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\title{Numerate decision makers don't use more effortful strategies unless it pays: A process tracing investigation of skilled and adaptive strategy selection in risky decision making}

\author{
Jakub Traczyk\thanks{Corresponding author Email: jtraczyk@swps.edu.pl}
  \thanks{Wroclaw Faculty of Psychology, SWPS University of
  Social Sciences and Humanities, Wroclaw, Poland.}
\and 
  Agata Sobkow\footnotemark[2]
\and
  Kamil Fulawka\footnotemark[2] % indicates same as 2nd thanks
\and
  Jakub Kus\footnotemark[2]
\and
  Dafina Petrova\thanks{Andalusian School of Public Health (EASP), Granada, Spain. Medical Research
Institute ibs.GRANADA, University Hospitals of Granada/University of
Granada, Granada, Spain.}
\and
  Rocio Garcia-Retamero\thanks{Mind, Brain, and Behavior Research Center, University of Granada, Spain;
Harding Center for Risk Literacy, Max Planck Institute for Human
Development, Berlin, Germany.}
}


\date{} % leave empty
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\newcommand{\jref}{http://journal.sjdm.org/vol13.4.html}
\newcommand{\jhead}{Judgment and Decision Making, Vol.~13, No.~4}
\newcommand{\jdate}{July 2018}
\pagestyle{myheadings} \markright{\protect\small \href{\jref}{\jhead}, \jdate \hfill Numeracy and strategy selection \qquad}
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\href{\jref}{\jhead}, \jdate, pp.\ 372-381
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{\small \href{\jref}{\jhead}, \jdate, pp.\ 372--381}
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%\begin{latexonly}
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\begin{onecolabstract}
%\end{latexonly}
  The present study investigated skilled and adaptive strategy
  selection in risky decision making. We proposed that people with
  high objective numeracy, a strong predictor of general decision
  making skill, would have a broad repertoire of choice strategies and 
adaptively select these strategies depending on the importance
  of the decision. Thus more objectively numerate people would
  maximize their effort (e.g., invest more time) in important,
  high-payoff decisions and switch to a simple, fast heuristic
  strategy in trivial decisions. Subjective numeracy would, by
  contrast, be more closely related to interest in problem solving for
  its own sake and would not yield such an effect of
  importance. Participants made twelve high-payoff choices and twelve
  low-payoff choices in binary two-outcome gambles framed as gains. We
  measured objective and subjective numeracy using standard
  measures. Results showed that people with high subjective numeracy
  generally maximized the expected value (EV) in all decisions. In
  contrast, participants with high objective numeracy maximized EV
  only when choice problems were meaningful (i.e., they could result
  in high payoffs). When choice problems were trivial
  (i.e., choosing the normatively better option would not result in a
  large payoff), more objectively numerate participants
  made choices consistent with faster, more frugal heuristic
  strategies. 

% some revisions

\smallskip
\noindent
Keywords: numeracy, strategy selection, risky choice, prospect theory, priority heuristic, skilled decision making, deliberation
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]
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{\renewcommand{\thefootnote}{}
\footnotetext{ % note blank lines above and below acknowledgment

  The research in this article was supported by the National Science
  Centre, Poland under grant 2015/17/D/HS6/00703 and the Foundation
  for Polish Science (FNP) under grant START (111.2016). The current
  research was also partially funded by the Ministerio de Economía y
  Competitividad (Spain) (PSI2014-51842-R).  Dafina Petrova is
  supported by a Juan de la Cierva Fellowship (FJCI-2016-28279) from
  the Spanish Ministry of Economy, Industry, and Competitiveness.

Copyright: \copyright\ 2018.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

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\section{Introduction}

In real-life financial situations, people use various choice
strategies and decision rules to deal with problems that involve
processing of outcomes and probabilities. For instance, if we use the
expected value (EV) maximization principle as a model of rational
behavior under risk, the decision regarding whether or not to buy a
lottery ticket for 3 EUR with a probability of winning a one-million
main prize at 0.0000072\%, should be preceded by optimization analyses
that include calculations comparing EV of the bet (i.e., +0.072 EUR)
with the certain loss of $-$3 EUR. However, comprehensive optimization
is almost impossible in the majority of everyday decision problems, as
they are usually very complex, dynamic and uncertain (people do not
know every possible alternative, precise values of probabilities,
etc.). Furthermore, apart from material and accountable consequences
(e.g., money), people also attach value to other resources such as
their effort and time spent on making a decision.

