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 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 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 equilibria in 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 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 A B C D E 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 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 1 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 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 are the equilibria if there is no pre ball communication Alice and Bob without talk Bob Go to A 2 3 Go to A Go to B Alice Alice Go to B 0 0 Go to A 1 1 Go to B 3 2 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 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 A symmetric 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 …