# Transforming expected utility functions

I am using the following theorem:

To better understand how I can transform expected utility functions.

An example with which to work:

I want to show that the preferences represented here satisfy the three axioms [A1,A2,A3] that characterize expected utility. [order, continuity, and independence]

Where $$E_{\pi} \mu = \sum_{z \in Z} u(z)\pi(z)$$

What I want to do is transform the given E.U.F. s.t.:

$$U(\pi) = h + k....$$ $$U(\pi)= k...$$

Then by multiplying by $\frac{1}{k}$ and using $ln$

$$U(\pi) = \sum_{i=1}^n \pi_i ln(\alpha_i)$$

EDIT

Perhaps this is actually the solution. Any feedback?

This is not Homework. I am studying because I do not understand this well

• It is better not to use the same notation $U$ for four different functions, maybe $U_1,\dots,U_4$. Sep 27 '20 at 10:10

The key idea here is that binary relations are necessarily ordinal relations, and do not contain cardial information. Taking (strictly positive) monotone transformations of representations preserves the preference structure. So, yes, you have shown $U(\cdot)$ (from the example) is related via a monotone transformation to an linear function with the index $ln(\alpha)$ and therefore, by the von Neumann Morgenstern theorem, satisfies the EU axioms.