manuscript.Rmd 32.1 KB
Newer Older
1
---
linushof's avatar
linushof committed
2
3
title: 'Sampling Strategies in Decisions from Experience'
author: "Linus Hof, Thorsten Pachur, Veronika Zilker"
4
5
6
7
8
9
bibliography: sampling-strategies-in-dfe.bib
output:
  html_document:
    code_folding: hide
    toc: yes
    toc_float: yes
linushof's avatar
linushof committed
10
    number_sections: no
linushof's avatar
linushof committed
11
12
13
  pdf_document:
    toc: yes
csl: apa.csl
linushof's avatar
linushof committed
14
15
16
editor_options: 
  markdown: 
    wrap: sentence
17
18
---

19
20
```{r}
# load packages
linushof's avatar
linushof committed
21
22
pacman::p_load(repro,
               tidyverse,
linushof's avatar
linushof committed
23
24
               knitr, 
               viridis)
25
26
```

linushof's avatar
linushof committed
27
# Note
28

29
This document was created from the commit with the hash `r repro::current_hash()`.
30

linushof's avatar
linushof committed
31
32
# Abstract

33
34
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.
35
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.
linushof's avatar
linushof committed
36
37
38

# Summary

39
Provide short summary of simulation results.
40

linushof's avatar
linushof committed
41
# Introduction
42

43
## Prospects as Probability Spaces
linushof's avatar
linushof committed
44

45
Let a prospect be a *probability space* $(\Omega, \mathscr{F}, P)$ [@kolmogorovFoundationsTheoryProbability1950; @georgiiStochasticsIntroductionProbability2008, for an accessible introduction].
linushof's avatar
linushof committed
46

47
$\Omega$ is the *sample space* containing an at most countable set of possible outcomes 
linushof's avatar
linushof committed
48

49
50
51
$$\begin{equation}
\omega_i = \{\omega_1, ..., \omega_n\} \in \Omega
\end{equation}$$ 
linushof's avatar
linushof committed
52

53
$\mathscr{F}$ is a set of subsets of $\Omega$, i.e., the *event space*
linushof's avatar
linushof committed
54

55
$$\begin{equation}
56
A_i = \{A_1, ..., A_n\} \in \mathscr{F} = \mathscr{P}(\Omega)
57
\end{equation}$$
linushof's avatar
linushof committed
58

59
$\mathscr{P}(\Omega)$ denotes the power set of $\Omega$. 
linushof's avatar
linushof committed
60

61
$P$ is a probability mass function that maps $\mathscr{F}$ to the set of real numbers in $[0, 1]$ 
62
63
64
65
66
67
68
69
70
71
72

$$\begin{equation}
P: \mathscr{F} \mapsto [0,1]
\end{equation}$$

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. 
73
In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for, i.e., choose either one of them. 
74
75
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 $(\Omega, \mathscr{F}, P)_j$.
76
For decision from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], the probability triples are not described but must be explored by the means of *sampling*. 
77

78
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. 
79
80
81
82
83
84
85
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 $P$. 
86
87
88
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 
89
90
91
92
93

$$\begin{equation}
X: (\Omega, \mathscr{F})  \mapsto (\Omega', \mathscr{F'})
\end{equation}$$

94
95
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 $A'_i \in \mathscr{F'}$ 
96
97

$$\begin{equation}
98
A'_i \in  \mathscr{F'} \Rightarrow X^{-1}A'_i \in \mathscr{F}
99
100
\end{equation}$$

101
102
103
104
105
106
107
108
[cf. @georgiiStochasticsIntroductionProbability2008].

(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. 

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.
109
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. 
110
111
112
113
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.

