Children’s application of decision strategies in a compensatory environment

Abstract

Adaptive actors must be able to use probabilities as decision weights. In a computerized multi-attribute task, the authors examined the decisions of children (5–6 years, n = 44; 9–10 y., n = 39) and adults (21–22 y., n = 31) in an environment that fosters the application of a weighted-additive strategy that uses probabilities as weights (WADD: choose option with highest sum of probability-value products). Applying a Bayesian outcome-based strategy classification procedure from adult research, we identified the utilization of WADD and several other strategies (lexicographic, equal weight, naïve Bayes, guessing, and saturated model) on the individual level. As expected based on theory, the prevalence of WADD-users in adults was high. In contrast, no preschoolers could be classified as users of probability-sensitive strategies. Nearly one-third of third-graders used probability-sensitive strategies.

1 Introduction

Individuals use different strategies when searching and evaluating information for decision making. The economist Herbert Simon introduced this notion to decision research in the mid-20th century (Simon, 1955). Still, decision researchers did not begin to study the application of decision strategies systematically until some years later. A groundbreaking development in research methodology paved the way for such studies – information-board technology (Payne, 1976). In an important paper, Payne, Bettman and Johnson (1988) introduced the Mouselab, in which monetary gambles were displayed in a matrix on a computer screen that crossed options (the choice alternatives) with attribute dimensions. Dimensions differed with regard to the probability that outcomes (certain amounts of money) could occur. Individuals could inspect the outcomes by opening cells in the matrix with the computer mouse. The gathering of information was tracked by the computer. These search data were subsequently used to identify strategies of information acquisition.

Using the Mouselab, decision processes are inferred from search movements, i.e., the course and amount of inspected information. However, inspecting a certain piece of information does not necessarily imply that it is actually used in subsequent preference formation and choice. With the advance of formal modelling, it became possible to identify strategies without search data (Glöckner & Witteman, 2010, for overviews). In outcome-based strategy classification (Bröder, 2010), the researcher varies patterns of information in such a way that each decision strategy predicts a distinct choice sequence over a series of decision trials. Participant’s choices can then be compared to the strategy predictions. These and other sophisticated techniques have been frequently applied in decision research on adults (e.g., Bröder 2003; Glöckner & Betsch, 2008; Jekel, Glöckner, Bröder & Maydych, 2014; Lee, 2016) and adolescents (e.g., van Duijvenvoorde, Jansen, Visser & Huizenga, 2010) and have significantly advanced our understanding of strategy application in adults. For children, however, empirical evidence is scarce. In most studies using an information-board, classification of strategies rarely reached the standards of research on adults (but see Mata, Helversen & Rieskamp, 2011). In this paper, we apply the outcome-based classification method to preschoolers and compare their performance with third-graders and adults in order to gain insight into the development of strategy application in a probabilistic, multi-dimensional environment. Specifically, we focus on the application of a weighted-additive (WADD) strategy that uses probabilities as weights for values.

1.1 Strategies

Subjective expected utility theory (Edwards, 1954) implies that actors should choose the option with the highest expected utility. Accordingly, for each outcome of an option, the decision maker assesses its subjective value and subjective probability of occurrence. Then, each value is weighted by its probability. Those products are then summed over all outcome dimensions to determine the expected utility for a particular option. The option with the highest expected utility is then chosen. The manner in which information is integrated can be described by a so-called weighted-additive rule (WADD, Payne et al., 1988).¹ WADD is one of the most complex decision making strategies (Shah & Oppenheimer, 2008) because it is responsive to the relative importance of outcomes (via weighting) and considers all available information (adding all value-probability products). It underlies the integration rule in the utility maximization approach in economic theory (e.g., von Neumann & Morgenstern, 1947). As such, it is part of an approach that is considered a normative model (Simon, 1983). Consequently, WADD has often been used as a comparison standard in early process tracing studies and modelling of adaptive decision making (Payne, Bettmann & Johnson, 1993).

It is important to note, however, that WADD, as a rule for information search and integration, is not the rational or normative benchmark per se. There are other rules rooted in probability theory that also allow individuals to improve or even optimize decision accuracy. Naïve Bayes, for example, also takes the relative importance of all outcomes into account. In contrast to WADD, naïve Bayes transforms probabilities of outcomes into log-odds. For each option, the model then sums the log-odds of outcomes that support the option. The model finally chooses the option with the highest sum of log-odds. Naïve Bayes is the optimal solution when options are equally likely a-priori and cues are conditionally independent. However, due to the complex transformations of the outcome probabilities required, naïve Bayes is likely not a psychologically plausible model of human decision making. Still, proxy models may exist that closely mimic naïve Bayes (e.g., Jekel, Glöckner, Fiedler & Bröder, 2012; Lee & Cummins, 2004).

A simpler way to reach a decision is to apply an equal-weight rule (EQW, Payne et al., 1988), which does not require the weighting of outcomes because it ignores probabilities. Formally, this can be expressed by setting weights equal to one. Outcome values are summed for each option; and the option with highest sum is chosen. Lexicographic strategies (LEX, Fishburn, 1974; see also the take-the-best strategy, e.g., Gigerenzer, 2004) ignore portions of the outcome information and do not require integration (weighting, adding). A LEX user begins by inspecting the most important attribute dimension. In a probabilistic environment, the rank order of the dimensions follows their probability. Accordingly, the most likely outcome values are compared first. The option with the highest value on that probability dimension is then chosen. Only in the case of ties does the individual inspect outcomes on the subordinate probability dimension. In this research, we focus on WADD but also assess a number of other strategies such as EQW, LEX, and naïve Bayes (NB). Moreover, we include some lower benchmark strategies in the classification such as guessing and a saturated model (Hilbig, 2011) that allows for testing whether none of the other models in the set can best describe participants’ choices. The saturated model allows us to detect systematic decision making that is not captured by one of the other models.

