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In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory:

$\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$

Where $\theta$ is the state of nature from the set of possible states of nature $\Theta$, $a$ is the action chosen from the set of possible actions $\mathscr{A}$, and $\pi^*$ is the posterior distribution.

Question: how would I incorporate risk aversion into this function? In the above formulation, the loss function would be identical for a certain loss of 50 or a {0,100} space with 50/50 chance of either.

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    $\begingroup$ Risk aversion is part of the loss function $L(\theta,a)$. $\endgroup$
    – Xi'an
    Dec 7 '14 at 14:51
  • $\begingroup$ Doesn't the loss function have to operate on the probabilities $\pi^*$ to incorporate the risk of outcomes? Or is it that by having $L(\theta, a)$ reflect worse outcomes for higher wealth, loss aversion will result? $\endgroup$
    – MikeRand
    Dec 7 '14 at 16:29
  • $\begingroup$ Risk aversion refers to particular policy for choosing actions. You have not given a policy, only a loss function. If doesn't make sense to ask if a loss function is risk averse. If the implied policy is to choose the action that minimizes the expected loss, then this would not be risk averse. $\endgroup$
    – Tom Minka
    Dec 8 '14 at 9:38
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    $\begingroup$ Not sure I agree with that. Being risk averse implies that the utility of expected wealth $U(E[w])$ is greater than the expected utility of wealth $E[U(w)]$ (or the certainty-equivalent utility). Those seem to be directly related to the utility/loss function. $\endgroup$
    – MikeRand
    Dec 8 '14 at 22:27
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The concept of risk aversion enters into analysis when you have a measure of objective gain/loss that is a random variable, and you wish to obtain an overall measure of subjective gain/loss that accounts for the full distribution of this random variable. This occurs commonly in the context of economics where the objective gain/loss is a random amount of wealth, income, goods, etc., and the subjective measure is the expected utility corresponding to this. Within this context, risk aversion occurs when the utility is a concave function of the objective measure.

So, with this in mind, it makes sense to interpret the loss function as measuring the objective loss under some (random) state of nature $\theta \in \Theta$ and decision $a \in \mathscr{A}$. The posterior expected utility under any action $a$ would be written as:

$$\bar{u}(a) \equiv \mathbb{E}(u(-L(\theta, a))) = \int \limits_\Theta u(-L(\theta, a)) d\pi^*(\theta),$$

where $\pi^*$ is the posterior distribution for $\theta$, and $U: \mathbb{R} \rightarrow \mathbb{R}$ is a subjective utility function satisfying all the normal requirements of utility theory (i.e., it is non-decreasing, etc.). Assuming that this utility is twice-differentiable, risk aversion occurs in the case where $u'' < 0$ so that utility is concave.

The most commonly used form of utility function to represent risk-aversion is the CES utility with constant-relative-risk-aversion parameter $\rho > 0$. This utility function is given by:

$$u(c) = \begin{cases} \ln c & \rho = 1 \\[8pt] \frac{c^{1-\rho}-1}{1-\rho} & \rho \neq 1 \end{cases}$$

In the text you are looking at by Berger, his expression is showing the expected loss, and so he is not including any risk aversion. Minimisation of expected loss is optimal in the case of a risk-neutral utility function. If you want to add risk-aversion, you simply transform the negative loss (i.e., the gain) through any concave utility function, whether that be the CES function or another one.

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