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. 
114
Thus, we may start rather abstractly by defining a decision variable $D$ 
115
116

$$\begin{equation}
117
D := f((\Omega, \mathscr{F}, P)_j)
118
119
\end{equation}$$

120
first without any further assumption on which information of the probability triple $f$ utilizes and how.
121
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.
122
Since we have defined the stochastic mechanism for generating such sequences, we write
123

124
125
126
127
$$\begin{equation}
D := f((X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}))_j)
\end{equation}$$

128
where $\Omega_j = \Omega'_j$. 
129

130
For n prospects, we write 
131
132

$$\begin{equation}
133
D := f(X_1, ..., X_j, ..., X_n)
134
135
\end{equation}$$

136
137
138
139
140
141
142
143
144
145
146
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.   

### Formalizing sampling and decision policies in the SSM

Hills and Hertwig [-@hillsInformationSearchDecisions2010] ... .

147
In the following, we consider the case of a choice between two prospects, their respective random variables we denote as $X$ and $Y$. 
148
149
150
151
152
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, 

we want $f$ to be a measurable function that maps the comparison of $X_1$ and $X_2$ to the measure space $\mathscr{D}$.
153
Because $X_1$ and $X_2$ are themselves measurable, we write their sample means as a fraction 
154

155
156
157
158
$$\begin{equation}
f: \frac{\overline{X_1}} {\overline{X_2}} = 
\frac{\frac{1}{N_1} \sum_{i=1}^{N_1} \omega_{i1}}{\frac{1}{N_2} \sum_{i=1}^{N_2} \omega_{i2}} \mapsto \mathscr{D}
\end{equation}$$
linushof's avatar
linushof committed
159

160
161
162
163
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 $\mathscr{F} = \{\frac{\overline{X_1}}  {\overline{X_2}} > 0, \frac{\overline{X_1}}  {\overline{X_2}} \leq 0\}$ 
164

165
166
167
$$\begin{equation}
f: (\Omega, \mathscr{F})  \mapsto (\Omega', \mathscr{D})
\end{equation}$$
168
169


linushof's avatar
linushof committed
170
171
# Method

linushof's avatar
linushof committed
172
## Test set
173

linushof's avatar
linushof committed
174
175
176
177
178
179
180
181
182
Under each condition, i.e., strategy-parameter combinations, all gambles are played by 100 synthetic agents.
We test a set of gambles, in which one of the prospects contains a safe outcome and the other two risky outcomes (*safe-risky gambles*).
Therefore, 60 gambles from an initial set of 10,000 are sampled.
Both outcomes and probabilities are drawn from uniform distributions, ranging from 0 to 20 for outcomes and from .01 to .99 for probabilities of the lower risky outcomes $p_{low}$.
The probabilities of the higher risky outcomes are $1-p_{low}$, respectively.
To omit dominant prospects, safe outcomes fall between both risky outcomes.
The table below contains the test set of 60 gambles.
Sampling of gambles was stratified, randomly drawing an equal number of 20 gambles with no, an attractive, and an unattractive rare outcome.
Risky outcomes are considered *"rare"* if their probability is $p < .2$ and *"attractive"* (*"unattractive"*) if they are higher (lower) than the safe outcome.
183

linushof's avatar
linushof committed
184
185
186
```{r message=FALSE}
gambles <- read_csv("data/gambles/sr_subset.csv")
gambles %>% kable()
187
188
```

linushof's avatar
linushof committed
189
## Model Parameters
190

linushof's avatar
linushof committed
191
**Switching probability** $s$ is the probability with which agents draw the following single sample from the prospect they did not get their most recent single sample from.
linushof's avatar
linushof committed
192
$s$ is varied between .1 to 1 in increments of .1.
193

linushof's avatar
linushof committed
194
The **boundary type** is either the minimum value any prospect's sample statistic must reach (absolute) or the minimum value for the difference of these statistics (relative).
linushof's avatar
linushof committed
195
Sample statistics are sums over outcomes (comprehensive strategy) and sums over wins (piecewise strategy), respectively.
196

linushof's avatar
linushof committed
197
198
For comprehensive integration, the **boundary value** $a$ is varied between 15 to 75 in increments of 15.
For piecewise integration $a$ is varied between 1 to 5 in increments of 1.
199