1.2 How the environment affects strategy use

In the 1970s (e.g., Payne, 1976), psychologists began to study how the environment and features of the decision task influence the way in which individuals make decisions. To date, a large body of evidence indicates that (adult) humans are adaptive decision makers, i.e., they tune information search and processing to context demands. For the purpose of this research, it is sufficient to briefly discuss two factors – weight structure and feedback in the environment. Both factors are of paramount importance in terms of understanding context-contingent decision making (e.g., Payne et al., 1993).

In probabilistic environments, weights are represented by the probabilities of outcomes or the validity of cues. Cues, for instance, can be testers that predict the quality of products (e.g., Glöckner & Betsch, 2008). The validity represents the probability that the tester makes a correct prediction (good vs. poor quality) regarding the outcome of an option (product). In non-probabilistic environments, in which outcomes occur with certainty, weights are determined by the decision maker’s goals and preferences. One attribute of an option (e.g., the color of a bike) may be more important than another (e.g., whether the bike has a bell). (Klayman, 1985, studied such a task in children.)

Decision tasks differ with respect to the dispersion of weights. If dispersion is high, weights differ relatively strongly. If dispersion is low, weights tend to converge. Weight dispersion has important implications for the selection of strategies. If dispersion is low, outcomes of an option can compensate for each other. In such a compensatory environment with low weight dispersion, individuals should use a compensatory strategy such as WADD that considers all relevant information. We will illustrate this briefly with the following example. Assume that an individual decides between two options. There are three cues that predict the outcomes of the options. The three cues differ with regard to their validity (p = .71; .71; .86), i.e., the probability that they predict outcomes correctly. Dispersion of these validities is low; thus, “low”-validity cues (p = .71) can compensate for the “high”-validity cue (p = .86). Compensation becomes evident if one reflects the arithmetic underlying the WADD rule. Assume, for example, that the high-validity cue predicts an outcome value 1 for option A and an outcome value 0 for option B. Assume also that the two low-validity cues jointly predict the opposite (0 for A, 1 for B). Applying WADD, we can calculate the overall expected values (EV) for the options as follows:

• Option A: EV = 0 + 0 + 1 × 0.86 = 0.86

• Option B: EV = 1 × 0.71 + 1 × 0.71 + 0 = 1.42

Due to its higher expected value, a WADD-user should choose Option B in this example, because the low-validity cues together compensate for the prediction of the high-validity cue.

It is not generally necessary to use WADD to achieve high decision accuracy. For instance, in an environment in which the dispersion of probabilities is so high that the low-validity cues cannot compensate for the predictions of a high-validity cue, even simple rules such as focusing on only the high-validity cue (i.e., LEX) can yield ideal results (Payne et al., 1988; see also Gigerenzer & Gaissmaier, 2011).

Another important factor is the presence and structure of feedback. Feedback, i.e., the actual outcome experienced after choosing an option, may reinforce choice options (Betsch & Haberstroh, 2005; Betsch et al., 2001), information search strategies, and one’s ultimate choice (Bröder & Schiffer, 2006; Rieskamp & Otto, 2006) or even all three (Bröder et al., 2013). Results show that adult decision makers (e.g., Rieskamp & Otto, 2006) and older children (Mata et al., 2011) are sensitive to feedback and can be trained to routinely apply a certain strategy contingent upon the reinforcement schedule.

In this research, we used a compensatory environment (fostering use of WADD) and additionally reinforced WADD application by using feedback.

1.3 WADD application in children

In general, WADD describes a rule for information integration. On the formal side, the arithmetic equivalents are multiplication and addition. According to folk wisdom and empirical evidence, mental arithmetic is exhausting. Its mastery requires years of school education and practice. Unfortunately, some individuals continue to have difficulties with arithmetic throughout adulthood. Thus, the backbone position of the bounded rationality approach appears to be a truism, assuming that the application of WADD requires working knowledge of arithmetic. Especially when the amount of relevant information is large, WADD could overtax our thinking abilities. Due to such limitations, decision makers are assumed to instead apply simple strategies that reduce or even avoid such effortful integration of information (Gigerenzer & Gaissmaier, 2011; Shah & Oppenheimer, 2008). From the viewpoint of bounded rationality, it seems rather odd to expect young children to apply WADD because they suffer from cognitive constraints (e.g., age dependent limitations of working memory, Cowan, 2016) and lack pertinent formal knowledge (Piaget & Inhelder, 1951) that motivate them to weight values with probabilities.

Acknowledging findings from other fields of cognition, however, we face a strikingly different notion. Research on, for instance, spatial perception (e.g., Cheng, Shettleworth, Huttenlocher & Rieser, 2007), categorization of biological motion (Troje, 2002), understanding irony (e.g., Gibbs & Colston, 2007), implicit attitude formation (Betsch, Plessner, Schwieren & Gütig, 2001), and intuitive decision making (Betsch & Glöckner, 2010) provide evidence that the human mind is capable of performing integration and weighting operations without noticeable effort in an astoundingly narrow time frame (see Betsch, Ritter, Lang & Lindow, 2016, for an overview).

Several developmental studies provide additional support for this notion. For instance, Streri, Coulon and Guellaï (2012) reviewed evidence from studies on face-voice integration in infants and concluded that integration abilities are already developed at birth. Ebersbach (2009) investigated whether children are able to combine width, height, and length when estimating the volume of objects and found that even five- to six-year-olds were able to integrate these dimensions in a multiplicative-like fashion. Schlottmann (2001) studied evaluative judgments in four- to six-year-old children and showed that children integrate probability and value in accordance with WADD. Finally, in a multidimensional, non-probabilistic decision task, Lindow, Lang and Betsch (2017) demonstrated that the majority of children (6–12 years old) applied a WADD rule for information integration. Notably, the prevalence of WADD use appears to decrease with age. Younger children (around 9 years) are more likely to apply complex integrative rules than older children (up to 17 years), who in turn begin to apply simpler rules and heuristics (Jansen, van Duijvenvoorde & Huizenga, 2012; Mata, et al., 2011).

