Mixing Task Strategies in Economic Decisions: Evidence from Fixed-point Property

UNIVERSITY OF AMSTERDAM

RESEARCH PROJECT II

---Mixing Task Strategies in Economic

Decisions: Evidence from Fixed-point

Property

dhr. dr. L. (Leendert) van Maanen

Date: August 31, 2015 Student ID: 10551395 E-mail: joaquinacouto@gmail.com Research Institute: University of Amsterdam Master’s programme: Brain and Cognitive Sciences

Track: Cognitive Science Number of EC: 42

Abstract

Different strategies of responding may be adopted when facing risky decisions. For instance, decisions can involve complex processes, in which people tend to evaluate all the informa t io n available and make trade-offs among its attributes. In contrast, they can also involve simpler processes, in which people tend to eliminate options and attend to only a subset of the availab le information. Despite the general consensus on the existence of these two modes of responding, there is no evidence, so far, whether these two processes (or strategies of responding) can dynamically coexist and jointly account for individual decisions. The present study has as its main purpose to investigate whether these two processes co-occur in a classical instantia t io n of repeated risky economic decisions.

Introduction

Value-based decision-making occurs whenever a choice is made on the basis of preferences and values. Examples of value-based decisions vary from choosing between different cake flavors at a restaurant to choosing between different financial investments in the stock market. Both examples require the interpretation of preferences/values of a sensory input (e.g. do I prefer chocolate cake or walnut cake? Do I make more money by investing in company stock x or in company stock y?) and their conversion into motor outputs (e.g. order walnut cake in the first case and invest money in company stock x in the second case). Value-based decisions can be divided into three basic computations (Rangel, Camerer, & Montague, 2008; Sugrue, Corrado, & Newsome, 2005) (see Fig.1). First, a representation of the

decision-problem is constructed. At this stage, internal states, external states and potential courses of action are considered by the decision-maker. Imagine someone deciding whether or not to play the lottery. Some possible considerations would be: do I need money (internal states)? Are my friends playing too (external states)? What numbers to bet (courses of action)? Second, a decision is computed. At this stage, the valuation process takes place - values are assigned to the available options/behavioral responses and compared. Whether values are assigned to options or behavioral responses is still an open question (Padoa-Schioppa, 2011; Rangel et al., 2008), but the most important point to keep in mind is that values are considered and compared at this stage. Third, a single behavioral response is selected and executed. Importantly, all decisions involve some degree of uncertainty which can affect the valuation process - the computation and comparison of values. Different types of uncertainty have been mentioned in the literature (Glimcher & Fehr, 2013). Risk and ambiguity are two of them (Levy, Snell, Nelson, Rustichini, & Glimcher, 2010). In risk conditions, the consequences of possible outcomes, although probabilistic, are known. For example, when someone bets on the outcome of the toss of a coin. In ambiguity conditions, however, the consequences of possible outcomes are not known. For example, when someone bets on the outcome of a football game. In this article, we focus on risk conditions, how probability (risk) is incorporated in the assignment of value.

Fig. 1. Basic computations of a value-base d decision. Value-based decisions are made on the basis of

preferences and values. They can be broken into three basic processes: 1. Representation of the decision-problem. 2.Valuation of the different options/actions . 3. Selection and execution of one of the actions. Some

factors (e.g. risk) can affect the valuation of the different options/actions.

Value-based Decisions

Theories of neoclassical economics, also known as value-based (VB) decision theories, have for decades accounted for how the valuation process incorporates probability (risk) in the assignment of value. According to these theories, mathematical calculations are executed by decision-makers. Expected-value (EV) theory (originally developed by Pascal and Fermat in 1654) assumes that, when facing several alternatives (x1, x2, …xk), a

decision-maker would 1) for each alternative (xk), calculate the sum of the product of value and

probability associated with all possible states of the world (si) arising from choosing that

alternative, such that, 𝐸𝑉(𝑥_𝑘) = ∑ p(si) * si; 2) compare the resulting values, for example,

𝐸𝑉(𝑥₁) - 𝐸𝑉(𝑥₂); and 3) select the option with the higher EV (Glimcher & Fehr, 2013). Considering that the decision-maker was facing alternatives 1) “25% chance of getting 8€

(and nothing otherwise)” or 2) “75% chance of getting 4€ (and nothing otherwise)”, he

would calculate “0.25 * 8 + (1 - 0.25) * 0 = 2” for option 1, “0.75 * 4 + (1 - 0.75) * 0 = 3” for option 2, and decide by option 2, once it is the option with the higher EV. Decision-makers employing this calculation are said to be “risk neutral” as they evaluate options by their objective value. Expected-value theory fails, however, in explaining averse attitudes towards risky options by the decision- makers. It cannot explain, for example, why a person would prefer the option with the lower amount but a higher chance of getting the amount (option 1), even when the option with the higher amount but a lower chance (option 2) is the option with higher EV. Expected-utility (EU) theory (Bernoulli, 1954; Neumann &

Morgenstern, 1947), in this respect, advanced with a proper explanation by assuming that people weight values differently in each alternative, depending on its utility, such that, 𝐸𝑈(𝑥_𝑘) = ∑ p(si) * u(si). Considering the same alternatives above, one cannot objectively

identify which option would be selected by the decision-maker as it depends on his utility function. According to this value-based decision theory, a decision-maker would be risk averse, prefer to receive 4€ with a 75% chance than to receive 8€ with a 25% chance, if the utility of receiving 4€ was more than half of the utility of receiving 8€. In contrast, he would

be risk seeking, prefer to receive 8€ with a 25% chance than to receive 4€ with a 75% chance, if the utility of receiving 8€ was more than half of the utility of receiving 4€. Despite some disagreements on how computations are done, value-based decision theories agree upon the assumption that evaluating all the available information (value and probability) and making trade-offs is necessary for rational choice. Those theories are normative in orientation and describe how decision-makers should decide (Kurz-Milcke & Gigerenzer, 2007). Since high values on some attributes (e.g. value) can compensate for low values on others (e.g.

probability), decision-makers are required to confront and resolve conflicts by making trade-offs. This requires a great cognitive effort and involves substantial computational processing of the information (Bettman, Johnson, & Payne, 1991).

