Cheap Talk
When can cheap talk be believed?
• We have discussed costly signaling models like
educational signaling.
• In these models, a signal of one’s type is
credible if the cost of a signal differs between
types and it doesn’t pay to send a false signal.
• But what can be learned if there is no cost to
anyone from sending a signal.
• When will senders tell the truth and receivers
believe what they are told?
Signaling intent
• Consider a simultaneous game in which one or
more players are allowed to say how they are
going to play.
• Will they tell the truth?
• Will others pay attention to what they say?
Example:
• In Rock, Paper, Scissors, Bart gets to say what
he is going to do on the next play, then gets to
choose what to do.
• What would Bart do?
• How would Lisa respond?
Babbling Equilibrium
• Message sender sends a completely
uninformative message.
• Receiver ignores it.
• In a pure conflict game, like RPS, this is the
only equilibrium.
• If sender’s signal was at all informative, it
would be used to his disadvantage.
Common interest games
• In some games, the players have a common
interest.
• If Player A gets a higher payoff when Player B
knows how he will move than when Player B
does not, it is in the interest of A to correctly
inform B of what he will do and in the interest
of B to believe A.
Dressing for the ball
The story
• Players are the Countess and the Duchess
• They are going to a formal ball.
• Each has two favorite dresses, a red dress or a
blue dress.
• Problem is they use the same designer. Their
red dresses are identical and so are their
blues.
• Both would be humiliated if they wore
identical dresses.
A common interest game:
Dressing for the Ball
Duchess
Red Dress
Blue Dress
Red Dress
-10, -10
20, 20
Blue Dress
20, 20
-10,-10
Countess
Nash equilibrium
• There are two asymmetric equilibria in pure
strategies.
• But if they play only once and don’t
communicate, how could they possibly find
an equilibrium.
• In single shot play for this game, a symmetric
mixed-strategy equilibrium seems more likely.
• Lets look for a symmetric Nash equilibrium in
mixed strategies.
If the countess wears a red dress with
probability ¾, the best response for
the duchess is to wear a red dress with
probability:
A)
B)
C)
D)
1/4
3/4
1/2
0
Red Dress
Blue Dress
Red Dress
Blue Dress
-10, -10
20, 20
20, 20
-10,-10
Symmetric equilibrium?
There is a symmetric Nash equilibrium in which
duchess and countess each play the mixed
strategy wear-a-red-dress with probability p
A) For any p less than 1/2
B) For any p greater than 1/2
C) Only if p=1/2
D) Only if p=0
E) Either if p=1 or p=0.
What is the expected payoff to each
player if each flips a fair coin to decide
the color of her dress?
A)
B)
C)
D)
E)
15
5
12.5
10
-5
Red Dress
Blue Dress
Red Dress
Blue Dress
-10, -10
20, 20
20, 20
-10,-10
How about messages?
• What do you think would happen if only the
Countess can send a message?
• Countess decides what color to say she’s
wearing and also what she does wear.
• Countess sends a footman to deliver the
message to Duchess.
• Duchess reads Countess’s message and
decides what to wear.
Possible Pure Strategies
• For Countess:
Say Red, Wear Red
Say Red, Wear Blue
Say Blue, Wear Red
Say Blue, Wear Blue
• For Duchess:
Wear Blue if C says Red, Red if C says Blue
Wear Blue if C says Red, Blue if C says Blue
Wear Red if C says Red, Blue if C says Blue
Wear Red if C says Red, Red if C says Blue
Suggested exercise
• Draw an extensive form representation of this
game
• Write out the strategic form and find subgame
perfect Nash equilibria.
What do you say? Red or Blue?
Countess
Say Red and
Wear Red
Say Blue and
Wear Blue
Say Red
And Wear
Blue
Say Blue and
Wear Red
Duchess
Wear
Red
-10
-10
Wear
Blue
20
20
Duchess
Wear
Red
Wear
Wear Red Wear
Blue
Blue
Wear
Red
20
20
-10 -10 20
-10 -10 20
20
20
Wear
Blue
-10
-10
A Nash equilibria
• Countess plays: Say “I’ll wear Red” and she
wears Red
• Duchess plays: wear Blue if C says “I’ll wear
Red”, and wear Red if C says “I’ll wear Blue”.
• Show that this is a N.E.
• What other Nash equilibria can you find?
The forgetful Countess:
Another Nash equilibrium
• Countess says “I’ll wear red, then flips a coin
to decide what to wear.
• Duchess pays no attention to what Countess
says, flips a coin herself.