Yet, this is only half the story. In a non-probabilistic task, Bereby-Meyer, Assor and Katz (2004) showed that even 8–9 year olds preferred simple strategies (such as lexicographic rules) over more demanding ones. In a series of studies with a probabilistic inference task, Betsch and colleagues (Betsch et al., 2014, 2016) found that the majority of children under the age of ten were reluctant to use probabilities as weights in their decisions as required by the WADD rule. Preschoolers, in particular, tended to apply maladaptive strategies such as arbitrary switching between options and change after failure (Lang & Betsch, 2018). As a caveat, the environment employed by Betsch and colleagues did not encourage the application of WADD. The dispersion of weights was large and, hence, yielded a non-compensatory environment. Feedback reinforced not specific strategies but rather the validity of the cues (Betsch et al., 2014). Moreover, the design of the pay-off structure did not allow the authors to classify strategies on a level comparable to those regularly reached in decision studies with adults.

1.4 Research goal and hypothesis

As a crucial test for children’s capability to apply a WADD rule, we thus studied their decisions in a compensatory probabilistic environment in which WADD use was reinforced by the feedback distribution structure. In doing so, we applied strategy classification methods from adult research to a young age group (preschoolers) in a probabilistic environment. Our research goal was to obtain insights into the development of WADD application in children (5–6 y.; 9–10 y.) as compared to an adult control group.

It is necessary at this point to briefly discuss some aspects of our method. We applied an open information board in which all pieces of information can be inspected directly and remain visible until the participant has reached a decision (a so-called open board, Glöckner & Betsch, 2008). Accordingly, individuals do not have to store information in memory. Otherwise, children’s memory capacities would likely have been overtaxed – a condition that obstructs the use of complex strategies such as WADD. Moreover, there are no time constraints. Participants are free to ponder their decisions as long as they wish. Hence, the decision environment in our study is (i) compensatory (low dispersion of probabilities), (ii) reinforces WADD, and (iii) is characterized by the absence of external constraints.

According to models of adaptive decision making and strategy use, these are conditions that, together, increase the likelihood that (adult) decision makers will apply compensatory strategies that rely on multiple information and utilize probabilities as decision weights (e.g., the model of adaptive decision making, Payne et al., 1993; and Shah & Oppenheimer, 2008, for a discussion). WADD is not the only strategy that uses probabilities and aggregates information in a compensatory fashion. Naïve Bayes, for example, fulfills the same criteria and was also considered in our strategy classification approach. For simplicity, however, we predicted that WADD should be the dominant strategy applied by decision makers. This assumption represents the theoretically derived hypothesis that we tested in this research.

Due to mixed prior evidence, we can only speculate regarding the development of WADD use to date. Referring to the bounded rationality approach and Piaget’s model of cognitive development (Piaget & Inhelder, 1951), one may expect that the younger children are, the more they lack the computational abilities and conceptual prerequisites (e.g., a concept of chance and probability) necessary to perform WADD. Cognitive theories on information integration (e.g., Betsch et al., 2016, for an overview) and post-Piagetian developmental research (e.g., Schlottmann & Wilkening, 2012), however, jointly suggest that children, from an early age, have the ability to intuitively integrate information in a weighted-additive fashion. To our knowledge, it is not possible to come up with clear, theoretically derived predictions on the development of WADD use based on the literature to date. Therefore, we consider our research to be exploratory regarding the question of the age at which children begin to systematically use WADD in an environment that fosters its application.

2 Method

2.1 Participants

Children (German native speakers) were recruited at one elementary school and nine daycare centers, located in middle-class areas, which had previously signed contracts with the university agreeing to participate in research. Parents signed informed consent forms prior to our approaching the children. Adults were students with different majors at the University of Erfurt who were sampled from our lab subject pool via ORSEE (Greiner, 2004).² The study lasted approximately 45 minutes including the breaks between the different blocks.

The sample consisted of 44 preschoolers (61.4% female; M _age = 68.5 months, SD = 2.3), 39 elementary schoolers (53.8% female; M _age = 112.1, SD = 8.0), and 31 adults (87.1% female, M _age = 271.7 months, SD = 34.3). Five participants did not pass the manipulation check (i.e., they rated one of the low validity cues to be smarter than the high validity cue after the learning session). Following prior procedures (e.g., Betsch et al., 2014), they were excluded from further analyses. One additional participant was excluded from further analyses due to not following instructions. The final sample (n = 108) included 39 preschoolers (66.7% female; M _age = 68.6, SD = 2.2), 38 elementary schoolers (52.6% female; M _age = 112.3, SD = 8.1), and 31 adults (87.1% female; M _age = 271.7, SD = 34.3).

2.2 Information board approach

Since the introduction of the “Mouselab” to decision research (Payne et al., 1988), information boards have become the standard tool in adult research to study strategies of information search and decision making. In child decision research, information boards were most frequently applied in the domain of multi-attribute decisions under certainty (i.e., non-probabilistic). In such multi-attribute decisions, the options (e.g., bicycles; Klayman, 1985) are described on different attributes (e.g., color). Attribute weights vary with regard to their subjective importance. The majority of these studies assessed general tendencies in search behavior (e.g., attention, search focus, amount of information considered) and do not classify specific strategies (Avond, 1997; Ball, Mann & Stamm, 1994; Davidson, 1991a, 1991b, 1996; Davidson & Hudson, 1988; Gregan-Paxton & Roedder-John, 1995, 1997; Howse, Best & Stone, 2003; Klayman, 1985; but see Lindow, Lang & Betsch, 2017, for a strategy classification approach). Note that, in all these studies, tasks were non-probabilistic, i.e., the outcomes of the options on the multiple attribute dimensions always occurred with certainty.

