Cheap Talk When can cheap talk be believed

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 Countess Red Dress Blue Dress Red Dress -10, -10 20, 20 Blue Dress 20, 20 -10, -10

Nash equilibrium • There are two asymmetric equilibria in pure strategies. • But if they play only once, how do they find it? • For single shot play, symmetric 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 -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? A) B) C) D) E) 15 5 12. 5 10 -5 Red Dress Blue Dress -10, -10 20, 20 -10, -10

An “eccentric” 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 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? A) B) C) D) E) ½ 1 ¼ ¾ 2/3

A second message? • Suppose that if they 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 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 equilibria if there is no preball 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 Interests Red preferred Duchess Countess Red Dress Blue Dress Red Dress -10, -10 20, 0 Blue Dress 0, 20 -10, -10 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: -10 p+20(1 -p)=20 -30 p – Wearing blue 0 p-10(1 -p)= 10 p-10 • Countess will mix if 20 -30 p= 10 p-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 10 p-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, each wears red with probability ¾.

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. 5 p+20(1 -p)=20 -22. 5 p • Say “blue” 0 p-2. 5(1 -p)= 2. 5 p-2. 5 • These are equal when 20 -22. 5 p= 2. 5 p-2. 5 or 25 p=22. 5, which implies p=9/10.

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