type.
Bayes-Nash equilibrium
• Each player in a given type will play a best
strategy for that type, given the player’s
beliefs about the probabilities of other types.
Simple Example: Greg and Marcia
Two possible types of Marcia
=p
How many possible strategies does Marcia have?
A) 1 B) 2 C) 3 D) 4 E) 8
=1-p
How many strategies are possible for
Greg?
A)
B)
C)
D)
E)
1
2
3
4
8
Strategies for Marcia
•
•
•
•
Say yes, whether you like him or not
Say no, whether you like him or not
Say yes if you like him, no if you don’t
Say no if you like him, yes if you don’t
What does Greg do?
• Greg knows that whatever her type, Marcia’s
strategy will be her best one. This is
yes if she likes him, no if she doesn’t.
• If probability that she likes him is p, his
expected payoff from Invite is
10p+2(1-p)=2+8p
• If he doesn’t invite her, his payoff is 3
• He will invite her if 2+8p>3. That is, if p>1/8
Wyatt Earp and the Gun Slinger
The story
• Wyatt Earp is town marshall in Dodge City.
• A stranger comes to town.
• Earp doesn’t know if the stranger is a
gunslinger or an ordinary cowboy.
• Earp faces the stranger after a saloon brawl.
• Earp and the stranger must simultaneously
decide whether to draw their guns.
A Bayesian gunslinger game
What are the strategies?
• Earp
– Draw
– Wait
• Stranger
–
–
–
–
Draw if Gunslinger, Draw if Cowpoke
Draw if Gunslinger, Wait if Cowpoke
Wait if Gunslinger, Draw if Cowpoke
Wait if Gunslinger, Wait if Cowpoke
The gunfight game when the stranger is (a) a gunslinger
or (b) a cowpoke
Things to notice about SPNE
• If stranger is a gunslinger, he will always draw.
• In the subgame where stranger is a cowboy,
there are two Nash equilibria. In one of them,
Cowboy and Earp both Wait. In the other,
Cowboy and Earp both Draw.
When is there a “peaceful”
equilibrium?
• Suppose that Earp waits and the other guy draws
if he is a gunslinger, waits if he is a cowpoke.
– Stranger in either case is doing a best response.
– If stranger follows this rule, is waiting best for Earp?
• Let p be the probability that stranger is a
gunslinger
– Earp’s Payoff from waiting is px1+(1-p)x8=8-7p.
– Earp’s Payoff from drawing, given these strategies for
the other guys is px2+((1-p)x4=4-2p.
• Wait is a best response for Earp if 8-7p>4-2p,
which implies that p<4/5.
One equilibrium
• We conclude that if the probability that the
stranger is a gunslinger is less than 4/5, there
is a Bayes-Nash equilibrium in which Earp
waits, and in which the cowboy draws if he is
a gunslinger and waits if he is a cowboy.
Another equilibrium?
• Is there is an equilibrium in which everybody
draws.
• If Earp always draws, both cowpoke and
gunslinger are better off drawing.
• Let p be probability stranger is gunslinger.
• If both types always draw, payoff to Earp from
Draw is 2p+5(1-p)=5-3p and payoff to Earp
from Wait is p+6(1-p)=6-5p
• Now 5-3p>6-5p if p>1/2.
Two equilibria?
• We found that if p<4/5, there is a Bayes-Nash
equilibrium in which Earp waits and Stranger
draws if he is a gunslinger, waits if he is a Cowboy.
• If p>1/2, there is a Bayes-Nash equilibrium in
which Earp draws and Stranger draws, regardless
of his type.
• So if ½<p<4/5 there are two Bayes-Nash
equilibria.
• What if p<1/2? What if p>4/5?
Another gunfight: Bat and Curly
Problem 10.2
Both men have private information about
whether they are fast shooters or slow shooters.
Curly believes that probability Bat is fast is .65.
Bat believes that probability Curly is fast is .60.
Each must decide whether to Draw or Wait.