Heuristics

Although the assumptions of value-based decision theories have dominated economics since its birth in the eighteenth century, it is still in debate whether humans

actually compute and compare values/utilities. In this respect, there is accumulating evidence showing that decision-makers often rely on simpler ways of deciding, also known as

heuristics. According to this line of research, when facing several alternatives, a decision-maker would 1) select which attribute to evaluate (value or probability); 2) compare the values on the selected attribute associated with each alternative; and 3) select the option with higher value on the selected attribute (Brandstätter, Gigerenzer, & Hertwig, 2006; Pachur, Hertwig, Gigerenzer, & Brandstätter, 2013, but see Birnbaum, 2008). Considering the alternatives above again, if probability was selected by the decision-maker as the attribute to be evaluated, option 2 (“75% chance of getting 4€”) would be chosen over option 1 (“25%

chance of getting 8€”). In contrast, if value was selected by the decision-maker as the

attribute to be evaluated, option 1 (“25% chance of getting 8€”) would be chosen over option 2 (“75% chance of getting 4€”). In general, this way of deciding focus on evaluating only a subset of available information, while ignoring the remaining one. Since high values on non-selected attributes cannot compensate for low values on non-selected attributes, decision-makers do not confront conflicts and ignore trade-offs. This saves a lot of time and cognitive effort (Bettman et al., 1991). Importantly, heuristics were initially considered as a deviation from rational behavior by most literature in psychology and economics. According to this view, humans suffer from cognitive limitations (e.g. selective attention, limited memory) which prevent us from acting rationally and force us to rely on heuristics (Gigerenzer & Gaissmaier, 2011; Kurz-Milcke & Gigerenzer, 2007). Heuristics, in this sense, do not guarantee the best

choice, as value-based decisions which maximize EV/EU, but allow a satisfactory choice (Simon, 1955, 1990). The lack of ability and resources to achieve optimal choices is, consequently, considered to imply judgmental errors (McFadden, 1999). This view has recently been challenged, however, by recent research which has emphasized the efficiency of heuristics. Some studies have shown, for example, that focusing on a subset of information can lead to equal or more accurate judgments in a variety of environments than evaluating the whole information (what researchers named “less-is-more effect”) (Czerlinski, Gigerenzer, & Goldstein, 1999). Altogether, these findings have challenged the neoclassical view of

rationality and have suggested heuristics as a rational way of deciding too (Hertwig & Todd, 2003; Todd & Gigerenzer, 2003).

The Present Study

As a result of previous findings, a new framework of rationality is currently in place (see Fig. 2A) and it goes as follows. Rationality is bounded by both internal (mental) and external (environmental) constraints, and efficient decisions can be made by taking advantage of this fit. Under this assumption, the use of complex calculations (value-based decisions) may be an efficient solution (or ecologically rational) to a specific decision-problem. Nevertheless, simpler processes (heuristics) may also be an efficient solution to another decision-problem (Hertwig & Todd, 2003; Todd & Gigerenzer, 2003). Value-based decisions and heuristics are, thus, mostly considered discrete processes (Evans, 2008). Both processes account for individual decisions, but the use of one or the other process may depend on how effective they are to the decision-problem in question. Empirically speaking, if value-based decisions are an efficient solution to the decision-problem, then, behavioral correlates of this process would be observed. In contrast, if heuristics are an efficient solution to the decision-problem, then, behavioral correlates of this process would be observed instead. In this study, we go further in investigating whether these two processes can dynamically coexist and jointly account for individual decisions (see Fig. 2B), a neglected question so far. We hypothesize that both processes jointly and dynamically account for individual decisions when facing the same decision-problem. Under this assumption, value-based decisions and heuristics are considered as a combination of processes, rather than discrete processes. Empirically speaking, we hypothesize that, once both value-based decisions and heuristics account for individual decisions, observed behavioral correlates are the result of a mixture of these two processes. In order to test this hypothesis, we implemented a task containing a single decision-problem, while manipulating this presumed mixture with different levels of

time pressure. Under our hypothesis, a set of falsifiable predictions at the behavior level, in particular, choices and response times (RT), are made. We predict choice proportion (result of the mixture of processes) to differ depending on our manipulation, and RT distributions (again, result of the mixture of processes) to cross a common point, regardless of the relative proportion of one or another process in the different conditions of manipulation. Those

predictions are described in more detail in the article (see Experimental Predictions under the

Mixture in the Materials and Methods section).

Fig. 2. Current and alternative views of value-base d decisions and heuristics and respective hypotheses.

(A) In the current view, value-based decisions and heuristics are considered as discrete processes. Both processes account for individual choices but the use of one or another depends on how ecological rational they

are to the decision-problem. Behavioral correlates (e.g. choice proportion and RT distributions) correspond, thus, to the process in use. (B) We propose an alternative view where value-based decisions and heuristics are considered as a combination of process. Both processes jointly and dynamically account for individual choices when facing the same decision-problem. Behavioral correlates correspond to a mixture of the two processes (see

red line).