Many studies have reported that people with greater numerical
abilities are more likely to make better choices in similar problems,
that is, decisions more consistent with EV maximization (Cokely et al., 2018; Pachur \&
Galesic, 2013; Jasper, Bhattacharya, Levin, Jones \& Bossard, 2013;
Peters \& Bjalkebring, 2015). Nonetheless,
there is compelling evidence that superior decisions in more numerate
people do not simply result from complex computations of EVs but
rather they are driven by exhaustive considerations of multiple
aspects of the problem (i.e., elaborate heuristic processing, Cokely
\& Kelley, 2009; Ghazal, Cokely \& Garcia-Retamero, 2014; Jasper, Bhattacharya \& Corser,
2017). The ability to understand and use statistical and probability
information) is also related to superior decision making in both
health and financial domains (Galesic \& Garcia-Retamero, 2011; Garcia-Retamero, Andrade, Sharit \&
Ruiz, 2015; Ghazal et al., 2014, 2014; Reyna \&
Brnst-Renck, 2014) and is associated with greater personal wealth in
general (Estrada-Mejia, de Vries \& Zeelenberg, 2016). However, the
question of how people with high numeracy arrive at better decisions
seems to be complex.

Researchers argue that more (objectively) numerate individuals have
better understanding of the gist of decisions (Reyna, Nelson, Han \&
Dieckmann, 2009), deliberatively employ metacognitive heuristics
(Garcia-Retamero, Cokely \& Hoffrage, 2015; Ghazal et al., 2014), draw
precise affective meaning from numbers (Peters, 2012; Peters et al.,
2006), and are more likely to conduct explicit number operations
(Peters \& Bjalkebring, 2015).  Higher numeracy is also related to
more exact mental number-line mapping (Schley \& Peters, 2014) and
more linear transformations of objective numbers (Millroth \& Juslin,
2015), resulting in less pronounced distortions in processing outcomes
and probabilities (Patalano, Saltiel, Machlin \& Barth, 2015),
although other results raise questions about this conclusion (Peters
et al., 2006). Furthermore, people with higher objective numeracy seem
to be less prone to affective influences during decision making (e.g.,
Petrova, van der Pligt \& Garcia-Retamero, 2014; Traczyk \& Fulawka,
2016; Traczyk et al., 2018).

A question posed here is whether those with higher objective numeracy
are more inclined to adaptively change choice strategy depending on
the task requirements. Do they allocate their time and effort in
proportion to the importance of decisions? In particular, we
investigated how the structure of the environment (operationalized as
the difference in the magnitudes of expected payoffs) influences
strategy selection under risk and what is the role of numeracy in this
process.  The present study thus investigated whether highly numerate
individuals always engage more time and effort to compute EV, or
rather adaptively select their choice strategy depending on the
task and environment (Jasper et al., 2013).

Additionally, we examined the relative contribution of objective
(i.e., inferred from the numerical task performance) and subjective
(i.e., declared preference for numbers and number-related operations)
numeracy in predicting choices. Although a measure of subjective
numeracy has been shown to be a good proxy for objective numeracy
(e.g., it predicts performance in numerical tasks; Fagerlin et
al., 2007) and some researchers use objective and subjective numeracy scales
interchangeably (Zikmund-Fisher, Smith, Ubel \& Fagerlin, 2007), other
studies have demonstrated that these tests measure different traits
(Anderson, Obrecht, Chapman, Driscoll \& Schulkin, 2011; Miron-Shatz,
Hanoch, Doniger, Omer \& Ozanne, 2014; Peters \& Bjalkebring, 2015;
Petrova et al., 2017). For instance,
subjective numeracy predicts perceptions of health, whereas objective
numeracy predicts actual health in patients (Garcia-Retamero et al. 2015). 
Taking this into account, in the present study we
measured both subjective and objective numeracy to examine their
relative contribution to choice.

\subsection{Research problem and hypothesis}

In this study, we ask whether people with high objective numeracy
adapt their effort to the importance of the decision, or whether they
simply make better decisions regardless of the importance of the decision. 
On the one hand, more numerate subjects are likely to make more normatively
superior choices in financial problems as a result of greater
deliberation during decision making (Ghazal et al., 2014). This effect
can be presumably attributed to the elaborative heuristic search
posited by Cokely and Kelley (2009) --- a larger number of simple
considerations as well as better understanding of formal operations
with numbers. On the other hand, more numerate subjects could also
strategically allocate deliberation time and use different choice
strategies depending on the importance of a decision, quantified here
by the EV ratio between gambles. That is, people high in numeracy
would maximize their effort (e.g., invest more time and deliberation)
when the decision really matters (i.e., when differences between
expected payoffs are large), but when the decision is trivial (i.e.,
when differences between expected payoffs are small) they would switch
to a simple and fast heuristic strategy.

