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Probability theoretic definition and SSM

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A probability theoretic definition of sampling and a rough stochastic model of the random process underlying decisions from experience are proposed.
It is demonstrated how the stochastic 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.
Synthetic choice data is simulated and modeled in cumulative prospect theory to test these assumptions.
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Synthetic choice data is simulated and modeled in cumulative prospect theory to test these predictions.
# Introduction
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## Prospects as Probability Spaces
Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an accessible introduction].
## Random Processes in Sequential Sampling
$\Omega$ is the *sample space* containing an at most countable set of possible outcomes
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 $(\Omega, \mathscr{F}, P)$.
$\Omega$ is the sample space
$$\begin{equation}
\omega_i = \{\omega_1, ..., \omega_n\} \in \Omega
\omega_i = \{\omega_1, ..., \omega_n\} \in \Omega
\end{equation}$$
$\mathscr{F}$ is a set of subsets of $\Omega$, i.e., the *event space*
containing a finite set of possible outcomes, gains and/or losses respectively.
$\mathscr{F}$ is a set of subsets of $\Omega$, i.e., the event space
$$\begin{equation}
A_i = \{A_1, ..., A_n\} \in \mathscr{F} = \mathscr{P}(\Omega)
A_i = \{A_1, ..., A_n\} \in \mathscr{F} = \mathscr{P}(\Omega)
\; .
\end{equation}$$
$\mathscr{P}(\Omega)$ denotes the power set of $\Omega$.
$P$ is a probability mass function that maps $\mathscr{F}$ to the set of real numbers in $[0, 1]$
$P$ is a probability mass function
$$\begin{equation}
P: \mathscr{F} \mapsto [0,1]
P: \mathscr{F} \mapsto [0,1]
\end{equation}$$
by assigning each $\omega_i \in \Omega$ a probability of $0 \leq p_i \leq 1$ with $P(\Omega) = 1$.
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].
## Random Processes in Sequential Sampling
In research on the decision theory, a standard paradigm is the choice between $n \geq 2$ monetary prospects (hereafter indexed with j), where $\omega_{ij} \in \Omega_j$ are monetary outcomes, gains and/or losses respectively.
$P_j$ is then the probability measure which assigns each $\omega_{ij}$ a probability with which they occur.
In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for, i.e., choose either one of them.
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].
For decisions from description (DfD), agents are provided a full symbolic description of the triples $(\Omega, \mathscr{F}, P)_j$.
For decision from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], the probability triples are not described but must be explored by the means of *sampling*.
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.
Thus, if for each
$$\begin{equation}
\omega_{i} \in \Omega: p(\omega_{i}) \neq 1
\omega_{i} \in \Omega: p(\omega_{i}) \neq 1
\; ,
\end{equation}$$
we refer to the respective prospect as *"risky"*, where risky describes the fact that if agents would choose the prospect and any of the outcomes $\omega_{i}$ must occur, none of these outcomes will occur with certainty but according to the probability measure $P$.
It is acceptable to speak of the occurrence of $\omega_{i}$ as the realization of a random variable iff the following conditions a. and b. are met:
It is acceptable to speak of the occurrence of $\omega_{i}$ as the realization of a random variable iff the following conditions A and B are met:
(a) The random variable $X$ is defined as the function
A) The random variable $X$ is defined as the function
$$\begin{equation}
X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
\; ,
\end{equation}$$
where the image $\Omega'$ is the set of possible values $X$ can take and $\mathscr{F'}$ is a set of subsets of $\Omega'$.
I.e., $X$ maps any event $A_i \in \mathscr{F}$ to a subset $A'_i \in \mathscr{F'}$
I.e., $X$ maps any event $A_i \in \mathscr{F}$ to a subset $A'_i \in \mathscr{F'}$:
$$\begin{equation}
A'_i \in \mathscr{F'} \Rightarrow X^{-1}A'_i \in \mathscr{F}
\end{equation}$$
[cf. @georgiiStochasticsIntroductionProbability2008].
[@kolmogorovFoundationsTheoryProbability1950, p. 21].
(b) The image $X: \Omega \mapsto \Omega'$ must be such that $\omega_i \in \Omega = x_i \in \Omega'$.
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 $n \geq 2$ prospects as a sampling strategy [see also @hillsInformationSearchDecisions2010].
Because for a sufficiently large number of single samples from a given prospect the relative frequencies of $\omega_{i}$ approximate their probabilities in $p_i \in P$, sampling in principle allows to explore a prospect's probability space.
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.
So far, we used the probability triple of a prospect and conditions a. and b. solely to provide a probability theoretic definition of a single sample and sequences thereof.