Materials and Methods

Participants

The data reported in this article is from a pilot study including twelve subjects. The subjects were recruited from our laboratory's participant database

(https://www.lab.uva.nl/spt/) and had the opportunity to choose between a course credit (1 ECTS) or a financial compensation (base amount of 8€) with the possibility of getting an extra amount up to 4€, depending on their decisions. The additional money was implemented to ensure a high performance from participants during the task. Informed consents, with a

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general description of the experiment, were signed prior to the start of the task and a debriefing document, explaining in detail what was the true purpose of the experiment, was provided in the end. The experiment and procedures were approved by the ethics review board of the Department of Psychology, University of Amsterdam.

Behavioral Task and Procedure

A modified version of the monetary wheel of fortune (WOF) by Ernst and colleagues (2004) was used in the present study (see Fig. 3A). This is a two-choice decision-making task involving probabilistic monetary outcomes. Participants were presented with different

lotteries at each trial. In each lottery, they had to decide between two options (option 1 vs. option 2) which could result in winning a certain amount of money a (e.g. 4€), with

probability p (e.g.75%) or a higher amount A (e.g. 8€) with a probability 1-p (e.g. 25%). The task illustrates very well the example given in the introduction where a decision should be made between one option with high probability/low amount and other with low

probability/high amount. Importantly, as it comprises a single lottery type, it seems an useful tool to test how decision-makers decide according to value-based decisions and heuristics when facing the same decision-problem. The probabilities were presented as pie charts and the amounts as bars. The colored portions of the pie and bars, blue at the left and yellow at the right, represented the two options at stake. The side where each option was presented (blue at the left and yellow at the right) was randomized. Text regarding probabilities and amounts was also presented.

The experiment involved two phases - a training session and an experimental session (see Fig. 3B). During the training session, participants were instructed to select one of the options by pressing the left or right key, depending on where the color of the portion/bar chosen was located. After the option was selected, a pointer was rotated on the pie chart and the subject could fictitiously win the amount of money paired with the selected portion if the pointer stopped on the selected portion. Otherwise, the subject could not win the amount. Note that the higher the probability of the option chosen by the participant, the higher the probability of the pointer to stop on the selected portion. After the pointer was rotated, explicit feedback was given (e.g. “You win a €!”). In total, participants performed 30 trials in this session in a random order. In the experimental session, participants performed three blocks, each block containing 216 trials in a random order too. Participants were told that the experimental task operated under the same principle of the training task, except that the rotating pointer was not visible on the computer screen. No feedback was provided whether

participants won the amount of money attached to the option selected by them either. The absence of explicit feedback was mainly to avoid that participants modulated their decisions according to it. Participants’ decisions should be modulated solely by our manipulations (see

Experimental Design and Manipulations).

Fig. 3. Behavioral task and procedure. (A) After participants were presented with a fixation cross

point, they were presented with a lottery, where they h ad to choose between an option with high amount/low probability and another with low amount/high probability. Probabilities we re presented as pie charts and the amounts as bars. The colored portions of the pie and bars, blue at the left and yellow at the right, represented the

two options at stake. After participants selected an option, that option was highlighted and the next trial started. (B) All participants performed a training and experimental session. In the experimental session, participants

performed 3 blocks, each block containing 216 trials .

Experimental Design and Manipulations

In order to test whether there is a mixture of value-based decisions and heuristics, the experiment was designed and manipulated as follows. The main goal of the experimental design was to create an environment where the conditions for rationality postulated by both value-based decision and heuristic theories were met. For this purpose, we manipulated the experiment in a way to elicit a variation in the mixture of the two processes (see Fig. 4A). Manipulations occurred between blocks so that in block 1, the proportion of value-based decisions was higher than heuristics, in block 2, the proportion of both processes was relatively the same, and in block 3, the proportion of heuristics was higher than value-based decisions. Note that under this manipulation both processes account for individual decisions, however, in different proportions. Each block corresponded to a different level of time pressure. In block 1, time pressure was low. A slow response was, thus, possible and the use

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of value-based decisions was more likely due to its time-consuming properties. In block 2, time pressure was average. In this case, a trade-off between slow and fast response would take place, and the use of both processes was equally likely. In block 3, time pressure was high. A fast response was, thus, necessary and the use of heuristics was more likely due to its time-saving properties. Time pressure was manipulated by different instructions. In block 1, participants were instructed to be as “thoughtful as possible” in their choices. In block 2, to be as “thoughtful as possible as well as fast as possible”. And finally, in block 3, to be as “fast as possible”.

Experimental Predictions under the Mixture

By recording participants’ choices and RT under the manipulations above, we were able to investigate whether the proportion of value-based decisions and heuristics varied across blocks. In case of variation, a couple of observations were predicted (see Fig. 4B). Regarding participants’ choices, the proportion of safe/risky choices (result of the mixture of processes) was expected to differ, depending on the variation in the mixture. In the context of this experiment, safe choices correspond to when options with high probability/low amount were chosen by participants, and risky choices to when options with low probability/high amount were chosen, instead. If participants were using increasingly more heuristics across blocks and were mostly selecting probability as the attribute to be evaluated (see