In contrast, we present our participants with a probabilistic environment. Compared to research with adults, information board studies with probabilistic environments are rare in child research (Betsch et al., 2013; 2014; 2016; Lang & Betsch, 2018; Mata et al., 2011). “Mousekids”, the tool, which we describe next, is structurally equivalent to information boards from adult research on probabilistic inference (e.g., Bröder, 2003; Glöckner & Betsch, 2008; Newell & Shanks, 2003). Specifically, cues make binary predictions of the outcome of the options. The cues differ with regard to their validity, i.e., the probability that their predictions are correct. For example, Glöckner and Betsch (2008) presented participants with a task in which products (options) were described by testers (cues) which differed with regard to the probability (validity) that they predicted the quality of the product correctly.

2.3 Mousekids

Mousekids (Figure 1) is a computerized research tool for studying multiple-cue decision making in a child-friendly probabilistic environment (Betsch et al., 2016). There is open access to the software package online.³ Mousekids is an analog of the Treasure Hunt Game, a non-computerized information board used to study probabilistic inference decisions in children (Betsch & Lang, 2013; Betsch et al., 2014). In this study, we apply an open-board version of the tool. In an open board, all information contained in the matrix is visible right from the start of each decision trial. Compared to closed boards, which require the sequential opening of the cells in the matrix, open boards foster exhaustive consideration and integration of all given information (Glöckner & Betsch, 2008). Accordingly, an open board should facilitate application of WADD, especially in children.

Figure 1: Mousekids. The screenshot on the left shows the last trial of the learning session after all smart circles had been assigned to the animals. An animal received a smart circle if it made a correct prediction. Numbers of smart circles represent cue validities. In the example, the last cue’s prediction was correct because the predicted outcome (treasure) was actually contained in the house above. The screenshot on the right shows one trial from the test session with prediction pattern 4 (Figure 2). In this example, the participant has chosen the third option by opening the door of the house on the right at the top row, which contained a treasure as predicted by the high validity cue (horse, p = .86).

A Mousekid session consisted of two parts – learning and testing. In the test session (Figure 1), participants repeatedly chose between three houses (options) containing either a treasure or a spider. For each chosen house covering a treasure, participants received a treasure point. The goal of the game was to collect as many treasure points as possible. Participants based their decisions on the predictions of the animals representing the cues. The animals gave binary predictions: treasure or spider. The validity of each cue was depicted by a series of circles next to the picture of the animal. The relative number of circles equaled the validity of the cue. The animals in the top two rows scored five of seven circles, so that their cue validity was p = .71. The score of the animal in the bottom row was six, i.e., this cue’s validity was p = .86.

Prior to testing, participants were trained on cues’ validities in a learning session (left part of Figure 1). This session comprised seven learning trials per cue. In each trial, the animal predicted the content of the house (treasure vs. spider). Participants opened the house by touching the screen and then inspected its content. If the animal had predicted the content correctly, the participant assigned a “smart circle” (“Schlaupunkt” in the German instructions) to the animal by touching a blank circle, which then turned red on the computer screen. The number of colored circles was said to indicate “how smart” an animal was in terms of making correct predictions.

Prior to the learning session, the participant selected three out of a pool of eight animals (cat, dog, elephant, giraffe, lion, hippo, horse, mouse). The animal selected first was then placed in the top row of the board, the second in the middle, and the third in the bottom row. The animal in the bottom row always had the highest cue validity. The location of the animals was determined to prevent the confounding effects of liking and reading habits (see Betsch et al., 2014, for a discussion).

In the test session, participants worked on 80 target trials (in addition to three practice trials at the beginning). Target trials presented variations of six types (T1–6) of prediction patterns (Figure 2) designed to apply outcome-based measures of strategy classification. The systematic application of each strategy under consideration (WADD, EQW, LEX, NB, and benchmark models) predicts a unique distribution of choices across these patterns. For example, in T1 an individual would choose O1 if applying WADD (Expected Value O1 = 1 × 0.71 + 1 × 0.71 + 0 = 1.42; EV O2 = 0 + 1 × 0.86 + 0 = 0.86; EV O3 = 0 + 0 + 1 × 0.86 = 0.86), and O3 if applying LEX (i.e., following the predictions of the cue with the highest validity). An equal weight (EQW) rule neglects the validities of the cues and chooses the option with the highest sum of positive predictions. In T1, results of WADD and EQW application converge. A user of EQW would also choose O1 because two cues jointly suggest this choice, whereas only a single cue suggests choice of one of the other options. For naïve Bayes, the log-Odds for O3 (i.e., log(.86/.14) = 1.82) is higher than the sum of log-Odds for O1 (i.e., 2 × log(.71/.29) = 1.79) and the log-Odds for O2 (i.e., log(.71/.29) = 0.90). There is also the possibility that application of a strategy yields ties. E.g., in T3, WADD and EQW suggest choice of O1 or O3. In such types of patterns, application of WADD or EQW should result in an even distribution of choices over O1 and O3. Table 1 shows, as examples, choices expected for WADD, LEX, EQW, and NB over the six types of prediction patterns.

Figure 2: The six types of prediction patterns used in the decision trials of the study. Rows contain the predictions of the three cues differing in cue validity (p = .71; .71; .86). Each cue makes outcome predictions (1 = treasure; 0 = spider) for the three options depicted at the top of each column.

Table 1: Choices over the six types of prediction patterns for fourexample strategies.

