<p>Consider the following two utility functions:</p>
<p><span class="math-container">$EU(p)=\sum_i u_ip_i$</span></p>
<p><span class="math-container">$EU^2(p)=(\sum_i u_ip_i)^2$</span>.</p>
<p>In preference theory, <span class="math-container">$EU$</span> and <span class="math-container">$EU^2$</span> are <strong>equivalent</strong> because they represent the same preference. A preference is a complete and transitive binary relation over the set of alternatives.</p>
<p>However, obviously, in likelihood-estimation with logit/luce choice rule, <span class="math-container">$EU$</span> and <span class="math-container">$EU^2$</span> give different probabilities. For example, the probability of choosing <span class="math-container">$p$</span> over <span class="math-container">$q$</span> can be:</p>
<p><span class="math-container">$L(p)=\frac{EU(p)}{EU(p)+EU(q)}$</span></p>
<p>OR:</p>
<p><span class="math-container">$L(p)=\frac{EU^2(p)}{EU^2(p)+EU^2(q)}$</span></p>
<p>That is, for exactly the <strong>same preference</strong>, two <strong>different likelihood</strong> functions are viable.</p>
<p>How do econometricians justify the usage of <span class="math-container">$EU$</span> instead of <span class="math-container">$EU^2$</span>?</p>
<hr />
<p>Related questions:</p>
<p><a href="https://economics.stackexchange.com/questions/5226/does-concavity-of-the-utility-function-has-any-bite">Does concavity of the utility function has any bite?</a></p>
<p><a href="https://economics.stackexchange.com/questions/370/when-can-one-safely-talk-about-decreasing-marginal-utility">When can one safely talk about decreasing marginal utility?</a></p>
<p>The related posts are about cardinal utility or the strength of preference. However, those arguments cannot be simply applied here because:</p>
<blockquote>
<p>The &quot;cardinal utility&quot; applies to the lower case <span class="math-container">$u$</span>, which is unique up to an affine transformation, but it does not apply to EU as a whole, which is unique up to a positive monotonic transformation.</p>
</blockquote>

CC BY-SA 4.0
2022-04-09T02:23:07.737Z
Where do you see anyone use EU^2?   There are risk-loving utility functions but I think your question is fundamentally why we are maximizing utility instead of squared utility?
RegressForward
CC BY-SA 4.0
2022-04-09T03:51:28.810Z
@RegressForward (EU)^2 is equivalent to EU. The risk-loving utility you are mentioning is probably E(U^2).
dodo
CC BY-SA 4.0
2022-04-09T08:28:46.850Z
When one wants to rely on the ratios of utilities, one clearly cannot say that monotonic transformations do not matter. One can argue for *cardinal utility* values, one could say that the utility the choice model is using is merely a tool with nice theoritecial properties, or one could say that utility models are full of gas. So what exactly are you asking here? I feel like you should specify the exact model you are looking at, and perhaps also the situation you think the model would fare poorly in.
Giskard
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2022-04-09T12:11:54.100Z
Why do you claim that "$EU$ and $EU^2$ give different probabilities"? The first and second order conditions for both problems are the same, and so are their solutions. You can consider any even power, or any increasing transformation of $EU$.
Bertrand
CC BY-SA 4.0
2022-04-09T17:22:32.003Z
@Giskard For EU, the "cardinal utility" applies to the lower case u, which is unique up to an affine transformation, but it does not apply to EU as a whole, which is unique up to a positive monotonic transformation. Let me update the question. I will deeply appreciate you if there is a cardinal utility argument for EU as a whole.
dodo
CC BY-SA 4.0
2022-04-09T17:23:22.120Z
@Bertrand They give different probabilities while doing likelihood estimation. See the update.
dodo
CC BY-SA 4.0
2022-04-10T07:06:34.950Z
Can you define precisely what you mean by $\mathbb{E}(U)$ and $\mathbb{E}(U^2)$ are equivalent?
Richard Hardy
CC BY-SA 4.0
2022-04-10T23:18:00.977Z
@RichardHardy It is not about $\mathbb E(U^2)$. It is about $[\mathbb E(U)]^2$. They are equivalent in preference theory because they represent the same binary relation. I updated the precise definition in the question.
dodo
CC BY-SA 4.0
2022-04-11T05:33:01.053Z
I suspect they are not equivalent when the probabilities are fixed. It seems you are changing probabilities together with utilities to argue for equivalence. That does not seem reasonable to me, but perhaps I am missing something about preference theory (which I hardly know anything about).
Richard Hardy
<microeconomics><econometrics><decision-theory><theory>
How did econometricians justify the use of $EU$ instead of $EU^2$?
dodo
# How did econometricians justify the use of $EU$ instead of $EU^2$?

