<p>Quadratic loss is convenient because it provides a close form solution after investigating the first order conditions.</p>
<p>That said, if you're interested in belief updating, an absolute loss function can modify (slow down) the update when a signal happens to be a the tail of the distribution and look like an outlier.</p>
<p>You might also consider a composite function <span class="math-container">$L$</span> (for example a weighted average of a quadratic loss and absolute loss) to control how fast those updates are made. Let <span class="math-container">$x^*$</span> be the true value, and <span class="math-container">$x$</span> an estimate
<span class="math-container">$$L_{x^*}(x)=\alpha \left( x^* - x \right)^2 + \beta \left| x^* - x \right| $$</span>
where <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> could be any non-negative real number (e.g. <span class="math-container">$\alpha=\beta=0.5$</span>).</p>

CC BY-SA 4.0
2023-12-31T10:37:02.103Z
A weighted average of a quadratic loss and absolute loss? How would it look like? Could you write down the formula that you have in mind?
Oliver Queen
CC BY-SA 4.0
2023-12-31T22:56:39.323Z
It's done, please let me know if you have further questions.
Thomas Favory
CC BY-SA 4.0
2023-12-31T22:57:59.410Z
Thank you! I will check this out!
Oliver Queen
CC BY-SA 4.0
2023-12-31T10:02:40.067Z
2023-12-31T22:56:02.297Z
2023-12-31T22:56:02.297Z
Thomas Favory
<p>I have a complex question.</p>
<p>Is there any other way to model the preference misalignments of the sendres (in senders-receivers games) towards the desire of some specific state apart from the quadratic loss functions? Why not quadratic gains functions? I understand that it is a bit intuitive to use this model, but is there any other model that is also used for such situations?</p>

<microeconomics><mathematical-economics><game-theory><bayesian-game><information>
Beyond quadratic loss preferences?
Oliver Queen
# Beyond quadratic loss preferences?

(Asked by: username_0 on 2023-12-11)

I have a complex question.

Is there any other way to model the preference misalignments of the sendres (in senders-receivers games) towards the desire of some specific state apart from the quadratic loss functions? Why not quadratic gains functions? I understand that it is a bit intuitive to use this model, but is there any other model that is also used for such situations?

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, information

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## 1 answers

### Answer by username_1 on 2023-12-31. Score: 1 | Accepted answer

Quadratic loss is convenient because it provides a close form solution after investigating the first order conditions.

That said, if you're interested in belief updating, an absolute loss function can modify (slow down) the update when a signal happens to be a the tail of the distribution and look like an outlier.

You might also consider a composite function $L$ (for example a weighted average of a quadratic loss and absolute loss) to control how fast those updates are made. Let $x^\*$ be the true value, and $x$ an estimate
$$L\_{x^\*}(x)=\alpha \left( x^\* - x \right)^2 + \beta \left| x^\* - x \right| $$
where $\alpha$ and $\beta$ could be any non-negative real number (e.g. $\alpha=\beta=0.5$).

> **Comments:**
> **username_0**: A weighted average of a quadratic loss and absolute loss? How would it look like? Could you write down the formula that you have in mind?  
> **username_1**: It's done, please let me know if you have further questions.  
> **username_0**: Thank you! I will check this out!  

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