Sunil Simon Utility theory The axioms of utility theory Axiom 1 Continuity For

# Sunil simon utility theory the axioms of utility

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Sunil Simon Utility theory
The axioms of utility theory Axiom 1 – Continuity. For every triplet of outcomes x y z , there exists a θ [ 0 , 1 ] such that y [ θ ( x ) , ( 1 - θ )( z )] . Axiom 2 – Monotonicity. Let α,β [ 0 , 1 ] and suppose that x y then [ α ( x ) , ( 1 - α )( y )] [ β ( x ) , ( 1 - β )( y )] iff α β. Lemma. If a preference relation satisfies the Axioms of Continuity and Monotonicity, and if x y z and x z , then the value of θ defined in the Axiom of Continuity is unique. Corollary. If a preference relation satisfies continuity and monotonicity and if x k x 1 , then for each j = 1 ,.. ., k there exists a unique θ j such that x j [ θ j ( x k ) , ( 1 - θ j )( x 1 )] Sunil Simon Utility theory
The axioms of utility theory Axiom 3 – Simplification. Let ˆ L = [ q 1 ( L 1 ) , q 2 ( L 2 ) ,..., q m ( L m )] and for each j 1 j m let L j = [ p j 1 ( x 1 ) , p j 2 ( x 2 ) ,..., p j k ( x k )] . For each i = 1 ,..., k , let r i = q 1 p 1 i + q 2 p 2 i + ... + q m p m i . Consider the simple lottery L = [ r 1 ( x 1 ) , r 2 ( x 2 ) ,..., r k ( x k )] , then ˆ L L . Example L = [ 1 2 ( x 1 ) , 1 4 ( x 2 ) , 1 8 ( x 5 ) , 1 8 ( x 7 )] L 1 = [ 2 3 ( x 1 ) , 1 3 ( x 2 )] L 2 = [ 1 2 ( x 5 ) , 1 2 ( x 7 )] ˆ L = [ 3 4 ( L 1 ) , 1 4 ( L 2 )] Sunil Simon Utility theory
The axioms of utility theory Axiom 4 – Independence. Let ˆ L = [ q 1 ( L 1 ) , q 2 ( L 2 ) ,.. ., q m ( L m )] and M be a simple lottery. If L j M then ˆ L [ q 1 ( L 1 ) ,.. . q j - 1 ( L j - 1 ) , q j ( M ) , q j + 1 ( L j + 1 ) ,..., q m ( L m )] . Sunil Simon Utility theory
A characterization theorem Theorem. If the preference relation over ˆ L is complete, transitive and satisfies the four von Neumann-Morgenstern axioms then can be represented by a linear utility function. Sunil Simon Utility theory
A characterization theorem Proof. Assume that the most desired outcome x k x 1 . By Lemma 1, for each outcome x j we have x j [ θ j ( x k ) , ( 1 - θ j )( x 1 )] .

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