manuscript.Rmd 31.1 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
In summary, we have defined the decision variable $D$ as a function of the random variables associated with the prospects probability spaces.
In the following, we consider the case of a choice between two prospects, their respective random variables we denote as $X$ and $Y$. 
138
139
Since the decision variable $D$ serves as a measure for the evidence for one prospect over the other, 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 
140

141
142
143
144
$$\begin{equation}
f: \frac{\overline{X_1}} {\overline{X_2}} = 
\frac{\frac{1}{N_1} \sum_{i=1}^{N_1} \omega_{i1}}{\frac{1}{N_2} \sum_{i=1}^{N_2} \omega_{i2}} \mapsto \mathscr{D}
\end{equation}$$
linushof's avatar
linushof committed
145

146
147
148
149
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 $\mathscr{F} = \{\frac{\overline{X_1}}  {\overline{X_2}} > 0, \frac{\overline{X_1}}  {\overline{X_2}} \leq 0\}$ 
150

151
152
153
$$\begin{equation}
f: (\Omega, \mathscr{F})  \mapsto (\Omega', \mathscr{D})
\end{equation}$$
154
155


linushof's avatar
linushof committed
156
157
# Method

linushof's avatar
linushof committed
158
## Test set
159

linushof's avatar
linushof committed
160
161
162
163
164
165
166
167
168
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.
169

linushof's avatar
linushof committed
170
171
172
```{r message=FALSE}
gambles <- read_csv("data/gambles/sr_subset.csv")
gambles %>% kable()
173
174
```

linushof's avatar
linushof committed
175
## Model Parameters
176

linushof's avatar
linushof committed
177
**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
178
$s$ is varied between .1 to 1 in increments of .1.
179

linushof's avatar
linushof committed
180
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
181
Sample statistics are sums over outcomes (comprehensive strategy) and sums over wins (piecewise strategy), respectively.
182

linushof's avatar
linushof committed
183
184
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.
185

linushof's avatar
linushof committed
186
```{r message=FALSE}
187
188
189
190
191
192
193
194
# 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
195
choices <- read_csv("data/choices/choices.csv", col_types = cols)
196
197
```

linushof's avatar
linushof committed
198
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
199

linushof's avatar
linushof committed
200
# Results
201

linushof's avatar
linushof committed
202
203
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.
204
205

```{r}
linushof's avatar
linushof committed
206
207
208
# remove choices where prospects were not attended
choices <- choices %>%
  filter(!(is.na(a_ev_exp) | is.na(b_ev_exp)))
209
210
```

linushof's avatar
linushof committed
211
212
213
214
215
```{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
216

linushof's avatar
linushof committed
217
Removing the respective trials, we are left with `r nrow(choices)` choices.
linushof's avatar
linushof committed
218

linushof's avatar
linushof committed
219
## Sample Size
linushof's avatar
linushof committed
220

linushof's avatar
linushof committed
221
222
223
224
225
226
```{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")
227
228
```

linushof's avatar
linushof committed
229
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.
230

linushof's avatar
linushof committed
231
### Boundary type and boundary value (a)
232

linushof's avatar
linushof committed
233
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
234

linushof's avatar
linushof committed
235
236
```{r message=FALSE}
group_med <- samples_piecewise %>%
linushof's avatar
linushof committed
237
  group_by(boundary, a) %>% 
linushof's avatar
linushof committed
238
  summarise(group_med = median(n_med)) # to get the median across all s values
linushof's avatar
linushof committed
239

linushof's avatar
linushof committed
240
241
samples_piecewise %>%
  ggplot(aes(a, n_med, color = a)) + 
linushof's avatar
linushof committed
242
  geom_jitter(alpha = .5, size = 2) +
linushof's avatar
linushof committed
243
244
245
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
246
  labs(title = "Piecewise Integration",
linushof's avatar
linushof committed
247
       x ="a", 
linushof's avatar
linushof committed
248
       y="Sample Size", 
linushof's avatar
linushof committed
249
       col="a") + 
linushof's avatar
linushof committed
250
  theme_minimal()
linushof's avatar
linushof committed
251
```
linushof's avatar
linushof committed
252

linushof's avatar
linushof committed
253
254
```{r message=FALSE}
group_med <- samples_comprehensive %>%
linushof's avatar
linushof committed
255
  group_by(boundary, a) %>% 
linushof's avatar
linushof committed
256
  summarise(group_med = median(n_med)) 
linushof's avatar
linushof committed
257

linushof's avatar
linushof committed
258
259
samples_comprehensive %>%
  ggplot(aes(a, n_med, color = a)) + 
linushof's avatar
linushof committed
260
  geom_jitter(alpha = .5, size = 2) +
linushof's avatar
linushof committed
261
262
263
  geom_point(data = group_med, aes(y = group_med), size = 3) +
  facet_wrap(~boundary) + 
  scale_color_viridis() + 
264
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
265
       x ="a", 
linushof's avatar
linushof committed
266
       y="Sample Size", 
linushof's avatar
linushof committed
267
       col="a") + 
linushof's avatar
linushof committed
268
  theme_minimal()
269
270
```

linushof's avatar
linushof committed
271
### Switching probability (s)
272

linushof's avatar
linushof committed
273
274
275
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
276

linushof's avatar
linushof committed
277
278
```{r message=FALSE}
group_med <- samples_piecewise %>%
linushof's avatar
linushof committed
279
  group_by(boundary, s) %>% 
linushof's avatar
linushof committed
280
  summarise(group_med = median(n_med)) # to get the median across all a values
linushof's avatar
linushof committed
281