• This kind of equilibrium is known as a babbling
equilibrium.
An eccentric countess:
Another Nash equilibrium
• Duchess says “I’ll wear red”, then wears blue.
• Countess plays “Wear color that Duchess
claims she will wear.”
• This is an equilibrium. Duchess always “lies”
Countess believes that duchess will “lie” and
acts accordingly.
• What does it mean when Duchess says “Red”?
Simultaneous messages
• Why should one of them get to move first?
• Suppose that the day before the ball, the
duchess and the countess each have a
footman deliver a message to the other.
• Neither knows what the other’s message says
when she sends hers.
Single messages sent simultaneously
• A symmetric Nash equilibrium:
• Each flips a coin and sends a message “I will
wear red” or “I will wear blue” with
probability ½.
• If they said different colors, each wears what
she said she would. If they said the same
color, they each toss a coin to decide what to
wear.
• Check that this is a Nash equilibrium
If they each use the single message
strategy discussed in previous slide,
what is the probability that they wear
different colors to the ball?
A)
B)
C)
D)
E)
½
1
¼
¾
2/3
A second message?
• Suppose that if they say same color on first message,
they get a chance to send a second message in an
attempt to coordinate.
• If the messages say different colors, both wear what
they said they did.
• If messages said the same color, each flips a coin to
determine a color and has footman deliver a new
message with that color.
• If second messages have different colors, they wear
what they said they would.
• If second messages say same colors, each flips a coin to
decide what to wear.
Two message outcome
• With the symmetric strategies of previous slide, they
wear different colors if they sent different messages
the first time. This happens with probability ½.
• They also wear different colors if they sent the same
message the first time, but different messages the
second time. This happens with probability ¼.
• They also wear different colors if they sent same
messages both times, but their final coin flips came out
differently. Probability of this is 1/8.
• Probability they wear different colors is therefore
1 /2+1/4+1/8=7/8.
Partially Conflicting Interests
Red preferred
Duchess
Red Dress
Blue Dress
Red Dress
-10, -10
20, 0
Blue Dress
0, 20
-10,-10
Countess
What is the mixed strategy equilibrium if there is
no pre-ball communication?
Finding symmetric mixed equilibrium
• Payoff to countess if duchess wears red with
probability p
– Wearing red: -10p+20(1-p)=20-30p
– Wearing blue 0p-10(1-p)= 10p-10
• Countess will mix if 20-30p= 10p-10, so p=3/4.
• By symmetry, each will mix if the other wears red
with probability ¾.
• In this equilibrium, each gets a payoff of
10p-10= -2.5
Simultaneous message case
• Suppose each sends a message, “Red” or “Blue”.
– If messages are different, each wears what she said
– If messages are the same, game reverts to a subgame
that is the same as the no-message game. In
symmetric equilibrium for this subgame, each wears
red with probability ¾ and the expected payoff for
each is -2.5
Strategic form with one round of talk
Say Red
Say Blue
Say Red
-2.5, -2.5
20, 0
Say Blue
0, 20
-2.5,-2.5
• Expected payoffs to countess if duchess says “red” with
probability p,
• Say “red” -2.5p+20(1-p)=20-22.5p
• Say “blue” 0p-2.5(1-p)= 2.5p-2.5
• These are equal when 20-22.5p= 2.5p-2.5 or 25p=22.5,
which implies p=9/10.
The Social Climber and the
Duchess
Conflicting Interests
Dressing for the Ball
Duchess
Social
Climber
Red Dress
Blue Dress
Red Dress
10, -10
0, 10
Blue Dress
0, 10
10,-10
What are the Nash equilibria if there is no
pre-ball communication?
One player sends signal
Suppose Duchess sends a message to the social
climber saying what she will wear.
Can the duchess gain by lying? What will the social
climber make of what she says?
Is any informative message an equilibrium?
What about babbling?
Alice and Bob
Bob
Go to Movie A Go to Movie B
Alice
Go to Movie A 3,2
1,1
Go to Movie B
2,3
0,0
• Mixed strategy equilibrium:
Alice goes to A with probability p such that
2p= p+3(1-p), so p=3/4.
• Similar reasoning finds Bob goes to B with
probability 3/4
Alice and Bob without talk
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
Nash equilibrium
• Mixed strategy equilibrium. Bob goes to B
with p=3/4, Alice goes to A with probability
3/4.
• Probabilities: Meet at A 3/16: Meet at B 3/16
• Probability they find each other is only 3/8.
