<p>A <strong>necessary</strong> condition for a preference relation to be represented by a utility function is that the preference relation is rational (where a "utility function" is a real-valued function that assigns a higher or equal numerical value to bundle A than to bundle B, when A is weakly preferred to B).</p>

<p>Preferences are <strong>rational</strong> when they are <strong>complete</strong> (I can express my preferences for all conceivable bundles), and <strong>transitive</strong> (If I weakly prefer A to B and B to C, then I weakly prefer A to C).  </p>

<p>Given rationality, a <strong>sufficient</strong> condition is that the rational preference relation is <strong>continuous</strong>, i.e. if it is preserved under limits. </p>

<p>Then, the related utility function is also continuous.  </p>

<p>Monotonicity is a property where we go from quantities to preferences, after assuming desirability ("goods" rather than "bads"). Local nonsatiation is actually the weaker assumption that is required for most of the theory.</p>

CC BY-SA 4.0
2019-01-15T01:46:44.087Z
(+1) Two related results may be worth noting. First, if $\succsim$ is defined on a finite or countable set, then rationality of $\succsim$ alone is both necessary and sufficient for utility representation. Second, if $\succsim$ is defined on an uncountable set, then rationality and continuity together are necessary and sufficient for (continuous) utility representation.
Herr K.
CC BY-SA 4.0
2019-01-14T21:37:56.497Z
2019-01-14T22:43:34.567Z
2019-01-14T22:43:34.567Z
Alecos Papadopoulos
<p>It is well-known that a rational, continuous and monotone preference relation <span class="math-container">$\succeq$</span> defined on <span class="math-container">$\mathbb{R}^L$</span> admits a utility representation.</p>

<p>I would like to understand why monotonicity is required. In other words, what is an example of a rational and continuous but non-monotone preference relation that does not have a utility representation?</p>

CC BY-SA 4.0
2019-01-14T20:05:30.143Z
I don't think monotonicity is required for utility representation, at least in Debreu’s representation theorem.
Herr K.
CC BY-SA 4.0
2019-01-14T20:30:34.197Z
Thank you, you are right. I had never seen this result without the monotonicity axiom, which is probably used for simplifying the proof. I'll accept this answer if you post it.
Oliv
<consumer-theory><decision-theory>
Example of a rational and continuous preference relation that does not admit a utility representation
Oliv
# Example of a rational and continuous preference relation that does not admit a utility representation

(Asked by: username_0 on 2019-01-14)

It is well-known that a rational, continuous and monotone preference relation $\succeq$ defined on $\mathbb{R}^L$ admits a utility representation.

I would like to understand why monotonicity is required. In other words, what is an example of a rational and continuous but non-monotone preference relation that does not have a utility representation?

**Tags:** consumer-theory, decision-theory
> **Comments:**
> **username_1**: I don't think monotonicity is required for utility representation, at least in Debreu’s representation theorem.  
> **username_0**: Thank you, you are right. I had never seen this result without the monotonicity axiom, which is probably used for simplifying the proof. I'll accept this answer if you post it.  

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## 1 answers

### Answer by username_2 on 2019-01-14. Score: 3 | Accepted answer

A **necessary** condition for a preference relation to be represented by a utility function is that the preference relation is rational (where a "utility function" is a real-valued function that assigns a higher or equal numerical value to bundle A than to bundle B, when A is weakly preferred to B).

Preferences are **rational** when they are **complete** (I can express my preferences for all conceivable bundles), and **transitive** (If I weakly prefer A to B and B to C, then I weakly prefer A to C).

Given rationality, a **sufficient** condition is that the rational preference relation is **continuous**, i.e. if it is preserved under limits.

Then, the related utility function is also continuous.

Monotonicity is a property where we go from quantities to preferences, after assuming desirability ("goods" rather than "bads"). Local nonsatiation is actually the weaker assumption that is required for most of the theory.

> **Comments:**
> **username_1**: (+1) Two related results may be worth noting. First, if $\succsim$ is defined on a finite or countable set, then rationality of $\succsim$ alone is both necessary and sufficient for utility representation. Second, if $\succsim$ is defined on an uncountable set, then rationality and continuity together are necessary and sufficient for (continuous) utility representation.  

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