linushof's avatar
linushof committed
200
```{r message=FALSE}
201
202
203
204
205
206
207
208
# read choice data 
cols <- list(.default = col_double(),
             strategy = col_factor(),
             boundary = col_factor(),
             gamble = col_factor(),
             rare = col_factor(),
             agent = col_factor(),
             choice = col_factor())
linushof's avatar
linushof committed
209
choices <- read_csv("data/choices/choices.csv", col_types = cols)
210
211
```

linushof's avatar
linushof committed
212
In sum, 2 (strategies) x 60 (gambles) x 100 (agents) x 100 (parameter combinations) = `r nrow(choices)` choices are simulated.
linushof's avatar
linushof committed
213

linushof's avatar
linushof committed
214
# Results
215

linushof's avatar
linushof committed
216
217
Because we are not interested in deviations from normative choice due to sampling artifacts (e.g., ceiling effects produced by low boundaries), we remove trials in which only one prospect was attended.
In addition, we use relative frequencies of sampled outcomes rather than 'a priori' probabilities to compare actual against normative choice behavior.
218
219

```{r}
linushof's avatar
linushof committed
220
221
222
# remove choices where prospects were not attended
choices <- choices %>%
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp)))
223
224
```

linushof's avatar
linushof committed
225
226
227
228
229
```{r eval = FALSE}
# remove choices where not all outcomes were sampled
choices <- choices %>% 
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp) | a_p1_exp == 0 | a_p2_exp == 0))
```
linushof's avatar
linushof committed
230

linushof's avatar
linushof committed
231
Removing the respective trials, we are left with `r nrow(choices)` choices.
linushof's avatar
linushof committed
232

linushof's avatar
linushof committed
233
## Sample Size
linushof's avatar
linushof committed
234

linushof's avatar
linushof committed
235
236
237
238
239
240
```{r message=FALSE}
samples <- choices %>% 
  group_by(strategy, s, boundary, a) %>% 
  summarise(n_med = median(n_sample))
samples_piecewise <- samples %>% filter(strategy == "piecewise")
samples_comprehensive <- samples %>% filter(strategy == "comprehensive")
241
242
```

linushof's avatar
linushof committed
243
The median sample sizes generated by different parameter combinations ranged from `r min(samples_piecewise$n_med)` to `r max(samples_piecewise$n_med)` for piecewise integration and `r min(samples_comprehensive$n_med)` to `r max(samples_comprehensive$n_med)` for comprehensive integration.
244

linushof's avatar
linushof committed
245
### Boundary type and boundary value (a)
246

linushof's avatar
linushof committed
247
As evidence is accumulated sequentially, relative boundaries and large boundary values naturally lead to larger sample sizes, irrespective of the integration strategy.
linushof's avatar
linushof committed
248

linushof's avatar
linushof committed
249
250
```{r message=FALSE}
group_med <- samples_piecewise %>%
linushof's avatar
linushof committed
251
  group_by(boundary, a) %>% 
linushof's avatar
linushof committed
252
  summarise(group_med = median(n_med)) # to get the median across all s values
linushof's avatar
linushof committed
253

linushof's avatar
linushof committed
254
255
samples_piecewise %>%
  ggplot(aes(a, n_med, color = a)) + 
linushof's avatar
linushof committed
256
  geom_jitter(alpha = .5, size = 2) +
linushof's avatar
linushof committed
257
258
259
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
260
  labs(title = "Piecewise Integration",
linushof's avatar
linushof committed
261
       x ="a", 
linushof's avatar
linushof committed
262
       y="Sample Size", 
linushof's avatar
linushof committed
263
       col="a") + 
linushof's avatar
linushof committed
264
  theme_minimal()
linushof's avatar
linushof committed
265
```
linushof's avatar
linushof committed
266

