Master Thesis:
Formal Models of Decision Making: Integrative versus Sequential Decision Rules
Maria C. Olthof (10645772)
Research Master Psychology Major: Psychological Methods Supervisor: prof. dr. H.M. (Hilde) Huizenga Second assessor: prof. dr. M.E.J. (Maartje) Raijmakers
Developmental Psychology University of Amsterdam
Abstract
The aim of this study was to determine what different decision rules are used by children and
adolescents when faced with alternatives that differ on several dimensions (e.g. certain gain, frequency of loss), and whether children and adolescents evaluate values and probabilities objectively or
subjectively. Jansen, Van Duijvenvoorde and Huizenga (2012) found that six different decision rules are used. Some children use integrative decision rules (i.e. they integrate all dimensions for each alternative separately, before they compare alternatives), whereas other children use different kind of sequential rules (i.e. they compare alternatives on each dimension separately, and base their decision on the first dimension that distinguishes alternatives). To assess the specific rules used, Jansen et al. used deterministic models and only considered objective evaluation of probabilities and values. We reanalyzed data of Jansen et al., using Cumulative Prospect Theory (Tversky & Kahnemann, 1992) and Subjective Lexicographic Theory (Huizenga, Dekkers, Van Duijvenvoorde, Figner, & Jansen, in preparation), which are probabilistic models that allow for subjective evaluation of attributes. We found that children and adolescents did not use different decision rules, but integrated dimensions. Furthermore, they did evaluate values and probabilities subjectively. Use of the most objective integrative rule increased with age. We discuss results in light of several strengths and limitations of the current study.
Formal Models of Decision Making: Integrative versus Sequential Decision Rules
Decisions have to be made every day by children and adolescents; some are not important, but others are (e.g. the decision for a high school program). Often some degree of risk is involved. Alternatives may differ in sure positive outcomes, as well as in risks on negative outcomes. For example: Jane wants a summer job and got two offers: with job A she will earn $400, and with job B she will earn $600. As she is too young for either of these jobs she would have to take the risk of paying a penalty of $50 for job A and $400 for job B in case her age is discovered. She is told that this happens to 2 out of 10 adolescents in case of job A, and to 3 out of 10 adolescents in case of job B. Which job will Jane take, and more generally, how do children and adolescents decide in such situations? What decision rules do they use? Do they rationally compare the expected values (EVs)? Here that would mean that job B seems more attractive (EV of B: 600 − .3 ∙ 400 = 480 versus A: 400 − .2 ∙ 50 = 390). Or do they consider one aspect (‘dimension’) first, e.g. the risk of being punished, and base their decision on that aspect only? This would lead to a preference for job A.
The aim of this study is to determine what decision rules children and adolescents use when faced with alternatives that differ on several dimensions, and that involve risk. In general, we can distinguish 1) integrative and 2) sequential decision rules. People who use an integrative rule combine information of all dimensions, before comparing alternatives. People who use a sequential rule consider dimensions sequentially, and compare alternatives per dimension.
The most simple integrative rule is to select the alternative with the highest EV. That is, to multiply (‘integrate’) the possible gains or losses with their associated probabilities, to sum these per alternative, and compare alternatives on the resulting values. This is an example of an alternative based decision rule: each alternative is evaluated separately, before comparison with other
alternatives. It is very accurate, but also time- and effort consuming. Faster is an attribute-based rule, where alternatives are compared per dimension, and a decision is made before all dimensions are evaluated.
Theory of lexicographic decision making (Luce, 1987; Tversky, Sattath & Slovic, 1988)
dimension, and only evaluate the next dimension when the first does not sufficiently distinguish alternatives. When it does, the alternative is chosen that is most favorable on that first dimension. Sequential rules are mostly less accurate than integrative rules, but this is compensated by a large difference in effort (Payne, Bettman, & Johnson, 1988).
Empirical evidence has been found for both integrative and sequential decision rules in another context: that of the balance scale task (Siegler, Strauss & Levin, 1981). For this task, weights are placed at certain distances from the middle of a balance scale, and children have to predict whether, and to which side, it will move. Some children only consider weight (one-dimensional sequential rule), others consider distance as well, but only when there is no difference in weight
(two-dimensional sequential rule). And some do combine both dimensions (integrative rule) (Jansen & Van der Maas, 2002).
The integrative and sequential rules described above are somewhat simplistic. They assume that people evaluate probabilities and values objectively, whereas earlier research suggests that people are likely to evaluate probabilities and values subjectively. One example is ‘loss aversion’, which is the phenomenon that people give more weight to the prospect of a loss of a given amount than to the prospect of a gain of the same amount (Tversky & Kahneman, 1991). Other examples are the underestimation of high probabilities, and the underweighting of high absolute values. These effects might be different for gains than for losses (Trepel, Fox, & Poldrack, 2005).
Models that take into account subjective evaluation of attributes might describe decision behavior more accurately. Cumulative Prospect Theory (CPT, Tversky & Kahneman, 1992) describes integrative rule use, and allows for subjective evaluation of attributes. It assumes that people first edit alternatives, for example by omitting information, then subjectively evaluate all attributes per
alternative, and finally calculate for each alternative a subjective utility (by multiplying the subjective probability and subjective value). The alternative with the highest subjective utility is chosen.
A theory that combines sequential rules of Theory of lexicographic decision making with subjective evaluation of attributes, is Subjective Lexicographic Theory (SLT; Huizenga, Dekkers, Van Duijvenvoorde, Figner, & Jansen, in preparation). In this theory the subjective evaluation formulas from CPT are used.