However, since in the decision literature the (stochastic) occurrence of the raw outcomes in $\Omega$ is often treated as the event of interest, it should be justified to say that the stochastic model formulated under a. with the restriction b. is abundantly although implicitly assumed to underlie the evaluation processes of agents.
We do not contend that this model is not adequate but rather empirically warranted and mathematically convenient, not least because of the measurable nature of the monetary outcomes in $\Omega$.
However, in line with the literature that deviates from utility models and its derivatives [@heOntologyDecisionModels2020, for an ontology of decision models], we propose that the above restricted model is not the only suitable for describing the random processes agents are interested in, when building a preference between risky prospects, from sampling respectively.
## A Stochastical Sampling Model for DfE
We can construct an alternative stochastic sampling model (hereafter SSM) underlying DfE between risky prospects by starting from the assumption that agents do not make random choices but base their decisions on the information provided by the prospects, which is readily described by their probability triples.
Thus, we may start rather abstractly by defining a decision variable $D$
Consider a choice between $1,\, ...,\, j,\,...,\, n$ prospects, where $j \leq n \geq 2$.
To construct a rough stochastic sampling model (hereafter SSM) of the random process underlying DfE, it is assumed that agents base their decisions on the information provided by the prospects, which is in principle fully described by their probability triples.
Thus, a decision variable
$$\begin{equation}
D := f((\Omega, \mathscr{F}, P)_j)
\end{equation}$$
first without any further assumption on which information of the probability triple $f$ utilizes and how.
Although in principle many models for $f$ are proposed and tested in the decision literature, in DfE we can restrict the SSM to the case where decisions are based on sequences of single samples generated from the prospect triples.
Since we have defined the stochastic mechanism for generating such sequences, we write
is defined.
Since in DfE no symbolic descriptions of the triples are provided, the model is restricted to the case where decisions are based on sequences of single samples generated from the triples:
$$\begin{equation}
D := f((X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}))_j)
D := f((X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}))_j) = f(X_1, ..., X_j, ..., X_n)
\; ,
\end{equation}$$
where $\Omega_j = \Omega'_j$.
For n prospects, we write
Note that the decision variable $D$ is defined as a function $f$ of the random variables associated with the prospects' probability spaces, where $f$ can operate on any quantitative measure, or moment, related to these random variables.
Since decision models differ in the form of $f$ and the measures the latter utilizes [@heOntologyDecisionModels2020, for an ontology of decision models], we take the stance that these choices should be informed by psychological or other theory and empirical protocols.
For what do these choices mean?
They reflect the assumptions about the kind of information agents process and the way they do, not to mention the question of whether they are capable of doing so.
In the following section, it is demonstrated how such assumptions about the processing strategies that agents may apply in DfE can be captured by the SSM.
$$\begin{equation}
D := f(X_1, ..., X_j, ..., X_n)
\end{equation}$$
## Integrating sampling and decision strategies into the SSM
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.
Hills and Hertwig [-@hillsInformationSearchDecisions2010] discussed a potential link between the sampling and decision strategies of agents in DfE, i.e., a systematic relation between the pattern according to which sequences of single samples are generated and the mechanism of integrating and evaluating these sample sequences to arrive at a decision.
Specifically, the authors suppose that frequent switching between prospects in the sampling phase translates to a round-wise decision strategy, for which the evaluation process is separated into multiple rounds of ordinal comparisons between single samples (or small chunks thereof), such that the unit of the final evaluation are round wins rather than raw outcomes.
In contrast, infrequent switching is supposed to translate to a decision strategy, for which only a single ordinal comparison of the summaries across all samples of the respective prospects is conducted [@hillsInformationSearchDecisions2010, see Figure 1].
The authors assume that these distinct sampling and decision strategies lead to characteristic patterns in decision behavior and may serve as an additional explanation for the many empirical protocols which indicate that DfE differ from DfD [@wulffMetaanalyticReviewTwo2018, for a meta-analytic review; but see @foxDecisionsExperienceSampling2006].
### Formalizing sampling and decision policies in the SSM
In the following, choices between two prospects are considered to integrate the assumptions about the sampling and decision strategies from above into the SSM.
Hills and Hertwig [-@hillsInformationSearchDecisions2010] discussed a potential link between the sampling and decision policies of agents, i.e., a systematic relation between the pattern according to which sequences of single samples are generated and the mechanism of integrating and evaluating these sample sequences to arrive at a decision.
Specifically, the authors suppose that frequent switching between prospects in the sampling phase translates to a round-wise decision strategy, in which the evaluation process is separated into rounds of ordinal comparisons between single samples (or small chunks thereof), such that the unit of the final evaluation are round wins rather than raw outcomes.
