RMSC2001_Tutorial_3(1).pdf - RMSC2001 Introduction to Risk Management Tutorial 3 Utility Chen Sihan Oct 2 2019 1 1.1 Utility Outcomes Definition 1.1

RMSC2001_Tutorial_3(1).pdf - RMSC2001 Introduction to Risk...

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RMSC2001 - Introduction to Risk Management Tutorial 3: Utility Chen, Sihan Oct 2, 2019 1 Utility 1.1 Outcomes Definition 1.1. Suppose there are n < distinct outcomes x 1 , . . . , x n . Define X := { x 1 , . . . , x n } as the set of all possible outcomes. The finite setting can be released. Definition 1.2. On the set of outcomes X , a gamble G ( p 1 , . . . , p n ) := G ( p 1 , . . . , p n |X ) is a game that yields outcome x j with probability p j 0, for j = 1 , . . . , n , satisfying p 1 + · · · + p n = 1. The set of all possible gambles is defined as G := G ( X ) := { G := G ( p 1 , . . . , p n ) | p i 0 , n X i =1 p i = 1 } . Each gamble G ∈ G can be regarded as a (multinomial) probability distribution on the events (outcomes) in X : for gamble G = G ( p 1 , . . . , p n ) , x 1 ∈ X , P ( x 1 ) = p i . 1.2 Preference Definition 1.3. A preference relation is an order relation on X or G ( X ) . G 1 , G 2 ∈ G , if G 1 is indifferent , more preferable or less preferable to G 2 , then these relations are denoted 1
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respectively by: G 1 G 2 , G 1 < G 2 , G 1 4 G 2 . If the preferences are strict, then the latter two are denoted G 1 G 2 and G 1 G 2 , respec- tively. Axiom 1.1. (Axioms for Preference Relation) Let G be the set of all gambles on X . Preference relations on G has the following properties: 1. Completeness . For all G 1 , G 2 ∈ G , we have either G 1 < G 2 or G 1 4 G 2 . 2. Transitivity . For all G 1 , G 2 , G 3 ∈ G , if G 1 G 2 and G 2 G 3 , then G 1 G 3 . 3. Continuity . For all G 1 , G 2 , G 3 ∈ G with G 1 G 2 G 3 , there exists p [0 , 1] such that G 2 G ( G 1 , G 3 | p, 1 - p ). 4. Independence . For all G 1 , G 2 , G 3 ∈ G , p [0 , 1], G 1 4 G 2 if and only if G ( G 1 , G 3 | p, 1 - p ) 4 G ( G 2 , G 3 | p, 1 - p ). Let G = G ( G 1 , G 2 | p, 1 - p ) , p [0 , 1], it means P ( G = G 1 ) = p, P ( G = G 2 ) = 1 - p . Sup- pose G 1 = G ( p 11 , . . . , p 1 n ) , G 2 = G ( p 21 , . . . , p 2 n ), then G = G ( pp 11 +(1 - p ) p 21 , . . . , pp 1 n + (1 - p ) p 2 n ). 1.3 Utility Definition 1.4. A utility function is defined as a function: u : X → R such that x 1 , x 2 ∈ X , x 1 4 x 2 if and only if u ( x 1 ) u ( x 2 ). Theorem 1.1. (Von Neumann - Morgenstern Utility Theorem) Suppose the preference relation satisfies Axiom 1.1, then there exists a unique utility function u such that G 1 , G 2 ∈ G ( X ) , G 1 4 G 2 if and only if E ( u ( G 1 )) E ( u ( G 2 )), where E ( u ( G )) = n i =1 p i u ( x i ) if G = G ( p 1 , . . . , p n ).
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