On a daily basis people are faced with choices like running a yellow light, smoking a cigarette, or staying up late at night. These risky decisions are often not one-shot and deterministic, but made repeatedly and involve uncertainty. In laboratory environments, such decisions can be studied using behavioral tasks that involve repeated decisions in the face of uncertainty. Known as sequential risk-taking tasks, their sequential nature incorporates uncertainty in the decision process and thus may require learning and exploration (Busemeyer and Stout, 2002, Daw et al., 2006), making different demands on our cognitive systems than one-time risky choices (e.g., making a midlife career change). Therefore, sequential risk-taking decisions have the potential to tap multiple constructs underlying risk-taking propensity (Denrell, 2007, Haffke and Hübner, 2020). For example, we could learn how rational the decisions are during the task (van Ravenzwaaij et al., 2011), and how participants learn about and react to the underlying risk and adjust their strategies (Frey et al., 2015). The added complexity of sequential tasks could help us build a more complete profile of risk-taking behavior.

There are several prominent, representative sequential risk-taking tasks, including the Balloon Analogue Risk Task (BART; Lejuez et al., 2002), the Columbia Card Task (CCT; Figner et al., 2009), and the Iowa Gambling Task (IGT; Bechara et al., 1994). These tasks all involve sequential decisions with increased reward for every choice, with the potential reward going hand in hand with increased risk of losing. The behavioral measures of these tasks were shown to be associated with characteristics of risk-taking behavior, and are widely used in the field of decision making (Buelow, 2015, Buelow and Blaine, 2015, Buelow and Suhr, 2009, Cauffman et al., 2010, Lauriola et al., 2014, Lejuez et al., 2003). To gain more information from these tasks and depict the underlying cognitive processes, computational models have been developed to describe (simulate) task performance (e.g., Ahn et al., 2008, Busemeyer and Stout, 2002, Wallsten et al., 2005, Pleskac, 2008). Models assume different cognitive mechanisms and functional forms, and are compared using behavioral and simulated data. Model development is an important yet highly challenging step towards understanding the processes that underlie decision making.

In the modeling of sequential risk-taking decisions, most of the existing models assume a single response pathway (e.g., Steingroever et al., 2013, Wallsten et al., 2005) that is controlled and attention-demanding, as the theories on which the models are based describe deliberative decision processes (e.g., Brandstätter et al., 2006, Tversky and Kahneman, 1992). However, recent evidence suggests that an automatic and effortless response pathway could also emerge in sequential decisions, and having multiple response pathways could explain better the observed choices and response times (RTs) in these decisions (e.g., Pleskac and Wershbale, 2014, Simonovic et al., 2018). This reasoning is motivated by the prevalent idea that human cognition involves two main families of processes, formalized as dual-process theory. Despite different names, most dual-process theories have reached a consensus regarding the defining characteristics of the two information-processing systems, often referred to as System 1 and System 2 (e.g., Kahneman and Frederick, 2002, Stanovich and West, 2000). System 1 is described as fast, unconscious, automatic, intuitive, and experiential, while System 2 is characterized as slow, conscious, controlled, analytic, and rational (Evans, 2008, Kahneman, 2003, Mukherjee, 2010). Dual-process theories have been widely applied to explain phenomena observed in higher cognition, including reasoning, decision making, and social judgement (e.g., Chaiken and Trope, 1999, Reyna, 2004, Sloman, 1996). Therefore, the current state of modeling sequential risk-taking decisions could benefit from applying a dual-process framework and building computational dual-process models to account for both choice and RT behavior.

We sought to model the engagement of these two processes in the BART, which is one of the most commonly used behavioral measures of risk-taking propensity. We focused on this task for several reasons. The task is a representative instance of the class of sequential risk-taking tasks, in which participants are likely to use different response pathways depending on where they stand in the sequence of decisions in the experiment (e.g., beginning, middle, or end; Figner et al., 2009). A second reason is that behavior in the BART has been found to correlate with real-world risk-taking behavior (e.g., Bornovalova et al., 2005, Hopko et al., 2006), thus modeling the multiple processes underlying this behavior could shed light on whether and how the two processes (fast and automatic vs. slow and deliberate) used in the decisions contribute to different levels of risk-taking in everyday life. The third reason is that the BART has established single-process models that describe the decision process during the task, which serve as good foundations for developing dual-process models.

The BART is a computerized task for measuring one’s propensity for risky behavior. During the BART, participants are presented with a sequence of balloons and they need to decide whether to pump them. On each trial, as shown in Fig. 1, participants can either choose to take a risk to earn more money (to “pump”) or to settle on what they have already earned (to “collect”). These two actions simulate risky situations involving sequential decision making in daily life. Each successful pump adds 5 cents to a temporary bank (shown in the center of the balloon). If a pump makes the balloon explode, then the temporary bank is emptied, and a new trial begins. When participants choose to collect, which ends the current trial, all the money in the temporary bank becomes permanent and is counted towards the final total earnings (shown in the piggy bank). The goal of participants is to earn as much money as possible at the end of 30 trials.

