12.Signaling - GAMES OF INCOMPLETE INFORMATION -SIGNALING...

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GAMES OF INCOMPLETE INFORMATION -SIGNALING GAMES- IN A GAME OF INCOMPLETE INFORMATION AT LEAST ONE PLAYER LACKS INFORMATION ABOUT THE OBJECTIVE/MOTIVES OF HER OPPONENT(S). THAT IS, AT LEAST ONE PLAYER DOES NOT KNOW HER OPPONENT(S) PAYOFFS … A SIGNALING GAME IS AN INCOMPLETE INFORMATION GAME WITH THE STRUCTURE GIVEN IN THE EXAMPLE BELOW EXAMPLE : PLAYER 1 MOVES FIRST, CHOOSING BETWEEN ACTION X AND ACTION Y . PLAYER 2 OBSERVES PLAYER 1’S ACTION AND THEN CHOOSES BETWEEN LEFT AND RIGHT . THE GAME ENDS. PAYOFFS : SUPPOSE PLAYER 1’S PAYOFFS CAN TAKE ONLY TWO FORMS: AS GIVEN IN COLUMN “ PLAYER 1(a)” ( TYPE-a PLAYER 1 ) OR IN COLUMN “ PLAYER 1 (b)” ( TYPE-b PLAYER 1 ) . PLAYER 2’S PAYOFFS MAY ALSO DEPEND ON THE TYPE OF PLAYER 1. OUTCOME PLAYER 1 (a) PLAYER 2 PLAYER 1 (b) PLAYER 2 (X, LEFT) 5 0 5 6 (X, RIGHT) 0 6 10 0 (Y,LEFT) 6 10 6 10 (Y, RIGHT) 0 0 10 0 (This is not the usual strategic (normal) form; it is just a presentation of the payoffs) INFORMATION STRUCTURE : 1
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PLAYER 1 KNOWS HER PAYOFFS, THAT IS, HER TYPE ( a OR b ). PLAYER 2 DOES NOT KNOW WHETHER HE IS PLAYING AGAINST TYPE-a PLAYER 1 OR TYPE-b PLAYER 1. THAT IS, PLAYER 2 HAS PRIOR BELIEFS ABOUT PLAYER 1’S TYPE: Example : IT IS COMMON KNOWLEDGE THAT PLAYER 2 BELIEVES THAT PLAYER 1 IS OF TYPE-a WITH PROBABILITY 0.65, OF TYPE-b WITH PROBABILITY 0.35. ALTHOUGH THERE ARE TWO PLAYERS IN THIS GAME, WE THINK OF THE GAME AS IF IT WERE PLAYED BY THREE PLAYERS: Player 1-a, Player 1-b and Player 2. The extensive form representation of this game is as follows: Nature {0.65} {0.35} 1a 1b X Y Y X Player 2 Player 2 Left Right Left Right Left Right Left Right ( 5 ,0) ( 0 ,6) ( 6 ,10) ( 0 ,0) ( 6 ,10) ( 10 ,0) ( 5 ,6) ( 10 ,0) WE CAN ALSO REPRESENT THE SAME GAME AS FOLLOWS: 2
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( 5 ,0) ( 6 ,10) L {μ} 1a {β} L X Y ( 0 ,6) R R ( 0 ,0) {0.65} #2 Nature #2 {0.35} ( 5 ,6) ( 6 ,10) L X Y L {1- μ} 1b {1-β} R R ( 10 ,0) ( 10 ,0) REMARKS 1. Player 2 does not know whether he is playing against type-a or type-b of Player 1. This fact is represented by the information sets (broken lines connecting the two nodes after X is played and the two nodes after Y is played). 2. Suppose Player 2 has observed that action X is chosen by player 1 . Player 2 will now choose between “Left” and “Right”: We assume he has BELIEFS , represented by the PROBABILITY {μ} which he assigns to the possibility that he is facing type-a of Player 1. 3. When Player 2 finds herself at the information set following action X (chosen by Player 1) , we can compute Player 2’s expected payoffs as a function of his beliefs: If Player 2 plays “Left”, he gets: μ . (0) + (1 – μ) (6). 3
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If Player 2 plays “Right”, he gets: μ . (6) + (1 – μ) (0). Therefore Player 2 will play
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This note was uploaded on 03/16/2012 for the course FENS 101 taught by Professor Selçukerdem during the Fall '12 term at Sabancı University.

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12.Signaling - GAMES OF INCOMPLETE INFORMATION -SIGNALING...

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