linushof's avatar
linushof committed
267
268
```{r message=FALSE}
group_med <- samples_comprehensive %>%
linushof's avatar
linushof committed
269
  group_by(boundary, a) %>% 
linushof's avatar
linushof committed
270
  summarise(group_med = median(n_med)) 
linushof's avatar
linushof committed
271

linushof's avatar
linushof committed
272
273
samples_comprehensive %>%
  ggplot(aes(a, n_med, color = a)) + 
linushof's avatar
linushof committed
274
  geom_jitter(alpha = .5, size = 2) +
linushof's avatar
linushof committed
275
276
277
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
278
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
279
       x ="a", 
linushof's avatar
linushof committed
280
       y="Sample Size", 
linushof's avatar
linushof committed
281
       col="a") + 
linushof's avatar
linushof committed
282
  theme_minimal()
283
284
```

linushof's avatar
linushof committed
285
### Switching probability (s)
286

linushof's avatar
linushof committed
287
288
289
For piecewise integration, there is an inverse relationship between switching probability and sample size.
I.e., the lower s, the less frequent prospects are compared and thus, boundaries are only approached with larger sample sizes.
This effect is particularly pronounced for low probabilities such that the increase in sample size accelerates as switching probability decreases.
linushof's avatar
linushof committed
290

linushof's avatar
linushof committed
291
292
```{r message=FALSE}
group_med <- samples_piecewise %>%
linushof's avatar
linushof committed
293
  group_by(boundary, s) %>% 
linushof's avatar
linushof committed
294
  summarise(group_med = median(n_med)) # to get the median across all a values
linushof's avatar
linushof committed
295

linushof's avatar
linushof committed
296
297
298
299
300
301
samples_piecewise %>%
  ggplot(aes(s, n_med, color = s)) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
302
  labs(title = "Piecewise Integration",
linushof's avatar
linushof committed
303
       x ="s", 
linushof's avatar
linushof committed
304
       y="Sample Size", 
linushof's avatar
linushof committed
305
       col="s") + 
linushof's avatar
linushof committed
306
307
308
  theme_minimal()
```

linushof's avatar
linushof committed
309
310
311
For comprehensive integration, boundary types differ in the effects of switching probability.
For absolute boundaries, switching probability has no apparent effect on sample size as the distance of a given prospect to its absolute boundary is not changed by switching to (and sampling from) the other prospect.
For relative boundaries, however, samples sizes increase with switching probability.
linushof's avatar
linushof committed
312

linushof's avatar
linushof committed
313
314
```{r message=FALSE}
group_med <- samples_comprehensive %>%
linushof's avatar
linushof committed
315
  group_by(boundary, s) %>% 
linushof's avatar
linushof committed
316
  summarise(group_med = median(n_med)) # to get the median across all a values
linushof's avatar
linushof committed
317

linushof's avatar
linushof committed
318
319
320
321
322
323
samples_comprehensive %>%
  ggplot(aes(s, n_med, color = s)) + 
  geom_jitter(alpha = .5, size = 2) +
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
324
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
325
326
327
       x ="s",
       y = "Sample Size", 
       col="s") + 
linushof's avatar
linushof committed
328
329
330
  theme_minimal()
```

linushof's avatar
linushof committed
331
## Choice Behavior
linushof's avatar
linushof committed
332