(Asked by: username_0 on 2022-04-08)

Consider the following two utility functions:

$EU(p)=\sum\_i u\_ip\_i$

$EU^2(p)=(\sum\_i u\_ip\_i)^2$.

In preference theory, $EU$ and $EU^2$ are **equivalent** because they represent the same preference. A preference is a complete and transitive binary relation over the set of alternatives.

However, obviously, in likelihood-estimation with logit/luce choice rule, $EU$ and $EU^2$ give different probabilities. For example, the probability of choosing $p$ over $q$ can be:

$L(p)=\frac{EU(p)}{EU(p)+EU(q)}$

OR:

$L(p)=\frac{EU^2(p)}{EU^2(p)+EU^2(q)}$

That is, for exactly the **same preference**, two **different likelihood** functions are viable.

How do econometricians justify the usage of $EU$ instead of $EU^2$?

---

Related questions:

[Does concavity of the utility function has any bite?](https://economics.stackexchange.com/questions/5226/does-concavity-of-the-utility-function-has-any-bite)

[When can one safely talk about decreasing marginal utility?](https://economics.stackexchange.com/questions/370/when-can-one-safely-talk-about-decreasing-marginal-utility)

The related posts are about cardinal utility or the strength of preference. However, those arguments cannot be simply applied here because:

> The "cardinal utility" applies to the lower case $u$, which is unique up to an affine transformation, but it does not apply to EU as a whole, which is unique up to a positive monotonic transformation.

**Tags:** microeconomics, econometrics, decision-theory, theory

> **Comments:**
> **username_1**: Where do you see anyone use EU^2?   There are risk-loving utility functions but I think your question is fundamentally why we are maximizing utility instead of squared utility?  
> **username_0**: @username_1 (EU)^2 is equivalent to EU. The risk-loving utility you are mentioning is probably E(U^2).  
> **username_2**: When one wants to rely on the ratios of utilities, one clearly cannot say that monotonic transformations do not matter. One can argue for *cardinal utility* values, one could say that the utility the choice model is using is merely a tool with nice theoritecial properties, or one could say that utility models are full of gas. So what exactly are you asking here? I feel like you should specify the exact model you are looking at, and perhaps also the situation you think the model would fare poorly in.  
> **username_3**: Why do you claim that "$EU$ and $EU^2$ give different probabilities"? The first and second order conditions for both problems are the same, and so are their solutions. You can consider any even power, or any increasing transformation of $EU$.  
> **username_0**: @username_2 For EU, the "cardinal utility" applies to the lower case u, which is unique up to an affine transformation, but it does not apply to EU as a whole, which is unique up to a positive monotonic transformation. Let me update the question. I will deeply appreciate you if there is a cardinal utility argument for EU as a whole.  
> **username_0**: @username_3 They give different probabilities while doing likelihood estimation. See the update.  
> **username_4**: Can you define precisely what you mean by $\mathbb{E}(U)$ and $\mathbb{E}(U^2)$ are equivalent?  
> **username_0**: @RichardHardy It is not about $\mathbb E(U^2)$. It is about $[\mathbb E(U)]^2$. They are equivalent in preference theory because they represent the same binary relation. I updated the precise definition in the question.  
> **username_4**: I suspect they are not equivalent when the probabilities are fixed. It seems you are changing probabilities together with utilities to argue for equivalence. That does not seem reasonable to me, but perhaps I am missing something about preference theory (which I hardly know anything about).  

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## 0 answers

