<p>I don't particularly understand the question.</p>
<p>Start with any function <span class="math-container">$f:X \rightarrow \mathbb{R}$</span>.</p>
<p>Define <span class="math-container">$x \succeq y$</span> if <span class="math-container">$f(x) \geq f(y)$</span>.</p>
<p>We get a rational preference over <span class="math-container">$X$</span>.</p>
<p>By the way <span class="math-container">$\sin(x) + 1$</span> is a perfectly valid utility representation.</p>

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2020-10-30T00:24:47.537Z
2020-10-30T00:24:47.537Z
Walrasian Auctioneer
<p>Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.</p>
<p>I know rationality over <span class="math-container">$\preccurlyeq$</span> does not imply a utility function. But rationality and continuity over <span class="math-container">$\preccurlyeq$</span> implies a utility function. But what about the reverse direction?</p>
<p>For example, <span class="math-container">$u(x) = sin(x) + 1$</span>, is not rational, but is continuous, is it a utility function?</p>
<p>In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?</p>
<p><strong>My Answer</strong> A utility function is the representation of a preference relation <span class="math-container">$\preccurlyeq$</span>. All preference relations are by assumption (or definition), rational. Given a function, if there does not exist any rational preference relation, then it must not be a utilty function.</p>

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2020-10-29T21:03:05.630Z
Does this answer your question: https://economics.stackexchange.com/a/18234/42?
Herr K.
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2020-10-29T23:14:48.287Z
@HerrK. Not exactly, they are considering the existence of utility functions based on non-continuous preference relations. I am slightly asking about the existence of utility functions based on non-rational preferences. More directly I am asking, given a function with a non-rational preferences relation, can it still be a utility function?
Tsangares
CC BY-SA 4.0
2020-10-30T00:40:53.270Z
As my linked answer and the answer by @WalrasianAcutioneer suggest, any real-valued function can represent some rational preference. I think you might be confusing rationality of preferences with rationality of functions. A rational function is one that can be expressed as a ***ratio*** of two polynomial functions, so the word "rational" here is used as an adjective of "ratio". In contrast, a rational preference is one that embeds some intuitive sense of ***reasonableness***. Although the same word is used, but it has different meanings in the two cases.
Herr K.
<utility>
Can an irrational function be a utility function?
Tsangares
# Can an irrational function be a utility function?

(Asked by: username_0 on 2020-10-29)

Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.

I know rationality over $\preccurlyeq$ does not imply a utility function. But rationality and continuity over $\preccurlyeq$ implies a utility function. But what about the reverse direction?

For example, $u(x) = sin(x) + 1$, is not rational, but is continuous, is it a utility function?

In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?

**My Answer** A utility function is the representation of a preference relation $\preccurlyeq$. All preference relations are by assumption (or definition), rational. Given a function, if there does not exist any rational preference relation, then it must not be a utilty function.

**Tags:** utility

> **Comments:**
> **username_1**: Does this answer your question: https://economics.stackexchange.com/a/18234/42?  
> **username_0**: @HerrK. Not exactly, they are considering the existence of utility functions based on non-continuous preference relations. I am slightly asking about the existence of utility functions based on non-rational preferences. More directly I am asking, given a function with a non-rational preferences relation, can it still be a utility function?  
> **username_1**: As my linked answer and the answer by @WalrasianAcutioneer suggest, any real-valued function can represent some rational preference. I think you might be confusing rationality of preferences with rationality of functions. A rational function is one that can be expressed as a ***ratio*** of two polynomial functions, so the word "rational" here is used as an adjective of "ratio". In contrast, a rational preference is one that embeds some intuitive sense of ***reasonableness***. Although the same word is used, but it has different meanings in the two cases.  

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## 1 answers

### Answer by username_2 on 2020-10-30. Score: 4 | Accepted answer

I don't particularly understand the question.

Start with any function $f:X \rightarrow \mathbb{R}$.

Define $x \succeq y$ if $f(x) \geq f(y)$.

We get a rational preference over $X$.

By the way $\sin(x) + 1$ is a perfectly valid utility representation.


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