As this study focuses on children and adolescents, one might wonder how subjective
interpretation, and the use of different decision rules (integrative/rational versus sequential/heuristic) relates to age. There is not much literature on this topic, and results are inconclusive. But, what has been found is that most biases found in adult thinking have also been found in children’s and
adolescents’ thinking. Also, there seems to be a general trend to more rational thinking with increasing age (Stanovich, Toplak & West, 2008).
The question remains how to test which decision rule someone uses. One way to estimate this is with the ‘Gambling Machine Task’ (GMT; Jansen, Van Duijvenvoorde & Huizenga, 2012). Based on the Iowa Gambling Task (Bechara, Damasio, Damasio, & Anderson, 1994), the GMT is constructed in such a way that the use of different decision rules leads to unique answer patterns. In the Jansen et al. study, items (‘choose the most favorable of two alternatives’) were answered by 231 children and adolescents. Data were not averaged, but sorted on answer pattern by a latent class analysis, because different people may use different rules (Bröder, 2002). Six different answer patterns emerged, which indicated that there were six groups of children, each using a different decision rule. In order to find out which decision rules, Jansen et al. formulated 17 theoretical decision rules: one integrative, 15 sequential and a guessing rule. For each rule, an expected answer pattern on the GMT items was formulated in a deterministic way. That is, children were assumed to use a rule consistently; each rule matched perfectly with one theoretical answer pattern. Moreover, no subjective evaluation of attributes was assumed. To relate the six answer patterns to their most likely decision rule, the observed answer patterns were compared with the 17 theoretical answer patterns. The answer pattern of one group matched that of the integrative rule, that of another group matched that of a ‘guessing’ pattern, and those of four groups matched answer patterns of sequential rules: three three-dimensional and one two-dimensional sequential rule (for the exact rules and the group sizes, see Table 4, column 2).
Jansen et al. (2012) also investigated the relation between age and rule use. Multinomial logistic regression analyses, with rule use as dependent variable and age as predictor, showed that the use of the integrative rule as well as the use of the two-dimensional rule decreased with age, whereas the use of a three-dimensional rule increased with age.
Although Jansen et al. (2012) used a strong design to discriminate the use of different rules, the labeling of the rules used by each of the six groups, required the assumption of deterministic choice and objective evaluation. With respect to deterministic choice, children are unlikely to show perfectly consistent decision behavior. Probabilistic instead of deterministic models may be more realistic (Bröder, 2002). As discussed before, the assumption that children evaluate probabilities and values objectively, might also be unrealistic.
The aim of the current study was to find out what decision rules children and adolescents use. As in Jansen et al. (2012), we assumed that children stick to one decision rule during the task. By fitting the more sophisticated and realistic CPT and SLT models to the same six answer patterns that empirically emerged in Jansen et al., we again labeled the decision rule of each group. In this way we investigated whether children and adolescents use integrative or sequential decision rules, and whether they treat attributes objectively or subjectively. We expected to replicate the results of Jansen et al.: 1) The group of children/adolescents that seemed to use an integrative rule in Jansen et al. was expected to use an integrative rule; and 2) The four groups of children/adolescents that seemed to use different sequential rules, were expected to use different sequential rules. Moreover, we expected that all groups evaluated values and probabilities subjectively instead of objectively. Finally, we explored how subjective interpretation, and rule use, relate to age.
Method
We reanalyzed the data from Jansen et al. (2012), therefore we first briefly describe their participants and their task.
Participants
Participants were 231 children and adolescents, from the highest grades of elementary school and grades 2 and 4 of secondary school. Ages ranged from 8 to 17 years. Parents gave permission, and both the study of Jansen et al. (2012) and the present study were approved by the local Ethics Committee. For more details see Jansen et al..
Materials
A paper-and-pencil test called the Gambling Machine Task (GMT) was used. This test was developed by Jansen et al. (2012) and allows to distinguish between different decision rules. An example of an
item is presented in Figure 1. On each item, two machines were shown that differ on at least one of the dimensions 1) certain gain (CG), 2) frequency of loss (FL), and 3) amount of loss (AL). Participants had to decide whether machine A or machine B was most profitable. They could also decide that both were equally profitable. CG refers to the points that one will receive for sure when that machine is chosen. FL is the frequency that one will lose a given amount of points, and AL is the corresponding number of points. There are seven item types (see Table 1). In three item types, the alternatives differ on only one dimension, in three other item types they differ on two dimensions, and in another item type they differ on all three dimensions. Of each type there are two versions that differ in the specific values, but not in the structure of the item type (in Table 1 both versions are given at each row). Moreover, of each version there is one replicate included in the test. In those replicates only the order of alternatives is changed. This resulted in 7∙2∙2=28 items. The item types were constructed in such a way that the use of different decision rules (objective integrative or objective sequential) would lead to different response patterns. For example in case of the item type ‘FL_cg’, where the alternatives differ on FL and on CG, the use of an one-dimensional rule that focuses on FL would lead to a correct the answer (i.e. the alternative with the highest EV), whereas the use of an one-dimensional rule that focuses on CG would lead to an incorrect answer. This corresponds to the name of the item type where FL is printed in uppercase, and cg is printed in lowercase.
In the present study we scaled the values CG and AL, so that -50 became -1, and +50 became 1. FL values were scaled as probabilities.
Figure 1. Gambling Machine Task-item (item type ‘FL_cg’) where alternatives differ on dimensions frequency of loss and certain gain. When one focuses on frequency of loss only, one would select the correct answer (which has the highest EV), which is ‘Machine A’. Figure taken from Jansen et al. (2012).
