<p>Yes. What you are doing is a solution. </p>

<p>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.</p>

<p>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 <em>linear</em> representation. While this is clearly delineated in the theorem, I think it is conflated in many people's minds with generic uniqueness. </p>

<p>To summarize, by choosing a particular functional form (or normalization) it is possible to act <em>as if</em> 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.</p>

CC BY-SA 3.0
2016-06-03T19:23:55.173Z
2016-06-03T19:23:55.173Z
201p
<p>I would say that any monotonic transformation may be applied to utility functions, as they preserve the characteristics A1-A3; e.g. see <a href="https://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Monotonic_function</a>.</p>

CC BY-SA 4.0
2020-09-27T08:38:49.020Z
2020-09-27T08:38:49.020Z
aatr
<p>I am using the following theorem:</p>

<p><a href="https://i.sstatic.net/El17X.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/El17X.png" alt="enter image description here"></a></p>

<p>To better understand how I can transform expected utility functions. </p>

<p>An example with which to work: </p>

<p><a href="https://i.sstatic.net/5VNry.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/5VNry.png" alt="enter image description here"></a></p>

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

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

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

<p>$$U(\pi) = h + k....$$ 
$$U(\pi)= k...$$ </p>

<p>Then by multiplying by $\frac{1}{k}$ and using $ln$ </p>

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

<p><strong>EDIT</strong> </p>

<p>Perhaps this is actually the solution. Any feedback? </p>

<p><strong>This is not Homework. I am studying because I do not understand this well</strong></p>

CC BY-SA 4.0
2020-09-27T10:10:22.350Z
It is better not to use the same notation $U$ for four different functions, maybe $U_1,\dots,U_4$.
Bertrand
<microeconomics><mathematical-economics><utility><expected-utility><theory>
Transforming expected utility functions
123
# Transforming expected utility functions

(Asked by: username_0 on 2016-05-30)

I am using the following theorem:

[![enter image description here](https://i.sstatic.net/El17X.png)](https://i.sstatic.net/El17X.png)

To better understand how I can transform expected utility functions.

An example with which to work:

[![enter image description here](https://i.sstatic.net/5VNry.png)](https://i.sstatic.net/5VNry.png)

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**

**Tags:** microeconomics, mathematical-economics, utility, expected-utility, theory

> **Comments:**
> **username_1**: It is better not to use the same notation $U$ for four different functions, maybe $U_1,\dots,U_4$.  

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## 2 answers

### Answer by username_2 on 2016-06-03. Score: 2

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.


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### Answer by username_3 on 2020-09-27. Score: 0

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>.


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