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Review of abstract and implementation of Hills & Hertwig (2010) assumptions in the SSM

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# Note
# Author Note
This document was created from the commit with the hash `r repro::current_hash()`.
# Abstract
- Add information on how to reproduce the project.
- Add contact.
Synthetic choice data from so-called decisions from experience is generated by applying different strategies of sample integration to a series of choice problems between two prospects.
The synthetic data is explored for characteristic choice patterns produced by these strategies under varying structures of the environment (prospect features) and aspects of the sampling- and decision behavior.
We start our argument by giving a probability theoretic account of prospects, sampling, and sample integration and derive assumptions about the choice patterns that result from different integration strategies, if applied.
# Abstract
# Summary
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.
Provide short summary of simulation results.
# Introduction
## Prospects as Probability Spaces
Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an accessible introduction].
......@@ -75,7 +78,7 @@ It is common to make a rather crude distinction between two variants of this eva
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*.
To provide a formal definition of sampling in risky or uncertain choice, we make use of the mathematical concept of a random variable, a function that models the random processes decision theory is concerned with but which is rarely explicated.
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
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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.
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.
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.
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### Formalizing sampling and decision policies in the SSM
Hills and Hertwig [-@hillsInformationSearchDecisions2010] ... .
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].
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
S = \left\{\frac{\overline{X}} {\overline{Y}}\right\}^{\mathbb{N}}
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 $S$
In the following, we consider the case of a choice between two prospects, their respective random variables we denote 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, i.e., $X$ over $Y$ or vice versa.
$f$ should thus map the comparison of $X$ and $Y$ to a measurable space.
In principle,
\mathscr{D} = \left\{\frac{\overline{X}}{\overline{Y}} > 0, \frac{\overline{X}}{\overline{Y}} \leq 0 \right\}
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
The decision variable $D$ is thus the measurable function
f: \frac{\overline{X_1}} {\overline{X_2}} =
\frac{\frac{1}{N_1} \sum_{i=1}^{N_1} \omega_{i1}}{\frac{1}{N_2} \sum_{i=1}^{N_2} \omega_{i2}} \mapsto \mathscr{D}
D:= f: (S, \mathscr{D}) \mapsto (S', \mathscr{D'})
The decision variable $D$ is thus a function of the comparative measure $\frac{\overline{X_1}} {\overline{X_2}}$ of the random variables both defined on the probability spaces of their respective prospects.
We assume that the elements of $\mathscr{D}$ are the natural numbers $\{0, 1\}$, indicating that the ordinal comparison of $\overline{X_1}$ and $\overline{X_2}$ either provides evidence for a given prospect $\{1\}$ or not $\{0\}$.
Thus, $f$ itself can be defined as a random variable that maps the sample space $\Omega = \{\frac{\overline{X_1}} {\overline{X_2}}\}$ to the measurable space $\mathscr{D} = \{0, 1\}$.
However, since we are not interested in the measure $\frac{\overline{X_1}} {\overline{X_2}} = \mathbb{R}$ itself but in the ordinal comparison of $X_1$ and $X_2$, we introduce the event space $\mathscr{F} = \{\frac{\overline{X_1}} {\overline{X_2}} > 0, \frac{\overline{X_1}} {\overline{X_2}} \leq 0\}$
with the concrete mapping
f: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{D})
\left(\frac{\overline{X}} {\overline{Y}}\Bigg) \in S : D\Bigg(\frac{\overline{X}} {\overline{Y}}\right) =
1 & if & \frac{\overline{X}}{\overline{Y}} > 0 \in \mathscr{D} \\
0 & if & \frac{\overline{X}}{\overline{Y}} \leq 0 \in \mathscr{D}
# Method
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# Discussion
# Conclusion
# References
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