Table 1
Item characteristics of the GMT; table from Jansen et al. (2012);
Blocks 1 and 3 Blocks 2 and 4
FL (%) AL CG FL (%) AL CG Item type A B A B A B A B A B A B FL 10 50 -2 -2 2 2 50 10 -50 -50 4 4 AL 10 10 -10 -2 4 4 50 50 -50 -2 2 2 CG 10 10 -50 -50 4 2 50 50 -10 -10 4 2 FL_cg 10 50 -10 -10 2 4 50 10 -50 -50 4 2 AL_cg 10 10 -10 -50 2 4 50 50 -10 -2 4 2 fl_al 50 10 -2 -10 4 4 50 10 -10 -50 2 2 Al,CG_fl 50 10 -2 -10 4 2 50 10 -10 -50 4 2
Note. Blocks 3 and 4 contain exact replicates of block 1 and 2 (except that alternative A and B are interchanged); FL=frequency of loss, AL=amount of loss; CG=certain gain
Data analysis
To investigate whether children and adolescents use integrative or sequential decision rules, we fitted CPT and SLT models to the response pattern of each of the latent groups found in Jansen et al. (2012). We did this twice, once assuming objective evaluation of values and probabilities, and once assuming subjective evaluation of values and probabilities. As a result there are four ‘model-variants’:
‘objective-CPT’, ‘objective-SLT’, ‘subjective-CPT’, ‘subjective-SLT’. We expected that the
objective-CPT model-variant fits better than the objective-SLT model-variant to the response pattern of the group that seemed to use an integrative rule in Jansen et al., and also that the subjective-CPT model-variant fits better than the subjective-SLT model-variant to that response pattern. Similarly, we expected a reverse pattern for the four groups that seemed to use a sequential rule in Jansen et al.. To test the third hypothesis, whether children/adolescents treat attributes objectively or subjectively, the objective-CPT was compared with the subjective-CPT, and the objective-SLT with the subjective-SLT model-variant. It was expected that the subjective model-variants would fit better to the response patterns than their objective counterparts.
Each of the four model-variants was fitted to each of the six response patterns. However, each model-variant has many parameters and of each model-variant we wanted to reduce the number of unnecessary parameters, so that the resulting ‘optimal solution’ for that specific model-variant not only fits best to the response pattern of a specific group, but also is parsimonious. Therefore within a model-variant, each parameter could be either ‘on’ or ‘off’, i.e. freely estimated or fixed. Many combinations (‘parametrizations’) of fixed and free parameters were constructed, and each was fitted to each response pattern. For example, in case of model-variant ‘objective-SLT’ which has 8
parameters, 28=256 parametrizations were fitted to each response pattern. Selection of the best parametrization for that model-variant, and for that group, was based on BIC-values, as this indicator not only rewards good fit, but also rewards sparseness of models (Burnham & Anderson, 2004).
Below, both CPT and SLT are described in detail, followed by more details on the fit procedure. Cumulative Prospect Theory
For the formulas of the CPT, see equations 1 to 5 (adapted from Huizenga et al., in
refers to the value weighting function (Tversky & Kahneman, 1992). The resulting subjective FL and subjective AL are multiplied and subtracted from the subjective CG for each alternative. This results in a subjective utility (𝑠𝑠𝑠𝑠). Equation 5 shows how the difference in subjective utility of alternatives (A and B) influences the probability on choosing option A (Stott, 2006). Parameter 𝜙𝜙 indicates to what degree the decision for option A is influenced by the difference in subjective utility.
𝑠𝑠𝑠𝑠 = 𝑣𝑣(𝐶𝐶𝐶𝐶) − 𝑤𝑤(𝐹𝐹𝐹𝐹) ∙ 𝑣𝑣(𝐴𝐴𝐹𝐹) (1) 𝑣𝑣(𝐶𝐶𝐶𝐶) = [𝐶𝐶𝐶𝐶]𝛼𝛼𝑔𝑔 (2) 𝑤𝑤(𝐹𝐹𝐹𝐹) = 𝛿𝛿[𝐹𝐹𝐹𝐹]𝛾𝛾 𝛿𝛿[𝐹𝐹𝐹𝐹]𝛾𝛾+(1−[𝐹𝐹𝐹𝐹])𝛾𝛾 (3) 𝑣𝑣(𝐴𝐴𝐹𝐹) = −𝜆𝜆(−[𝐴𝐴𝐹𝐹])𝛼𝛼𝑙𝑙 (4) 𝑝𝑝(𝐴𝐴) = 1 1+exp{−𝜙𝜙[𝑠𝑠𝑠𝑠(𝐴𝐴)−𝑠𝑠𝑠𝑠(𝐵𝐵)]} (5)
The weighting functions contain four different subjectivity parameters. A value weighting parameter 𝛼𝛼 that is smaller than 1 indicates underweighting of high absolute values (see figure 2). As this might be different for gains than for losses, we estimated two 𝛼𝛼 parameters, one for gains (𝛼𝛼𝑔𝑔), and one for losses (𝛼𝛼𝑙𝑙). To test whether these are different from each other, we constrained these two 𝛼𝛼 parameters to be equal in half of the parametrizations. Evaluation of gain and amount of loss also depends on loss aversion parameter 𝜆𝜆, with 𝜆𝜆 larger than 1 indicating overweighting of losses compared to gains. Parameter 𝛿𝛿 indicates to what degree people have a general tendency to over- or underestimate probabilities, with 𝛿𝛿 smaller than 1 indicating underweighting and larger than 1 indicating
overweighting of probabilities (see figure 3). When this concerns loss probabilities, 𝛿𝛿 smaller than 1 indicate a risk averse tendency and 𝛿𝛿 larger than 1 indicates a risk seeking tendency, and reverse when this concerns gain probabilities. Evaluation of probabilities also depends on probability weighting
parameter 𝛾𝛾, where a 𝛾𝛾 smaller than 1 indicates overweighting of low and underweighting of high probabilities (Trepel, Fox & Poldrack, 2005). In the GMT, there was no variation in gain probability, therefore, in the present study, 𝛿𝛿 and 𝛾𝛾 refer to subjective evaluation of loss probabilities only.
