# Risk neutrality in single-agent choice problem under uncertainity

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?

• I don't think this says anything about risk aversion. Risk aversion depends on the functional form of $u_i(\cdot)$. – Art Oct 15 at 14:05
• 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 Oct 15 at 14:57
• @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 (faculty.chicagobooth.edu/emir.kamenica/documents/…) when talking about Bayesian persuasion (that is essentially one-player BCE) assumes risk neutrality. – user3285148 Oct 15 at 15:50
• 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 Oct 16 at 14:05

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