Prior research suggests two potential outcomes. Firstly, in the case
of trivial choice problems, more numerate individuals would behave
similarly as in meaningful problems. That is, they will deliberate more
on the problem, leading to more of their choices being consistent with
EV (Ghazal et al., 2014; Jasper et al., 2017). Secondly, because of
the adaptive sensitivity to changes in EV that is exhibited by more
numerate people (Jasper et al., 2013), their ability to get the gist
of numbers (Reyna et al., 2009) and to employ heuristic processing
(Cokely \& Kelley, 2009), they are expected to switch to a faster
heuristic-based strategy as it provides comparable performance in case
of decisions that seem trivial (i.e., low-payoff gambles with similar
EVs).

%Table 1
\begin{table*}[t!]
\caption{Twelve low-payoff (with EV ratios around 1) and twelve high-payoff (with EV ratios between 5 and 6) binary choice problems consisting of two-outcome gambles in the gain domain. The priority heuristic (PH) and cumulative prospect theory (CPT) predicted opposite choices. Each problem met criteria of nondominance.}
\begin{tabular}{L{1.4in}C{1.4in}C{.7in}C{.7in}C{.7in}C{.7in}C{.7in}}\toprule
Gamble A & Gamble B & EV Gamble A & EV Gamble B & EV ratio & Choice by PH & Choice by CPT \\ \midrule
5.40, 0.29; 0, 0.71 & 9.70, 0.17; 0, 0.83 & 1.57 & 1.65 & 1.05 & A & B \\
17.50, 0.17; 0, 0.83 & 3.00, 0.94; 0, 0.06 & 2.98 & 2.82 & 1.05 & B & A \\
9.70, 0.17; 0, 0.83 & 5.40, 0.29; 0, 0.71 & 1.65 & 1.57 & 1.05 & B & A \\
3.00, 0.29; 0, 0.71 & 5.40, 0.17; 0, 0.83 & 0.87 & 0.92 & 1.06 & A & B \\
31.50, 0.17; 0, 0.83 & 5.40, 0.94; 0, 0.06 & 5.36 & 5.08 & 1.05 & B & A \\
31.50, 0.29; 0, 0.71 & 56.70, 0.17; 0, 0.83 & 9.14 & 9.64 & 1.06 & A & B \\
9.70, 0.17; 0, 0.83 & 3.00, 0.52; 0, 0.48 & 1.65 & 1.56 & 1.06 & B & A \\
5.40, 0.17; 0, 0.83 & 3.00, 0.29; 0, 0.71 & 0.92 & 0.87 & 1.06 & B & A \\
3.00, 0.52; 0, 0.48 & 9.70, 0.17; 0, 0.83 & 1.56 & 1.65 & 1.06 & A & B \\
17.50, 0.52; 0, 0.48 & 56.70, 0.17; 0, 0.83 & 9.1 & 9.64 & 1.06 & A & B \\
9.70, 0.52; 0, 0.48 & 31.50, 0.17; 0, 0.83 & 5.04 & 5.36 & 1.06 & A & B \\
56.70, 0.17; 0, 0.83 & 17.50, 0.52; 0, 0.48 & 9.64 & 9.1 & 1.06 & B & A \\
3.00, 0.17; 0, 0.83 & 56.70, 0.05; 0, 0.95 & 0.51 & 2.84 & 5.56 & A & B \\
3.00, 0.94; 0, 0.06 & 31.50, 0.52; 0, 0.48 & 2.82 & 16.38 & 5.81 & A & B \\
56.70, 0.05; 0, 0.95 & 3.00, 0.17; 0, 0.83 & 2.84 & 0.51 & 5.56 & B & A \\
5.40, 0.94; 0, 0.06 & 56.70, 0.52; 0, 0.48 & 5.08 & 29.48 & 5.81 & A & B \\
31.50, 0.52; 0, 0.48 & 3.00, 0.94; 0, 0.06 & 16.38 & 2.82 & 5.81 & B & A \\
56.70, 0.52; 0, 0.48 & 5.40, 0.94; 0, 0.06 & 29.48 & 5.08 & 5.81 & B & A \\
3.00, 0.94; 0, 0.06 & 56.70, 0.29; 0, 0.71 & 2.82 & 16.44 & 5.83 & A & B \\
5.40, 0.52; 0, 0.48 & 56.70, 0.29; 0, 0.71 & 2.81 & 16.44 & 5.86 & A & B \\
31.50, 0.29; 0, 0.71 & 3.00, 0.52; 0, 0.48 & 9.14 & 1.56 & 5.86 & B & A \\
56.70, 0.29; 0, 0.71 & 5.40, 0.52; 0, 0.48 & 16.44 & 2.81 & 5.86 & B & A \\
3.00, 0.29; 0, 0.71 & 56.70, 0.09; 0, 0.91 & 0.87 & 5.1 & 5.87 & A & B \\
56.70, 0.09; 0, 0.91 & 3.00, 0.29; 0, 0.71 & 5.1 & 0.87 & 5.87 & B & A \\ \bottomrule
\mc{7}{p{6in}}{CPT and EV were always consistent in their choice predictions in these problems.}\\
\end{tabular}
\end{table*}