Mind that when all subjectivity parameters are constrained to 1, we are assuming objective evaluation, only parameter 𝜙𝜙 is left, and the subjective utility of each alternative is simply its EV. The only difference with the integrative rule used in Jansen et al. (2012) in that case, is that we used a probabilistic model (equation 5), whereas they used a deterministic model. When we assume subjective evaluation, the model has six parameters.
Figure 2. Value weighting parameter 𝛼𝛼 and loss aversion parameter 𝜆𝜆 indicate how gain and amount of loss are evaluated; here 𝛼𝛼𝑔𝑔 and 𝛼𝛼𝑙𝑙 are both equal to 𝛼𝛼
Figure 3. Parameter 𝛿𝛿 and probability weighting parameter 𝛾𝛾 indicate how probabilities are subjectively evaluated
Subjective Lexicographic Theory
SLT was formulated by Huizenga et al. (in preparation) for a different context, with two dimensions, one continuous and one dichotomous. We formulated a SLT model for the GMT, with three continuous dimensions (see equations 6 to 10). Here the probability of selecting alternative A, depends on the difference between FL probabilities (diffFL), on the difference between CG values (diffCG), on the difference between AL values (diffAL), and on their interactions. These FL, CG and AL values can be interpreted objectively or subjectively. Subjective values and probabilities are calculated using the same weighting functions as in CPT (equations 2-4). When we assumed objective
evaluation, then all subjectivity parameters were constrained to 1, and seven 𝛽𝛽-parameters were to be estimated (three ‘diff’ main effects, one intercept, and three interactions). When we assumed
subjectivity, then the subjectivity parameters were estimated as well, resulting in eleven parameters (four subjectivity and seven 𝛽𝛽-parameters). Because of the large amount of parameters compared to the amount of items in the dataset, we did not estimate separate 𝛼𝛼-parameters for gains and losses. Also because of the size of the model, and the corresponding large computation times, we had to reduce the number of parametrizations, by either fixating all four subjectivity parameters or freely estimating all four subjectivity parameters within a parametrization. This constraint reduced the number of parametrizations from 211=2048 to 256.
𝑝𝑝(𝐴𝐴) = 1 1+exp {𝑟𝑟𝑟𝑟𝑔𝑔𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑙𝑙} (6) 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = − �𝛽𝛽 𝛽𝛽0+ 𝛽𝛽1∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹+ 𝛽𝛽2∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐶𝐶𝐶𝐶+ 𝛽𝛽3∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴𝐹𝐹+ 4∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐶𝐶𝐶𝐶+ 𝛽𝛽5∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴𝐹𝐹+ 𝛽𝛽6∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐶𝐶𝐶𝐶∙ 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴𝐹𝐹� (7) 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹 =𝑤𝑤[𝐹𝐹𝐹𝐹](𝐴𝐴)− 𝑤𝑤[𝐹𝐹𝐹𝐹](𝐵𝐵) (8) 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐶𝐶𝐶𝐶 =𝑣𝑣[𝐶𝐶𝐶𝐶](𝐴𝐴)− 𝑣𝑣[𝐶𝐶𝐶𝐶](𝐵𝐵) (9) 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴𝐹𝐹=𝑣𝑣[𝐴𝐴𝐹𝐹](𝐴𝐴)− 𝑣𝑣[𝐴𝐴𝐹𝐹](𝐵𝐵) (10)
Part of the function is based on a regression model (equation 7) with an intercept, a main effect for each of the dimensions, and three two-way interactions of the combinations of dimensions.
Interpretation of the 𝛽𝛽-parameters is not straightforward, because they refer to effects of differences between alternatives. A significant intercept 𝛽𝛽0 would indicate a general preference for one alternative, regardless of the differences between the alternatives. A significant main effect of 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐹𝐹𝐹𝐹 would mean that decisions depend on a difference in FL. Similarly, main effects of 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐶𝐶𝐶𝐶 and 𝑟𝑟𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴𝐹𝐹 would indicate that decisions depend on differences in CG and in AL respectively.
A significant interaction term (e.g. 𝛽𝛽5) would indicate that the effect of the difference in one dimension (AL) depends on the difference in another dimension (FL). This could occur when someone only takes the difference in AL into account when the difference in FL is small. This would match a two-dimensional decision rule (i.e. the decision is based on a second dimension when the first dimension does not substantially distinguish alternatives).
The presence of two significant interaction terms, would mean that the evaluation of one dimension depends on differences in both other dimensions. For example, when the difference in CG is only decisive if both the difference in FL is small and the difference in AL is small. This is
congruent with a three-dimensional rule, where the decision maker only takes the 3rd dimension into account when the other two dimensions give inconclusive information.
