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Probability theoretic account for prospects and sampling

parent 688d806a
...@@ -26,50 +26,103 @@ pacman::p_load(repro, ...@@ -26,50 +26,103 @@ pacman::p_load(repro,
# Note # Note
- Some of the R code is folded but can be unfolded by clicking the `Code` buttons.
- This document was created from the commit with the hash `r repro::current_hash()`. - This document was created from the commit with the hash `r repro::current_hash()`.
# Abstract # Abstract
Synthetic choice data from decisions from experience is generated by applying different strategies of sample integration to choice problems of 2-prospects. 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 comprehensive and piecewise forms of sample integration under varying structures of the environment (gamble features) and aspects of the sampling- and decision behavior (model parameters). 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.
# Summary # Summary
Provide short summary of simulation study results. Provide short summary of simulation results.
# Introduction # Introduction
## Prospects ## Prospects as Probability Spaces
Let a single prospect be a *probability space* $(\Omega, \Sigma, P)$ [cf. @kolmogorovFoundationsTheoryProbability1950]. $\Omega$ is the *sample space* containing a finite set of possible outcomes $\{\omega_1, ..., \omega_n\}$. Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an more accessible introduction].
$\Sigma$ is a set of subsets of $\Omega$, i.e., the *event space*.
$P$ is then a *probability mass function* (PMF) which maps the event space to the set of real numbers in the interval between 0 and 1: $P: \Sigma \mapsto [0,1]$.
I.e., the PMF assigns each event $\varsigma_i$ a probability of $0 \leq p_i \leq 1$ with $\sum_{i=1}^{n} p(\varsigma_i) = 1$.
The PMF also fulfills the condition $P(\Omega) = 1$.
## Monetary Prospects as Random Variables $\Omega$ is the *sample space* containing a finite set of possible outcomes
We can define a random variable on the probability space of a prospect by defining a function that maps the sample space to a measurable space: $X: \Omega \mapsto E$, where $E = \mathbb{R}$. $$\begin{equation}
Hence, every subset of $E$ has a preimage in $\Sigma$ and can be assigned a probability. \omega_i = \{\omega_1, ..., \omega_n\} \in \Omega
\end{equation}$$
In choice problems, where agents are asked to make a decision between $n$ monetary prospects, the mapping $\Omega \mapsto E$ is often implicit since all elements of $\Omega$ are real numbered (monetary gains or losses) and usually equal to the elements in $\Sigma$. $\mathscr{F}$ is a set of subsets of $\Omega$, i.e., the *event space*
## Sampling in Decisions from Experience (DFE) $$\begin{equation}
A_i = \{A_1, ..., A_n\} \in \mathscr{F}
\end{equation}$$
In DFE [@hertwigDecisionsExperienceEffect2004], where no summary description of prospects' probability spaces are provided, agents can either first explore them before arriving to a final choice (*sampling paradigm*), or, exploration and exploitation occur simultaneously (*partial-* or *full-feedback paradigm*) [cf. @hertwigDescriptionExperienceGap2009]. where
Below, only the sampling paradigm is considered.
In the context of choice problems between monetary gambles, we define a *single sample* as an outcome obtained when randomly drawing from a prospect's sample space $\Omega$. $$\begin{equation}
Technically, a single sample is thus the realization of a discrete random variable $X$, which fulfills the conditions outlined above. \mathscr{F} \subset \mathscr{P}(\Omega)
\end{equation}$$
In general terms, we define a *sampling strategy* as a systematic approach to generate a sequence of single samples from a choice problem's prospects as a means of exploring their probability spaces. $\mathscr{P}(\Omega)$ denotes the power set of $\Omega$.
Single samples that are generated from the same prospect reflect a sequence of realizations of random variables that are independent and identically distributed.
### Sampling Strategies and Sample Integration $P$ is a *probability mass function* (PMF) which maps the event space to the set of real numbers in $[0, 1]$:
$$\begin{equation}
P: \mathscr{F} \mapsto [0,1]
\end{equation}$$
by assigning each $\omega_i \in \Omega$ a probability of $0 \leq p_i \leq 1$ with $P(\Omega) = 1$.
## 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.
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*.
To provide a formal definition of sampling in risky or uncertain choice, we make use of random variables, functions which build the foundation of the random processes decision theory is concerned with but which are rarely explicated.