Introduction), then, the proportion of safe choices was expected to increase considerably

across the blocks. However, if participants were mostly selecting value as the attribute to be evaluated, then, the proportion of risky choices was expected to increase across the blocks. To test this difference in choice type proportion, statistical analyses were performed by comparing the mean of safe choices of each block (1, 2 and 3) with a repeated-measure analysis of variance (ANOVA). As for RT, RT distributions (also, result of the mixture of processes) were expected to cross a common point, regardless of the relative proportion of one or another process in the different conditions. This last prediction is known as fixed-point property and, although quite precise, it has already been documented by some researchers (Falmagne, 1968; van Maanen, de Jong, & van Rijn, 2014). The fixed-point property is a very elegant way to test a mixture of processes. According to this method, as the probability

density of a binary mixture distribution is always the weighted sum of the densities of the two base distributions (each one corresponding to one process), then, there is one value that has the same density that is independent of the mixture proportions. In terms of RT mixture distributions, this means that there is one RT for which the probability of providing a

response at that particular time is equal for all mixtures. To test the presence of the fixed-point in the data, the continuous density functions were estimated for each condition (block 1, 2 and 3) and statistical tests were computed. For the fixed-point property to hold, the

probability density functions (PDFs) of the RT distributions of the three blocks should cross a common RT. That is, there should be evidence against a difference in crossing-points (e.g. no difference should be found between blocks). Or in other words, there should be evidence in favor of a null-hypothesis that there is no difference between crossing-points. This differs from the standard null-hypothesis significance test assumptions (e.g. the case of ANOVA), which typically quantify support against the null- hypothesis (not in favor, as it is in this case). In order to quantify support in favor of the null-hypothesis we used the Bayesian hypothesis testing (Rouder, Morey, Speckman, & Province, 2012). It quantifies the probability that the observed crossing-points are sampled from one underlying population (when the fixed-point property holds) or are sampled from multiple populations (when the fixed-point property does not hold). The fixed-point property was estimated and tested with the R package provided by van Maanen and colleagues (2014) (which is available on http://www.

leendertvanmaanen.com/fp).

Fig. 4. Experimental design and manipulations. (A)The experiment was manipulated in a way to elicit a

variation in the mixture of value-based decisions and heuristics. Manipulations occurred between blocks so that in block 1, the proportion of value-based decisions was higher than heuristics, in block 2, the proportion of both

processes was relatively the same, and in block 3, the proportion of heuristics was higher than value -based decisions. (B) In case of variation in the mixture, we expected choice proportion (result of the mixture of processes) to differ depending on those variations (e.g. the higher the use of heuris tics, the higher the safe choices) and RT distributions (again, result of the mixture of processes) to cross a common point, regardless of

the relative proportion of one or another process in the different variations (fixed-point property, see dashed line).

Results

Response Time Modulation under Time Pressure

Before analyzing the results of our predictions on participants’ choices and RT, it is important to analyze whether our manipulation of time pressure was successful. If this was

the case, we would observe an effect of time pressure (blocks) on RT, with participants responding increasingly faster across blocks. In this respect, a one-way repeated-measure ANOVA revealed a significant main effect of blocks, F(2, 22) = 11.88, p < .001, and post-hoc comparisons using Bonferroni revealed that RT in block 2 was significantly lower than in block 1 (p < .05), and that RT in block 3 was significantly lower than in block 1 (p < .05) and block 2 (p < .05), respectively (see Fig. 5A). This result is in agreement with what we were expecting and suggests that our manipulation of time pressure was successful. We further analyzed whether our manipulation of time pressure was successful for both types of choice. If this was the case, the decrease in RT observed above would not differ for safe/risky choices. This would be expected because if participants were increasingly using more heuristics across blocks, then, RT would decrease across blocks regardless of which attribute (probability or value) was being evaluated by participants. For this analyses, we performed a two-way repeated-measure ANOVA, with blocks and choice type as factors. In agreement with the previous analysis, there was a significant main effect of blocks, F(2, 22) = 17.40, p < .001. Post-hoc comparisons revealed, similarly, that RT in block 2 was significantly lower than in block 1 (p < .001), and that RT in block 3 was significantly lower than in block 1 (p < .001) and block 2 (p < .05), respectively (see Fig. 5B). Interestingly, there was a significant main effect of choice type, F(1, 11) = 5.18, p < .05, with RT in safe choices (blue lines in Fig.

5B) being significantly lower than in risky choices (red lines in Fig. 5B), overall. This

suggests that although our manipulations of time pressure had an effect on participants’ RT, the type of choice adopted by participants had an effect too. More importantly, though, no significant interaction between blocks and choice type was revealed, F(2, 22) = 1.87, p = .177. This suggests that the decrease in RT (result of time pressure across blocks) did not differ for risky and safe choices, so our manipulation of time pressure was successful for both types of choice. Although this result is in line with what we were expecting, we must agree that this is not very informative on how RT was modulated under our hypothesized mixture of processes nor how RT was potentially modulated across blocks under this mixture. This analysis is purely behavioral. Our hypotheses, however, are made at the process level. Indirect translations/interpretations from behavior to processes may, thus, be inaccurate.

Fig. 5. RT modulation under time pressure. Bars represent RT means (in milliseconds) and standard errors for the three blocks (A), and for the three blocks and choice type (B) – safe choices (0) and risky choices (1). Asterisks represent s ignificant difference (paired-sample t-test, p < 0.05). A one-way repeated-measure ANOVA revealed a significant main effect of blocks. Post-hoc comparisons revealed that RT in block 2 was significantly lower than in block 1, and that RT in block 3 was significantly lower than in block 1 and block 2. A two-way repeated-measure ANOVA revealed a significant main effect of blocks, a significant difference of choice type, but no significant interaction between blocks and choice type. Post-hoc comparisons revealed, similarly, that RT in block 2 was significantly lower than in block 1, and that RT in block 3 was significantly lower than in block 1

and block 2.

