Signaling and Preplay communication Felix Munoz-Garcia Strategy and Game Theory - Washington State University
Signaling Intentions - Preplay communication Reading: Harrington, Ch. 12 pp. 374-79 Now we assume that the environment is common knowledge to the players (no private information, unlike in signaling games), but... Players are still uncertain about what other players intentions are. We then analyze preplay communication, that allows players to coordinate their actions. Remember the battle of the sexes? We found two psNE. We now analyze if such preplay communication can help players to coordinate in one of the two psNE.
Signaling Intentions - Preplay communication Consider the following Battle of the Sexes game, where we just relabelled: Football game is now going to an Action movie ( Gladiator ). Opera is now going to an Chick ‡ick movie ( How to Lose a Guy in Ten Days ; not my example, Harrington’s!!)
Signaling Intentions - Preplay communication Action movie Chick ‡ick movie
Signaling Intentions - Preplay communication
Signaling Intentions - Preplay communication This game has two psNE and one msNE. Let’s …nd them.
Signaling Intentions - Preplay communication Hence, the game has two psNE. What about a msNE, where Matt (Fiona) goes to the Chick ‡ick movie with probability m ( f , respectively).
Signaling Intentions - Preplay communication Therefore, msNE �� 2 � � 1 �� 3 A , 1 4 A , 3 3 C , 4 C Therefore, the probability that they both end up in the same theater is 3 1 3 + 1 2 3 = 0 . 42 4 4 where the …rst term indicates the joint probability that they both go to the Chick ‡ick, and the second term the joint probability that they both go to see the Action movie.
Signaling Intentions - Preplay communication Their expected payo¤s in the msNE �� 2 � � 1 �� 3 A , 1 4 A , 3 3 C , 4 C are: 3 1 32 + 3 31 + 1 2 30 + 1 1 2 EU Matt = 33 = 1 . 5, and 4 4 4 4 3 1 33 + 3 2 30 + 1 1 3 ( � 1 ) + 1 2 EU Fiona = 32 = 1 4 4 4 4
Signaling Intentions - Preplay communication But then both players have lower payo¤s if they play the msNE of the game than if they play any of the psNE Even that in which they watch their least preferred movie, with a payo¤ of 2!.
Signaling Intentions - Preplay communication What if we allow pre-play communication? For instance, what if players can simultaneously send a text message before decidicing what theater to go to? Then the previous BoS game becomes the following …gure.
Signaling Intentions - Preplay communication
Signaling Intentions - Preplay communication Note that there is no private information here (no types, no di¤erent production costs, etc.) There is only imperfect information, since Matt and Fiona: choose their text messages simultaneously (without knowing what is the other player texting), and choose what movie to go to simultaneously (but after observing the other player’s text message).
Signaling Intentions - Preplay communication Consider the following strategy: Matt : text message "let’s go to the chick ‡ick" with probability M , and text message "let’s go to the action movie" with probability 1 � M . if they both messaged "let’s go to the chick ‡ick", then go to the chick ‡ick. if they both messaged "let’s go to the action movie", then go to the action movie. if they messaged di¤erent movies, then go the chick ‡ick with probability m = 1 3 . behave as in the msNE with no preplay communication.
Signaling Intentions - Preplay communication Fiona : text message "let’s go to the chick ‡ick" with probability F , and text message "let’s go to the action movie" with probability 1 � F . if they both messaged "let’s go to the chick ‡ick", then go to the chick ‡ick. if they both messaged "let’s go to the action movie", then go to the action movie. if they messaged di¤erent movies, then go the chick ‡ick with probability f = 3 4 . behave as in the msNE of the game with no preplay communication.
Signaling Intentions - Preplay communication In summary: coordinate on the same …lm when they both send the same text message, and use their msNE strategies when they send di¤erent messages.
Signaling Intentions - Preplay communication Let us …rst note the proper subgames of this extensive-form game:
Signaling Intentions - Preplay communication We can then solve for these four proper subgames, and obtain a reduced-form game in which we focus on the "texting" part alone This is clearer!
Signaling Intentions - Preplay communication Hence, the previous game reduces to the following simultaneous-move game:
Signaling Intentions - Preplay communication We can represent this simultaneous-move game in its normal form: Hence, the game has two psNE where players text: ("Let’s go to the chick ‡ick","Let’s go to the chick ‡ick"), and ("Let’s go to the action movie","Let’s go to the action movie")
Signaling Intentions - Preplay communication What about msNE of the previous normal form game? Matt sends message "let’s go to the chick ‡ick" with probability M that makes Fiona indi¤erent, ) M = 1 M 3 + ( 1 � M ) 1 = M 1 + ( 1 � M ) 2 = 3 Similarly, Fiona sends message "let’s go to the chick ‡ick" with probability F that makes Matt indi¤erent, ) F = 3 F 2 + ( 1 � F ) 1 . 5 = F 1 . 5 + ( 1 � F ) 3 = 4
Signaling Intentions - Preplay communication What is the probability that they end up in the same theater? That would happen if... They both send each other the same text message, or They send di¤erent text messages, but they are lucky enough to end up in the same theather (msNE). First, the probability that they both send the same text message is 3 1 + 1 2 = 5 4 3 4 3 12 |{z} |{z} chick ‡ick action
Signaling Intentions - Preplay communication Second, the probability that they send di¤erent messages is simply 1 � 5 12 = 7 12 In that case, they play msNE, where the probability that they both end up in the same theather is 3 1 + 1 2 = 5 4 3 4 3 12 |{z} |{z} chick ‡ick action Hence, the joint probability that they send di¤erent messages and, despite that, they end up in the same theather is 7 12 = 35 5 12 144
Signaling Intentions - Preplay communication Therefore, the total probability that they end up watching the same movie is the sum of the two previous cases: 12 + 35 5 144 = 95 144 = 0 . 66 Recall that, when there was no possibility of pre-play communication, the probability that the players end up in the same theather was that given by just playing the msNE: 3 1 3 + 1 2 3 = 0 . 42 . 4 4 Now, by engaging in preplay communication, the players increased the chances that they end up in the same theather, from 0 . 42 to 0 . 66.
Signaling Intentions - Preplay communication Is this pattern con…rmed in actual experiments? Yes, the theoretical predictions of the previous model are close to actual behavior of players playing this game. Vince Crawford: "A Survey of Experiments on Communication via Cheap Talk," Journal of Economic Theory , 1998. Coordination rose: From 33% without preplay communication... to 41% with preplay communication.
Signaling Intentions - Preplay communication Is this pattern con…rmed in actual experiments (cont’d)? A twist on the experiments: What if only one player is allowed to send/not send a message? Then coordination (attending the same theather) rose to 95%!!! See Harrington, pp. 379-380, for more experimental references.
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