<p>No, risk neutrality is not assumed. Risk preferences are given by the concavity of  <span class="math-container">$u_i$</span> which is arbitrary in this setup.</p>

CC BY-SA 4.0
2019-11-20T19:16:56.467Z
Could you give a look to this question? https://economics.stackexchange.com/questions/32855/exact-definition-of-one-player-bayesian-correlated-equilibrium
Star
CC BY-SA 4.0
2019-10-16T14:02:27.667Z
2019-10-16T14:02:27.667Z
Regio
<p>Consider the following static single-agent choice problem under uncertainty. </p>

<p>Let <span class="math-container">$V$</span> be the state of the world with support <span class="math-container">$\mathcal{V}$</span> and probability distribution <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span>. First, let nature draw a realisation <span class="math-container">$v$</span> of <span class="math-container">$V$</span> from <span class="math-container">$P_V$</span>. 
Then, let the decision maker (DM) choose an action <span class="math-container">$y\in \mathcal{Y}$</span>, with <span class="math-container">$\mathcal{Y}$</span> finite, without observing <span class="math-container">$v$</span>. Upon the decision has been made, the DM gets a payoff <span class="math-container">$u(y,v)$</span>.</p>

<p>An optimal strategy of the decision problem above is <span class="math-container">$P_Y\in \Delta(\mathcal{Y})$</span> such that, <span class="math-container">$\forall y\in \mathcal{Y}$</span> such that <span class="math-container">$P_Y(y)&gt;0$</span> and <span class="math-container">$\forall \tilde{y}\neq y$</span>, 
we have that 
<span class="math-container">$$
\sum_{v\in \mathcal{V}} u_i(y,v)P_V(v)\geq \sum_{v\in \mathcal{V}} u_i(\tilde{y},v)P_V(v)
$$</span></p>

<p>Additionally, one can imagine that the DM can process some information structure <span class="math-container">$S$</span> to update her prior before choosing an action. In such case, we can use the notion of one-player BCE in Bergemann and Morris (2013,2016) to characterise the set of probability distributions of <span class="math-container">$(Y,V)$</span> that are predicted by the model while remaining agnostic about <span class="math-container">$S$</span>. </p>

<p><strong>Question:</strong> </p>

<p>Does the narrative above (and, hence, also the Bergemann and Morris framework) assume that the DM is risk neutral? Can it be reformulated also for the case in which the DM is risk averse?</p>

CC BY-SA 4.0
2019-10-15T14:05:25.770Z
I don't think this says anything about risk aversion. Risk aversion depends on the functional form of $u_i(\cdot)$.
Art
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2019-10-15T14:57:15.900Z
Is "v" a wealth payoff? Risk aversion depends on the function form of utility with respect to wealth but not necessarily with respect to other parameters.
BKay
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2019-10-15T15:50:53.900Z
@Art Thanks. The fact that they do not say anything means that her result works with risk aversion or risk neutrality? I.e., they work for any form of the utility function? I'm asking this because Kamenica and Gentzkow (https://faculty.chicagobooth.edu/emir.kamenica/documents/bayesianPersuasion.pdf) when talking about Bayesian persuasion (that is essentially one-player BCE) assumes risk neutrality.
Star
CC BY-SA 4.0
2019-10-16T14:05:01.237Z
Kamenica and Gentskow do not assume risk neutrality. You can read in the text "The prosecutor (Sender) gets utility 1 if the judge convicts and utility 0 if the judge acquits, regardless of the state." if they were assuming risk neutrality, the previous statement would say something like " The prosecutor  gets $1 dollar [...]".
Regio
<game-theory><decision-theory><bayesian-game><choice-theory>
Risk neutrality in single-agent choice problem under uncertainity
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# Risk neutrality in single-agent choice problem under uncertainity

(Asked by: username_0 on 2019-10-15)

Consider the following static single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P\_V\in \Delta(\mathcal{V})$. First, let nature draw a realisation $v$ of $V$ from $P\_V$.
Then, let the decision maker (DM) choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the DM gets a payoff $u(y,v)$.

An optimal strategy of the decision problem above is $P\_Y\in \Delta(\mathcal{Y})$ such that, $\forall y\in \mathcal{Y}$ such that $P\_Y(y)>0$ and $\forall \tilde{y}\neq y$,
we have that
$$
\sum\_{v\in \mathcal{V}} u\_i(y,v)P\_V(v)\geq \sum\_{v\in \mathcal{V}} u\_i(\tilde{y},v)P\_V(v)
$$

Additionally, one can imagine that the DM can process some information structure $S$ to update her prior before choosing an action. In such case, we can use the notion of one-player BCE in Bergemann and Morris (2013,2016) to characterise the set of probability distributions of $(Y,V)$ that are predicted by the model while remaining agnostic about $S$.

**Question:**

Does the narrative above (and, hence, also the Bergemann and Morris framework) assume that the DM is risk neutral? Can it be reformulated also for the case in which the DM is risk averse?

**Tags:** game-theory, decision-theory, bayesian-game, choice-theory

> **Comments:**
> **username_1**: I don't think this says anything about risk aversion. Risk aversion depends on the functional form of $u_i(\cdot)$.  
> **username_2**: Is "v" a wealth payoff? Risk aversion depends on the function form of utility with respect to wealth but not necessarily with respect to other parameters.  
> **username_0**: @username_1 Thanks. The fact that they do not say anything means that her result works with risk aversion or risk neutrality? I.e., they work for any form of the utility function? I'm asking this because Kamenica and Gentzkow (https://faculty.chicagobooth.edu/emir.kamenica/documents/bayesianPersuasion.pdf) when talking about Bayesian persuasion (that is essentially one-player BCE) assumes risk neutrality.  
> **username_3**: Kamenica and Gentskow do not assume risk neutrality. You can read in the text "The prosecutor (Sender) gets utility 1 if the judge convicts and utility 0 if the judge acquits, regardless of the state." if they were assuming risk neutrality, the previous statement would say something like " The prosecutor  gets $1 dollar [...]".  

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## 1 answers

### Answer by username_3 on 2019-10-16. Score: 1 | Accepted answer

No, risk neutrality is not assumed. Risk preferences are given by the concavity of $u\_i$ which is arbitrary in this setup.

> **Comments:**
> **username_0**: Could you give a look to this question? https://economics.stackexchange.com/questions/32855/exact-definition-of-one-player-bayesian-correlated-equilibrium  

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