<p>If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom  of independence.</p>

<p>For example in <a href="https://en.wikipedia.org/wiki/Prospect_theory" rel="nofollow noreferrer">prospect theory</a>, consumers display loss aversion, which  means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected utility form, but the utility function is still continuous. This preferences violate the independence axiom.</p>

<p>Similarly, you could have preferences that do not admit the expected utility form, and satisfy the independence axiom, but not continuity. </p>

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2019-10-22T15:22:36.400Z
2019-10-22T15:22:36.400Z
Regio
<p>Jonathan Levin in <a href="https://web.stanford.edu/~jdlevin/Econ%20202/Uncertainty.pdf" rel="nofollow noreferrer">"Choice under Uncertainty"</a> wrote in Theorem 1 " A complete and transitive preference relation on a set of lotteries P satisfies continuity and independence if and only if it admits as expected utility representation". So it seems that if we have an expected utility function, preferences will definitely be independent and continuous, and vice-versa. Am I correct? But I am unsure that if preferences don't have expected utility form, will they necessarily not be continuous and independent?</p>

<p>i.e. if preferences are inconsistent with expected utility, does that necessarily imply that the independence axiom is violated?</p>

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2019-10-22T02:09:50.587Z
Would be helpful to show us what you know and how far have you gone in your thinking so far.
Art
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2019-10-22T02:32:06.563Z
I have edited the question. hope it clarifies. An intuitive detailed explanation would be helpful
Frodo Baggins
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2019-10-22T02:56:35.660Z
Let $p$ be "$P$ satisfies continuity and independence" and $q$ be "$P$ admits expected utility representation." The link you provided shows $p \iff q$, which necessarily means $p \to q$. By modus tollens, this imples $\neg q \to \neg p$.
Art
<microeconomics><expected-utility><measuring-utility>
Relationship between expected utility and independence axiom
Frodo Baggins
# Relationship between expected utility and independence axiom

(Asked by: username_0 on 2019-10-22)

Jonathan Levin in ["Choice under Uncertainty"](https://web.stanford.edu/~jdlevin/Econ%20202/Uncertainty.pdf) wrote in Theorem 1 " A complete and transitive preference relation on a set of lotteries P satisfies continuity and independence if and only if it admits as expected utility representation". So it seems that if we have an expected utility function, preferences will definitely be independent and continuous, and vice-versa. Am I correct? But I am unsure that if preferences don't have expected utility form, will they necessarily not be continuous and independent?

i.e. if preferences are inconsistent with expected utility, does that necessarily imply that the independence axiom is violated?

**Tags:** microeconomics, expected-utility, measuring-utility

> **Comments:**
> **username_1**: Would be helpful to show us what you know and how far have you gone in your thinking so far.  
> **username_0**: I have edited the question. hope it clarifies. An intuitive detailed explanation would be helpful  
> **username_1**: Let $p$ be "$P$ satisfies continuity and independence" and $q$ be "$P$ admits expected utility representation." The link you provided shows $p \iff q$, which necessarily means $p \to q$. By modus tollens, this imples $\neg q \to \neg p$.  

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## 1 answers

### Answer by username_2 on 2019-10-22. Score: 1

If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom of independence.

For example in [prospect theory](https://en.wikipedia.org/wiki/Prospect_theory), consumers display loss aversion, which means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected utility form, but the utility function is still continuous. This preferences violate the independence axiom.

Similarly, you could have preferences that do not admit the expected utility form, and satisfy the independence axiom, but not continuity.


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