\section{Method}
\subsection{Subjects}
One hundred and thirty-nine volunteers from the general population
(age range: 18–58 years; $M_{age}$ = 29.21; $SD$ = 8.57; 52\% female)
participated in an online study for a 30 PLN (Polish Zloty)
compensation (equivalent to approximately 8 EUR).\footnote{One hundred
  and thirty-three subjects completed all tasks.} Subjects
were recruited via an advertisement published on a local
webpage. Subjects received explicit information stating that the
study examined cognitive abilities and were assured that payment was
not related to their performance. Volunteers gave informed consent
before the study.
% The study was approved by the departmental Ethical
% Board of the SWPS University of Social Sciences and Humanities.

%Table 2
\begin{table*}[t!]\centering
\caption{An example of cognitive operations and choice predictions according to cumulative prospect theory (CPT) and priority heuristic (PH).}
\begin{tabular}{@{}L{2.7in}L{1.9in}L{1.9in}@{}}\toprule
& Gamble A: 29\% probability to win 5.40 EUR or 71\% probability to win nothing (0 EUR) & Gamble B: 17\% probability to win 9.70 EUR or 83\% probability to win nothing (0 EUR) \\ \midrule
Cumulative prospect theory (CPT) &  &  \\
Step 1: compute the CPT value of a gamble & \mc{2}{L{4in}}{Using equation \(V = \;\mathop \sum \limits_{i = 1}^n w\left( {{p_i}} \right)v\left( {{x_i}} \right)\), where $w$ is the decision weight for probability ${p_i}$, and ${v}$ is a value of an outcome ${x_i}$ with standard CPT parameters (0.88 for the value function and 0.61 for the probability weighting function in gain domain), CPT values for Gamble A and B equal 1.38 and 1.78, respectively.}  \\
Decision: select the gamble with a higher CPT value & \mc{2}{L{4in}}{Select Gamble B because of the higher CPT value.}  \\ \midrule
Priority heuristic (PH) &  &  \\
Step 1: is the difference in minimum gains larger than 10\% of minimum gain? & \mc{2}{L{4in}}{No. Minimum gain in Gamble A (0 EUR) equals to the minimum gain in Gamble B (0 EUR)} \\
Step 2: is the difference in the probability of the minimum gain larger than 10\% in the probability scale? & \mc{2}{L{4in}}{Yes. 71\% in Gamble A vs. 83\% in Gamble B (the difference of 12\% is larger than 10\%). Thus, considering the last step posited by PH (i.e., the difference between maximum gains) is not necessary.} \\
Decision: stop the process and select the gamble with a lower probability of minimum gain & \mc{2}{L{4in}}{Select Gamble A.} \\ \bottomrule
\mc{3}{@{}l}{CPT and EV were always consistent in their choice predictions in these problems.}\\
\end{tabular}
\end{table*}

\subsection{Design and materials}

The study was designed to investigate the relationship between
numeracy and choices under risk. We measured both subjective and
objective numeracy as main predictors in this study as well as fluid
intelligence and need for cognition. We also introduced a
within-subjects manipulation of choice payoff (i.e., 12 low-payoff vs.\ 12
high-payoff binary two-outcome choice problems). The dependent
variable was subjects’ choices in these problems. Response
latencies in each problem were used to operationalize deliberation
time and enhance our understanding of the cognitive processes
underlying decision making.

Subjects completed the following measures:

\subsubsection{The Berlin Numeracy Test}
The Berlin Numeracy Test (BNT; Cokely, Galesic, Schulz, Ghazal \&
Garcia-Retamero, 2012) is a psychometric instrument that measures
statistical numeracy, risk literacy and comprehension of the concept
of probability. The BNT is widely used as an efficient research tool
to measure objective numerical abilities (Cokely et al., 2012). In the
current study, we used a computerized version of the BNT consisting of
four items presented to subjects in a fixed order. Subjects
were not forced to provide any response to complete the items.

\subsubsection{The Subjective Numeracy Scale}
Subjects completed the Subjective Numeracy Scale (SNS; Fagerlin et
al., 2007), an 8-item self-assessment scale that measures subjective
numeracy and includes two subscales referring to perceived numerical
abilities (e.g., “How good are you at working with percentages?”) and
preference for numerical and statistical information (e.g., “How often
do you find numerical information to be useful?”). Subjects
completed the SNS by answering each question using a 6-point scale.