linushof's avatar
linushof committed
333
Below, in extension to Hills and Hertwig [-@hillsInformationSearchDecisions2010], the interplay of integration strategies, gamble features, and model parameters in their effects on choice behavior in general and their contribution to underweighting of rare events in particular is investigated.
linushof's avatar
linushof committed
334
335
336
337
338
339
340
341
342
343
344
345
We apply two definitions of underweighting of rare events: Considering false response rates, we define underweighting such that the rarity of an attractive (unattractive) outcome leads to choose the safe (risky) prospect although the risky (safe) prospect has a higher expected value.

```{r message=FALSE}
fr_rates <- choices %>% 
  mutate(ev_ratio_exp = round(a_ev_exp/b_ev_exp, 2), 
         norm = case_when(ev_ratio_exp > 1 ~ "A", ev_ratio_exp < 1 ~ "B")) %>% 
  filter(!is.na(norm)) %>% # exclude trials with normative indifferent options
  group_by(strategy, s, boundary, a, rare, norm, choice) %>% # group correct and incorrect responses
  summarise(n = n()) %>% # absolute numbers 
  mutate(rate = round(n/sum(n), 2), # response rates 
         type = case_when(norm == "A" & choice == "B" ~ "false safe", norm == "B" & choice == "A" ~ "false risky")) %>% 
  filter(!is.na(type)) # remove correct responses
linushof's avatar
linushof committed
346
347
```

linushof's avatar
linushof committed
348
Considering the parameters of Prelec's [-@prelecProbabilityWeightingFunction1998] implementation of the weighting function [CPT; cf. @tverskyAdvancesProspectTheory1992], underweighting is reflected by decisions weights estimated to be smaller than the corresponding objective probabilities.
linushof's avatar
linushof committed
349

linushof's avatar
linushof committed
350
### False Response Rates
linushof's avatar
linushof committed
351

linushof's avatar
linushof committed
352
353
354
```{r message=FALSE}
fr_rates_piecewise <- fr_rates %>% filter(strategy == "piecewise")
fr_rates_comprehensive <- fr_rates %>% filter(strategy == "comprehensive")
linushof's avatar
linushof committed
355
```
356

linushof's avatar
linushof committed
357
The false response rates generated by different parameter combinations ranged from `r min(fr_rates_piecewise$rate)` to `r max(fr_rates_piecewise$rate)` for piecewise integration and from `r min(fr_rates_comprehensive$rate)` to `r max(fr_rates_comprehensive$rate)` for comprehensive integration.
linushof's avatar
linushof committed
358
However, false response rates vary considerably as a function of rare events, indicating that their presence and attractiveness are large determinants of false response rates.
linushof's avatar
linushof committed
359

linushof's avatar
linushof committed
360
361
362
363
364
365
```{r message=FALSE}
fr_rates %>% 
  group_by(strategy, boundary, rare) %>% 
  summarise(min = min(rate),
            max = max(rate)) %>% 
  kable()
linushof's avatar
linushof committed
366
367
```

linushof's avatar
linushof committed
368
The heatmaps below show the false response rates for all strategy-parameter combinations.
linushof's avatar
linushof committed
369
370
Consistent with our - somewhat rough - definition of underweighting, the rate of false risky responses is generally higher, if the unattractive outcome of the risky prospect is rare (top panel).
Conversely, if the attractive outcome of the risky prospect is rare, the rate of false safe responses is generally higher (bottom panel).
linushof's avatar
linushof committed
371
As indicated by the larger range of false response rates, the effects of rare events are considerably larger for piecewise integration.
372

linushof's avatar
linushof committed
373
374
375
376
377
378
379
380
381
382
383
384
385
386
```{r message=FALSE}
fr_rates %>% 
  filter(strategy == "piecewise", boundary == "absolute") %>% 
  ggplot(aes(a, s, fill = rate)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_tile(colour="white", size=0.25) + 
  scale_x_continuous(expand=c(0,0), breaks = seq(1, 5, 1)) +
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1)) +
  scale_fill_viridis() + 
  labs(title = "Piecewise Integration | Absolute Boundary",
       x = "a", 
       y= "s", 
       fill = "% False Responses") + 
  theme_minimal() 
387
388
```

linushof's avatar
linushof committed
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
```{r message=FALSE}
fr_rates %>% 
  filter(strategy == "piecewise", boundary == "relative") %>% 
  ggplot(aes(a, s, fill = rate)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_tile(colour="white", size=0.25) + 
  scale_x_continuous(expand=c(0,0), breaks = seq(1, 5, 1)) +
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1)) +
  scale_fill_viridis() + 
  labs(title = "Piecewise Integration | Relative Boundary",
       x = "a", 
       y= "s", 
       fill = "% False Responses") + 
  theme_minimal() 
```
linushof's avatar
linushof committed
404

