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Game Theory_lecture4_08_handouts

Game Theory_lecture4_08_handouts - EF4484_Game...

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EF4484_Game Theory--Lecture 4 9/22/08 1 9/22/08 EF4484_Game Theory--Lecture 4 1 Static (or Simultaneous- Move) Games of Complete Information Mixed Strategy Equilibrium 9/22/08 EF4484_Game Theory--Lecture 4 2 Outline of Static Games of Complete Information Introduction to games (chapter 1) Normal-form (or strategic-form) representation (chapter 3) Iterated elimination of strictly dominated strategies (chapters 6 & 7) Nash equilibrium (chapter 9) Review of concave functions, optimization Applications of Nash equilibrium (chapter 10) Mixed strategy equilibrium (chapter 11) 9/22/08 EF4484_Game Theory--Lecture 4 3 Today’s Agenda Mixed strategies Mixed strategy Nash equilibrium Use best response to find mixed strategy Nash equilibrium 9/22/08 EF4484_Game Theory--Lecture 4 4 Matching pennies Head is Player 1’s best response to Player 2’s strategy Tail Tail is Player 2’s best response to Player 1’s strategy Tail Tail is Player 1’s best response to Player 2’s strategy Head Head is Player 2’s best response to Player 1’s strategy Head Hence, NO (Pure Strategy) Nash equilibrium -1 , 1 1 , -1 1 , -1 -1 , 1 Player 1 Player 2 Tail Head Tail Head
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EF4484_Game Theory--Lecture 4 9/22/08 2 5 Background In order to predict outcomes for games without (pure) Nash equilibria or with multiple equilibria, we need an extension of the concepts of strategies and equilibria. randomization of moves mixed strategies The need for randomizing moves in the play of a game usually arises when one player prefers a coincidence of actions, while his rival prefers to avoid it. Each player would like to outguess the other. 9/22/08 EF4484_Game Theory--Lecture 4 6 Background Examples: The matching pennies game Baseball games Tennis matches In all these games, players want to take advantage of the element of surprise . They want to be unpredictable . There is a skill to being unpredictable that skill requires understanding and being able to find the mixed-strategy equilibria of these games. 9/22/08 EF4484_Game Theory--Lecture 4 9/22/08 EF4484_Game Theory--Lecture 4 7 Solving matching pennies Randomize your strategies to surprise the rival Player 1 chooses Head and Tail with probabilities r and 1- r, respectively. Player 2 chooses Head and Tail with probabilities q and 1- q, respectively. Player 2 Head Tail Player 1 Head -1 , 1 1 , -1 Tail 1 , -1 -1 , 1 r 1-r q 1-q 9/22/08 EF4484_Game Theory--Lecture 4 8 Solving matching pennies Mixed Strategy: Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. A mixed strategy for Player 1 is a probability distribution ( r , 1 - r ) , where r is the probability of playing Head, and 1 - r is the probability of playing Tail.
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