\subsubsection{The Need for Cognition Scale}
Individual differences in epistemic motivation were assessed using the
Need for Cognition Scale (NCS; Cacioppo \& Petty, 1982). The Polish
adaptation of the scale (Matusz, Traczyk \& Gąsiorowska, 2011)
includes 36 items and measures "the tendency for an individual to
engage in and enjoy thinking" (Cacioppo \& Petty, 1982, p. 116). The
main goal of the NCS is to assess the individual’s level of joy and
satisfaction emerging from the processes of thinking, resolving
problems, learning new things, etc. Subjects used a 5-point scale
to indicate the extent they agree with each statement (e.g., “Thinking
is not my idea of fun”).

\subsubsection{The Raven’s Advanced Progressive Matrices}
To measure individual differences in fluid intelligence we used the
Raven’s Advanced Progressive Matrices test (RAPM; Raven, 2000). The
RAPM is a nonverbal test that is typically employed to assess basic
cognitive functioning such as abstract reasoning. Problems included in
this test are presented in the form of matrices (e.g., 3 x 3 elements)
with one missing piece. The task is to figure out the rule underlying
the uncovered elements and select the missing element that satisfies
the appropriate rule. In the present study, we used a short form of
the RAPM consisting of two training and six test matrices displayed
with ascending difficulty.

%\renewcommand{\baselinestretch}{1.03} \large\normalsize

\subsubsection{Choice problems}
We used twenty-four binary choice problems consisting of two-outcome
gambles from a study by Pachur, Hertwig, Gigerenzer and Brandstätter
(2013, Experiment 2). The outcomes of the choice problems were framed
as gains. Choice problems differed in their EV ratios (Table 1), which
we used to define high- and low-payoff problems. If the EVs of two
gambles were similar (dividing their EVs led to outcomes around 1),
then we classified them as low-payoff problems in comparison to high-payoff 
problems in which the EVs of the two gambles differed
substantially (dividing their EVs led to outcomes between 5 and 6). In
other words, low-payoff problems (with EVs ratios around 1) can be
regarded as trivial, because playing them repeatedly would lead to
relatively small differences in payoffs irrespective of the chosen
gamble (i.e., it does not really matter which option you should
select, because in the long-run a decision maker would get a similar
payoff). In contrast, high-payoff problems (with EVs ratios between 5
and 6) are more meaningful, because their EVs differ and choosing the
gamble with the higher EV will lead to much higher payoffs (i.e., it
pays to choose the gamble with higher EV because differences in
payoffs are large in the long run).

The problems were generated in such a way that two competing models of
choice under risk --- the priority heuristic (PH; Brandstätter,
Gigerenzer \& Hertwig, 2006) and cumulative prospect theory (CPT;
Tversky \& Kahneman, 1992) --- always led to different predictions on
choice. According to CPT, a decision-maker multiplicatively combines
the subjectively transformed outcomes and their weighed probabilities
to arrive at an overall value of a gamble. Next, a gamble with a
higher value is chosen. In contrast, the PH posits that a decision
maker relies on a series of sequential steps aimed at comparing
gambles according to their minimum gains, the probabilities of these
minimum gains, and the maximum gains (in fixed order). The sequence of
comparative operations is stopped resulting in a choice when the
difference between minimum (maximum) gains is larger than 10\% of the
minimum (maximum) gain, or if the difference in probabilities of
minimum gains is larger than 10\% of the probability scale (Table 2).

Let us illustrate the difference between CPT and PH in the following
two gambles presented in the first row of Table 1, (Gamble A: 5.40,
0.29; 0, 0.71 and Gamble B: 9.70, 0.17; 0, 0.83). PH predicts that a
decision maker will choose Gamble A because the difference in minimum
gain probabilities is larger than 10\% of the probability scale (i.e.,
0.71 vs. 0.83). In contrast, CPT with standard parameters from Tversky
and Kahneman (1992) predicts that a decision maker will choose Gamble
B because of its greater CPT value (i.e., 1.38 vs. 1.78).

The choice of CPT and PH for assessing the gambles is not meant to
imply that subjects conform to either theory. Rather it was intended
to reflect the difference between simple heuristics and more
compensatory approaches to decision making. In general, we expect CPT
predictions to be the same as EV predictions.

%\renewcommand{\baselinestretch}{1.03} \large\normalsize

\subsection{Procedure}
Subjects were instructed that they should complete the procedure
individually during one session. Subjects also confirmed that they
would focus on the task at hand and turn off other applications/music
that could distract their attention. In reference to the BNT they were
informed that they were not allowed to use calculators, but they could
take notes on paper. 

Tasks were administered with the InquisitWeb (2016) software and ran
in pseudorandom order. Text was displayed in a black font on a light
gray background. All subjects started by reading instructions and
completing a demographic survey. They then completed a block of tasks
including the BNT, RAPM, and choice problems (presented in random
order), followed by another block which included the self-report
measures of SNS and NCS (also presented in random
order). Additionally, twelve low- and twelve high-payoff choice
problems were mixed and presented to each subject in a different
random order.  At the end of the study, subjects declared whether
we could include their data in the analyses. The entire procedure took
approximately 40 minutes (although there were no time
constraints). There was no immediate feedback regarding
performance. However, after the study, subjects received
information on their results in comparison to average scores
calculated from the study sample.

