11.IncompInfoIntrod

# 11.IncompInfoIntrod - GAMES OF INCOMPLETE INFORMATION: IN...

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GAMES OF INCOMPLETE INFORMATION: IN AN INCOMPLETE INFORMATION GAME, AT LEAST ONE PLAYER DOES NOT KNOW THE PAYOFFS (PREFERENCES) OF ANOTHER PLAYER. INCOMPLETE INFORMATION IS PERVASIVE IN THE GAMES WE PLAY IN REAL LIFE. EXAMPLE: Driving on a country road, you spot a person X who asks for a ride. You will decide on whether to offer the ride. But you don’t know what kind of person X is. If you offer the ride, X will choose between (H) harming you and (N) not harming you. Suppose X can have two TYPES, “ BAD ” and GOOD ”. A “ BAD ” X gets the payoff 5 from H, 0 from N A “ GOOD ” X gets the payoff –3 from H, 2 from N. If X is BAD and chooses H, your payoff is –10. If X is GOOD and chooses H, your payoff is –5. If X chooses N, your payoff is 10 no matter the type of X. IF YOU KNEW THE TYPE OF X, WHAT IS THE GAME THAT YOU ARE PLAYING (UNDER COMPLETE INFORMATION)? EXTENSIVE-FORM: (There are two cases) 1

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1. X IS “GOOD ”: You Don’t offer offer ( 0 ,0) X H N (-5 , -3) ( 10 , 2) In the SPE, you offer the ride and good X chooses N. 2. X IS “BAD”: You Don’t offer offer ( 0 ,0) X H N (-10 , 5) ( 10 , 0) In the SPE, you don’t offer the ride, for if you do, the “bad” X will choose H. IF YOU DON’T KNOW THE TYPE OF X, WHAT IS THE GAME THAT YOU ARE PLAYING? X can be “bad” or “good”: This is determined by NATURE. You are the uninformed player . 2
What is COMMON knowledge ? (i) That you don’t know the type of X; (ii) that X knows his own type; but (iii) that you have prior beliefs about the type of X; (iv) the payoffs of both types of X and your payoffs, as well as the structure of the game. YOUR PRIOR BELIEFS ABOUT THE TYPE OF X IS REPRESENTED BY A PROBABILITY DISTRIBUTION OF THE TYPES OF X: TYPES OF X: { GOOD , BAD } A probability distribution over types of X: Suppose X is “good” with probability μ = 0.6, “bad” with probability 1 - μ = 0.4. EXTENSIVE-FORM REPRESENTATION OF THIS GAME: Nature Good-X Bad-X { μ = 0.6} You {1- μ = 0.4} Offer Don’t Offer Don’t good-X bad-X H N H N -5 10 0 -10 10 0 -3 2 0 5 0 0 3

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PLAYERS: YOU , THE “GOOD” X and THE “BAD” X . STRATEGIES: YOU: Choose an action at the information set (beliefs: μ, 1- μ) GOOD X: Choose H or N when offered a ride. BAD X: Choose H or N when offered a ride. PRINCIPLES: 1. Every player’s strategy must be individually optimal GIVEN (i) his beliefs whenever he moves, and (ii) the strategies of other players. 2. When a player does not know which node he is at, he computes the expected payoffs from each action using his beliefs. STRATEGIES THAT SATISFY THESE TWO PRINCIPLES
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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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11.IncompInfoIntrod - GAMES OF INCOMPLETE INFORMATION: IN...

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