• Expected payoff to each is (3/16)3+(3/16)2+
(9/16)1+(1/16)0=3/2
Talking it over
Suppose Bob gets to say where he is going and
Alice doesn’t get to say anything.
What do you think would be the outcome?
Two-way conversation, single message
• Each gets to send the other a single message,
suggesting which movie to go to, then decide
where to go.
• Suggested equiibrium: If both say same movie,
they both go there. If they name different
movies, they play original mixed strategy
game.
• Draw extensive form tree.
Game of simultaneous messages
• Pure strategies, at first decision node
– Say I am going to A
– Say I am going to B
• After hearing other person’s message (and
one’s own) go to one movie or the other.
• Sample strategy for Bob
Say A, If Alice says A, go to A. If Alice says B, go
to to B with probability ¾.
A Nash equilibrium
in mixed strategies
• With probability p, say “I am going to A” and
with probability (1-p) say “I am going to B.”
• If both say they are going to same place, they
both go there. If they say different things,
they ignore the conversation and play mixed
strategy for which movie to attend.
Talking game: Abbreviated payoff
matrix
Bob
Alice Say Movie A
Say Movie B
Say Movie A
Say Movie B
3, 2
3/2,3/2
3/2,3/2
2,3
If both say same movie, they both go there. If
they say different movies,
They play original mixed strategy game.
Mixed strategy equilibrium for this
game
• If Alice says Movie A with probability p,
Then Bob’s payoff from saying “movie A” is
2p+(3/2)(1-p) and his payoff from saying “Movie
B” is 3(1-p)+(3/2)p.
These are equal if 3/2+1/2p=3-(3/2)p, which
implies p=3/4.
Bob
Alice
Payoffs
• With probability 3/16, they both say A and go
to A
• with probability 3/16, they both say B and go
to B.
• With probability 10/16, they say different
things from each other and play original mixed
strategy equilibrium.
• Expected payoffs
:3(3/16)+2(3/16)+1.5(10/16)=15/8>3/2.
Mixed strategy equilibrium for Talking
Game
• If Bob says “movie A” with probability q, when
will Alice be willing to use a mixed strategy?
• Her expected payoff from saying Movie A is
3q+3/2(1-q) and her expected payoff from saying
B is 3/2q+2(1-q).
• These are equalized when q=1/4. In a mixed
strategy equilibrium, Bob says A with probability
¼ and B with probability ¾.
• Symmetric argument shows that Alice says A with
probability ¾ and B with probability ¼.
• Probability they both say the same thing is
therefore 3/16+3/16=3/8.
What is probability they get together?
• With probability 3/8, they agree on where to
go. If they don’t agree, then they play the no
communications mixed strategy equilibrium
and meet with probability 3/8.
• So probability they meet is
3/8+5/8(3/8)=39/64
Simple talk helped, but didn’t completely solve
the problem.
Would more talk help?
Adding further rounds of discussion
• Suppose that if first set of messages do not
say same place, they try again.
• Then if second set do not coincide they try yet
again, and so on.
• Is this a reasonable model of an argument?
Things to think about
•
•
•
•
Why bother to talk? Only pays if others listen.
Why listen if all you hear is nonsense or lies.
Why do politicians lie?
Do some voters pay attention to what they
say?
• How did language evolve. Prevalence of
common interest games?
• Why don’t more animals have more language?
Aesop’s Reason for Truth-telling
• There was once a young Shepherd Boy who
tended his sheep at the foot of a mountain
near a dark forest. It was rather lonely for him
all day, so he thought upon a plan by which he
could get a little company and some
excitement. He rushed down towards the
village calling out “Wolf, Wolf,” and the
villagers came out to meet him, and some of
them stopped with him for a considerable
time.
• This pleased the boy so much that a few days
afterwards he tried the same trick, and again
the villagers came to his help.
• But shortly after this a Wolf actually did come
out from the forest, and began to worry the
sheep, and the boy of course cried out “Wolf,
Wolf,” still louder than before. But this time
the villagers, who had been fooled twice
before, thought the boy was again deceiving
them, and nobody stirred to come to his help.
The moral of the story
•
. So the Wolf made a good meal off
the boy’s flock, and when the boy
complained, the wise man of the
village said:
“A liar will not be believed, even when
he speaks the truth.”
Lesson for Game theory
A truth-telling equilibrium is more difficult to
find in games that are played only once.
In the wolf story, the reason for being truthful
when it is not too costly is that you are more
likely to be believed when it is very important to
be believed.
That’s all for today…