% random order?

%Table 3
\begin{table*}[t]\centering
\caption{Descriptive statistics for measures used in the study. BNT – the Berlin Numeracy Test, NCS – the Need for Cognition Scale, RAPM – the Raven’s Advanced Progressive Matrices, SNS – the Subjective Numeracy Scale. CPT/EV-consistent choices in low- and high-payoff problems refer to the number of expected value choices in these problems.}
\begin{tabular}{rC{.6in}C{.6in}C{.6in}C{.6in}C{1.6in}C{1.6in}}\toprule
 & BNT & SNS & RAPM & NCS & CPT/EV-consistent choices in high-payoff problems & CPT/EV-consistent choices in low-payoff problems \\ \midrule
Mean & 1.75 & 34.87 & 3.71 & 131.9 & 8.8 & 4.41 \\
SEM & 0.11 & 0.67 & 0.13 & 1.66 & 0.27 & 0.27 \\
Median & 2 & 36 & 4 & 133 & 10 & 4 \\
SD & 1.27 & 7.88 & 1.48 & 19.38 & 3.17 & 3.15 \\
Minimum & 0 & 11 & 0 & 75 & 0 & 0 \\
Maximum & 4 & 48 & 6 & 171 & 12 & 11 \\
N & 139 & 139 & 136 & 136 & 138 & 138 \\\bottomrule
\end{tabular}
\end{table*}

%Table 4
\begin{table*}[t]\centering
\caption{Pearson zero-order correlation coefficients for the relationships between measures used in the study. CPT/EV-consistent choices in low- and high-payoff problems refer to the number of expected value choices in these problems. BNT – the Berlin Numeracy Test, NCS – the Need for Cognition Scale, RAPM – the Raven’s Advanced Progressive Matrices, SNS – the Subjective Numeracy Scale. Deliberation time is log-transformed median choice latency.}
\begin{tabular}{l*{8}{d{2.4}}}\toprule
 & \mc{1}{c}{{ ~ }1} & \mc{1}{c}{{ ~ }2} & \mc{1}{c}{{ ~ }3} & \mc{1}{c}{{ ~ }4} & \mc{1}{c}{{ ~ }5} &
 \mc{1}{c}{{ ~ }6} & \mc{1}{c}{{ ~ }7} & \mc{1}{c}{{ ~ }8} \\\midrule
1. BNT & . & .503^{***} & .529^{***} & .214^{*} & .344^{***} & .281^{***} & -.085 & .301^{***} \\
2. SNS &  & . & .372^{***} & .428^{***} & .315^{***} & .196^{*} & .200^{*} & .323^{***} \\
3. RAPM &  &  & . & .238^{**} & .178^{*} & .121 & -.077 & .241^{**} \\
4. NCS &  &  &  & . & .237^{**} & .163 & .182^{*} & .137 \\
\mc{5}{l}{5. Deliberation time in low-payoff problems (log)} & . & .807^{***} & .144 & .519^{***} \\
\mc{6}{l}{6. Deliberation time in high-payoff problems (log)}  & . & -.106 & .389^{***} \\
\mc{6}{l}{7. CPT/EV choices in low-payoff problems}  &  & . & .340^{***} \\
\mc{6}{l}{8. CPT/EV choices in high-payoff problems}  & &  & . \\\bottomrule
\mc{9}{l}{$^* p < .05$, $^{**} p < .01$, $^{***} p < .001$.}
\end{tabular}
\end{table*}

\begin{figure}[t]
\bigskip
\includegraphics[width=\columnwidth]{fig1.eps}\smallskip
\caption{Proportion of choices consistent with expected value (CPT/EV) as
  a function of payoff (high/low, dashed lines are high) and numeracy
  (subjective in dark blue, objective in orange). Subjective numeracy (SNS)
  scores are displayed in quintiles (as equal as possible), so that
  the number of subjects in each point --- represented by the area of
  each point --- are roughly comparable to those for objective
  numeracy.}\smallskip
\end{figure}