Fit procedure
The CPT and SLT functions above give the probability of selecting alternative A. Participants could also select alternative B, or they could state that both were equally profitable. This third option was not often selected, therefore, whenever this option was chosen, we replaced this answer randomly by a choice for alternative ‘A’ or ‘B’. When 𝑝𝑝(𝐴𝐴) is larger than .5, we expected participants to select option A and when it is smaller than .5, we expected them to select option B.
We fitted models to the response pattern of each latent group separately. We fitted the CPT and the SLT model with each parametrization, with the observed response pattern as data. For each parametrization, parameter values are optimized by minimizing the negative maximum likelihood function, using the R optimization function ‘auglag’ (Alabama package; Varadhan, 2015). For each parametrization this was done 500 times with different starting values to prevent local minima. For the
estimation of parameter values, a range was specified with lower and upper bounds. Whenever a parameter was estimated at or very close to its upper- or lowerbound within the 5 best
parametrizations for that group and model-variant, the bounds were adapted and the whole set of parameterizations was fitted again. When after two new trials, no parameters were estimated at boundaries within the three best parametrizations, we did not fit models again, and used the results of the optimal parametrization (for the exact bounds used, see Appendix).
So, for each group, one optimal parametrization (‘optimal solution’) per model-variant was selected that explained the response pattern best. This procedure was repeated for the response patterns of each of the six latent groups. This resulted in four optimal model-variant outcomes for each latent group. By comparing the optimal solution of the objective-CPT model-variant with that of the objective-SLT model-variant and by comparing the optimal solution of the subjective-CPT with the optimal solution of the subjective-SLT model-variant, we could evaluate whether that group used an integrative or a sequential rule. And by comparing the objective-CPT with the subjective-CPT model and the objective-SLT with the subjective-SLT model, we could conclude whether a group evaluates attributes objectively or subjectively. Moreover, a comparison of all four BIC-values leads to one final best model-variant per group. Details about the specific subjective evaluation used, and about the specific sequential rules used, can be derived from the subjectivity parameter estimates and the beta parameter estimates respectively.
To improve the speed and the accuracy of the optimization procedure, we calculated 1st order gradients of both the CPT and the SLT maximum likelihood functions, and Jacobian matrices of constraints used, and added these as input to the ‘auglag’ function.
Results
For most groups and model-variants, a few parametrizations fitted almost as good as the optimal solution; on average 2.5 and 6 parametrizations had a BIC closer than 2 points from the optimal solution for the SLT and CPT model respectively.
The first hypothesis stated that group A would use an integrative rule and group B, C, D, and E would use different sequential rules, as was found by Jansen et al. (2012). This was tested twice: when assuming objectivity, and when assuming subjectivity.
‘Objective’ CPT versus ‘objective’ SLT
Considering the ‘objective’ model-variants, i.e. when possible subjectivity is ignored, the lower BIC values of SLT compared to CPT indicate that all groups seem to use a sequential rule (Table 2, column 2 and 3), including group A (that used an integrative rule according to the results of Jansen et al., (2012)) and group F (that guessed according to Jansen et al.).
The parameter estimates for these optimal solutions (Table 3) indicate the type of sequential rules used. According to Jansen et al. (2012), group B, C, D, and E used either a two- or a three-dimensional rule. The absence of estimated intercepts indicates that the children/adolescents did not have a general preference for one alternative, independent of the characteristics of the alternatives. The negative 𝛽𝛽1estimates indicate that all groups evaluate FL to some extent, and have a preference for the option with the lowest FL (when the difference FLA-FLB is negative). Similarly, the positive 𝛽𝛽2 and 𝛽𝛽3estimates indicate that all groups do evaluate AL and CG to some extent, and have a preference for
the alternative with the lower AL and the higher CG.
Only for group D, an interaction term (𝛽𝛽5) between FL and Al was estimated. This indicates that this group let the effect of the difference in FL on their final decision, depend on the difference in AL, or the other way around. Therefore this indicates the use of a two-dimensional rule: AL-FL, or FL-AL. However, the use of this rule is not very strict, as 𝛽𝛽3 suggests that this group does not ignore the third dimension of CG totally. These results are not very different from those of Jansen et al. (2012) who found that this group used a three-dimensional rule with the order AL-FL-CG.
For group B, C, and E, all the interaction-betas are zero which indicates that these groups do not use more-dimensional rules. But they do not seem to use one-dimensional rules either because more than one of the main effects (𝛽𝛽1, 𝛽𝛽2 or 𝛽𝛽3) is estimated and differs from 0. It seems that these groups do evaluate all three dimensions, but without really integrating them (i.e. calculate the EV), which is indicated by a better fit of the SLT compared to the CPT model.