linushof's avatar
linushof committed
405
406
```{r message=FALSE}
fr_rates %>% 
linushof's avatar
linushof committed
407
  filter(strategy == "comprehensive", boundary == "absolute") %>% 
linushof's avatar
linushof committed
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
  ggplot(aes(a, s, fill = rate)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_tile(colour="white", size=0.25) + 
  scale_x_continuous(expand=c(0,0), breaks = seq(15, 75, 15)) +
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1)) +
  scale_fill_viridis() + 
  labs(title = "Comprehensive Integration | Absolute Boundary",
       x = "a", 
       y= "s", 
       fill = "% False Responses") + 
  theme_minimal() 
```

```{r message=FALSE}
fr_rates %>% 
linushof's avatar
linushof committed
423
  filter(strategy == "comprehensive", boundary == "relative") %>% 
linushof's avatar
linushof committed
424
425
426
427
428
429
430
431
432
433
434
  ggplot(aes(a, s, fill = rate)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_tile(colour="white", size=0.25) + 
  scale_x_continuous(expand=c(0,0), breaks = seq(15, 75, 15)) +
  scale_y_continuous(expand=c(0,0), breaks = seq(.1, 1, .1)) +
  scale_fill_viridis() + 
  labs(title = "Comprehensive Integration | Relative Boundary",
       x = "a", 
       y= "s", 
       fill = "% False Responses") + 
  theme_minimal() 
435
436
```

linushof's avatar
linushof committed
437
#### Switching Probability (s) and Boundary Value (a)
linushof's avatar
linushof committed
438

linushof's avatar
linushof committed
439
As for both piecewise and comprehensive integration the differences between boundary types are rather minor and of magnitude than of qualitative pattern, the remaining analyses of false response rates are summarized across absolute and relative boundaries.
linushof's avatar
linushof committed
440

linushof's avatar
linushof committed
441
Below, the $s$ and $a$ parameter are considered as additional sources of variation in the false response pattern above and beyond the interplay of integration strategies and the rarity and attractiveness of outcomes.
linushof's avatar
linushof committed
442

linushof's avatar
linushof committed
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
```{r message=FALSE}
fr_rates %>% 
  filter(strategy == "piecewise") %>% 
  ggplot(aes(s, rate, color = a)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_jitter(size = 2) + 
  scale_x_continuous(breaks = seq(0, 1, .1)) +
  scale_y_continuous(breaks = seq(0, 1, .1)) +
  scale_color_viridis() + 
  labs(title = "Piecewise Integration",
       x = "s", 
       y= "% False Responses", 
       color = "a") + 
  theme_minimal() 
```
458

linushof's avatar
linushof committed
459
460
```{r message=FALSE}
fr_rates %>% 
linushof's avatar
linushof committed
461
  filter(strategy == "comprehensive") %>% 
linushof's avatar
linushof committed
462
463
464
465
466
467
  ggplot(aes(s, rate, color = a)) + 
  facet_grid(type ~ fct_relevel(rare, "attractive", "none", "unattractive"), switch = "y") +
  geom_jitter(size = 2) + 
  scale_x_continuous(breaks = seq(0, 1, .1)) +
  scale_y_continuous(breaks = seq(0, 1, .1)) +
  scale_color_viridis() + 
468
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
469
470
471
472
       x = "s", 
       y= "% False Responses", 
       color = "a") + 
  theme_minimal() 
473
474
```