Note: WADD = weighted additive, LEX = lexicographic, EQW = equal weight, NB = naïve Bayes. T1 to T6: Types of prediction patterns (Figure 2). Cell entries indicate predicted choices of option 1,2 or 3 in Figure 2.

The classification method, described in Appendix A, compared expected and observed choices to determine the posterior probability that the data was produced by an individual using a certain strategy. Each pattern was shown in three variants, so that each of the options was favored by the predictions equally often. The type of patterns and their variants were parallelized over blocks of trials and positions, so that each occurred four or five times in each block in counterbalanced positions. Moreover, we ensured that none of the patterns or their variations occurred in two successive trials. None of the patterns were used for the three non-target trials at the beginning of the test session. These free-shot trials used patterns in which cues made joint predictions – for example, all cues predicted that option 1 hides the treasure.

Just as in recent research applying the Mousekid tool (e.g., Betsch et al., 2016; Lang & Betsch, 2018), individuals received feedback after each choice. Specifically, they opened the chosen house by touching its symbol on the screen, after which either a treasure or a spider appeared. Importantly, and in contrast to the prior studies, we reinforced strategies instead of cues. Specifically, the structure of the feedback was designed to reinforce the application of WADD in order to provide ideal conditions for the use of this rule. Strategy reinforcement through feedback is a technique that is widely used in adult research on adaptive decision making (e.g. Rieskamp & Otto, 2006; Bröder & Schiffer, 2006) but rarely in research on child decision making (but see Mata et al., 2011). Due to arithmetic constraints, it is not possible to reinforce a strategy and, at the same time, ideally maintain the described validity of the cues in the feedback distribution. We only briefly illustrate this problem in an example with prediction pattern Type 1. First assume that feedback would reinforce the validities of cues. Accordingly, the participant should actually find a treasure (the outcome referred to by “1” in Figure 2) in 86% of her O3 choices. This is what happened in prior studies (e.g., Betsch et al., 2014, 2016). In the present research, however, we wish to strongly reinforce WADD application. In T1, WADD application should result in O1 choices. Accordingly, the O1 choices should yield treasures more often than choices of the other options. Thus, dominantly reinforcing WADD entails that the reinforcement rate of the cues will diverge from their absolute initial validity.

In our case, we structured the feedback to reach a reinforcement rate of 90% for WADD, 73% for LEX, 65% for EQW. This reinforcement of WADD entails a decrease in the absolute validity of the cues in experienced feedback. Nevertheless, we made sure that relative validity, i.e. the rank order of the cues’ validity, was reproduced. Over the entire set of 80 test trials, the validity of the cues in the feedback was p = .59 for the first and the second cue and p = .66 for the third, high validity cue. We ensured that the described validity of the cues learned by participants in the learning session was visually present in the form of smart circles next to the cues in each decision trial (Figure 1).

2.4 Procedure for children

The experimenter met the child at his or her primary school or daycare center. Both sat down next to each other at a table in front of a touch screen monitor (19”) connected to an IBM compatible laptop in a separate, quiet room. The experimenter first explained that the purpose of the game is to find treasures, and then stated: “But you don’t have to find the treasures on your own; somebody will be helping you. Look, do you see the animals on the screen? You are allowed to choose three of them to help you find the treasures. Now, please choose three animals that you would like to play the game with. To do so, tap the chosen animal on the screen and then you tap the grey box.” The experimenter supervised the child in the operation of the touchscreen computer throughout the session.

After choosing the animals, the experimenter continued with the learning session and told the child: “Now I will explain to you how we can find out how smart the animals are. Do you see the house up there? Maybe there is a treasure in the house and maybe not. The animals will tell you whether there is a treasure hidden there or not. But the animals are not always right. Therefore, we are going to check how often they are right.”

Following these instructions, the experimenter began the 21 learning trials with the animal at the top. In each trial, the child opened the box next to the animal’s picture by sliding the box to the right with their finger. A picture of either a treasure or a spider was visible behind the box. The child then opened the door to the house in the same manner to determine whether the prediction of the cue was correct. If this was the case (i.e., the animal said “spider”; and the house actually contained a spider), the child was instructed to bestow a smart circle to the animal by tapping one of the circles in the row to the left of the animal. By doing so, the touched circle on the screen turned red. After seven trials, the experimenter pointed out the number of smart circles gained by the animal (e.g., 5 out of 7, if p = .71) and continued to the next animal. After having finished the learning session with all three animals, the child was asked which of the animals is the smartest. This question served as the manipulation check for learning the cue validities.

Then, the experimenter proceeded to the next screen, which depicted the test session board and explained the goal of the game (find as many treasures as possible), pay-offs (one house contains a treasure), and actions (coloring treasure points after success). During three warm-up trials, which did not count toward a participant’s overall performance, the experimenter verbalized the prediction of the animals in different order and checked whether the child understood the board. After the child had inspected the animals’ predictions, the experimenter asked the child to make a choice (i.e., by opening one house at the top of the board). If the chosen house contained a treasure, the experimenter said: “Oh great, you got a treasure point. Now, you can tap one circle at the top of the screen to award yourself the treasure point.” If the house contained a spider, she said: “Ugh, a spider. Well, let’s see whether you find a treasure next time.”

The test session comprised four blocks of trials consisting of 20 trials each (80 trials total). Breaks between blocks, in which children were allowed to have a drink of juice, lasted 4-min each. During the break following the 40th trial, the experimenter engaged the child in a motoric game. After the break, the child was reminded of the game’s goal, i.e., to find as many treasures as possible, and that animals’ predictions should be considered before making a choice. In addition, the child was again asked to state how many smart circles were assigned to each animal.

After completing the 80th trial, the experimenter commended the child for “earning so many treasure points,” irrespective of the actual yield. The child then received a personalized certificate depicting their performance on the treasure hunt game as a reward.