linushof's avatar
linushof committed
282
283
284
285
286
287
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() + 
288
  labs(title = "Piecewise Integration",
linushof's avatar
linushof committed
289
       x ="s", 
linushof's avatar
linushof committed
290
       y="Sample Size", 
linushof's avatar
linushof committed
291
       col="s") + 
linushof's avatar
linushof committed
292
293
294
  theme_minimal()
```

linushof's avatar
linushof committed
295
296
297
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
298

linushof's avatar
linushof committed
299
300
```{r message=FALSE}
group_med <- samples_comprehensive %>%
linushof's avatar
linushof committed
301
  group_by(boundary, s) %>% 
linushof's avatar
linushof committed
302
  summarise(group_med = median(n_med)) # to get the median across all a values
linushof's avatar
linushof committed
303

linushof's avatar
linushof committed
304
305
306
307
308
309
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() + 
310
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
311
312
313
       x ="s",
       y = "Sample Size", 
       col="s") + 
linushof's avatar
linushof committed
314
315
316
  theme_minimal()
```

linushof's avatar
linushof committed
317
## Choice Behavior
linushof's avatar
linushof committed
318

linushof's avatar
linushof committed
319
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
320
321
322
323
324
325
326
327
328
329
330
331
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
332
333
```

linushof's avatar
linushof committed
334
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
335

linushof's avatar
linushof committed
336
### False Response Rates
linushof's avatar
linushof committed
337

linushof's avatar
linushof committed
338
339
340
```{r message=FALSE}
fr_rates_piecewise <- fr_rates %>% filter(strategy == "piecewise")
fr_rates_comprehensive <- fr_rates %>% filter(strategy == "comprehensive")
linushof's avatar
linushof committed
341
```
342

linushof's avatar
linushof committed
343
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
344
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
345

linushof's avatar
linushof committed
346
347
348
349
350
351
```{r message=FALSE}
fr_rates %>% 
  group_by(strategy, boundary, rare) %>% 
  summarise(min = min(rate),
            max = max(rate)) %>% 
  kable()
linushof's avatar
linushof committed
352
353
```

linushof's avatar
linushof committed
354
The heatmaps below show the false response rates for all strategy-parameter combinations.
linushof's avatar
linushof committed
355
356
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
357
As indicated by the larger range of false response rates, the effects of rare events are considerably larger for piecewise integration.
358

linushof's avatar
linushof committed
359
360
361
362
363
364
365
366
367
368
369
370
371
372
```{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() 
373
374
```

linushof's avatar
linushof committed
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
```{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
390

linushof's avatar
linushof committed
391
392
```{r message=FALSE}
fr_rates %>% 
linushof's avatar
linushof committed
393
  filter(strategy == "comprehensive", boundary == "absolute") %>% 
linushof's avatar
linushof committed
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
  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
409
  filter(strategy == "comprehensive", boundary == "relative") %>% 
linushof's avatar
linushof committed
410
411
412
413
414
415
416
417
418
419
420
  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() 
421
422
```

linushof's avatar
linushof committed
423
#### Switching Probability (s) and Boundary Value (a)
linushof's avatar
linushof committed
424

linushof's avatar
linushof committed
425
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
426

linushof's avatar
linushof committed
427
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
428

linushof's avatar
linushof committed
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
```{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() 
```
444

linushof's avatar
linushof committed
445
446
```{r message=FALSE}
fr_rates %>% 
linushof's avatar
linushof committed
447
  filter(strategy == "comprehensive") %>% 
linushof's avatar
linushof committed
448
449
450
451
452
453
  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() + 
454
  labs(title = "Comprehensive Integration",
linushof's avatar
linushof committed
455
456
457
458
       x = "s", 
       y= "% False Responses", 
       color = "a") + 
  theme_minimal() 
459
460
```

linushof's avatar
linushof committed
461
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
462
Accordingly, switching probability is positively related to false response rates.
linushof's avatar
linushof committed
463
464
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.
465

linushof's avatar
linushof committed
466
467
468
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
469
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.
470

linushof's avatar
linushof committed
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
### 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.

495
#### Piecewise Integration
linushof's avatar
linushof committed
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529

```{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)) + 
530
  geom_path() +
linushof's avatar
linushof committed
531
532
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  facet_wrap(~a) + 
533
534
535
536
537
  labs(title = "Piecewise Integration: Weighting functions",
       x = "p",
       y= "w(p)",
       color = "Switching Probability") + 
  scale_color_viridis() +
linushof's avatar
linushof committed
538
539
540
541
542
543
544
545
546
547
548
549
550
551
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
  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() 
```

614
#### Comprehensive Integration
linushof's avatar
linushof committed
615
616
617
618

##### Weighting function w(p)

We start by plotting the weighting curves for all parameter combinations under piecewise integration.
linushof's avatar
linushof committed
619

linushof's avatar
linushof committed
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
```{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", 
646
647
       y= "w(p)") + 
  facet_wrap(~a) + 
linushof's avatar
linushof committed
648
649
650
651
652
  theme_minimal() 
```

```{r}
cpt_curves_comprehensive %>% 
653
654
  ggplot(aes(p, w, color = s)) + 
  geom_path() +
linushof's avatar
linushof committed
655
656
657
  geom_abline(intercept = 0, slope = 1, color = "red", size = 1) +
  labs(title = "Comprehensive Integration: Weighting functions",
       x = "p", 
658
659
660
       y= "w(p)", 
       color = "Switching Probability") + 
  scale_color_viridis() +
linushof's avatar
linushof committed
661
662
663
664
665
666
667
668
  theme_minimal() 
```

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

```{r}
cpt_curves_comprehensive %>% 
680
  filter(s >= .7) %>% 
linushof's avatar
linushof committed
681
682
683
684
685
686
687
688
689
690
691
692
693
694
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
  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
751

linushof's avatar
linushof committed
752
# References