Lecture 8 Expected Utility Theory 2010

Lecture 8 Expected Utility Theory 2010 - ECON 4109H LECTURE...

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Unformatted text preview: ECON 4109H LECTURE 8 Expected Utility Theory October 6, 2010 ECON 4109H LECTURE 8 Expected Utility Theory Finite set X = { 1, 2, . . . , n } . lottery = a probability distribution over X . But what is a probability distribution? ECON 4109H LECTURE 8 Expected Utility Theory lottery = a vector ( p 1 , p 2 , . . . , p n ) p k = probability of k . Assume: 1 p k > 0 for all k in X . 2 ∑ n k = 1 p k = 1. Assumptions imply p k 6 1 for all k . ECON 4109H LECTURE 8 Expected Utility Theory Δ = set of all lotteries on X . Notation: δ = ( p 1 , . . . , p n ) δ = ( p 1 , . . . , p n ) δ 00 = ( p 00 1 , . . . , p 00 n ) ECON 4109H LECTURE 8 Expected Utility Theory For lotteries δ = ( p 1 , . . . , p n ) and δ = ( p 1 , . . . , p n ) and α ∈ [ 0, 1 ] : Form a compound lottery αδ + ( 1- α ) δ . αδ +( 1- α ) δ means: with probability α , δ happens and with probability 1- α , δ happens. Formally: αδ + ( 1- α ) δ = ( α p 1 + ( 1- α ) p 1 , . . . , α p n + ( 1- α ) p n ) ECON 4109H LECTURE 8 Expected Utility Theory Theorem If δ and δ are lotteries, then so is αδ +( 1- α ) δ for any α ∈ [ 0, 1 ] . ECON 4109H LECTURE 8 Expected Utility Theory Theorem If δ and δ are lotteries, then so is αδ +( 1- α ) δ for any α ∈ [ 0, 1 ] . We must verify the properties of a probability distribution: 1 p 00 k > 0 for all k in X . 2 ∑ n k = 1 p 00 k = 1. ECON 4109H LECTURE 8 Expected Utility Theory Theorem If δ and δ are lotteries, then so is αδ +( 1- α ) δ for any α ∈ [ 0, 1 ] . α p k + ( 1- α ) p k > This verifies the first property. n X k = 1 [ α p k +( 1- α ) p k ] = α n X k = 1 p k +( 1- α ) n X k = 1 p k = α +( 1- α ) = 1 This verifies the second property. ECON 4109H LECTURE 8 Axioms for Expected Utility Theory Completeness Either δ 4 δ or δ 4 δ ....
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This note was uploaded on 01/19/2012 for the course ECON 4109H taught by Professor Notsure during the Fall '09 term at Minnesota.

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Lecture 8 Expected Utility Theory 2010 - ECON 4109H LECTURE...

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