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 -10, -10 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: 1/4 3/4 1/2 Red Dress Blue Dress -10, -10 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? 15 5 12.5 10 -5 Red Dress Blue Dress -10, -10 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 Duchess Wear Blue Wear Red Wear Red Wear Blue Wear Red Wear Blue Wear Blue Wear Red -10 20 20 -10 -10 20 20 -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? 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 -10, -10 20, 0 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 -2.5, -2.5 20, 0 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 Red Dress Blue Dress 10, -10 0, 10 10,-10 Social Climber 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 Mixed strategy equilibrium: Go to Movie A Go to Movie B Go to Movie A 3,2 1,1 Go to Movie B 0,0 2,3 Alice 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 Go to A Go to B Alice Alice Go to B Go to A Go to A Go to B 2 3 1 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 Say Movie A Say Movie B Say Movie A 3, 2 3/2,3/2 Say Movie B 2,3 Alice 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…