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Mathematical Game Theory (Von Neumann Morganstern Utility)

  1. Dec 8, 2014 #1
    1: If u: omega---> reals is a Von Neumann Morganstern Utiliy function and L is a lottery, prove that expectation E is "linear" ie: E(Au(L)+B)=AEu(L)+B


    2. Given none:


    3. The attempt at a solution: My attempt at a solution has gone nowhere. I found a stanford and princeton game theory notes that went into it, but I could exactly follow.

    I found in a book that if E[v(c)]=v(E[c]) the person is risk netural and they're risk neutral iff VNM Utility function is linear.

    I'm really grasping at straws here though.

    Here is where I've found my information, but I haven't been able to translate anything into a formal proof.
    https://www.princeton.edu/~dixitak/Teaching/EconomicsOfUncertainty/Slides&Notes/Notes03.pdf
    http://web.stanford.edu/~jdlevin/Econ 202/Uncertainty.pdf
    and finally this book which seems to be the best (see theorem 3.9.1)
    http://books.google.com/books?id=nv...orgenstern utility function is linear&f=false

    I would love a shove in the right direction. thx
     
  2. jcsd
  3. Dec 8, 2014 #2

    RUber

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    ## L=\sum p_i C_i ## where ## \sum p_i = 1 ## would be the expected value for the Lottery L with probabilities p_i corresponding to possible payouts C_i.
    Then,
    ##E(Au(L)+B) ## is the expected value ... which is ##\sum p_i (Au(C_i)+B)##, where A and B are constants.
    You should be able to change this to the form you were looking for using basic properties of sums and products.
     
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