Table 2
BIC values of the optimal solution of the two models CPT and SLT, when subjectivity parameters were not estimated (‘obj’) and when they were estimated (‘subj’); between brackets the number of estimated parameters
Group CPT - obj SLT - obj CPT - subj SLT - subj
A 1492.02 (1) 1477.71 (3) 1088.80 (3) 1110.15 (7) B 1329.88 (1) 1236.13 (3) 845.61 (3) 847.93 (8) C 1406.23 (1) 988.59 (3) 896.59 (4) 917.34 (8) D 631.17 (1) 516.21 (4) 344.97 (3) 358.12 (8) E 924.50 (1) 655.17 (3) 622.33 (3) 635.46 (8) F 532.73 (1) 509.38 (3) 504.43 (3) 513.93 (7) Table 3
Parameter estimates of the ‘objective’ SLT model; subjectivity parameters 𝛼𝛼, 𝛾𝛾, 𝛿𝛿, and 𝜆𝜆 were fixed at 1 Group 𝛽𝛽0 intercept 𝛽𝛽1 main*difFL 𝛽𝛽2 main difAL 𝛽𝛽3 main difCG 𝛽𝛽4 difFL∙difCG 𝛽𝛽5 difFL∙difAL 𝛽𝛽6 difCG∙difAL A 0 -7.22 45.42 4.65 0 0 0 B 0 -8.77 34.27 3.19 0 0 0 C 0 -10.60 63.74 2.25 0 0 0 D 0 -4.83 40.17 5.87 0 2.81 0 E 0 -8.43 13.97 2.12 0 0 0 F 0 -1.43 15.84 1.52 0 0 0
Note. Parameter estimates of 0 indicate that the parameter was not freely estimated but fixed at 0 *main= main effect of the difference in FL between both alternatives
‘Subjective’ CPT versus ‘subjective’ SLT
The first and second hypothesis, about the rule use of each group, was also tested when subjectivity parameters were included. Comparison between the optimal solutions of the subjective-CPT and subjective-SLT model-variants (Table 2, column 4 and 5) shows that the CPT model fits better to the response patterns, for each of the 6 groups. So, in contrast with the conclusions of the ‘objective’ model-variants, now all groups seem to integrate dimensions. The parameter estimates of the subjective- CPT model-variants (Table 4) consist of five subjectivity parameters and weighting parameter 𝜙𝜙. This last parameter indicates to what extent groups, after integrating dimensions into a subjective utility for each alternative, base their decision on the difference between the subjective utilities. All 𝜙𝜙 estimates are positive which indicates that each group prefers the alternative with the highest subjective utility. Group E acts most consistent with this difference, and group F (the guessing group according to Jansen et al. (2012)) and B act least consistent with this difference.
Subjective evaluation versus objective evaluation
Before interpreting the subjectivity parameter estimates of the CPT model-variants, let us evaluate the third hypothesis, which stated that each group evaluates values and probabilities subjectively instead of objectively. Overall, this is confirmed by comparison of the BICs of the objective-CPT and subjective-CPT model-variants (Table 2, column 2 and 4) and by comparison of BICs of objective-SLT and subjective-SLT model-variants (column 3 and 5). Only for the last group the use of objective sequential rules are more likely than the use of subjective sequential rules. Mind that this is the guessing group according to Jansen et al. (2012).
By combining the results of the three hypotheses we can also evaluate which model-variant (objective-CPT, objective-SLT, subjective-CPT, subjective-SLT) is most likely for each group: comparison of all BICs of Table 2 per group shows that all groups are most likely subjectively integrating dimensions.
How each group subjectively integrates can be deduced from the subjectivity parameter estimates (Table 4 and figures 4 to 9). Group A, that objectively integrated dimensions according to Jansen et al. (2012), evaluates gains objectively but overweights small loss amounts and loss
small loss amounts and loss probabilities, but to a lesser extent. This group does handle gains as subjectively as losses. Overall, this group seems to handle attributes the least subjective of all groups, and comes closest to objective integration of dimensions. Group C (three-dimensional sequential rule according to Jansen et al.) is the only group that underestimates large loss probabilities, which is indicative of a risk seeking style. As group A, this groups handles gains objectively and
underestimates loss amounts. Group D (three-dimensional sequential rule according to Jansen et al.) has a similar subjectivity profile as group A, but overestimates loss probabilities even more. Group E (two-dimensional sequential rule according to Jansen et al.) and F (guessing according to Jansen et al.) do distinguish gains from losses but not the specific amounts of gains/losses. They also seem to ignore the loss probabilities. Whereas group E handles loss probabilities as if they are all 1, group F handles loss probabilities as if they are all .5, which might indeed indicate guessing.
Table 4
Parameter estimates of the optimal solutions of the ‘subjective’ CPT model; Group label Jansen et
al. (2012) (group size)
Value parameters Probability parameters Weighting parameter No. of free parameters subjectivity effect* 𝛼𝛼𝑔𝑔 𝛼𝛼𝑙𝑙 𝜆𝜆 𝛿𝛿 𝛾𝛾 𝜙𝜙 AL CG FL
A obj integration (29%) 1 0.06 1 76.86 1 51.72 3 ++ obj. ++
B seq: FL-AL-CG (23%) 0.23 0.23 1 10.18 1 22.44 3 + + +
C seq: FL-CG-AL (19%) 1 0.09 1 0.32 0.23 65.19 4 ++ obj. - D seq: AL-FL-CG (10%) 1 0.06 1 105.04 1 54.90 3 ++ obj. +++
E seq: FL-AL (13%) 0 0 1 936.48 1 415.11 3 / / /
F guessing (6%) 0.04 0.04 1 1 0.04 32.32 3 / / /
Note. Parameter estimates of 1 indicate that the parameter was fixed at 1, and not freely estimated * summary of interpretation of parameter estimates: symbols indicate whether amounts of loss (AL), amount of gain (CG) and loss probabilities (FL) are overestimated (+ or ++), underestimated (-), objectively evaluated (‘obj.’) or ignored (/)
Figure 4. Subjective evaluation of gains/loss amounts and probabilities of group A (that used an integrative rule according to Jansen et al. (2012))
Figure 5. Subjective evaluation of gains/loss amounts and probabilities of group B (that used the sequential rule FL-AL-CG according to Jansen et al.)
Figure 6. Subjective evaluation of gains/loss amounts and probabilities of group C (that used the sequential rule FL-CG-AL according to Jansen et al.)
Figure 7. Subjective evaluation of gains/loss amounts and probabilities of group D (that used the sequential rule AL-FL-CG according to Jansen et al.)