Payoff table
Curly
Bat
FD
FW
SD
SW
FD
20,20
30,-40
30,-40
30,-40
FW
-40,30
50,50
20,20
50,50
SD
-40,30
20,20
20,20
40,-30
SW
-40,30
50,50
-40,30
50,50
Table shows payoffs to Bat and Curly conditional on their actions and type.
Note: This is not a strategic form representation. A strategy takes the form
x/y where x is what you do if you are fast and y is what you do if you are slow.
Is D/D for both players a SPNE?
Need to show that if Curly always draws, Bat will prefer to draw if he is fast
and also if he is slow.
Also need to show that if Bat always draws, Curly will prefer to draw if he is fast
also if he is slow.
Suppose Curly always draws…
If Bat draws when he is fast his expected payoff is .6 x 20 +.4 x
30=24.
If Bat waits when he is fast, his expected payoff is .5x(40)+.4x20=-12
So if Bat is fast, Bat’s best response is draw.
If Bat draws when he is slow, his expected payoff is .6x(40)+.4x20=-16.
If Bat waits when he is slow, his expected payoff is .6 x(40)+.4x(-40)=-40.
So if Bat is slow, his best response is draw.
Thus if Curly always draws, Bat’s best response is to always
draw.
Suppose Bat always draws
• Suppose Curly is fast.
– If he draws, his expected payoff is .65x 20 +.35x30=25.5.
– If he waits, his expected payoff is
.65x(-40)+.35x20=-19.
-So if Bat always draws, and if
Curly is fast, his best response is Draw.
• Suppose Curly is slow.
– If he draws, his expected payoff is .65x(-40)+.35x20.
– If he waits, his expected payoff is .65x(-40)+.35x(-30).
So if Curly is slow, is best response is Draw
• Conclusion: If Bat always draws, Curly’s best response
is to always draw.
Some conclusions
• We showed that there is a Bayes Nash
equlibrium where both always draw.
• A similar exercise will show that there is also
an equilibrium where both always wait.
• Another similar exercise will show that there is
also an equilibrium where each of them will
draw if and only if he is slow.
Problem 10.5 Find Bayes-Nash equilibrium
How many strategies are there for Players 1 and 2?
A) 4 for 1, 2 for 2 B) 4 for each player C) 2 for each player
D) 2 for 1, 8 for 2 E) 4 for 1, 8 for 2
Problem 10.5 Find Bayes-Nash equilibrium
If Player 2 plays a, best response for Player 1 is x if High type, y if low type.
If Player 2 plays b, best response for Player 1 is y for either type.
We found that:
1) If Player 2 plays a, best response for Player 1 is x if High type, y if low
type.
2) If Player 2 plays b, best response for Player 1 is y for either type.
Now we want to know whether:
A) if a is the best response for Player 2 when Player 1 plays x if High type and y if
low type.
And whether:
B) b is the best response for Player 2 when Player 1 always plays y.
A) If Player 1 plays x if High type and y if low type, and if p is the probability that Player 1
is a high type, then the expected payoff to Player 2 from playing a is px1+(1-p)x1=1.
And the expected Payoff to 2 from playing b is px3 +0. Playing a is the best response if
1>3p or p<1/3.
B) If Player 1 always plays y, then the expected payoff to Player 2 from playing a is
1xp+1x(1-p)=1 and the expected payoff to Player 2 from playing b is 2xp+0=2p. Thus
for Player 2, b is the best response when Player 1 always plays y if 2p>1, which
means p >1/2.
Summing up
• If p<1/3, there is a Bayes-Nash equilibrium in
which Player 1 plays x if high and y if low and
Player 2 plays a.
• There is another Bayes-Nash equilibrium in
which Player 1 always plays y and Player 2
plays b.
The Perils of Alice and Bob:
Revisited
Does she or doesn’t she
• Bob doesn’t know whether Alice likes him or
not. Whether or not she likes him, he likes to
be with her.
• Bob and Alice must each decide on which of
two movies to go to.
