In summary, we have defined the decision variable $D$ as a function of the random variables associated with the prospects probability spaces.

In summary, we have defined the decision variable $D$ as a function of the random variables associated with the prospects' probability spaces.

As such, $f$ is allowed to operate on any quantitative measure related to these random variables.

We have already pointed out that decision theories will differ in the form of $f$ and the measures, or moments, it utilizes and we take the stance that these choices should be informed by the theory and data of the psychological and other sciences.

For what do these choices mean?

We think they reflect the assumptions about the kind of information agents process and the way they do, notwithstanding the question of whether they are capable of doing so.

In the following section, we show how different processing assumptions for DfE, outlined by Hills and Hertwig [-@hillsInformationSearchDecisions2010], can be captured by the SSM.

### Formalizing sampling and decision policies in the SSM

Hills and Hertwig [-@hillsInformationSearchDecisions2010] ... .

In the following, we consider the case of a choice between two prospects, their respective random variables we denote as $X$ and $Y$.

Since the decision variable $D$ serves as a measure for the evidence for one prospect over the other, we want $f$ to be a measurable function that maps the comparison of $X_1$ and $X_2$ to the measure space $\mathscr{D}$.

By definition, we require the decision variable $D$ to be a measure of the evidence for one prospect over the other, i.e., $X$ over $Y$ or vice versa.

$f$ should thus map the comparison of $X$ and $Y$ to a measurable space.

In principle,

we want $f$ to be a measurable function that maps the comparison of $X_1$ and $X_2$ to the measure space $\mathscr{D}$.

Because $X_1$ and $X_2$ are themselves measurable, we write their sample means as a fraction