% table 5
\begin{table}[t]\centering
\caption{Loadings on canonical variates (pairs of linear combinations
  of X and Y variables), CV1 and CV2.}
\begin{tabular}{ld{2.2}d{2.2}}\toprule
 & \mc{1}{c}{CV1} & \mc{1}{c}{CV2}\\\midrule
\mc{3}{l}{Four X's}\\
BNT & 0.92 & 0.09 \\ 
SNS & 0.46 & 0.88 \\ 
RAPM & 0.76 & 0.08 \\ 
NCS & 0.03 & 0.61 \\\midrule
\mc{3}{l}{Y's with four X's}\\
CPT/EV choices & 0.23 & 0.97 \\ 
Payoff sensitivity & 0.97 & -0.23 \\ \midrule\midrule
\mc{3}{l}{Two X's}\\
BNT & 0.92 & -0.38 \\ 
SNS & 0.78 & 0.62 \\\midrule
\mc{3}{l}{Y's with two X's}\\
CPT/EV choices & 0.61 & 0.79 \\ 
Payoff sensitivity & 0.79 & -0.61 \\ \bottomrule
\end{tabular}
\end{table}

\section{Results}

Descriptive statistics for the measures used in the study and the
correlations between them are presented in Table 3 and Table 4
respectively. Higher scores in the BNT, SNS, and RAPM as well as
longer deliberation time were correlated with more choices consistent
with CPT/EV in high-payoff problems. In low-payoff problems, only
SNS and NCS, but not BNT or RAPM were positively correlated with
CPT/EV-consistent choices. Of interest is the large difference between
objective and subjective numeracy in predicting choices in
low-payoff problems. In high-payoff problems, both objective and
subjective numeracy strongly predicted CPT/EV choices, explaining almost
twice as much variance as RAPM or NCS.

Figure 1 shows the different roles of subjective and objective
numeracy by plotting a proportion of CPT/EV choices as a function of numeracy for high- and 
low-payoff conditions, for both subjective and objective numeracy. Note
that the CPT/EV choice proportion increases with subjective numeracy for both
payoff conditions, but the choice proportion increases with objective
numeracy only for the high-payoff condition. Thus, higher objective
numeracy (BNT) appears to lead to greater sensitivity to payoffs
(greater difference between top and bottom lines in Figure 1).

To ask whether subjective and objective numeracy predict different
aspects of performance, we calculated the mean proportion of
CPT/EV-consistent choices for each subject and the difference
between high- and low-payoff conditions in  CPT/EV-consistent choices, to
assess sensitivity to payoff.

Next, we asked whether these two components, CPT/EV choices and payoff
sensitivity, were differentially affected by measures of individual differences used in the study. 
To do this, we used canonical correlation (fit by the R package yacca: Butts,
2012). The idea is to find a linear combination of the ``X'' variables
(BNT, SNS, RAPM, NCS) that correlate with a linear combination of the
``Y'' variables (CPT/EV choices and payoff sensitivity), optimizing the
weights for the two combinations so as to get the highest
correlation. Then (in essence) repeat the process on the residuals, to
see if a different pattern of weights can account for additional
variance (here we can have only two of these canonical variates
because we have only two Y variables). Table 5 shows the loadings
(analogous to factor loadings) on the two canonical variates.

The top half of Table 5 shows the weights for all four X
variables. The two objective X variables (BNT and RAPM) load mostly on
CV1, which is most strongly related to Payoff-sensitivity, while the
two subjective X variables (SNS and NCS) load on CV2, which is related
to CPT/EV choices. The bottom half shows similar results using only the two
numeracy measures as X variables. For both of these analyses (with four and two X variables), the
second canonical variate was significant ($p = .0024$ and $p = .0004$,
respectively, by Bartlett's $\chi^2$ test), indicating that the second
canonical variate accounted for significant variance. Thus, the
results cannot be explained in terms of a single underlying factor
affecting all X and Y variables.

Finally, we investigated the relationships among payoff, choices and response times. 
Log response times were slower for the low-payoff condition,
presumably because the decision was more difficult, and slower for the
choice of the better option (CPT/EV-consistent choice), presumably because the choice was made
after more thought. The latter effect was greater in the low-payoff
condition.\footnote{These effects were tested with a multi-level model
  with subjects and choices as crossed random effects. All effects,
  including the interaction, had 95\% confidence intervals that
  excluded zero. The use of such a model was necessary because
  subjects differed in the number of choices of the better option in
  the two payoff conditions, so that choice and payoff were not
  orthogonal.} Means (in msec, derived from the mean logs) were: 3631
for high payoff, CPT/EV choice; 3280 for high payoff, PH choice; 6542 for
low payoff, CPT/EV choice; and 3364 for low payoff, PH choice. Mean log
response times increased with both BNT ($r=.32$, $p<.001$) and SNS
($r=.25$, $p=.003$), but the two measures of numeracy did not differ
significantly in any analysis, and, if anything, show effects opposite
to those that might be expected given other results. Likewise, the two
correlations did not differ for just the low-payoff condition ($r=.33$,
$p<.001$ for BNT, $r=.30$, $p=.001$ for SNS; calculated with logs
taken before averaging).