Figure 8. Subjective evaluation of gains/loss amounts and probabilities of group E (that used the sequential rule FL-AL according to Jansen et al.)
Figure 9. Subjective evaluation of gains/loss amounts and probabilities of group F (that was guessing according to Jansen et al.)
Age, rule use, and subjectivity
Exploratively, we related age to rule use, by reinterpreting the developmental analyses of Jansen et al. (2012) (Figure 10). The rule of group A, highly subjective integration of dimensions, was used most by the youngest children. Use of the rule of group B, integration with the least subjectivity, increased with age. This rule was used most by the oldest adolescents. Group C underestimated loss frequencies while integrating dimensions. This rule is most used by 11/12 year old children, and least by the youngest age group. Although the subjectivity pattern of group D is similar to that of group A, the age distribution is not. This integrative rule, where loss amounts are underestimated and loss probabilities are overestimated, is most used by the oldest age group. Use of the rule of group E, where differences in gains, losses and loss probabilities are almost ignored, decreases with increasing age. The rule of group F was not much used, and least by the oldest adolescents. This rule was characterized by the ignoring of differences in all three dimensions.
Figure 10. Age distribution of the groups that age use a different decision rule (figures copied from Jansen et al., 2012)
Discussion
This study combined strengths of the design of the study by Jansen et al. (2012), with a formal approach to detect the specific characteristics of decision rules used by children and adolescents. Data came from Jansen et al., who had used a strict empirical way to distinguish different decision rules. We analyzed the resulting groups by using probabilistic, and therefore realistic, formal models to label these different decision rules. Using the CPT and SLT model, integrative rules could be distinguished from sequential rules, and a detailed subjective evaluation profile could be made of each group. Whereas Jansen et al., who assumed deterministic choice and objective evaluation of attributes, found
that four groups used different sequential rules and one group integrated dimensions (i.e. calculated the EVs), we found that all groups integrated dimensions, but with (large) subjectivity. Exploratory, we related rule use to age and found that, in contrast with Jansen et al., use of the least subjective integrative rule increases with age.
More specifically, Jansen et al. (2012) found that the largest group (29%, group A) integrated dimensions objectively., i.e. that they calculated the EV. This was surprising because this rule was most used by the youngest group and it seems unlikely that children of eight years old are able to calculate EVs. In the present study, we also found that this group integrates dimensions, but with highly subjective evaluation of loss amounts and loss probabilities. It remains unclear how these children exactly integrate dimensions, but this extreme subjectivity makes it unlikely that they calculate EVs. The group that, according to our results, comes closest to rational integrating of
dimensions, is the second largest group (23%, group B, that used a three-dimensional rule according to Jansen et al.). This group did evaluate attributes subjectively, but to a lesser extent than the other groups. Only of this rule, the use clearly increased over age, which is consistent with the findings of Stanovich, Toplak, and West (2008) that more rational thinking increases with increasing age.
Our characterization of the last group (group F, 6%) seems to be consistent with the labeling of Jansen et al. (2012) that stated that this group was guessing. We did not fit a ‘guessing’ model, but the subjectivity profile of this group indicates that differences on each of the dimensions are ignored. Moreover, the weighting parameter 𝜙𝜙 of this group was relatively small, indicating that the difference in subjective utility (the value that results from integrating dimensions) between both alternatives does only partly influence what decision is made.
This study was the first to use the SLT model in a context with more than two dimensions, and with dimensions that had more than two levels. This also brought some complications. One is the size of the SLT model when subjectivity parameters are included. To find the most parsimonious variant of the model, ideally all combinations of fixed and free parameters should be fitted to the six data
patterns. Because of the large expected computation time (1.5 to 2 months), we reduced the number of variants by either fixating or estimating all four subjectivity parameters. As a consequence, this model had a disadvantage compared to the CPT model. This is because most likely only some of the
subjectivity parameters were needed in the model and the others unnecessarily were included and raised the BIC. Especially for the group where BIC of the subjective CPT and subjective SLT models were close (group B), this might have affected the final conclusion that the SLT model explained the response pattern worse than the CPT model.
Another practical issue with the SLT model relates to its complex interpretation of the interaction terms and our handling of the data. When objectivity was assumed, the SLT model
explained the response patterns best, suggesting the use of sequential rules. The parameter estimates of these SLT results however were not typical for sequential rules. For most groups only main effects of all three dimensions appeared, whereas the use of an one-dimensional rule would lead to only one main effect, and the use of more-dimensional rules would lead to interaction effects of the dimensions. However, we cannot be sure that the absence of interaction terms truly means absence of the use of more-dimensional rules. It might have been an artifact of our coding of the data, as is outlined next.
This has to do with the interpretation of the interaction terms. When we have a
two-dimensional rule, say CG-FL, we expect people to base their decision on CG whenever the alternatives differ in their CGs, and we expect them to base their decision on FL whenever the alternatives do not differ in their CGs. That is, the effect of FL on the final choice should be 0 when CGs do differ, and this effect should not be 0 when alternatives do not differ in their CG-values. This means that the interaction term (of FL∙CG) has to make sure that the total effect of FL on the final choice disappears whenever CGs do differ. So, in that case it has to compensate the main effect of FL in order to result in a final effect of 0. For example, with a difference in CG between the alternatives of .04, and a main effect of FL of -10, the total effect of FL on the final choice should be 0 because the difference in CG is large enough to prevent the person from evaluating the second dimension FL. Therefore the interaction beta term of FL∙CG should compensate the main effect of FL of -10, with a value of -250 (−10 + .04 ∙ −250 = 0).
