<p>A preference such that <span class="math-container">$p\succeq q$</span> for all lotteries <span class="math-container">$p$</span> and <span class="math-container">$q$</span> clearly does (!) have a cardinal utility representation: The utility function over outcomes must be constant, and this is clearly also necessary.</p>
<p>In Savage's theory, the axiom plays a different role. If a decision-maker is indifferent between all outcomes, a subjective expected utility representation still exists. For example, a constant function on outcomes together with an arbitrary probability distribution over states. In that case, there is clearly no unique probability distribution (with at least two states), and even the Bernoulli function need not be unique up to positive affine transformations.</p>
<p>For Debreu's theorem, a similar assumption is needed to ensure that there are really three factors in a substantive way. His result <a href="https://mathoverflow.net/questions/156881/a-categorical-analogue-of-debreus-independent-factors-theorem#comment401621_156881">does not work with only two factors</a>, and a third factor that does not matter is like having only two factors.</p>

CC BY-SA 4.0
2024-06-26T10:29:19.700Z
First paragraph: your constant utility function seems not unique up to a positive affine transformation. For example, both $U=-1$ and $U=1$ works. Usually, a "cardinal utility" function is something unique up to positive linear or positive affine transformation. If it is only unique up to a positive monotonic transformation, then it is called an "ordinal utility".
dodo
CC BY-SA 4.0
2024-06-26T11:49:59.617Z
The two functions you gave are related by a positive affine transformation: $1=-1+2$
Michael Greinecker
CC BY-SA 4.0
2024-06-26T08:13:50.283Z
2024-06-26T08:13:50.283Z
Michael Greinecker
<p>The vNM theorem suggests that weak-ordering, continuity, and independence is equivalent to the existence of expected utility, unique up to an affine transformation.</p>
<p>In Savage's axioms of expected utility, the P5 is the non-nullity assumption: there exists two outcomes <span class="math-container">$x,y$</span> such that <span class="math-container">$x\succ y$</span>. Also, in Debreu's theorem of cardinal utility, there is an essentiality assumption serving a similar role.</p>
<p>My question: does the vNM axioms imply P5?</p>
<p>If not, let's consider a preference <span class="math-container">$p\succsim q$</span> for all <span class="math-container">$p,q$</span>, this preference clearly does not have a cardinal utility representation.</p>

<microeconomics><preferences><decision-theory><expected-utility>
Non-nullity assumption in vNM theorem of cardinal utility
dodo
# Non-nullity assumption in vNM theorem of cardinal utility

(Asked by: username_0 on 2024-06-26)

The vNM theorem suggests that weak-ordering, continuity, and independence is equivalent to the existence of expected utility, unique up to an affine transformation.

In Savage's axioms of expected utility, the P5 is the non-nullity assumption: there exists two outcomes $x,y$ such that $x\succ y$. Also, in Debreu's theorem of cardinal utility, there is an essentiality assumption serving a similar role.

My question: does the vNM axioms imply P5?

If not, let's consider a preference $p\succsim q$ for all $p,q$, this preference clearly does not have a cardinal utility representation.

**Tags:** microeconomics, preferences, decision-theory, expected-utility


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## 1 answers

### Answer by username_1 on 2024-06-26. Score: 2 | Accepted answer

A preference such that $p\succeq q$ for all lotteries $p$ and $q$ clearly does (!) have a cardinal utility representation: The utility function over outcomes must be constant, and this is clearly also necessary.

In Savage's theory, the axiom plays a different role. If a decision-maker is indifferent between all outcomes, a subjective expected utility representation still exists. For example, a constant function on outcomes together with an arbitrary probability distribution over states. In that case, there is clearly no unique probability distribution (with at least two states), and even the Bernoulli function need not be unique up to positive affine transformations.

For Debreu's theorem, a similar assumption is needed to ensure that there are really three factors in a substantive way. His result [does not work with only two factors](https://mathoverflow.net/questions/156881/a-categorical-analogue-of-debreus-independent-factors-theorem#comment401621_156881), and a third factor that does not matter is like having only two factors.

> **Comments:**
> **username_0**: First paragraph: your constant utility function seems not unique up to a positive affine transformation. For example, both $U=-1$ and $U=1$ works. Usually, a "cardinal utility" function is something unique up to positive linear or positive affine transformation. If it is only unique up to a positive monotonic transformation, then it is called an "ordinal utility".  
> **username_1**: The two functions you gave are related by a positive affine transformation: $1=-1+2$  

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