The following risks are given: 1. X 3 </msub> 2. X 2 </msub>

abbracciopj

abbracciopj

Answered question

2022-06-24

The following risks are given:
1. X 3
2. X 2 B i n ( 15 , 1 3 )
3. X 3 15 I ,   where   I B e r n o u l l i ( 1 3
Do any docaiou makers with (increasing) ulility function agree aboul prceferring risk X 1 to X 2 ? given an exponential utility function which risk does a decision makerpreder: X 1 or X 3 ?
The last task is related to eponential ordering of risk denoted by e.

Answer & Explanation

Jake Mcpherson

Jake Mcpherson

Beginner2022-06-25Added 23 answers

The expectations of X 1 and X 2 are equal: E ( X 1 ) = E ( X 2 ). So there would be no reason to prefer X 1 or X 2 on this basis.
The question is asking if there is any increasing utility function u ( x ) such that
E ( u ( X 1 ) ) > E ( u ( X 2 ) ) x = 0 10 u ( x ) P r ( X 1 = x ) > x = 0 15 u ( x ) P r ( X 2 = x ) x = 0 10 u ( x ) ( 10 x ) 0.5 x 0.5 1 x > x = 0 15 u ( x ) ( 15 x ) ( 1 / 3 ) x ( 2 / 3 ) 1 x
Note that X 1 's probability of success, 1/2, is greater than X 2 's probability of success, 1/3. But X 2 still has the same expectation as X 1 because with X 2 the maximum number of successes you can get, 15, is greater than the maximum number of successes you can get for X 1 , 10. This suggests that if such a utility function exists, it is one which disproportionately rewards a lower number of successes.

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