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# slides3_4_inc - Lecture 3 Representation of Games 14.12...

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Lecture 3 Representation of Games 14.12 Game Theory Muhamet Yildiz Road Map 1. Cardinal representation – Expected utility theory 2. Quiz 3. Representation of games in strategic and extensive forms 4. Dominance; dominant-strategy equilibrium 1

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Cardinal representation – definitions Z = a finite set of consequences or prizes. • A lottery is a probability distribution on Z. • P = the set of all lotteries. • A lottery: \$1M .00001 .99999 \$0 Cardinal representation Z z Z z ) ( ) ( ) ( ) ( f U( p ) U( q ) A lottery (in P) Expected value of u under p Von Neumann-Morgenstern representation: z q z u z p z u q p 2
VNM Axioms Axiom A1: is complete and transitive. VNM Axioms Axiom A2 ( Independence ): For any p,q,r P, and any a (0,1], a p + (1- a )r a q + (1- a )r p q. p q \$1000 .5 .00001 .99999 .5 .5 \$1M \$0 .5 \$100 .5 .5 A trip to Florida A trip to Florida 3

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VNM Axioms Axiom A3 ( Continuity ): For any p,q,r P, if p q r, then there exist a , b (0,1) such that a p + (1- a )r q b p + (1- b ) r. Theorem – VNM-representation A relation on P can be represented by a VNM utility function u : Z R iff satisfies Axioms A1-A3. u and v represent iff v = a u + b for some a > 0 and b R. 4
Exercise Consider a relation among positive real numbers represented by VNM utility function u with u ( x ) = x 2 .

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