• Suppose that Bob chooses which movie to go
to and, as both know, Alice has a spy who will
tell her where he is going before she chooses
which to go to.
She loves me, she loves me not?
(Bob moves before Alice)
Nature
She scorns him
She loves him
Bob
Go to A
Bob
Go to B
Go to A
Alice
Alice
Alice
Go to A
2
3
Go to B
0
0
Go to A
1
1
Go to B
Go to B
3
2
Go to A
2
1
Alice
Go to B
Go to A
0
2
1
3
Go to B
3
0
What are their strategies?
• For Bob
– Go to A
– Go to B
• Alice has four information sets.
–
–
–
–
I Love him and he’s at A
I Love him and he’s at B
I Scorn him and he’s at A
I Scorn him and he’s at B
• In each information set, she can go to either A or
B. This gives her 2x2x2x2=16 possible strategies.
A weakly dominant strategy for Alice
• Go to A if you love him and he goes to A. Go
to B if you love him and he goes B. Go to B if
you scorn him and he goes to A. Go to A if you
scorn him and he goes to B. (We write this as
A/B/B/A)
• This is weakly dominant but not strictly
dominant. Explain.
• Let’s look for a subgame perfect Nash
equilibrium where Alice goes A/B/B/A
Checking equilibrium
• Suppose Alice goes where Bob is if she loves him
and goes where he is not if she scorns him.
(A/B/B/A)
• Payoff to Bob from A is 2p. Payoff from B is
3p+1(1-p)=2p+1. Since 2p+1>2p, for all p>=0, B
is his best response to (A/B/B/A).
• Also A/B/B/A is a best response for Alice to Bob’s
B.
• So we have a Bayes-Nash equilibrium.
Simultaneous choice
• The story gets more complex (and interesting)
if Bob and Alice must choose their movies
simultaneously, so neither knows what the
other did when choosing a movie.
Does she or doesn’t she?
Simultaneous Play
Nature
She scorns him
She loves him
Bob
Go to A
Bob
Go to B
Go to A
Alice
Alice
Alice
Go to A
2
3
Go to B
0
0
Go to A
1
1
Go to B
Go to B
3
2
Go to A
2
1
Alice
Go to B
Go to A
0
2
1
3
Go to B
3
0
Alice’s (pure) strategies
• Alice doesn’t know what Bob did, so she can’t
make her action depend on his choice. She can go
to either A or B.
• She does know whether she loves him or scorns
him when she chooses.
• She has 4 possible strategies
–
–
–
–
A if love, A if scorn
A if love, B if scorn
B if love, A if scorn
B if love, B if scorn
Bayes’ Nash equilibrium
• Is there a Bayes’ Nash equilibrium where Bob
goes to B and Alice goes to B if she loves Bob,
and to A if she scorns him?
– This is a best response for both Alice types.
– What about Bob?
Bob’s Calculations
If Bob thinks the probability that Alice loves him
is p and Alice will go to B if she loves him and A
if she scorns him:
– His expected payoff from going to B is
3p+1(1-p)=1+2p.
– His expected payoff from going to A is
2(1-p)+0p=2-2p.
Going to B is Bob’s best response to the strategies of
the Alice types if 1+2p>=2-2p. Equivalently p>=1/4.
What we know so far
• If Bob goes to B for sure, then Alice’s best
response is to go to B if she loves Bob and A if
she scorns Bob.
• If Alice goes to B if she loves Bob and A if she
scorns him, then Bob’s best response is to go to B
for sure if and only if p≥1/4.
• So if p≥1/4, there will be a Nash equilibrium
where Bob uses the pure strategy go to B, but if
p<1/4, there is no Nash equilibrium where Bob
goes to B.
A more surprising pure strategy N.E.
• Suppose that Bob goes to movie A for sure.
• The best response for Alice is to go to movie A if
she loves Bob and movie B if she scorns him.
