Continuous Cheap Talk Felix Munoz-Garcia Strategy and Game Theory - PowerPoint PPT Presentation

Continuous Cheap Talk Felix Munoz-Garcia Strategy and Game Theory Washington State University

Environment set up The sate of the world θ is uniformly distributed on [ 0 , 1 ] Player 1’s action a 1 , player 2’s action a 2 , a 1 , a 2 2 [ 0 , 1 ] Player 2’s payo¤: v 2 ( a 2 , θ ) = � ( a 2 � θ ) 2 Player 1’s payo¤: v 1 ( a 2 , θ ) = � ( a 2 � ( θ + b )) 2 ( where b > 0 is the bias of player 1 ) Implies player 2’s optimal choice is a 2 = θ , player 1’s optimal choice is a 2 = θ + b ( ! …gure )

Payo¤s in the continuous Cheap Talk

No fully truthful equilibrium Claim: As in …nite game, There can never be a fully truthful equilibrium in this continuous cheap talk game. Proof: If player 2 believes that player 1 reports the true θ , then player 2’s best response is a 2 = θ . But if player 2 takes this action, player 1 will report a 1 = θ + b to get higher utility .

Babbling perfect Bayesian equilibrium Claim: There exists a babbling perfect Bayesian equilibrium in which player 1’s message reveals no information and player 2 chooses an action to maximize his expected utility given his prior belief.

Babbling perfect Bayesian equilibrium Proof: Let player 1’s strategy be a 1 = a B 1 2 [ 0 , 1 ] regardless of θ . This means player 1’s message is completely uninformative. Player 2 believes that θ is distributed uniformly on [ 0 , 1 ] Z 1 0 � ( θ � a 2 ) 2 d θ = � 1 3 + 2 a 2 � a 2 max a 2 Ev 2 ( a 2 , θ ) = 2 which is maximized when a 2 = 1 2 . Let player 2’ o¤-equilibrium-path beliefs be � � � = 1 , so that his best response to any � a 1 6 = a B θ = 1 Pr � � 1 2 a 1 6 = a B is a 2 = 1 other message 2 as well 1 From player 1’s payo¤ function, player 1 is indi¤erent between any of his messages and hence best response is a 1 = a B 1 .

Two message perfect Bayesian equilibrium Since there is no truthful equilibrium and there is always a babbling equilibrium, how much information can player 1 credibly transmit to player 2? We begin by constructing a perfect Bayesian equilibrium in which player 1 uses one of two message, a 0 1 and a 00 1 , and player 2 chooses a di¤erent action following each message, a 2 ( a 0 1 ) < a 2 ( a 00 1 )

Two message perfect Bayesian equilibrium Claim: In a two-message equilibrium, with the condition a 2 ( a 0 1 ) < a 2 ( a 00 1 ) , player 1 must use a threshold strategy as follows: if 0 � θ � θ � he chooses a 0 1 , whereas if θ � � θ � 1 he chooses a 00 1 .

Two message perfect Bayesian equilibrium Proof: player 1’s payo¤s from a 0 1 and a 00 1 are as follows: � � � � � � � � b � θ � 2 a 0 a 0 v 1 a 2 , θ = � a 2 1 1 � � � � � � � � b � θ � 2 a 00 a 00 v 1 a 2 , θ = � a 2 1 1

Two message perfect Bayesian equilibrium Extra gain from choosing a 00 1 over a 0 1 is equal to � � � � b � θ � 2 + � � � � b � θ � 2 a 00 a 0 ∆ v 1 ( θ ) = � a 2 a 2 1 1 � � � � a 2 � �� > 0 for a 2 � � > a 2 � � ∂ ∆ v 1 ( θ ) a 00 a 0 a 00 a 0 = 2 a 2 1 1 1 1 ∂θ which implies if type θ prefers a 00 1 to a 0 1 , which means ∆ v 1 ( θ ) > 0 , 1 for every θ 0 > θ , if type θ prefers a 0 then a 00 1 � a 0 1 to a 00 1 , which 1 for every θ 0 < θ . means ∆ v 1 ( θ ) < 0 , then a 0 1 � a 00 Hence, there must be some threshold type θ = θ � , player 1 is indi¤erent between sending two messages.

Two message perfect Bayesian equilibrium Graph proof: 2 ( θ � + b ) and 1 2 θ � is indi¤erent As we can see, when θ = θ � , 1 for player 1, when θ �� > θ � , player 1 prefers 1 2 ( θ � + b ) to 1 2 θ � , when θ �� < θ � , player 1 prefers 1 2 ( θ � + b ) . 2 θ � to 1

Two message perfect Bayesian equilibrium Now we know the restrictions applying to player 1’s strategy in a two-message equilibrium. What about player 2’s best strategy. Claim: In any two-message perfect Bayesian equilibrium in which player 1 is using a threshold θ � strategy as described in 1 ) = θ � last slide, player 2’s equilibrium best response is a 2 ( a 0 2 1 ) = 1 � θ � and a 2 ( a 00 2 .