Response Time Predicted by Value-based Decisions and Heuristics

In order to explicitly test how participants’ RT was modulated under our hypothesized mixture of processes and how participants’ RT was potentially modulated across blocks under this mixture, we modelled individual choices by dissociating the contribution of the two processes (value-based decisions and heuristics) to RT. To do so, we used a simple regression which can be specified as follows: Regression(RT) = B0 + B1*(abs(P - p)) +

B2*(abs(EV1 - EV2). B0 corresponds to the RT baseline, that is, RT without taking into

consideration any of the heuristic and value-based decision predictions. B1, one of the

parameters of interest, corresponds to the heuristic predictions and models the absolute difference in probability between options. B2, the other parameter of interest, corresponds to

the value-based decision predictions and models the absolute EV difference between options. For analysis purposes, the simple regression was fit in each individual and each block.

Parameters were, then, tested at the population-level. T-tests against zero were performed first to test whether RT was significantly predicted by our parameters of interest. If this was the case, as the mixture of processes hypothesizes, repeated-measure ANOVAs were further performed to test those predictions across blocks. T-tests revealed that neither B1 nor B2 were

significantly different from zero, t(8) = 1.396, p = .200, t(8) = 0.936, p = .376, respectively (see Fig. 6). In disagreement with the mixture of processes hypothesis, this result suggests

that RT was not significantly predicted by heuristic and value-based decision parameters of the model.

Fig. 6. RT predicted by value-base d decisions and heuristics. Bars represent parameter estimates and

standard errors for the three model parameters . B0 models the RT baseline, RT without taking into consideration any of the heuristic and value-based decision predictions . B1 models the absolute probability difference and corresponds to heuristic predictions . B2 models theabsolute EV difference and corresponds to value-based decision predictions. One-sample t-tests revealed that parameters of interest (B1 and B2) were notsignificantly

different from zero.

Choice Modulation under Time Pressure

Our predictions on participants’ choices are analyzed in this subsection. Remember that we were expecting an effect of time pressure (blocks) on participants’ choices (see

Experimental Predictions under the Mixture). To our surprise, though, an one-way repeated-measure ANOVA revealed no significant main effect of blocks, F(2, 22) = 1.37, p = .274 (see

Fig. 7). This result suggests that there was no effect of time pressure on participants’ choices, with participants being consistent on their decision type during the whole experiment.

Although our prediction on participants’ choices was not corroborated, we should be careful in concluding that there was no mixture of value-based decisions and heuristics, and that this mixture did not vary according to our manipulation. Similarly to the analysis on RT

modulation under time pressure, this analysis is purely behavioral. Our hypothesized mixture, however, is made at the process level. The fact that there was no effect of time pressure on participants’ choices does not necessarily mean that there was no effect on participants’ processes/strategies.

Fig. 7. Choice modulation under time pressure. Bars represent proportion of safe choices and standard errors for the three blocks. A one-way repeated-measure ANOVA revealed no significant main effect of blocks.

Choices Predicted by Value-based Decisions and Heuristics

In order to explicitly test how participants’ choices were modulated under our

hypothesized mixture of processes and how participants’ choices were potentially modulated across blocks under this mixture, we modelled again individual choices but by dissociating the contribution of the two processes to left/right choices. To do so, we used a logistic regression: Logistic regression(Choicelr) = B0 + B1*(Pl > Pr) + B2*(EVl - EVr). B0, in this

case, corresponds to an overall bias towards left or right choices, regardless of heuristic and value-based decision predictions. B1 corresponds to heuristic predictions as in the previous

model, but in this case, it models whether the higher probability it is on the left or on the right side. B2 corresponds to value-based decision predictions as in the previous model too, but in

this case, it models the EV difference between left and right options specifically. As before, the logistic regression was fit in each individual and each block. Parameters were, then, tested at the population-level, using t-tests and repeated-measure ANOVAs. T-tests revealed that both B1 and B2 were significantly different from zero, t(8) = 4.038, p < .05 , t(8) = 3.196, p <

.05, respectively (see Fig. 8A). In agreement with the mixture of processes hypothesis, this suggests that left/right choices were significantly predicted by both heuristic and value-based decision parameters. We further performed a two-way repeated-measure ANOVA to test whether left/right choices were differently predicted by parameters over the blocks. This would be expected because if participants were using increasingly more heuristics across blocks, participants’ choices would be increasingly more predicted by B1 while increasingly

less by B2. In this respect, a significant main effect of parameters was revealed, F(2, 32) =

9.99, p < .05, and post-hoc comparisons comparing B1 and B2 (the parameters of interest), in

(see Fig. 8A). No significant effect of blocks was revealed, F(2, 32) = .82, p = .457, however. More importantly, no significant interaction between parameters and blocks was revealed either, F(4, 32) = 1.68, p = .178 (see Fig. 8B). In disagreement with the mixture of processes modulation hypothesis, this result suggests that, left/right choices were not differently

predicted by parameters over the blocks.

Fig. 8. Choices predicted by value-base d decisions and heuristics. Bars represent parameter estimates and

standard errors for the three model parameters (A) and for the three model parameters and three blocks (B). Asterisks represent significant difference (paired-sample t-test, p < 0.05). B0 models the overall bias towards left or right choices , regardless of heuristic and value-based decision predictions. B1 models whether the higher probability it is on the left or on the right and corresponds to heuristic predictions. B2 models the EV difference between left and right options and corresponds to value-based decision predictions. One-sample t-tests revealed that parameters of interest (B1 and B2) were significantly different from zero. A two-way repeated-measure

ANOVA revealed a significant main effect of model parameters, no significant effect of blocks , and no significant interaction between model parameters and blocks . Post-hoc comparisons revealed that B1 predicted

significantly more left/right choices than B2.