Thus, if for each
$$\begin{equation}
\omega_{i} \in \Omega: p(\omega_{i}) \neq 1
\end{equation}$$
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 latter is defined as the function
$$\begin{equation}
X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
\end{equation}$$
where $(\Omega', \mathscr{F'})$ is a measurable image of $(\Omega, \mathscr{F})$.
I.e., $X$ maps any event $A_i \in \mathscr{F}$ to a quantity $A'_i \in \mathscr{F'}$ and we denote the latter as the realization of the random variable $X$
$$\begin{equation}
A_i \in \mathscr{F}: X(A_i) \Rightarrow A'_i \in \mathscr{F'}
\end{equation}$$
However, to allow $\omega_{i}$ to be a realization of a random variable defined on $(\Omega, \mathscr{F})$, we must also set
$$\begin{equation}
\Omega = \mathscr{F} = \mathscr{F'}
\end{equation}$$
Given this restriction, we define a realization of the described random variable as a *single sample* 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 prospects probability space.
So far, we used the random variable defined on a prospect's triple $(\Omega, \mathscr{F}, P)$ and the restriction $\Omega = \mathscr{F'}$ solely to provide a probability theoretic definition of a single sample.
However, since in the decision literature the stochastic occurrence of the raw outcomes in $\Omega$ is often the event of interest, it should be justified to state that this restricted stochastic model is abundantly but implicitly assumed to underlay the evaluation processes of agents.
We do not contend that this model is not adequate but rather empirically warranted and mathematically convenient because of the measurable nature of monetary outcomes.
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.
How to construct alternative stochastic models underlying choices between risky prospects?
The above definition of a random variable allows us to depart from the standard model of the random process
... ...
# Method # Method
## Test set ## Test set
...@@ -409,7 +462,7 @@ gel_92 <- max(estimates$Rhat) # get largest scale reduction factor (Gelman & Rub ...@@ -409,7 +462,7 @@ gel_92 <- max(estimates$Rhat) # get largest scale reduction factor (Gelman & Rub
The potential scale reduction factor $\hat{R}$ was $n \leq$ `r round(gel_92, 3)` for all estimates, indicating good convergence. The potential scale reduction factor $\hat{R}$ was $n \leq$ `r round(gel_92, 3)` for all estimates, indicating good convergence.
#### Piecewise Integration #### Piecewise Integration
```{r} ```{r}
# generate subset of all strategy-parameter combinations (rows) and their parameters (columns) # generate subset of all strategy-parameter combinations (rows) and their parameters (columns)
...@@ -441,36 +494,20 @@ cpt_curves_piecewise %>% ...@@ -441,36 +494,20 @@ cpt_curves_piecewise %>%
theme_minimal() theme_minimal()
``` ```
Similarly to the false response rates, the patterns of the weighting function do not differ for the boundary types.
```{r} ```{r}
cpt_curves_piecewise %>% cpt_curves_piecewise %>%
ggplot(aes(p, w)) + ggplot(aes(p, w)) +
geom_path(size = .5) + geom_path() +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~boundary)
labs(title = "Piecewise Integration: Weighting functions",
x = "p",
y= "w(p)") +
theme_minimal()
```
Regarding the boundary value, we observe a distinct pattern for the smallest boundary, i.e. a = 1.
```{r}
cpt_curves_piecewise %>%
ggplot(aes(p, w)) +
geom_path(size = .5) +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) + geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
labs(title = "Piecewise Integration: Weighting functions",
x = "p",
y= "w(p)") +
facet_wrap(~a) + facet_wrap(~a) +
labs(title = "Piecewise Integration: Weighting functions",
x = "p",
y= "w(p)",
color = "Switching Probability") +
scale_color_viridis() +
theme_minimal() theme_minimal()
``` ```
As a general trend we find that with decreasing switching probabilities, probability weighting becomes more linear.
```{r} ```{r}
cpt_curves_piecewise %>% cpt_curves_piecewise %>%
ggplot(aes(p, w, color = s)) + ggplot(aes(p, w, color = s)) +
...@@ -484,8 +521,6 @@ cpt_curves_piecewise %>% ...@@ -484,8 +521,6 @@ cpt_curves_piecewise %>%
theme_minimal() theme_minimal()
``` ```
This trend holds for different boundary values.