Two message perfect Bayesian equilibrium Proof: In equilibrium, player 2’s posterior belief following a 0 1 is that θ is uniformly distributed on the interval [ 0 , θ � ] and his posterior belief following a 00 1 is that θ is uniformly distributed on the interval [ θ � , 1 ] Hence, player 2 plays a best response if and only if he sets ( 1 ] = θ � a 2 ( a 0 1 ) = E [ θ j a 0 2 a 2 ( a 1 ) = 1 ] = 1 � θ � a 2 ( a 00 1 ) = E [ θ j a 00 2

Two message perfect Bayesian equilibrium Claim: A two-message perfect Bayesian equilibrium exists if and only if b < 1 4

Two message perfect Bayesian equilibrium Proof: When θ = θ � , player 1 is indi¤erent between a 0 1 and a 00 1 � � � , θ � � = v 1 � � � , θ � � a 0 a 00 v 1 a 2 a 2 1 1 � θ � � 2 � 1 � θ � � 2 2 � b � θ � � b � θ � � = � 2 For fact that θ � 2 < θ � < 1 � θ � 2 , � � θ � � b � θ � θ � � b � 1 � θ � 2 = � 2

Two message perfect Bayesian equilibrium Solve θ � = 1 4 � b . In order to have θ � > 0 , b < 1 4 . To complete the speci…cation of o¤-the-equilibrium-path beliefs, let player 2’s beliefs be n o θ = θ � = 1 , so that he chooses a 2 = θ � 2 f a 0 1 , a 00 Pr 2 j a 1 / 1 g 2 . Which causes player 1 is indi¤erent between a 0 1 and any other 2 f a 0 1 , a 00 message a 1 / 1 g . So his threshold strategy is a best response.

Two message perfect Bayesian equilibrium Two message perfect Bayesian equilibrium is better than babbling perfect Bayesian equilibrium for both player 1 and player 2. For player 2, in babbling perfect Bayesian equilibrium, she takes the action 1 2 in all states, so that her payo¤ is � 1 � 2 . In two message perfect Bayesian equilibrium, her � 2 � θ � 1 � 2 for θ 2 [ 0 , θ � ] , � � 1 � 2 for 2 θ � � θ 2 ( θ � + 1 ) � θ payo¤ is � θ 2 [ θ � , 1 ] . ????prove bigger utility than no informative equilibrium????? For player 1, in babbling perfect Bayesian equilibrium her � 1 � 2 . In two message perfect Bayesian payo¤ is � 2 � θ � b � 1 � 2 for θ 2 [ 0 , θ � ] , 2 θ � � θ � b equilibrium, her payo¤ is � � 1 � 2 for θ 2 [ θ � , 1 ] . ????prove bigger 2 ( θ � + 1 ) � θ � b � utility than no informative equilibrium?????

More informative equilibrium What will happen if b becomes even smaller, i.e., b < 1 4 ? Similar to the two message perfect Bayesian equilibrium. Consider that case that player 1 makes one of K reports. When θ 2 [ 0 , θ 1 ] , she reports a 11 ; when θ 2 [ θ 1 , θ 2 ] , she reports a 12 ; when θ 2 [ θ 2 , θ 3 ] , she reports a 13 ; . . . ; when θ 2 [ θ K � 1 , θ K ] , she reports a 1 K . For convenience, let θ 0 = 0 , θ K = 1 .

More informative equilibrium Similar to two message perfect Bayesian equilibrium, player 2 plays a best response is 1 2 ( θ K � 1 + θ K ) if she observes a 1 K ..

More informative equilibrium From the …gure ( an example of three message equilibrium), the strategy is also optimal for player 1. θ 1 , θ 2 are the threshold points.

More informative equilibrium When θ = θ 1 , 1 2 θ 1 and 1 2 ( θ 1 + θ 2 ) are indi¤erent to player 1; When θ = θ 2 , 1 2 ( θ 1 + θ 2 ) and 1 2 ( θ 2 + 1 ) are indi¤erent to player 1; When θ < θ 1 , 1 2 θ 1 � 1 2 ( θ 1 + θ 2 ) ; when θ > θ 1 , 1 2 ( θ 1 + θ 2 ) � 1 2 θ 1 ; When θ < θ 2 , 1 2 ( θ 1 + θ 2 ) � 1 2 ( θ 2 + 1 ) ; when θ > θ 2 , 1 2 ( θ 2 + 1 ) � 1 2 ( θ 1 + θ 2 ) . The value of a 1 K do not matter as long as no two are same. It is just a signal of the state between θ K � 1 and θ K .

More informative equilibrium So, we have the general condition: When θ = θ K , 1 2 ( θ K � 1 + θ k ) and 1 2 ( θ K + θ K + 1 ) are indi¤erent. By symmetry of player 1’s utility function, � 1 � θ K + b = 1 2 ( θ K � 1 + θ k ) + 1 2 ( θ K + θ K + 1 ) 2 θ K + 1 � θ K = θ K � θ K � 1 + 4 b That is the interval between states θ K + 1 and θ K is longer by 4 b than the interval between θ K and θ K � 1 .

More informative equilibrium Therefore, we have θ 1 + ( θ 1 + 4 b ) + � � � + ( θ 1 + 4 ( K � 1 ) b ) = 1 K θ 1 + 4 b ( 1 + 2 + � � � + ( K � 1 )) = 1 | {z } 2 bK ( K � 1 ) K θ 1 + 2 bK ( K � 1 ) = 1 If b is small enough that 2 bK ( K � 1 ) < 1 , there is a positive value of θ 1 that satis…es the equation. Our three message equilibrium example: 24 , θ 1 = 1 1 3 � 4 b = 1 6 , θ 2 = 2 3 � 4 b = 1 b = 2 . q � � From 2 bK ( K � 1 ) < 1 , we know K < 1 1 + 2 1 + . 2 b

More informative equilibrium Player 2’s response a 2 as a function of θ :

More informative equilibrium If b were any smaller, more information equilibrium would exist. The greater the di¤erence between player 1’s and player 2’s preference, b , is, the coarser the information transmitted in the equilibrium. The most informative equilibrium in which both player 1 and player 2 get the highest payo¤. (proof by yourself)