The Fixed-point Property

Our predictions on participants’ RT are analyzed in this subsection. Remember that we were expecting RT distributions of the three blocks (each consisting of the mixture of processes) to cross a common fixed-point, regardless of the relative proportion of one or another process in the different blocks. Or putting in another way, pairwise differences of blocks to be equal when the difference is zero (see Fig. 9A and B). This is known as fixed-point property (Falmagne, 1968; van Maanen et al., 2014) and is a very elegant way to test a mixture of processes. Although this a behavioral analysis, as in the case of RT mean and choice proportion, the assumptions behind the fixed-point are at the process level and, particularly, regarding a hypothesized mixture of processes. Interpretations from these results can, thus, be interpreted at the process level, without indirect and uncertain translations. To

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test the presence of the fixed-point, the continuous density functions were estimated for each block, and statistical tests were computed. As referred above, we used the Bayesian

hypothesis testing (Rouder et al., 2012), instead of the standard hypothesis testing (see

Experimental Predictions under the Mixture). It quantifies the probability that the observed

crossing-points are sampled from one underlying population (when the fixed-point property holds) or are sampled from multiple populations (when the fixed-point property does not hold). In this respect, a Bayesian within-subjects ANOVA revealed a Bayes factor of ? in favor of multiple populations of crossing-points. This suggests that the data is _ times more likely to be generated by a model assuming multiple populations than by a model assuming one true population (see Fig. 9C). This is not in agreement with the fixed-point property, and by extension, with the mixture of processes hypothesis.

Fig. 9. Fixed-point property. (A) Averaged density for the three different blocks , (B) density difference curves

for each blocks pair and (C) boxplots for the distributions of crossing points per block pair. A Bayesian within-subjects ANOVA revealed no fixed-point in the data.

Discussion

In the present study, we investigated whether value-based decisions and heuristics can dynamically coexist and jointly account for individual decisions when facing the same

decision-problem. We hypothesized that, once both value-based decisions and heuristics account for individual decisions, observed behavioral correlates are the result of a mixture of these two processes. In order to test this hypothesis, we implemented a task containing a single decision-problem, while manipulating this presumed mixture with different levels of time pressure. The task was a modified version of the monetary wheel of fortune (WOF)

(Ernst et al., 2004) and consisted of a single lottery type (there was always one option with high probability/low amount and other with low probability/high amount). Time pressure was manipulated across blocks. In block 1, time pressure was low so that slow responding was possible and the use of value-based decisions was more likely. In block 2, time pressure was average so that the use of both processes was equally likely. Finally, in block 3, time pressure was high so that fast responding was necessary and the use of heuristics was more likely. In case of variation in the mixture of the two processes, a couple of observations were predicted at the behavioral level, on participants’ choices and RT. On the one hand, the proportion of safe/risky choices (result of the mixture of processes) was expected to differ between blocks. On the other hand, RT distributions (also, result of the mixture of processes) were expected to cross a common point (fixed-point property), regardless of the relative proportion of one or another process in the different blocks (Falmagne, 1968; van Maanen et al., 2014). Before discussing the results regarding these predictions, it is important to discuss the results about RT modulation under time pressure. Results suggest that our manipulation of time pressure successfully modulated participants’ RT, with participants responding faster in the last block (s), when time pressure was higher. Importantly, this decrease in RT did not differ for risky and safe choices, meaning that our manipulation of time pressure was successful for both types of choice. This would be expected under our hypothesized mixture of processes

modulation because if participants were increasingly using more heuristics across blocks (due to a higher time pressure), then, RT would decrease across blocks regardless on which

information (probability or value) participants were attending to. In general terms, this is in line with some studies investigating time pressure in risky decisions, which have identified two strategies of responding to cope with time pressure (Maule, Hockey, & Bdzola, 2000; Payne, Bettman, & Luce, 1996; Zur & Breznitz, 1981). According to these, complex

processes of information processing (e.g. value based decisions) are used in situations where time constraints are low, whereas simpler processes (e.g. heuristics) are used in situations where time constraints are high. Curiously, though, further analysis testing explicitly the contribution of value-based decisions and heuristics to RT suggests that participants’ RT was not successfully predicted by value-based decision nor heuristic parameters. This is not in line with our mixture of processes hypothesis. Regarding our predictions on participants’ choices, results suggest that our manipulation of time pressure did not successfully modulate participants’ choices. Most of participants’ choices corresponded to the safe option and participants were consistent on their decisions throughout the whole experiment. Although this is in line with some studies showing no impact of time pressure on choices (Wegier &

Spaniol, 2015), this is not in line with our hypothesized mixture of processes modulation. Imagine that participants were using increasingly more heuristics across blocks and were mostly selecting probability as the attribute to be evaluated, then, the proportion of safe choices was expected to increase considerably across the blocks (the opposite could also be the case). In this respect, though, there is evidence showing that value-based decisions and heuristics are mathematically equivalent, and consequently, may produce the same choices (Piantadosi & Hayden, 2015). This could very well explain why we did not observe any effect on participants’ choices and why we should be careful in drawing conclusions from this behavioral analysis. With this in mind, further analysis (at the process level) testing explicitly the contribution of value-based decisions and heuristics to choices were carried out. Results suggest that, although participants’ choices were successfully predicted by value-based decision and heuristic parameters, these predictions did not vary significantly across blocks according to our manipulation of time pressure. This is not in line with our

hypothesized mixture of processes modulation. Despite this conclusion, it is relevant to point out that although the results were not significant, they went on the expected direction

(decrease in value-based decisions and increase in heuristics). A lack of statistical power, could, then, be behind the absence of results. As for our predictions on participants’ RT, results suggest that RT distributions of the three blocks (each consisting of the mixture of the two processes) did not cross a common fixed-point. This is not in line with the fixed-point property, and by extension, with the mixture of processes hypothesis.