This works well for one item, but each item had a replicate in the GMT where the order of the two alternatives was changed. The SLT model is formulated in terms of chances on selecting the first alternative, and therefore the interaction beta terms are likely to cancel each other out over the whole item set. In the current example: in another item the options A and B are switched, resulting in a CG
difference of -.04, and now an interaction beta term of FL∙CG of +250 would be needed to compensate the main effect of FL. So, this is an alternative explanation for the absent interaction terms found. Till in half of the items, the alternatives have been relabeled, and till all analyses have been performed again, we cannot be sure that the zero’s found, truly represent absence of the use of more-dimensional sequential rules.
Moreover, the items used were especially constructed by Jansen et al. (2012) to disentangle 17 specific deterministic rules, and not to distinguish different kinds of subjectivity. We did use the same items and same data as Jansen et al., but we tried to deduce much more from the observed response patterns. Items with a larger range of values and probabilities might be required to give a more precise and more reliable subjectivity pattern. For example, the figures 4 till 9 are only based on two different probabilities, two gain values and three loss values. For more confidence in the subjectivity results, and also in the discrimination between the CPT and SLT models, this study should be performed again with a larger item set that has a wider range in values and probabilities.
A smaller practical issue with both the CPT and SLT models is that both are binomial models. Therefore, we had to ignore the third answer option of the GMT, which says that both alternatives are equally good. We divided the choices for this option randomly across choices for alternative A and B. This was not an optimal approach, because some data is ignored, and error is added to the data. In the future, the CPT and SLT models might be extended to allow for dependent variables with more than two levels.
Two small practical limitations of the procedure, are the absence of confidence intervals and the use of BIC values. Because of the complexity of the models, we did not estimate confidence intervals around the parameter estimates. Therefore we cannot be sure that the point estimates are reliable. For some parameters we did see that estimates were very similar among variants that fitted almost as good as the optimal variant, which does give confidence that these estimates were reliable.
For model comparison, we used BIC-values. These are not comparable across groups, and therefore we only know for each group which rule was most likely used. It might be that for some of the groups neither of the models fit well, and that these groups still use other rules, not assessed here.
Looking at the subjectivity profiles of the groups, one may ask how specific the CPT model explains integration of dimensions. When subjectivity is extreme, like in group E and F, then
mathematically we can still explain the decision behavior with the CPT model, but one may wonder to what extent this is justified. For example, if no distinction is made between high and low values, or high and low probabilities, this is not integration in the sense of calculating EVs anymore. The subjectivity parameters seem to be able to account for very diverse decision behavior. At some point the subjectivity itself may have a larger effect on the decisions made than the integration of
dimensions has. This means that one should be careful with translating a good match of the CPT model to data directly into labeling a rule as an ‘integrative’ rule.
This study aimed at reaching a deeper understanding concerning the reasoning that children and adolescents use when making decisions. One reason why such an understanding is important, is that it may guide discussions about the extent to which children should be involved in making important decisions. For example, when it concerns (dis)continuation of medical treatment of a seriously ill child. But also in more common situations like parental divorce, where it has to be decided with which parent the child will stay. When children of a specific age mostly use simplistic one-dimensional rules, we might want to give them less influence, than when these children consider more dimensions and reason in similar ways as adults do.
Concluding, this study brings us closer to an understanding of children’s and adolescents’ reasoning when confronted with alternatives that differ on several dimensions and that involve some degree of risk. In contrast with the results of Jansen et al. (2012), our results did not show that children and adolescents use different decision rules. They all seem to integrate dimensions after subjective evaluation of the dimensions. This subjectivity plays a large role, especially for the younger children. What might seem the use of a sequential rule when assuming objectivity, may actually be the
subjective integration of dimensions. In future research one should be aware that children and adolescents do evaluate attributes highly subjectively, and that this strongly influences results of the assessment of integrative versus sequential rule use.
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Appendix
Boundary settings used to estimate parameters
CPT 𝛼𝛼𝑔𝑔 𝛼𝛼𝑙𝑙 𝜆𝜆 𝛿𝛿 𝛾𝛾 𝜙𝜙 Group A, B, C, F upbound 5 5 10 100 5 1000 lbound .001 .001 .001 .001 .001 0 Group D upbound 5 5 10 300 5 2000 lbound .001 .001 .001 .001 .001 0 Group E upbound 5 5 10 1000 5 2000 lbound .001 .001 .001 .001 .001 0 SLT 𝛼𝛼 𝜆𝜆 𝛿𝛿 𝛾𝛾 𝛽𝛽0 𝛽𝛽1 𝛽𝛽2 𝛽𝛽3 𝛽𝛽4 𝛽𝛽5 𝛽𝛽6 Group A, F upbound 5 10 100 5 100 100 200 250 100 100 100 lbound .01 .01 .01 .01 -100 -100 -100 -100 -100 -100 -100 Group B upbound 5 150 150 10 100 100 500 250 100 100 100 lbound .01 .01 .01 .01 -100 -100 -100 -100 -100 -100 -100 Group C upbound 5 100 100 10 100 100 500 250 100 100 100 lbound .01 .01 .01 .01 -100 -100 -100 -100 -100 -100 -100 Group D upbound 5 50 100 5 100 100 500 250 100 100 200 lbound .01 .01 .01 .01 -100 -100 -100 -100 -100 -100 -100 Group E upbound 5 25 300 5 100 100 200 600 100 200 100 lbound .01 .01 .01 .01 -100 -100 -100 -100 -100 -100 -100