2.5 Procedure for adults

As in prior studies (Betsch et al., 2014; 2016), all procedures and instructions were identical across age groups with two exceptions. First, we told adult participants that they served as a control group to children who participated in the same game. Second, the breaks between the blocks of trails in the test session did not involve motoric games. Instead, adults were asked to walk through the room to pick up an object from another table. Adults’ payment was contingent upon performance (number of treasures found) and resulted in an average payment of 4 Euros.

3 Results

We applied a Bayesian outcome-based strategy classification method (Lee, 2016) to classify the decision strategy of each participant individually (Appendix A).⁴ This outcome-based classification method compares choice-predictions of strategies (WADD, EQW, LEX, NB, Guessing, and Saturated) for each pattern type (Figure 2) to individuals’ observed choices. An individual is classified as a user of a certain strategy if the posterior probability that his/her choices accord with that particular strategy exceeds the posteriors for the other strategies – with the restriction that the total error likelihood of the parameter priors (i.e., the likelihood of choosing one of the two options not predicted by the strategy) is not greater than .33. Note that this restriction requires the individual to be quite systematic in the application of only one strategy. The likelihood that an individual will be sorted into the ‘unclassified’ (i.e., saturated model) or ‘guessing’ category increases with the number of errors in strategy application, switches between multiple strategies, and choices that follow a strategy that is not considered in classification.

According to models of adaptive decision making, we hypothesized that a compensatory strategy such as WADD should be the dominant strategy in our compensatory environment.⁵ In support of this hypothesis, 74% of adults are classified as WADD users. Moreover, 13% of adults could be classified as NB users, which is also a compensatory strategy (Figure 3). For preschoolers, the hypothesis is clearly not supported: Not a single preschooler used WADD or NB systematically. For elementary schoolers, 39% are classified either as WADD users (21%) or NB users (18%). Thus, the use of a compensatory strategy that is sensitive to validities increases with age group. Thirteen percent of elementary schoolers are classified as users of the non-compensatory LEX strategy. Although LEX is sensitive to probabilities (because it prioritizes cues according to their validity), LEX is a maladaptive strategy in our compensatory environment.

Figure 3: Percentage of participants classified by strategy for each age group (i.e., preschoolers, elementary schoolers, and adults) according to a Bayesian outcome-based strategy classification (Lee, 2016). Guess = guessing, EQW = equal weight, LEX = lexicographic strategy, NB = naïve Bayes, WADD = weighted addititive, Saturated = saturated model. For details on posterior probabilities of classifications, see Appendix C.

Many preschoolers (44%) guess between options. A high percentage of preschoolers (41%) can be best described by a saturated model, which indicates that those children use one or several strategies that were not included in the model comparison (see Lang & Betsch, 2018, for a classification of non-adaptive strategies that preschoolers tend to use). An additional detailed analysis shows that 31% of those preschoolers can be described by an equal weight model that considers the position of the information on the screen (see Appendix B for details). Altogether, these results show that adults dominantly apply an appropriate compensatory strategy that is sensitive to validities (WADD or NB) in a compensatory environment, whereas young children below the age of seven and 61% of the older children fail to do so. We provide more detailed descriptive statistics on the posterior probabilities for strategy-classifications and posterior distributions of the probability of an application error in strategy use in Appendix C.

A competent decision maker is capable of using probabilities as decision weights. Post-Piagetian developmental research suggests that the utilization of probabilities begins at an early age. In the present study, however, no preschoolers could be classified as users of a probability sensitive strategy (WADD, LEX or NB). Notably, 52% of our 9-10 year old elementary schoolers systematically used a probability sensitive strategy (LEX:13%; WADD: 21%; NB: 18%). This is the highest portion we ever found in studies with the Mousekids tool (e.g., approximately 30% of individuals who followed the high validity cue, e.g., Betsch et al., 2016).

Another important observation refers to children’s capability of integrating information from multiple sources. Even when probabilities are neglected, children may base their decision on the entire set of predictions available in the information board, for instance, by applying an EQW rule. Although 35% of all children are classified as EQW users, only 18% of preschoolers are classified as such. Notably, EQW is the only systematic strategy detected in preschoolers in this study. It is still possible that preschoolers used other strategies in a systematic fashion. Lang & Betsch (2018) and Betsch and colleagues (2018) identified a number of such strategies – however, they were all maladaptive from a normative perspective (e.g., switching among options, change after failure, etc.) and were, therefore, not considered in this study. In comparison to prior studies with the Mousekids tool, we extended the number of trials from less than 30 (e.g., Betsch et al., 2016) to 80 in order to increase the reliability of the classification procedure. Thus, one may argue that decision accuracy might have decreased over trials in children, and the youngest age group in particular, due to decreasing motivation over trials.

As a measure of motivation, we determined accuracy scores (number of treasure points) for each of the four decision blocks. Inspection of Table 2 reveals that accuracy does not notably decrease over blocks, indicating that participants did become less motivated across trials.⁶ Thus, we were justified in using the entire set of choices for classification.

Table 2: Accuracy scores (number of treasure points) for each of the four decision blocks. (Standard deviations are in parentheses.)

To further illustrate the striking differences between age groups, we more closely examined the types of prediction patterns in our task set separately. In two additional analyses, we focused on patterns T4 and T6.

