# How did econometricians justify the use of $EU$ instead of $EU^2$?

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?

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.

• 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? Commented Apr 9, 2022 at 2:23
• @RegressForward (EU)^2 is equivalent to EU. The risk-loving utility you are mentioning is probably E(U^2).
– dodo
Commented Apr 9, 2022 at 3:51
• 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. Commented Apr 9, 2022 at 8:28
• 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 $EU$. Commented Apr 9, 2022 at 12:11
• @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
Commented Apr 9, 2022 at 17:22