In contrast, infrequent switching is supposed to translate to a decision strategy, in which only a single comparison of the summaries across all single samples of the respective prospects is conducted [see Figure 1, @hillsInformationSearchDecisions2010].
The authors assume, that these distinct sampling and decision process lead to differences in the decision behavior and may serve as an additional explanation for the many empirical protocols which indicate that DfE differ from DfD [@barronSmallFeedbackbasedDecisions2003; @weberPredictingRiskSensitivity2004; @hertwigDecisionsExperienceEffect2004; @wulffMetaanalyticReviewTwo2018, for a meta-analytic review].
Let $X$ and $Y$ be random variables related to the prospects with the probability spaces $(\Omega, \mathscr{F}, P)_X$ and $(\Omega, \mathscr{F}, P)_Y$.
By definition, the decision variable $D$ should quantify the accumulated evidence for one prospect over the other, which Hills and Hertwig [-@hillsInformationSearchDecisions2010] describe 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:
In the following, we consider the case of choices between two prospects and show how the assumptions of Hills and Hertwig [-@hillsInformationSearchDecisions2010] on specific sampling and decision policies can be integrated into the SSM.
We will demonstrate, how this allows us to formulate testable predictions about the decision behavior that results from these processes, or, the functional forms of algebraic models commonly used to describe it.
We denote the random variables of the two respective prospects as $X$ and $Y$.
By definition, we require the decision variable $D$ to be a measure of the evidence for one prospect over the other.
$f$ should thus map the comparisons of $X$ and $Y$-one for the summary strategy and multiple for the round-wise strategy-to a measure space that enables us to quantify the accumulated evidence for both prospects.
Since in Hills and Hertwig [-@hillsInformationSearchDecisions2010] accumulated evidence is described in units of won comparisons, the respective measure space contains the natural numbers $\{0, 1\}$, indicating the possible outcomes of a single comparison.
I.e., $f$ is a function that maps the possible outcomes of a comparison of quantitative measures related to $X$ and $Y$, hereafter the sampling space $S = \mathbb{R}$, to the measure space $S' = \{0,1\}$.
As the authors assume that the comparisons of prospects are based on sample means, we define $S$ as the set
$$\begin{equation}
D:= f: S \mapsto S'
\; .
\end{equation}$$
Since Hills and Hertwig [-@hillsInformationSearchDecisions2010] assume that comparisons of prospects are based on sample means, $S$ is the set
$$\begin{equation}
S = \left\{\frac{\overline{X}} {\overline{Y}}\right\}^{\mathbb{N}}
S =
\left\{
\frac{\frac{1}{N_X} \sum\limits_{i=1}^{N_X} x_i}
{\frac{1}{N_Y} \sum\limits_{j=1}^{N_Y} y_j}
\right\}^{\mathbb{N}}
=
\left\{
\frac{\overline{X}} {\overline{Y}}
\right\}^{\mathbb{N}}
\; ,
\end{equation}$$
where $\mathbb{N}$ denotes the number of comparisons between prospects.
To indicate that the comparison of prospects on the ordinal rather than the metric scale is of primary interest, we define the event space as a set of subsets of $S$
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.
To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define
$$\begin{equation}
\mathscr{D} = \left\{\frac{\overline{X}}{\overline{Y}} > 0, \frac{\overline{X}}{\overline{Y}} \leq 0 \right\}
\end{equation}$$
The decision variable $D$ is thus the measurable function
as a set of subsets of $S$ and the decision variable as the measure
$$\begin{equation}
D:= f: (S, \mathscr{D}) \mapsto (S', \mathscr{D'})
D:= f: (S, \mathscr{D}) \mapsto S'
\end{equation}$$
with the concrete mapping
with the mapping
$$\begin{equation}
\left(\frac{\overline{X}} {\overline{Y}}\Bigg) \in S : D\Bigg(\frac{\overline{X}} {\overline{Y}}\right) =
D:=
\left(
\frac{\overline{X}} {\overline{Y}}
\right)
\in S :
f
\left(
\frac{\overline{X}} {\overline{Y}}
\right)
=
\begin{cases}
1 & if & \frac{\overline{X}}{\overline{Y}} > 0 \in \mathscr{D} \\
0 & if & \frac{\overline{X}}{\overline{Y}} \leq 0 \in \mathscr{D}
\end{cases}
1 & if & \frac{\overline{X}}{\overline{Y}} > 0 \in \mathscr{D} \\
0 & if & \frac{\overline{X}}{\overline{Y}} \leq 0 \in \mathscr{D}
\end{cases}
\; .
\end{equation}$$
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# Method
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