linushof's avatar
linushof committed
475
For piecewise integration, switching probability is naturally related to the size of the samples on which the round-wise comparisons of prospects are based on, with low values of $s$ indicating large samples and vice versa.
linushof's avatar
linushof committed
476
Accordingly, switching probability is positively related to false response rates.
linushof's avatar
linushof committed
477
478
I.e., the larger the switching probability, the smaller the round-wise sample size and the probability of experiencing a rare event within a given round.
Because round-wise comparisons are independent of each other and binomial distributions within a given round are skewed for small samples and outcome probabilities [@kolmogorovFoundationsTheoryProbability1950], increasing boundary values do not reverse but rather amplify this relation.
479

linushof's avatar
linushof committed
480
481
482
For comprehensive integration, switching probability is negatively related to false response rates, i.e., an increase in $s$ is associated with decreasing false response rates.
This relation, however, may be the result of an artificial interaction between the $s$ and $a$ parameter.
Precisely, in the current algorithmic implementation of sampling with a comprehensive integration mechanism, decreasing switching probabilities cause comparisons of prospects based on increasingly unequal sample sizes immediately after switching prospects.
linushof's avatar
linushof committed
483
Consequentially, reaching (low) boundaries is rather a function of switching probability and associated sample sizes than of actual evidence for a given prospect over the other.
484

linushof's avatar
linushof committed
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
### Cumulative Prospect Theory

In the following, we examine the possible relations between the parameters of the *choice-generating* sampling models and the *choice-describing* cumulative prospect theory.

For each distinct strategy-parameter combination, we ran 20 chains of 40,000 iterations each, after a warm-up period of 1000 samples.
To reduce potential autocorrelation during the sampling process, we only kept every 20th sample (thinning).

```{r}
# read CPT data
cols <- list(.default = col_double(),
             strategy = col_factor(),
             boundary = col_factor(),
             parameter = col_factor())
estimates <- read_csv("data/estimates/estimates_cpt_pooled.csv", col_types = cols)
```

#### Convergence

```{r}
gel_92 <- max(estimates$Rhat) # get largest scale reduction factor (Gelman & Rubin, 1992) 
```

The potential scale reduction factor $\hat{R}$ was $n \leq$ `r round(gel_92, 3)` for all estimates, indicating good convergence.

509
#### Piecewise Integration
linushof's avatar
linushof committed
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543

```{r}
# generate subset of all strategy-parameter combinations (rows) and their parameters (columns)
curves_cpt <- estimates %>% 
  select(strategy, s, boundary, a, parameter, mean) %>% 
  pivot_wider(names_from = parameter, values_from = mean)
```

##### Weighting function w(p)

We start by plotting the weighting curves for all parameter combinations under piecewise integration.

```{r}

cpt_curves_piecewise <- curves_cpt %>% 
  filter(strategy == "piecewise") %>% 
  expand_grid(p = seq(0, 1, .1)) %>% # add vector of objective probabilities
  mutate(w = round(exp(-delta*(-log(p))^gamma), 2)) # compute decision weights (cf. Prelec, 1998)

# all strategy-parameter combinations 

cpt_curves_piecewise %>% 
  ggplot(aes(p, w)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Piecewise Integration: Weighting functions",
       x = "p", 
       y= "w(p)") + 
  theme_minimal() 
```

```{r}
cpt_curves_piecewise %>% 
  ggplot(aes(p, w)) + 
544
  geom_path() +
linushof's avatar
linushof committed
545
546
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  facet_wrap(~a) + 
547
548
549
550
551
  labs(title = "Piecewise Integration: Weighting functions",
       x = "p",
       y= "w(p)",
       color = "Switching Probability") + 
  scale_color_viridis() +
linushof's avatar
linushof committed
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
  theme_minimal() 
```

```{r}
cpt_curves_piecewise %>% 
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Piecewise Integration: Weighting functions",
       x = "p", 
       y= "w(p)", 
       color = "Switching Probability") + 
  scale_color_viridis() +
  theme_minimal() 
```

```{r}
cpt_curves_piecewise %>% 
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  facet_wrap(~a) + 
  labs(title = "Piecewise Integration: Weighting functions",
       x = "p",
       y= "w(p)",
       color = "Switching Probability") + 
  scale_color_viridis() +
  theme_minimal() 
```