T4 is the only pattern in which each cue makes only one positive prediction and all predictions differ (Figure 2). Thus, this is an ideal pattern with which to check whether individuals are sensitive to differences in cue validities. Normatively, a decision maker should weight predictions or select cues in accordance with the rank order of cue validities. The high-validity cue predicts a treasure hidden in the third house (prediction “1” for O3 in Figure 2). Consequently, one should expect that decision makers choose O3. Figure 4 depicts mean choice-frequencies separately for each age group. Adults almost always choose O3. Elementary schoolers less frequently choose O3 compared to adults; still, O3 is the dominantly preferred choice in this age group. Preschoolers, however, differ strikingly in their choices. They choose O3 less frequently than O1 and O2. According to this observation, a GLM analysis produced a moderately strong age-by-choice interaction effect (F _Pillai-Spur (4, 210) = 29.37, p < .01, η²_partial = .36). Moreover, the main effect for the repeated-measurement factor choice was also strong, reflecting the observation that frequencies of O1-choices differ markedly between age groups (F _Pillai-Spur (2, 104) = 98.32, p < .01, η²_partial = .65; note that the between effect for age cannot be computed because the sum of choice frequencies is constant.)

Figure 4: Choice frequencies in type 4 prediction pattern. Error bars show 95% CI.

T6 (Figure 5) is another interesting candidate for assessing probabilistic decision making. In contrast to patterns discussed above, this pattern is quite complex. Some cues make multiple positive predictions, resulting in ties. Nevertheless, the application of WADD, NB, and LEX⁷ will jointly result in O1-choices. An EWQ-rule would result in an even distribution of O1 and O3 choices. None of the strategies, however, would lead to a dominance of O2 or O3 choices. Figure 4 displays the mean frequencies of choices by age groups. Evidently, adults show a strong preference for O1. O1 is also the dominantly chosen option in elementary schoolers, although they less frequently choose O1 as compared to adults. This pattern reverses for preschoolers, who predominantly choose O3 – a behavioral tendency that is not predicted by any of the three strategies. In a GLM-ANOVA, these observations manifest themselves in a strong interaction effect between age and choice (FPillai-Spur (4, 210) = 28.31, p < .01, η²_partial = .35). Again, the main effect for the repeated-measurement factor “choice” was also strong (FPillai-Spur (2, 104) = 93.48, p < .01, η²_partial = .64).

Figure 5: Choice frequencies in in type 6 prediction pattern. Error bars show 95% CI.

A high proportion of preschoolers’ choices (41%) did not resemble our tested models, indicating that we did not hit all potential variants of strategies (Hilbig, 2011) that might be used by children. A closer post-hoc inspection of the choices of some preschoolers revealed that they might have applied a strategy that is sensitive to prediction patterns in the matrix. We re-ran the Bayesian analysis as described in Appendix A with a new strategy (EQW*). Analyses revealed that differential application of EQW* could account for choices in approximately 36% of all preschoolers (see Appendix B for the more detailed post-hoc explanation). It is possible that children used other strategies that are not usually considered in the decision-making literature. Lang and Betsch (2018) found that preschoolers are able to systematically apply a couple of non-adaptive strategies (e.g., switching between options). It is surely a promising line of future research to consider the use of non-adaptive strategies. This research, however, focuses on WADD application, and, hence we do not consider such variants further.

4 Discussion

Models of adaptive decision making (e.g., Payne et al., 1993) predict that the application of linear strategies of information integration such as the weighted additive rule (WADD) should be most prevalent in compensatory environments. The absence of constraints that hinder information acquisition (e.g., time pressure, access costs) and strategy contingent feedback should further promote WADD application. To investigate the development of WADD use, we exposed children and adults to a compensatory, probabilistic decision environment in which the dispersion of cue validities (i.e., the probabilities that cue-predictions are correct) was low. Decision time was unconstrained. An open information board format further encouraged individuals to use WADD, as all cue predictions could be directly inspected and did not have be stored in memory before making a decision. Moreover, we reinforced WADD by feedback. Altogether, these environmental characteristics set up a task that ideally fosters the application of WADD.

In support of our hypothesis, the majority of adult participants were classified as WADD users (74%). In striking contrast to this finding, WADD was not the dominant strategy in children. No preschoolers (6 y.) and only 21% of the elementary schoolers (9–10 y.) could be classified as users of the WADD rule. More than half of the elementary schoolers (52%) were systematically sensitive to differences in probabilities in their decisions as evidenced by the sum of WADD, NB, and LEX users. In preschoolers, however, we were unable to identify individuals who based their decisions on probabilistic information. The only systematic decision behaviour in the six-year olds in our sample was consistent with an equal weight rule (EQW), which bases decisions on all prediction values without weighting.⁸

Why did only so few children apply a WADD rule in our study? One might be tempted to attribute this finding to global deficits in children, such as insensitivity to probabilistic information, deficits in learning from probabilistic outcome distributions (feedback), immature general cognitive abilities (e.g., IQ, executive functioning), specific cognitive deficits in applying rules of weighted aggregation of information, or simply a lack of task comprehension. There is evidence, however, that children reveal a sensitivity to probabilities from an early age on (e.g., Denison & Xu, 2014). Preschoolers as young as 4 year old are sensitive to probabilistic outcome distributions (e.g., Pasquini, Corriveau, Koenig & Harris, 2007). In a series of studies, Lehmann and Betsch (2018) did not find support for the notion that performance in the Mousekids tool covaries with various measures of cognitive ability (e.g., memory span, executive functioning, selective attention, etc.). Most research from various areas of cognition suggests that humans from an early age on are capable of integrating multiple pieces of information in a linear fashion, as required by the WADD rule (see Betsch, Ritter, Lang & Lindow, 2016, for an overview; but see, e.g., Jansen & van der Maas, 2002, for conflicting evidence). In spatial perception and categorization, for example, children and even animals (e.g., snails, Gallistel, 1980) are capable of performing weighted integration procedures “in a near optimal fashion” (Cheng, Shettleworth, Huttenlocher & Rieser 2007, p. 625). In an evaluative judgment task, preschoolers (5–6 y.) were able to integrate probabilities and values in line with the predictions of models involving multiplication (Schlottmann, 2001) similar to the WADD rule. In other studies using a paradigm similar to our study, preschoolers could also apply weighting operations – unfortunately, they did not use probabilities as weights but rather an experimentally induced “lure” that was irrelevant from a normative perspective (Betsch & Lang, 2013; Betsch et al., 2014, Exp. 2). Finally, the possibility that children do not understand the treasure hunt task in our Mousekid tool has been ruled out by applying measures of task comprehension (Betsch, Lehmann, Lindow & Buttelmann, 2018). Hence, none of these factors alone provides a plausible account for the observed reluctance in children to apply a WADD rule.

