Transforming expected utility functions

Question

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

Answer 1

Yes. What you are doing is a solution.

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.

Now, there is often a point of confusing here, because of the emphasis on uniqueness with regards to EU representation. But the EU representation is not the unique representation, it is the unique linear representation. While this is clearly delineated in the theorem, I think it is conflated in many people's minds with generic uniqueness.

To summarize, by choosing a particular functional form (or normalization) it is possible to act as if the preferences have a cardinal relation. But, this is just an interpretation, and not something inherent about a binary relation, even one that satisfies the independence axiom.

Answer 2

I would say that any monotonic transformation may be applied to utility functions, as they preserve the characteristics A1-A3; e.g. see https://en.wikipedia.org/wiki/Monotonic_function.