##### Value function v(x)

```{r}

cpt_curves_piecewise <- curves_cpt %>% 
  filter(strategy == "piecewise") %>% 
  expand_grid(x = seq(0, 20, 2)) %>% # add vector of objective outcomes
  mutate(v = round(x^alpha, 2)) # compute decision weights (cf. Prelec, 1998)

# all strategy-parameter combinations 

cpt_curves_piecewise %>% 
  ggplot(aes(x, v)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Piecewise Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  theme_minimal() 
```

```{r}
cpt_curves_piecewise %>% 
  ggplot(aes(x, v, color = s)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Piecewise Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  scale_color_viridis() + 
  theme_minimal() 
```

```{r}
cpt_curves_piecewise %>% 
  ggplot(aes(x, v, color = s)) + 
  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)") + 
  scale_color_viridis() + 
  theme_minimal() 
```

628
#### Comprehensive Integration
linushof's avatar
linushof committed
629
630
631
632

##### Weighting function w(p)

We start by plotting the weighting curves for all parameter combinations under piecewise integration.
linushof's avatar
linushof committed
633

linushof's avatar
linushof committed
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
```{r}

cpt_curves_comprehensive <- curves_cpt %>% 
  filter(strategy == "comprehensive") %>% 
  expand_grid(p = seq(0, 1, .1)) %>% # add vector of objective probabilities
  mutate(w = round(exp(-delta*(-log(p))^gamma), 2)) # compute decision weights (cf. Prelec, 1998)

# all strategy-parameter combinations 

cpt_curves_comprehensive %>% 
  ggplot(aes(p, w)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Weighting functions",
       x = "p", 
       y= "w(p)") + 
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
  ggplot(aes(p, w)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Weighting functions",
       x = "p", 
660
661
       y= "w(p)") + 
  facet_wrap(~a) + 
linushof's avatar
linushof committed
662
663
664
665
666
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
667
668
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
linushof's avatar
linushof committed
669
670
671
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Weighting functions",
       x = "p", 
672
673
674
       y= "w(p)", 
       color = "Switching Probability") + 
  scale_color_viridis() +
linushof's avatar
linushof committed
675
676
677
678
679
680
681
682
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
683
  facet_wrap(~a) + 
linushof's avatar
linushof committed
684
  labs(title = "Comprehensive Integration: Weighting functions",
685
686
       x = "p",
       y= "w(p)",
linushof's avatar
linushof committed
687
688
689
690
691
692
693
       color = "Switching Probability") + 
  scale_color_viridis() +
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
694
  filter(s >= .7) %>% 
linushof's avatar
linushof committed
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  facet_wrap(~a) + 
  labs(title = "Comprehensive Integration: Weighting functions",
       x = "p",
       y= "w(p)",
       color = "Switching Probability") + 
  scale_color_viridis() +
  theme_minimal() 
```

##### Value function v(x)

```{r}

cpt_curves_comprehensive <- curves_cpt %>% 
  filter(strategy == "comprehensive") %>% 
  expand_grid(x = seq(0, 20, 2)) %>% # add vector of objective outcomes
  mutate(v = round(x^alpha, 2)) # compute decision weights (cf. Prelec, 1998)


# all strategy-parameter combinations 

cpt_curves_comprehensive %>% 
  ggplot(aes(x, v)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  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(~a) + 
  labs(title = "Comprehensive Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
  ggplot(aes(x, v, color = s)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  scale_color_viridis() + 
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
  ggplot(aes(x, v, color = s)) + 
  geom_path(size = .5) +
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  facet_wrap(~a) + 
  labs(title = "Comprehensive Integration: Value functions",
       x = "p", 
       y= "w(p)") + 
  scale_color_viridis() + 
  theme_minimal() 
```
linushof's avatar
linushof committed
765

linushof's avatar
linushof committed
766
# References