Deficits in the application of WADD thus presumably originate elsewhere. In the following, we suggest an explanation that presumes an interaction between task features, mental representation, and the status of conceptual knowledge in children. Specifically, we suspect that in our task the formation of a subjectively meaningful mental representation that properly reflects the informational structure of the given information is impaired in children due to their lack of developed explicit conceptions of probability.

Several features of the stimulus environment can either help or hinder the formation of representations that foster accuracy in task performance. In an illuminating paper, Wohlwill (1968) described three important task dimensions: redundancy, selectivity, and contiguity. We will briefly illustrate these dimensions using the example of the marble task – which Schlottmann (2001) used to demonstrate that even preschoolers can integrate probabilities and values in a multiplicative-like fashion, as predicted by utility theory.

Redundancy can be conceptualized as the degree of inter-correlation of cues predicting a criterion. In her marble task, Schlottmann (2001) visualized the probability and value of outcomes in the following manner (see also Schlottmann & Wilkening, 2012, p. 62). A marble was shaken in a tube with two clusters of colored segments (e.g., blue, yellow). Above each cluster, the potential gain was depicted (crayons). Value was manipulated by varying the number of crayons above each cluster (e.g., 6 for blue, 1 for yellow). Probability (e.g., 80% chance of winning if the marble stops in the blue cluster) was manipulated by varying the number of segments in a cluster (e.g., 4 segments in the blue, 1 segment in the yellow cluster). In this particular example, probability representation is redundant. Two cues, cluster size and the number of segments in the cluster, are correlated. Thus, these two cues jointly reference probability. Selectivity refers to the amount of irrelevant information that is contained in the task and should not affect the response. The graphical presentation in the marble task contains only relevant information – i.e., the number of crayons representing value and cluster size/segment number representing probability. Contiguity refers to the spatial and temporal separation of relevant information. In the marble task, contiguity between probability and value is high. The crayons are depicted directly above the right and left cluster in the tube. The blue and the yellow clusters represent the two potential outcomes of the task. As such, the outcomes contain all relevant information in a contiguous arrangement – values (number of crayons) and probability (size/number of segments).

According to Wohlwill (1968), perceptual tasks are characterized by a high degree of redundancy and contiguity and a low demand for selectivity. Individuals can respond to such tasks intuitively without the need of having formal conceptions and processes properly developed. The less a task contains such facilitative features (absence of redundancy, low contiguity, high selectivity), the more the individual must rely on conception to respond in a coherent manner. Although our paradigm contains many child-friendly features (e.g., an enjoyable task, animals as advice givers, learning and graphical presentation of cue validities, etc.) it is not a purely perceptual task. Most importantly, the matrix format, typical for all information board studies, spreads out decision criteria and options and hence decreases contiguity. Unlike Schlottmann’s marble task, in which options, outcomes, and probabilities are depicted closely together and function somewhat as a combined cue, our task sorts options and probabilities into margins of the board and outcomes into the grid. This presentation format requires some active structuring to link options, outcomes, and probabilities together and form an accurate mental representation. Another feature relates to selectivity. Advice givers (animals) do not simply make predictions regarding the target outcome (e.g., house A contains a treasure). Instead, they also make non-focal predictions (e.g., house B and house C contain a spider). Although the non-focal information is not irrelevant per-se, it is not necessary to make correct decisions, although it does increase the informational input. A more selective environment that depicts only the information necessary and sufficient for making a decision might enhance the likelihood that children form accurate representations of the stimulus input.

Analysing our task on Wohlwill’s dimensions gives rise to another interpretation that is, admittedly, a bit farfetched given the current state of empirical evidence. Mousekids is not purely a perceptual task. Thus, it cannot be coherently solved without the help of conception. As such, deficits in performance (i.e., application of WADD) may allow us to draw conclusions regarding the state of conception in the children in our sample. Failing to classify any preschoolers as users of a probability-sensitive strategy (WADD, NB, LEX) may be taken as evidence that conceptions of probability and their role as weights in decision making are insufficiently developed in this age group. In our task, these conceptions might be necessary to properly encode probabilistic information and form a meaningful representation that associates outcome values with weights. The still low percentage of WADD, NB, and LEX users in nine to ten year-olds indicates that it takes a long time for these conceptions to be consolidated.

From what age are children capable of adapting to probabilistic environments? With our discussion above, we aimed at showing that empirical evidence should take into account the interaction of levels of cognitive processing and task features. Cognitive ability can be assessed on different levels. Mastering a perceptual task does not necessarily imply that the individual is capable of solving a structurally similar task that lacks the facilitative features fitting into formats of intuitive processing. Comparing evidence across studies cannot be achieved by statistical means only as, for instance, by subjecting data to meta-analysis. Rather, we must analyse tasks and paradigms on the qualitative level to detect potential origins of improved or impaired performance. Insights from these analyses should result in research endeavours that systematically compare and vary features of our paradigms to determine when decision makers in the course of their cognitive development overcome what Schlottmann and Wilkening (2012, p. 77) have nicely termed being “at the mercy of circumstance”.

Taken together, our research indicates that, before school age, children fail to apply a WADD rule to a task in which multiple probabilities and values are presented in a dissociated fashion.