The traditional index of risk-taking propensity in the BART is the adjusted score, calculated as the average number of pumps on unexploded balloons. The adjusted score of the BART has established external validity in many naturalistic risk-taking behaviors, such as smoking, substance use, and unprotected intercourse (e.g., Bornovalova et al., 2005, Lejuez et al., 2003). However, this composite measure glosses over the complexities of decision making, neglecting useful information in the data, such as changes in pumping behavior over trials as participants explore different strategies to maximize performance (Schmitz et al., 2016). The repetitive pumping required in the task helps to provide information about the decision making strategy, and formal computational models of the BART have been developed to extract more information from the data in order to depict the decision processes underlying this task.

Many past experiments using the BART analyzed only choice data, and models were developed assuming a single response pathway where the participant carries out a simple pumping strategy (i.e., whether to pump or to collect given an opportunity). Models were proposed with varying mental representations and processes, and then model comparison studies were conducted to select the model that best describes observed choice data. In this section we review the single-process models that emerged as the best performing model to describe decision making in the BART.

Wallsten et al. (2005) made the first attempt to model behavior in the BART and favored a model (“Model 3”) that fitted the data better than nine competing models, which was later referred to as the Bayesian sequential risk-taking model (BSR; Pleskac, 2008). The BSR model is the first successful attempt to model choice behavior in the BART that provides meaningful parameters correlated with unhealthy and unsafe risky behaviors (Wallsten et al., 2005). However, some parameters in the BSR model showed low identifiability (van Ravenzwaaij et al., 2011), suggesting that it could be further improved.

Recently, efforts have been made to improve BSR and provide a better depiction of the psychological processes underlying the BART. Park et al. (2021) proposed the Exponential-Weight Mean-Variance (EWMV) model that utilizes prospect theory (Kahneman & Tversky, 2013) and mean–variance analysis (Markowitz, 1952). The EWMV model aims at improving parameter identifiability and providing intuitive interpretation of the learning parameters. Zhou, Myung, and Pitt (2021) proposed the Scaled Target Learning (STL) model, which focuses on the trial-by-trial learning in the BART and uses a relatively simple form of learning from experience.

These three models (i.e., BSR, EWMV, and STL) have been found to outperform competitors in multiple model comparison tests and explain observed risk-taking behavior the best (Park et al., 2021, Wallsten et al., 2005, Zhou et al., 2021). Nonetheless, they are all single-process models, assuming a single response process where participants carry out a distance-to-target evaluation or a calculation of subjective utility on each pumping opportunity. Such an assessment is assumed to be controlled and attention-demanding (System 2). While these models provide insight into sequential risky decision making, they can account only for deliberative responses, not automatic pumping that is surely also involved in the BART and possibly other sequential decision making tasks.

In the field of judgment and decision making, many phenomena are often attributed to the cooperation between the two systems proposed by dual-process theory, such as the framing effect, where people’s choices are influenced by the framing (i.e., wording) of a problem (Tversky & Kahneman, 1981), and biases that are linked with heuristics like representativeness and availability (Kahneman and Tversky, 1972, Kahneman and Tversky, 1973). In sequential risk-taking decisions like the pumping behavior in the BART, participants often need to learn from past experiences to determine the optimal strategy (Busemeyer and Stout, 2002, Wallsten et al., 2005), and this learning process demands different levels of attention and memory throughout the task that may invoke responses from different systems (Barron & Erev, 2003). For example, the BART can involve repetitive pumping when the perceived risk is low, which could elicit the development of automatic responses within each trial. On the other hand, as risk increases, participants might become more cautious and deliberative in their pumping. These features of the task make it likely the two systems influence performance and should be included in a model of the BART.

Pleskac and Wershbale (2014) were the first to explore modeling of the BART assuming multiple response pathways. They collected RT data in the BART and found support for the existence of two response pathways. They proposed a non-assessment process which is a learned, automatic response and an assessment process which is a controlled, deliberative response. The automatic pump response could be a learned response, as their data showed that the faster pump responses appeared more often with task exposure. However, it could also be a default process as dual-process theory assumes, as RTs became fast and remained so after only two or three pump opportunities in the first few trials. They also laid the groundwork for modeling the multiple response pathways in the BART by developing a formal computational model, named the Dual Response Bayesian Sequential Risk-taking model (drBSR), and demonstrated that it can account for observed choices and RTs in the BART better than models that include only a single response pathway, supporting the superiority of incorporating multiple response pathways in modeling sequential risk-taking decisions.

While introduction of the drBSR model is an important first step in developing a dual-process model of the BART, it also raises questions about the model configuration. For instance, is the formulation of BSR the best way to describe the distance-to-target evaluation in the assessment pumps? Should the non-assessment pump be modeled with an ex-Gaussian distribution assumed by drBSR? Should the two systems be modeled sequentially, in parallel, or in some other way? Diederich and Trueblood (2018) provide guidance on this last question. They proposed a framework to computationally model and test the interaction between the two systems, and found the two-stage models to be significantly better than the one-stage models in predicting and describing data patterns. The drBSR model combined with the dual-process framework of Diederich and Trueblood (2018) provided a starting point for taking on these questions to advance modeling of the BART.

Our approach involved exploring the pros and cons of alternative model configurations and their implications for sequential risky decision making. We first developed a set of 24 BART models that varied in the evaluation model (e.g., BSR vs STL) and how the two processes are configured. Then, we compared the quality of the proposed models using both simulations (i.e., parameter identifiability and model recovery) and fitting them to behavioral data (i.e., predictive accuracy). We evaluated how well the dual-process models account for observed behavior in the BART and identified a best-performing model.