Commit 62f86319 authored by linushof's avatar linushof
Browse files

Reformulation of single samples

parent 250bd86c
...@@ -33,17 +33,17 @@ This document was created from the commit with the hash `r repro::current_hash() ...@@ -33,17 +33,17 @@ This document was created from the commit with the hash `r repro::current_hash()
# Abstract # Abstract
A probability theoretic definition of sampling and a rough stochastic model of the random process underlying decisions from experience are proposed. A rough stochastic model of the random processes underlying decisions from experience is 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. 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.
Synthetic choice data is simulated and modeled in cumulative prospect theory to test these predictions. Synthetic choice data is simulated and modeled in cumulative prospect theory to test these predictions.
# Introduction # Introduction
... ...
## Random Processes in Sequential Sampling ## Random Processes in Decisions from Experience
In research on the decision theory, a standard paradigm is the choice between at least two prospects. 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 $(\Omega, \mathscr{F}, P)$. Let a prospect be a probability space $(\Omega, \mathscr{F}, P)$.
$\Omega$ is the sample space $\Omega$ is the sample space
...@@ -68,40 +68,25 @@ P: \mathscr{F} \mapsto [0,1] ...@@ -68,40 +68,25 @@ P: \mathscr{F} \mapsto [0,1]
that assigns each outcome $\omega$ a probability $0 < p(\omega) \leq 1$ with $P(\Omega) = 1$ [ @kolmogorovFoundationsTheoryProbability1950, pp. 2-3]. that assigns each outcome $\omega$ a probability $0 < p(\omega) \leq 1$ with $P(\Omega) = 1$ [ @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. 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. @hertwigDescriptionexperienceGapRisky2009]. It is common to make a distinction between two variants of this evaluation process [cf. @hertwigDescriptionexperienceGapRisky2009].
For decisions from description (DfD), agents are provided a full symbolic description of the prospects. For decisions from description (DfD), agents are provided a full symbolic description of the prospects.
For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], prospects are not described but must be explored by the means of sampling. For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], prospects are not described but must be explored by the means of sampling.
To provide a formal definition of sampling in risky choice, we make use of the mathematical concept of a random variable and start by referring to a prospect as *"risky"* in the case where $p(\omega) \neq 1$ for all $\omega \in \Omega$. To provide a formal definition of sampling in risky choice, we make use of the mathematical concept of a random variable and start by referring to a prospect as *"risky"* in the case where $p(\omega) \neq 1$ for all $\omega \in \Omega$.
Here, risky describes the fact that if agents would choose a prospect and any of its outcomes in $\Omega$ must occur, none of these outcomes will occur with certainty. Here, risky describes the fact that if agents would choose a prospect and any of its outcomes in $\Omega$ must occur, none of these outcomes will occur with certainty.
It is acceptable to speak of the occurrence of $\omega$ as a realization of a random variable $X$ defined on a prospect iff the following conditions A and B are met: It is acceptable to speak of the occurrence of $\omega$ as a realization of a random variable $X$ defined on a prospect iff the following conditions (1) and (2) are met:
A) $X$ is a measurable function (1) $X$ is a measurable function $$\begin{equation} X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}) \; , \end{equation}$$ where $\Omega'$ is a set of real numbered values $X$ can take and $\mathscr{F'}$ is a set of subsets of $\Omega'$. I.e., $\Omega$ maps into $\Omega'$ such that correspondingly each subset $A' \in \mathscr{F'}$ has a pre-image $X^{-1}A' \in \mathscr{F}$, which is the set $\{\omega \in \Omega: X(\omega) \in A'\}$ [@kolmogorovFoundationsTheoryProbability1950, p. 21].
$$\begin{equation} (2) The mapping is such that $X(\omega) = x \equiv \omega$.
X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
\; ,
\end{equation}$$
where $\Omega'$ is a set of real numbered values $X$ can take and $\mathscr{F'}$ is a set of subsets of $\Omega'$.
I.e., $\Omega$ maps into $\Omega'$ such that correspondingly each subset $A' \in \mathscr{F'}$ has a pre-image
$$\begin{equation}
X^{-1}A' \in \mathscr{F} \; ,
\end{equation}$$
which is the set $\{\omega \in \Omega: X(\omega) \in A'\}$ [@kolmogorovFoundationsTheoryProbability1950, p. 21].
B) The mapping is such that $X(\omega) = \omega$.
Given conditions A and B, 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 single samples of a sequence originating from 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 such a sequence [cf. @bernoulliArsConjectandiOpus1713].
Thus, large sample sequences in principle allow to obtain the same information about a prospect by sampling as by symbolic description.
## A Stochastical Sampling Model for DfE 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].
Thus, long sample sequences in principle allow to obtain the same information about a prospect by sampling as by symbolic description.
Consider a choice between prospects $1, ..., k$. Consider now a choice between prospects $1, ..., k$.
To construct a stochastic sampling model (hereafter SSM) of the random process 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 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
$$\begin{equation} $$\begin{equation}
D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k) D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k)
...@@ -111,26 +96,26 @@ D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k) ...@@ -111,26 +96,26 @@ D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k)
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: 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:
$$\begin{equation} $$\begin{equation}
D := f(X_{i1}, ..., X_{ik}) D := f(x_{i1}, ..., x_{ik}) \quad \text{for} \; x \equiv{\omega}
\; , \; ,
\end{equation}$$ \end{equation}$$
where $i = 1, ..., N$ denotes a sequence of length $N$ of random variables that are iid. where $i = 1, ..., N$ denotes a sequence of length $N$ of random variables that are iid and $x$ their realizations.
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 - not to mention whether they are capable of doing so - and that these choices should be informed by psychological or other theory [@heOntologyDecisionModels2020, for an ontology of decision models] and empirical protocols. 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 captured by the SSM. 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.
## Integrating sampling and decision strategies into the SSM ## Example: Formulating Sampling and Decision Strategies as Stochastic Processes
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. Hills and Hertwig [-@hillsInformationSearchDecisions2010] discussed a potential link between sampling and decision strategies in DfE.
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. 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, 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]. 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].
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]. We now consider choices between two prospects to integrate these assumptions into 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. 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.
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: 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:
$$\begin{equation} $$\begin{equation}
...@@ -144,7 +129,7 @@ $$\begin{equation} ...@@ -144,7 +129,7 @@ $$\begin{equation}
S = S =
\left\{ \left\{
\frac{\frac{1}{N_X} \sum\limits_{i=1}^{N_X} x_i} \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} {\frac{1}{N_Y} \sum\limits_{i=1}^{N_Y} y_i}
\right\} \right\}
= =
\left\{ \left\{
...@@ -153,7 +138,7 @@ S = ...@@ -153,7 +138,7 @@ S =
\; , \; ,
\end{equation}$$ \end{equation}$$
where $x_i$ and $y_j$ are single samples and $N_X$ and $N_Y$ denotes the number of single samples within a comparison. where $N_X$ and $N_Y$ denotes the number of single samples within a comparison.
To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define
$$\begin{equation} $$\begin{equation}
...@@ -193,7 +178,7 @@ D \sim B\left( p \left( \frac{\overline{X}_{N_X = 1}}{\overline{Y}_{N_Y = 1}} > ...@@ -193,7 +178,7 @@ 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). where $n$ is the number of comparisons (see *Proof 1* in Appendix).
## Predicting Choices From the SSM ### Predicting Choices From the SSM
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. 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? How does the current version of the SSM support this proposition?
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment