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.
A common interest game: Dressing for the Ball Duchess Countess What is the symmetric equilibrium if there is no pre-ball communication? What happens if they can each send a message before the ball?
Nash equilibrium • There are two asymmetric equilibriain pure strategies. • But if they play only once, how do they find it. • There is also a symmetric Nash equilibrium in mixed strategies. • Each wears red or blue with probability ½. • Check that this is a Nash equilibrium
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
How to model messages? • What do you think would happen if only the Countess can send a message. • What if they send messages simultaneously?
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
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.
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 odd 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 • Suppose that the duchess and the countess each get to send one 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 tells the other “I will wear red” or “I will wear blue” with probability ½. If they each said a different color, they wear what they said they 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. • What would a symmetric equilibrium look like? • What would be the chances of wearing different dresses?
Conflicting InterestsDressing for the Ball Duchess Social Climber What are the 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?
Partially Conflicting InterestsRed preferred Duchess Countess What are the equilibria if there is no pre-ball communication?
Alice and Bob without talk Bob Go to A Go to B Alice Alice Go to A Go to B Go to A Go to B 2 3 0 0 1 1 3 2
Alice and Bob Bob Alice • Symmetric 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
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 an equilibrium?
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 symmetric Nash equilibriumin 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 If both say same movie, they both go there. If they say different movies, They play original mixed strategy game.
Symmetric 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.
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 7/16, they say different things from each other and play original mixed strategy equilibrium. • Expected payoffs :3(3/16)+2(3/16)+1.5(7/16)=51/32>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.