• If this is Alice’s strategy, then Bob’s expected
payoff from going to A is 2p+0(1-p)=2p and his
expected payoff from going to B is p+3(1-p)=3-2p.
• Going to A will be his best response if
2p≥3-2p, which means p≥3/4.
So there will be a pure strategy N.E. in which Bob
goes to movie A for sure if and only if p≥3/4.
When is there a pure strategy N.E.?
• We have discovered that there is a pure
strategy N.E. where Bob goes to B if and only
if p≥1/4 and there will be a pure strategy N.E
where Bob goes to B if and only if p≥3/4.
• It follows that if p<1/4 there is no pure
strategy N.E. and if p≥3/4, there are two pure
strategy N.E.
Is there any Bayes-Nash equilibrium in
pure strategies if p<1/4?
A) Yes, Alice goes to B if she loves Bob and A if
she scorns him and Bob goes to B.
B) Yes, Alice goes to A if she loves Bob and B if
she scorns him and Bob goes to B.
C) Yes there is one, where Alice always goes to
A.
D) No there is no Bayes-Nash equilibrium in
pure strategies.
What If p<1/4
• We found that there is no pure strategy N.E in
this case.
• What about a mixed strategy NE?
Mixed strategy equilbrium:
Bob the stalker
• If Bob thinks it likely that Alice scorns him,
then if he uses a pure strategy, he knows that
if Alice scorns him, she will avoid him.
• If he uses a mixed strategy, he would catch her
sometimes.
• Let’s look for a mixed strategy for Bob such
that Alice, if she scorns Bob would be
indifferent between Movies A and B.
What about a mixed strategy
equilibrium?
• If p<1/4, can we find a mixed strategy for Bob
such that Alice is indifferent
• Let’s check out what would happen if Bob
knows Alice scorns him?
• Consider the Alice type who scorns Bob. If Bob
goes to movie A with probability q, When will
Alice be indifferent between going to the two
movies?
The game if Alice scorns Bob
Bob
Alice
A
B
A
B
1,2
2,0
3,1
0,3
Making Scornful Alice indifferent
• If Bob goes to Movie A with probability q and
Alice Scorns Bob:
– Alice’s payoff from A is 1q+3(1-q) =3-2q
– Alice’s payoff from B is 2q+0(1-q)=2q
– Alice will be indifferent if 3-2q=2q, which implies
q=3/4.
When will Bob do a mixed strategy?
• Note that if Bob goes to A with probability ¾,
and if Alice loves him, her best response is to
go to Movie A.
• If there is an equilibrium where Bob uses a
mixed strategy, he must be indifferent
between going to A and going to B.
• Can we find a mixed strategy for Alice to use if
she scorns him so that Bob will be indifferent
between A and B?
Making Bob indifferent
• Let r be the probability that Alice goes to Movie A
if she scorns Bob and suppose that Alice always
goes to A if she loves Bob.
• Expected payoffs for Bob are
– If he goes to A, 2p+(1-p)(2r+0(1-r))=2p-2pr+2r
– If he goes to B, 1p+(1-p)(1r+3(1-r))=2p+3+2pr-2r
– He is indifferent between A and B if these are equal.
• This implies r=(3-4p)/4-4p.
• Now r is between 0 and 1 if and only if p≤3/4
Summing up
• We have learned that if p<1/4, there is no pure strategy
Nash equilibrium, but there is a mixed strategy
equilibrium.
• When 1/4≤p≤3/4, there is one pure strategy N.E. and
one mixed strategy Bayes-Nash equilibrium. In the pure
strategy equilibrium, Bob always goes to movie B.
• When ¾≤p≤1 there is no mixed strategy equilibrium,
but there are two pure strategy Nash equilibrium. In
one pure strategy equilibrium, Bob goes to B and in the
other he goes to A.
• In the mixed strategy equilibria, Bob goes to movie A
with probability ¾, Alice goes to Movie A if she loves
Bob and she goes to movie A with probability r=(34p)/(4-4p) if she scorns him.
Maybe, later?