# Mathematical Game Theory (Von Neumann Morganstern Utility)

Tags:
1. Dec 8, 2014

### ctownballer03

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)

I would love a shove in the right direction. thx

2. Dec 8, 2014

### RUber

$L=\sum p_i C_i$ 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.