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Review definition of decision variable

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...@@ -33,8 +33,8 @@ This document was created from the commit with the hash `r repro::current_hash() ...@@ -33,8 +33,8 @@ This document was created from the commit with the hash `r repro::current_hash()
# Abstract # Abstract
A probability theoretic definition of prospects and a rough stochastic model for decisions from experience is proposed. A probability theoretic definition of prospects and a rough stochastic sampling 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. 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 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. Synthetic choice data is simulated and modeled in cumulative prospect theory to test these predictions.
# Introduction # Introduction
...@@ -86,7 +86,7 @@ Note that, since random variables defined on the same prospect are independent a ...@@ -86,7 +86,7 @@ Note that, since random variables defined on the same prospect are independent a
Thus, long sample sequences in principle allow to obtain the same information about a prospect by sampling as by symbolic description. 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 $1, ..., k$. Consider now a choice between prospects $1, ..., k$.
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: To construct a stochastic sampling 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:
$$\begin{equation} $$\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)
...@@ -103,19 +103,20 @@ D := f(X_{i1}, ..., X_{ik}) ...@@ -103,19 +103,20 @@ D := f(X_{i1}, ..., X_{ik})
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.
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. 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. Taking the case of different sampling and decision strategies previously assumed to play a role in DfE, the following section demonstrates how such assumptions can be explicated in a stochastic model that builds on the sampling approach outlined so far.
## A Stochastic Model Capturing Differences in Sampling and Decision Strategies ## A Stochastic Sampling Model Capturing Differences in Sampling and Decision Strategies
Hills and Hertwig [-@hillsInformationSearchDecisions2010] discussed a potential link between sampling and decision strategies in DfE. 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). 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]. 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.
We now consider choices between two prospects and the assumptions of Hills and Hertwig [-@hillsInformationSearchDecisions2010] in more detail to build the respective stochastic sampling model for DfE.
Let $X$ and $Y$ be random variables defined on the prospects $(\Omega, \mathscr{F}, P)_X$ and $(\Omega, \mathscr{F}, P)_Y$. 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. Hills and Hertwig [-@hillsInformationSearchDecisions2010] suggest that any two sample 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 Let thus $C = \mathbb{R}$ be the set of all possible outcomes of such a mean comparison for given sequence lengths $N_X$ and $N_Y$ and
$$\begin{equation} $$\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\} \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\}
...@@ -123,70 +124,25 @@ $$\begin{equation} ...@@ -123,70 +124,25 @@ $$\begin{equation}
\end{equation}$$ \end{equation}$$
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. 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.
The outcome of such an ordinal comparison can be regarded as evidence for or against a prospect and the number of wins over a series of independent ordinal comparisons as accumulated evidence.
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. To integrate the concept of evidence accumulation into the current model, we let $D$ be a measurable function that maps the possible outcomes of a mean comparison in $C$ onto a measure space $C' = \{0,1\}$, with $0$ ($1$) indicating a lost (won) 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:
$$\begin{equation}
D:= f: S \mapsto S'
\; .
\end{equation}$$
Since Hills and Hertwig [-@hillsInformationSearchDecisions2010] assume that comparisons of prospects are based on sample means, $S$ is the set
$$\begin{equation} $$\begin{equation}
S = D(c \in C) =
\left\{
\frac{\frac{1}{N_X} \sum\limits_{i=1}^{N_X} x_i}
{\frac{1}{N_Y} \sum\limits_{i=1}^{N_Y} y_i}
\right\}
=
\left\{
\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right\}
\; ,
\end{equation}$$
where $N_X$ and $N_Y$ denotes the number of single samples within a comparison.
To indicate that the comparison of prospects on the ordinal scale is of primary interest, we define
$$\begin{equation}
\mathscr{D} = \left\{\frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} > 0, \frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} \leq 0 \right\}
\end{equation}$$
as a set of subsets of $S$, i.e., the *event space*, and the decision variable as the measure
$$\begin{equation}
D:= f: (S, \mathscr{D}) \mapsto S'
\end{equation}$$
with the mapping
$$\begin{equation}
D:=
\left(
\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right)
\in S :
f
\left(
\frac{\overline{X}_{N_X}} {\overline{Y}_{N_Y}}
\right)
=
\begin{cases} \begin{cases}
1 & \text{if} & \frac{\overline{X}_{N_X}}{\overline{Y}_{N_Y}} > 0 \in \mathscr{D} \\ 1 & \text{for} & \{c \in C: (\overline{X}_{N_X} - \overline{Y}_{N_Y} > 0) \in \mathscr{C} \} \\
0 & \text{else}. 0 & \text{else}.
\end{cases} \end{cases}
\end{equation}$$ \end{equation}$$
It can be shown that for the case $N_X = N_Y = 1$, $D$ is a random variable that follows the Bernoulli distribution It can be shown that for fixed sequence lengths $N_X$ and $N_Y$, a sequence $D_i = D_1, ..., D_n$ is a Bernoulli process following the binomial distribution
$$\begin{equation} $$\begin{equation}
D \sim B\left( p \left( \frac{\overline{X}_{N_X = 1}}{\overline{Y}_{N_Y = 1}} > 0\right), n\right) \; , D \sim B\left( p \left(\overline{X}_{N_X} - \overline{Y}_{N_Y} > 0\right), n\right) \; ,
\end{equation}$$ \end{equation}$$
where $n$ is the number of comparisons (see *Proof 1* in Appendix). where $p$ is the probability of $X$ winning a single mean comparison and $n$ is the number of comparisons (see [Appendix]).
However, although $p$ can in principle be determined, it becomes intractable with increasing elements in $\Omega$ and growing sequence lengths.
## Predicting Choice Behavior in DfE ## Predicting Choice Behavior in DfE
...@@ -587,7 +543,7 @@ cols <- list(.default = col_double(), ...@@ -587,7 +543,7 @@ cols <- list(.default = col_double(),
strategy = col_factor(), strategy = col_factor(),
boundary = col_factor(), boundary = col_factor(),
parameter = col_factor()) parameter = col_factor())
estimates <- read_csv("data/estimates/estimates_cpt_pooled.csv", col_types = cols) estimates <- read_csv("data/estimates/estimates_cpt_pooled_goldstein-einhorn-87.csv", col_types = cols)
``` ```
#### Convergence #### Convergence
...@@ -861,18 +817,19 @@ cpt_curves_comprehensive %>% ...@@ -861,18 +817,19 @@ cpt_curves_comprehensive %>%
# Appendix # Appendix
Let $X_n$ and $Y_m$ be independent and discrete random variables of the sequences Let
$$\begin{equation} $$\begin{equation}
X_1, ..., X_n, ..., X_{N_X} X_1, ..., X_{N_X}
\end{equation}$$ \end{equation}$$
and and
$$\begin{equation} $$\begin{equation}
Y_1, ..., Y_m, ..., Y_{N_Y} \; . Y_1, ..., Y_{N_Y}
\end{equation}$$ \end{equation}$$
be sequences of random variables that are iid.
Then Then
$$\begin{equation} $$\begin{equation}
......
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