slides3_4_inc - Lecture 3 Representation of Games 14.12...

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Unformatted text preview: 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 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 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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This note was uploaded on 11/28/2011 for the course ECONOMICS - taught by Professor Muhammadyildiz during the Spring '05 term at University of Massachusetts Boston.

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slides3_4_inc - Lecture 3 Representation of Games 14.12...

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