```{r} ```{r}
cpt_curves_piecewise %>% cpt_curves_piecewise %>%
ggplot(aes(p, w, color = s)) + ggplot(aes(p, w, color = s)) +
...@@ -521,30 +556,6 @@ cpt_curves_piecewise %>% ...@@ -521,30 +556,6 @@ cpt_curves_piecewise %>%
theme_minimal() theme_minimal()
``` ```
```{r}
cpt_curves_piecewise %>%
ggplot(aes(x, v)) +
geom_path(size = .5) +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~boundary) +
labs(title = "Piecewise Integration: Value functions",
x = "p",
y= "w(p)") +
theme_minimal()
```
```{r}
cpt_curves_piecewise %>%
ggplot(aes(x, v)) +
geom_path(size = .5) +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~a) +
labs(title = "Piecewise Integration: Value functions",
x = "p",
y= "w(p)") +
theme_minimal()
```
```{r} ```{r}
cpt_curves_piecewise %>% cpt_curves_piecewise %>%
ggplot(aes(x, v, color = s)) + ggplot(aes(x, v, color = s)) +
...@@ -570,7 +581,7 @@ cpt_curves_piecewise %>% ...@@ -570,7 +581,7 @@ cpt_curves_piecewise %>%
theme_minimal() theme_minimal()
``` ```
#### Comprehensive Integration #### Comprehensive Integration
##### Weighting function w(p) ##### Weighting function w(p)
...@@ -600,22 +611,23 @@ cpt_curves_comprehensive %>% ...@@ -600,22 +611,23 @@ cpt_curves_comprehensive %>%
ggplot(aes(p, w)) + ggplot(aes(p, w)) +
geom_path(size = .5) + geom_path(size = .5) +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) + geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~boundary)
labs(title = "Comprehensive Integration: Weighting functions", labs(title = "Comprehensive Integration: Weighting functions",
x = "p", x = "p",
y= "w(p)") + y= "w(p)") +
facet_wrap(~a) +
theme_minimal() theme_minimal()
``` ```
```{r} ```{r}
cpt_curves_comprehensive %>% cpt_curves_comprehensive %>%
ggplot(aes(p, w)) + ggplot(aes(p, w, color = s)) +
geom_path(size = .5) + geom_path() +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) + geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
labs(title = "Comprehensive Integration: Weighting functions", labs(title = "Comprehensive Integration: Weighting functions",
x = "p", x = "p",
y= "w(p)") + y= "w(p)",
facet_wrap(~a) + color = "Switching Probability") +
scale_color_viridis() +
theme_minimal() theme_minimal()
``` ```
...@@ -624,9 +636,10 @@ cpt_curves_comprehensive %>% ...@@ -624,9 +636,10 @@ cpt_curves_comprehensive %>%
ggplot(aes(p, w, color = s)) + ggplot(aes(p, w, color = s)) +
geom_path() + geom_path() +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) + geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~a) +
labs(title = "Comprehensive Integration: Weighting functions", labs(title = "Comprehensive Integration: Weighting functions",
x = "p", x = "p",
y= "w(p)", y= "w(p)",
color = "Switching Probability") + color = "Switching Probability") +
scale_color_viridis() + scale_color_viridis() +
theme_minimal() theme_minimal()
...@@ -634,6 +647,7 @@ cpt_curves_comprehensive %>% ...@@ -634,6 +647,7 @@ cpt_curves_comprehensive %>%
```{r} ```{r}
cpt_curves_comprehensive %>% cpt_curves_comprehensive %>%
filter(s >= .7) %>%
ggplot(aes(p, w, color = s)) + ggplot(aes(p, w, color = s)) +
geom_path() + geom_path() +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) + geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
...@@ -668,18 +682,6 @@ cpt_curves_comprehensive %>% ...@@ -668,18 +682,6 @@ cpt_curves_comprehensive %>%
theme_minimal() theme_minimal()
``` ```
```{r}
cpt_curves_comprehensive %>%
ggplot(aes(x, v)) +
geom_path(size = .5) +
geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
facet_wrap(~boundary) +
labs(title = "Comprehensive Integration: Value functions",
x = "p",
y= "w(p)") +
theme_minimal()
```
```{r} ```{r}
cpt_curves_comprehensive %>% cpt_curves_comprehensive %>%
ggplot(aes(x, v)) + ggplot(aes(x, v)) +
......
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