\section{Discussion}

In the present study, we demonstrated that more statistically numerate
decision makers are able to strategically invest sufficient
deliberation to make adaptive choices. When decisions really
mattered and could result in high payoffs, people with high objective
numeracy made normatively better choices predicted by CPT and
EV. However, when decisions were trivial and did not matter because
the payoffs of the available options were similar, those with high
objective numeracy made choices predicted by a heuristic strategy
(i.e., PH). These PH-consistent choices were also “better”, because
they resulted in getting an equally good payoff without wasting
additional time and effort that could be of more value than the negligible
difference in payoff earned by selecting the CPT/EV-consistent option.

At first glance, it may seem counterintuitive that those with high
objective numeracy did not always maximize EV. However, after deeper
consideration, we can conclude that decision makers with high
objective numeracy exhibited adaptive rationality as they tried to
maximize payoffs when choice problems were meaningful and, at the same
time, minimized effort when choice problems were trivial. What is
a potential cognitive mechanism underlying these effects?

Accordingly, our findings are a straightforward demonstration that
more numerate subjects adaptively selected their choice strategy
on more important choices. Nevertheless, it is not clear whether they
performed EV-like computations (i.e., multiplying outcomes and
probabilities) or rather used sophisticated heuristic processing
(i.e., transforming and comparing probabilities and outcomes,
reframing the problem, and so forth). Although the choice problems we
used in our study clearly distinguish between two choice models, the
lack of more sophisticated process-tracing measures (e.g., eye
tracking) does not allow us to conclude that subjects processed
gambles accordingly. For instance, Cokely and Kelley (2009)
demonstrated that choices consistent with EV rarely resulted from EV
computations but they were driven by elaborative heuristic processing
instead. Moreover, Pachur et al. (2013) also demonstrated that using
simple heuristics (e.g., the equiprobable heuristic, the equal-weight
heuristic, or the better-than-average heuristic, Brandstätter et al., 2006) can lead to the same choices as those
predicted by CPT or EV models. Indeed, Pachur et al. (2013)
documented that these trade-off heuristics reveled the same
performance as CPT in problems with both similar and different EV
ratios, but at the same time lead to the opposite predictions to
PH. Therefore, different cognitive processes from those posited by
CPT/EV can be responsible for CPT/EV-consistent choices. This research
problem can be addressed in future studies using process-tracing
measures like think-aloud protocols or Mouselab-type methods in order
to reveal the underlying cognitive process.

\subsection{The role of subjective and objective numeracy}

Apart from a measure of objective numeracy, we also used scales
measuring subjective numeracy as well as need for cognition and fluid
intelligence. These measures were related to choices. In particular,
SNS was associated with more CPT/EV choices in both low and high
payoff problems. NCS correlated with choices in low-payoff problems,
and RAPM was positively related to more choices that maximized EV in
high-payoff problems. While objective numeracy was a marker of adaptive
strategy selection, SNS was a persistent marker of CPT/EV
choices. These results are in line with recent advances in numeracy
theory and they show that objective and subjective numeracy (although
often used interchangeably) map different numerical competencies,
consequently eliciting different implications for decisions (Peters \&
Bjalkebring, 2015). Objective numeracy has been more strongly linked
to number comparisons, operations, and calculations. Subjective
numeracy, on the other hand, has been primarily linked to emotional
reactions to numbers, preference for them as well as motivation and
confidence in numeric tasks (Peters \& Bjalkebring, 2015). This
differentiation allows for qualitatively different predictions for
subjective and objective numeracy that can help understand the role of
these constructs in risky choices.

In light of our findings, it seems that objective numeracy is a marker
of adaptive and meta-cognitive strategy selection related to numerical
processing. Consistent with previous research (Ghazal et al., 2014;
Petrova, Garcia-Retamero, Catena \& van der Pligt, 2016), subjects
with higher objective numeracy spent more time deliberating on
problems, but they were also able to switch to a faster strategy when
choice problems became more trivial (i.e., the differences in payoffs
were small).

Higher subjective numeracy was related to EV maximization processes
irrespectively of the problem payoff. Subjective numeracy may, in a
sense, indicate additional utility from trying to solve a problem. If
so, subjects with high subjective numeracy would be less sensitive to
the lack of extrinsic reward. They may have simply found the task
interesting enough so that they tried to find the best option even
when it was clear that the two options did not differ much in their
expected outcomes.

\subsection{Conclusions}
To summarize, we demonstrated that subjective and objective numeracy play different roles in choices under risk. 
While people with high subjective numeracy tried to maximize every decision irrespectively of its payoff, objectively numerate, skilled decision makers did not waste extra time thinking harder and longer about trivial problems. Importantly, these people were able to assess which problem is meaningful and adapt their choice strategy to maximize payoff. 


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\end{hangparas}

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