Limitations

The fact that we did not observe the fixed-point may be interpreted according to a couple of possible explanations (see Fig. 10). Assumptions under the fixed-point are very restrictive and small deviations from these assumptions may lead to the absence of the fixed-point. One of the assumptions of the fixed-point property is that only the RT mixture

distributions should change depending on our manipulations. The two RT base distributions (each one corresponding to one process) that form that mixture should not change, however (van Maanen et al., 2014). One possible explanation for the absence of the fixed-point in the data is, then, that one of the RT base distributions may have changed depending on our manipulation of time pressure. For example, if one of the processes was speeded up (e.g. value-based decisions), then, the RT base distribution of this process would be shifted towards the left (faster RT). As a result, the RT mixture distribution would not be the weighted sum of the same RT base distributions anymore, and consequently, no fixed-point

property would be held in the data (see Fig. 10A). Another example is if one of the processes was non-randomly applied. Participants may have used more heuristics when value-based decisions were more difficult, and consequently, when value-based decisions were longer (tail of RT distribution in value-based decisions). If this was the case, then, the shape of RT base distribution of value-based decisions would be distorted, and it would become a truncated RT base distribution. Again, as a result, the RT mixture distribution would not be the weighted sum of the same RT base distributions anymore, and consequently, no fixed-point property would be held in the data (see Fig. 10B). Another assumption of fixed-fixed-point property is that only binary RT mixture distributions converge to the fixed-point (van Maanen et al., 2014). Another possible explanation for the absence of the fixed-point in the data is, then, that it may have been a mixture of more than two processes (e.g. value-based decisions plus two more heuristics). Again, if this was the case, no fixed-point property would be held in the data (see Fig. 10C). In last resort, a possible explanation for the absence of the fixed-point is that it may have not been a mixture of processes at all (see Fig. 10D). For example, participants may have used the same process (either value-based decisions or heuristics) throughout the whole experiment, regardless of our manipulations of time pressure. Altogether, these hypothetical situations serve to show how difficult it can be for the assumptions of the fixed-point property to be met. If, on the one hand, the assumptions are themselves very restrictive, on the other hand, there are innumerous factors (confounding variables) that may influence the results of the fixed-point. Importantly, the factors in

question may be of subjective nature and result from individual differences. Consider the situation where participants may have used more heuristics when value-based decisions were more difficult. Although choice difficulty can be controlled by controlling for the difference in EV between the two options of the lottery (see Appendix), choice difficulty is a subjective concept and may be estimated differently by participants. For example, a participant may consider a particular lottery difficult, but other participant may not. Future studies should take these subjective factors into account and should be carefully designed and analyzed in order to minimize the effects of these factors.

D C B

Fig. 10. Possible explanations for the absence of the fixed-point. (A) If value-based decisions were speeded

up, its RT base distribution would be shifted towards the left (see dashed RT distribution). (B) If heuristics was non-randomly applied (e.g. more used when value-based decisions were more difficult/longer, see red circle and

row), its RT base distribution shape would be distorted (e.g. truncated RT base distribution, see dashed RT distribution), (C) If there was a mixture of more than two processes (e.g. value-based decisions plus two more heuristics). (D) If there was no mixture of processes at all. In all cases, the assumptions of fixed -point are not me

and its presence is compromised (see red row).

The fact that we did not observe differences in choice proportion and choice

predictions across blocks may be interpreted according to a couple of possible explanations too (see Fig. 11). As referred in the introduction, there are different value-based decision theories. On the one hand, EV theory assumes that decision-makers ought to choose the option that offers the highest EV (Glimcher & Fehr, 2013). The calculation of EV

maximization is very straightforward in the sense that decision-makers evaluate options by their objective value. Consequently, the resulting choice is also very straightforward.

Considering the alternatives in the introduction: “25% chance of getting 8€ or “75% chance

of getting 4€”, a decision-maker would prefer to receive 4€ with a 75% chance than to

receive the 8€ with a 25% chance, because the EV of the option 2 is higher than the EV of the option 1. This preference is said to be risk-neutral and EV maximization only accounts for this neutral attitude towards risk (Glimcher & Fehr, 2013). On the other hand, EU theory assumes that decision-makers ought to choose the option that offers the highest EU, not the highest EV. According to this value-based decision theory, decision- makers do not evaluate options by their objective value, but rather, by their utility (Bernoulli, 1954; Glimcher & Fehr, 2013; Neumann & Morgenstern, 1947). The calculation of EU maximization is, thus, not as straightforward as the calculation of EV maximization and the resulting choice may differ, depending on the utility that a value has for the decision- maker. Considering the same alternatives above, a decision-maker may prefer to receive 4€ with a 75% chance than to receive 8€ with a 25% chance, if the utility of receiving 4€ was more than half of the utility of receiving 8€. However, other decision-maker may refer to receive 8€ with a 25% chance than to receive 4€ with a 75% chance, if the utility of receiving 8€ was more than half of the utility of receiving 4€. These preferences are said to be risk-averse and risk-seeking,

respectively, and EU maximization accounts for both aversive and seeking attitudes towards risk (Glimcher & Fehr, 2013). As for the heuristics, it is assumed that decision-makers often rely on simpler ways of deciding, for example, by focusing on evaluating only a subset of available information (Brandstätter et al., 2006; Gigerenzer & Gaissmaier, 2011; Pachur et al., 2013). Considering the alternatives above again, a decision-maker would choose option 2

