that assigns each $\omega_i \in \Omega$ a probability of $0 \leq p_i \leq 1$ with $P(\Omega) = 1$ [cf. @kolmogorovFoundationsTheoryProbability1950, pp. 2-3].

In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for either one of them.

It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionExperienceGap2009].

It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionexperienceGapRisky2009].

For decisions from description (DfD), agents are provided a full symbolic description of the triples $(\Omega, \mathscr{F}, P)_j$, where j denotes a prospect.

For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], the probability triples are not described but must be explored by the means of sampling.

To provide a formal definition of sampling in risky or uncertain choice, we make use of the mathematical concept of a random variable.

B) The image $X: \Omega \mapsto \Omega'$ must be such that $\omega_i \in \Omega = x_i \in \Omega'$.

Given conditions A and B, we denote any realization of a random variable defined on the triple $(\Omega, \mathscr{F}, P)$ as a *"single sample"* of the respective prospect and any systematic approach to generate a sequence of single samples from multiple prospects as a sampling strategy [see also @hillsInformationSearchDecisions2010].

Because for a sufficiently large number of single samples *n* from a given prospect, i.e., $\lim_{n \to \infty}$, the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$ [@bernoulliOpusPosthumumAccedit1713], sampling in principle allows to explore a prospect's probability space.

Because for a sufficiently large number of single samples *n* from a given prospect, i.e., $\lim_{n \to \infty}$, the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$ [@bernoulliArsConjectandiOpus1713], sampling in principle allows to explore a prospect's probability space.

## A Stochastical Sampling Model for DfE

...

...

@@ -159,16 +159,23 @@ S =

\right\}^{\mathbb{N}}

=

\left\{

\frac{\overline{X}} {\overline{Y}}

\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}

\right\}^{\mathbb{N}}

\; ,

\end{equation}$$

where $\mathbb{N}$ is the number of comparisons, $x_i$ and $y_j$ are the realizations of the respective random variables, i.e., the single samples, and $N_X$ and $N_Y$ are the numbers of single samples within a comparison.

where $\mathbb{N}$ is the number of comparisons, $x_i$ and $y_j$ are the realizations of the respective random variables, i.e., the single samples, and $N_X$ and $N_Y$ are the numbers of single samples within a comparison.

For the elements of $S$ to be defined, however, the condition

$$\begin{equation}

P(\overline{Y} = 0) = 0 \; .

\end{equation}$$

must be fulfilled.

To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define

Hills and Hertwig [-@hillsInformationSearchDecisions2010] proposed the two different sampling strategies in combination with the respective decision strategies, i.e., piecewise sampling and round-wise comparison vs. comprehensive sampling and summary comparison, as an explanation for different choice patterns in DfE.

How does the current version of the SSM support this proposition?

Given prospects $X$ and $Y$, the sample spaces $S = \left\{\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}\right\}^{\mathbb{N}}$ can be varied by changes on three parameters, i.e., the number of comparisons $\mathbb{N}$ and the sample sizes $N_X$ and $N_Y$ on which these comparisons are based.

First, only considering the pure cases formulated by the above authors, the following restrictions are put the parameters:

$$\begin{equation}

\mathbb{N}

=

\begin{cases}

1 & \text{if} & \text{Summary} \\

\geq 1 & \text{if} & \text{Round-wise}

\end{cases}\\

\end{equation}$$

and

$$\begin{equation}

N_X \, \text{and} \, N_Y

=

\begin{cases}

\geq 1 & \text{if} & \text{Summary} \\

1 & \text{if} & \text{Round-wise}

\end{cases}\\

\end{equation}$$

For the summary strategy, the following prediction is obtained:

I.e., for the summary strategy, we assume that for increasing sample sizes $N_X$ and $N_Y$, the prospect with the larger expected value is chosen almost surely.

For the round-wise strategy, the following prediction is obtained: