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Review description of stochastic sampling model

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# Abstract
A rough stochastic model of the random processes underlying decisions from experience is proposed.
It is demonstrated how the model can be used a) to explicate assumptions about the sampling and decision strategies that agents may apply and b) to derive predictions about the resulting decision behavior in terms of function forms and parameter values.
A probability theoretic definition of prospects and a rough stochastic model for decisions from experience is proposed.
It is demonstrated how the model can be used a) to explicate assumptions about the sampling and decision strategies that agents may apply and b) to derive function forms and parameter values that describe the resulting decision behavior.
Synthetic choice data is simulated and modeled in cumulative prospect theory to test these predictions.
# Introduction
...
## Random Processes in Decisions from Experience
## Sampling in Decisions from Experience
In research on the decision theory, a standard paradigm is the choice between at least two (monetary) prospects.
Let a prospect be a probability space $(\Omega, \mathscr{F}, P)$.
......@@ -81,39 +81,49 @@ It is acceptable to speak of the occurrence of $\omega$ as a realization of a ra
(2) The mapping is such that $X(\omega) = x \equiv \omega$.
In (2), $x \equiv \omega$ means that the realization of a random variable $X(\omega) = x$ is numerically equivalent to its pre-image $\omega$.
Given conditions (1) and (2), we denote any occurrence of $\omega$ as a *"single sample"*, or realization, of a random variable defined on a prospect and the process of generating a sequence of single samples as *"sampling"*.
Note that, since random variables defined on the same prospect are independent and identically distributed (iid), the weak law of the large number applies to the relative frequency of occurrence of an outcome $\omega$ in a sequence of single samples originating from the respective prospect [cf. @bernoulliArsConjectandiOpus1713].
Given conditions (1) and (2), we denote any observation of $\omega$ as a *"single sample"*, or realization, of a random variable defined on a prospect and the act of generating a sequence of single samples in discrete time as *"sequential sampling"*.
Note that, since random variables defined on the same prospect are independent and identically distributed (iid), the weak law of the large number applies to the relative frequency of occurrence of an outcome $\omega$ in a sequence of single samples originating from the same prospect [cf. @bernoulliArsConjectandiOpus1713].
Thus, long sample sequences in principle allow to obtain the same information about a prospect by sampling as by symbolic description.
Consider now a choice between prospects $1, ..., k$.
To construct a stochastic sampling model (hereafter SSM) of the random processes underlying DfE, we assume that agents base their decision on the information related to these prospects and define a decision variable as a function
To construct a stochastic model for DfE, we assume that agents base their decision on the information related to these prospects and define a decision variable as a function of the latter:
$$\begin{equation}
D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k)
\;.
\end{equation}$$
Since in DfE no symbolic descriptions of the prospects are provided, we restrict the model to the case where decisions are based on sequences of single samples originating from the respective prospects:
Now, since in DfE no symbolic descriptions of the prospects are provided, the model must be restricted to the case where decisions are based on sequences of single samples originating from the respective prospects:
$$\begin{equation}
D := f(x_{i1}, ..., x_{ik}) \quad \text{for} \; x \equiv{\omega}
D := f(X_{i1}, ..., X_{ik})
\; ,
\end{equation}$$
where $i = 1, ..., N$ denotes a sequence of length $N$ of random variables that are iid and $x$ their realizations.
where $i = 1, ..., N$ denotes a sequence of length $N$ of random variables that are iid.
Concerning the form of $f$ and the measures it utilizes, it is quite proper to say that they reflect our assumptions about the exact kind of information agents process and the way they do and that these choices should be informed by psychological or other theory [@heOntologyDecisionModels2020, for an ontology of decision models] and empirical protocols.
In the following section, it is demonstrated how such assumptions about the processing strategies that agents may apply in DfE can be integrated into the SSM.
Concerning the form of $f$ and the measures it utilizes, it is quite proper to say that they reflect our assumptions about the exact kind of information agents process and the way they do and that these choices should be informed by psychological theory and empirical protocols.
In the following section, it is demonstrated how a stochastic model can be used to explicate assumptions about the sampling and decision strategies that agents may apply in DfE.
## Example: Formulating Sampling and Decision Strategies as Stochastic Processes
## A Stochastic Model Capturing Differences in Sampling and Decision Strategies
Hills and Hertwig [-@hillsInformationSearchDecisions2010] discussed a potential link between sampling and decision strategies in DfE.
Specifically, the authors suppose that if single samples originating from different prospects are generated in direct succession (piecewise sampling), the evaluation of prospects is based on multiple ordinal comparisons of single samples (round-wise decisions).
In contrast, if single samples originating from the same prospect are generated in direct succession (comprehensive sampling), it is supposed that the evaluation of prospects is based on a single ordinal comparison of long sequences of single samples [summary decisions; @hillsInformationSearchDecisions2010, Figure 1 for a graphical summary].
We now consider choices between two prospects to integrate these assumptions into the SSM.
In contrast, if single samples originating from the same prospect are generated in direct succession (comprehensive sampling), it is supposed that the evaluation of prospects is based on a single ordinal comparison of long sequences of single samples (summary decisions) [@hillsInformationSearchDecisions2010, Figure 1 for a graphical summary].
We now consider choices between two prospects to build a stochastic model that captures these assumptions.
Let $X$ and $Y$ be random variables defined on the prospects $(\Omega, \mathscr{F}, P)_X$ and $(\Omega, \mathscr{F}, P)_Y$.
Hills and Hertwig [-@hillsInformationSearchDecisions2010] suggest that any two sequences $X_i$ and $Y_i$ are compared by their means.
Let $C = \mathbb{R}$ be the set of all possible outcomes of such a mean comparison for given lengths $N_X$ and $N_Y$ of the sample sequences and
$$\begin{equation}
\mathscr{C} = \left\{ \{c \in C: \overline{X}_{N_X} - \overline{Y}_{N_Y} > 0\}, \{ c \in C: \overline{X}_{N_X} - \overline{Y}_{N_Y} \leq 0\}
\right\}
\end{equation}$$
be a set of subsets of $C$, indicating that comparisons of prospects on the ordinal (rather than on the metric) scale are of primary interest.
Let $X$ and $Y$ be random variables defined on the prospects $(\Omega, \mathscr{F}, P)_X$ and $(\Omega, \mathscr{F}, P)_Y$ and
$X(\omega \in \Omega_X) = x \equiv \omega$ and $Y(\omega \in \Omega_Y) = y \equiv \omega$ be single samples.
By definition, the decision variable $D$ should quantify the accumulated evidence for one prospect over the other, which is described in units of won comparisons.
Hence, $f$ should map the possible outcomes of a comparison of quantitative measures related to $X$ and $Y$, hereafter the sampling space $S = \mathbb{R}$, to a measure space $S' = \{0,1\}$, indicating the possible outcomes of a single comparison:
......@@ -178,7 +188,7 @@ D \sim B\left( p \left( \frac{\overline{X}_{N_X = 1}}{\overline{Y}_{N_Y = 1}} >
where $n$ is the number of comparisons (see *Proof 1* in Appendix).
### Predicting Choices From the SSM
## Predicting Choice Behavior in DfE
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?
......
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