(“75% chance of getting 4€”) over option 1 (“25% chance of getting 8€”) if probability was the attribute selected by the decision-maker to be evaluated. In contrast, a decision-maker would choose option 1 (“25% chance of getting 8€”) over option 2 (“75% chance of getting 4€”) if value was the attribute selected. Similarly to the EU maximization, the resulting choice may differ, depending on the heuristic employed (e.g. the attribute selected) by the decision-maker at the moment. The fact that both EU maximization and heuristics predict the same behavior (although under different processes) (see also Piantadosi & Hayden, 2015) is problematic in the context of our experiment and it may be one possible explanation why we did not observe differences in choice proportion. Participants may have chosen safe options throughout the whole experiment because they were being too risk-averse (EU maximization) and/or because they were simply choosing the option with the higher probability (heuristics). In addition, the model used in the study to test explicitly the contribution of value-based decisions and heuristics to choices consisted of an EV maximization rather than an EU maximization. The fact that EV maximization does not account for individual risk

preferences (as opposed to EU maximization) is also problematic and it may be one possible explanation why we did not observe differences in choice predictions across blocks. Choices predictions may have not differed significantly because one of the parameters of the model (value-based decision parameter) was inaccurate, and consequently, the model predictions across blocks too. Importantly, this may also be the case for RT predictions. In any case (at the behavior and process level), we were not able to accurately disentangle whether

participants switch between values-based decisions and heuristics across blocks. Future studies are necessary to fully investigate this question, and importantly, these studies should consider participants’ risk preferences (see also Krajbich, Bartling, Hare, & Fehr, 2015).

A

Fig. 11. Comparison between free model analyses/model analysis used in the study and ideal free model analyses/model analysis to test the mixture of value-based decisions and heuristics. (A) Analyses of choice

proportion and choice predictions were done according to the EV theory which does not account for individual risk preferences. Not considering individual ris k differences, does not allow to fully investigate whether participants were deciding according to the EU theory (e.g. being too risk-averse) and/or deciding according to

the heuristics theory (e.g. choosing the higher probability). (B) Ideal free-model and model analysis should be done according to the EU theory which accoun t for these individual risk preferences.

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Appendix

Stimulus Generation

Modulations in participants’ choices and RT across blocks should be purely due to the variation in the mixture of processes, not due to a variation in choice difficulty. In order to ensure this, a couple of precautions were taken when building the stimulus set. There are two main factors which may influence choice difficulty. One of them is the difference in expected value (EV) between the two options of the lottery. The higher the EV difference, the easier the choice. In contrast, the lower the EV difference, the more difficult the choice. Consider the alternatives presented in the introduction and the calculations carried out by decision-makers according to EV theory. In this case, it would be relatively easy for a person to

choose option 2 because the EV difference of the two alternatives is 1 (remember that EV(x1)

= 0.25 * 8 + (1 - 0.25) * 0 = 2 and EV(x2) = 0.75 * 4 + (1 - 0.75) * 0 = 3). Consider now,

however, the following alternatives: “25% chance of getting 6€ or “75% chance of getting

2€”. In this case, it would be very difficult for a person to select one of the alternatives

because the EV difference of the two is 0 (EV(x1) = 0.25 * 6 + (1- 0.75) * 0 = 1.5 and

EV(x2) = 0.75 * 2 + (1 - 0.75) * 0 = 1.5). The other factor which may influence choice

difficulty is the familiarity that a participant has with the options of the lottery. The higher the familiarity, the easier the choice. On contrary, the lower the familiarity, the more difficult the choice. Imagine someone facing alternatives that were presented before. As the EV of both options was previously calculated and compared, it would be relatively easy to him to make a choice. Imagine now someone facing alternatives that were not presented before. As the EV of both options was not previously calculated and compared, it would be more difficult for him to make a choice. In order to control for EV difference and familiarity (and consequently for choice difficulty), the stimulus set was generated so that all lotteries would be unique (an unique set of lotteries was generated per block) and that all lotteries would have similar EVs (fixed levels of EV difference were set). The mathematical derivation of the stimulus set went as follows:

In each lottery, there was always one option with high probability/low amount (p*a) and other with low probability/high amount (1-p*A), where:

a) 0.5 < p < 1, b) a > 0, c) A > 0, d) a < A.

In order to generate lotteries with a fixed EV difference (∆EV), we fixed ∆EV, a and

p, an almost full- factorial design (∆EV * p * a), and deduced A.

1) ∆EV = (1 - p) * A – p * a. An easy way to deduce A was if: e) A = a + d, and

f) d > 0.

In this way, conditions b), e), and f) satisfied conditions c) and d). We could write, then, that:

2) ∆EV = (1 - p) * (a + d) – p * a, so that, 3) d = (∆EV + a * (2p - 1)) / (1-p) > 0, and 4) a > - ∆EV / (2p - 1).

If ∆EV > 0, 4) was already implied by condition b), so there was no extra constraint on a. However, if ∆EV < 0, 4) was not implied by condition b), so an extra constraint here was that a > - ∆EV / (2.* min(p) - 1).

With the conditions and equations above, we were able to choose a, and to compute A for a lottery with specific ∆EV and p:

5) A = a + (∆EV + a * (2p - 1)) / (1-p)

Importantly, while ∆EV and p were fixed across the experiment, a, and consequently