In the cognitive and decision sciences, much research suffers from a lack of formalism. This is particularly the case for qualitative accounts of behavior proposed, for instance, within the heuristics-and-biases framework (Kahneman et al. Reference Kahneman, Slovic and Tversky1982), or within related dual process theories of cognition (Sloman Reference Sloman1996). We applaud Pothos & Busemeyer's (P&B's) attempt to promote a formal framework that contributes to remedying this shortcoming and that has a high potential for being innovative and useful. With that being said, we take issue with three aspects of the quantum probability (QP) program.

First, we posit that outcome models should be complemented by process models. What level of description do P&B envision for QP models? One of the central goals of many psychological theories is to describe cognitive processes. In contrast, behavioral economists and cognitive scientists working with, for example, Bayesian models (e.g., Griffiths et al. Reference Griffiths, Kemp, Tenenbaum and Sun2008) focus on predicting the outcomes of behavior, without necessarily aspiring to provide plausible accounts of the underlying processes (Berg & Gigerenzer Reference Berg and Gigerenzer2010). We worry that the QP program falls into this class of outcome-oriented (or as-if) models, banishing algorithmic-level accounts of memory, perceptual, motivational, and decisional processes into the behaviorist's black box. We challenge P&B to demonstrate how the QP framework can contribute to developing process models such as those proposed by the adaptive toolbox program (Gigerenzer & Selten Reference Gigerenzer and Selten2001). The repertoire of fast-and-frugal heuristics developed in this program includes algorithmic rules that specify how the cognitive system searches for information, when it will stop searching, and how it will combine the acquired pieces of information in order to make a decision. Consider, for instance, the priority heuristic (Brandstätter et al. Reference Brandstätter, Gigerenzer and Hertwig2006). This simple lexicographic strategy for making risky choices between gambles is composed of three rules that operate on the probability and outcome vectors constituting the gambles. For example, the first rule prescribes in what order the information contained in these vectors is searched. The rules of these and other heuristics have been used to motivate process predictions about reaction times (Bröder & Gaissmaier Reference Bröder and Gaissmaier2007), eye movements (Reisen et al. Reference Reisen, Hoffrage and Mast2008), functional magnetic resonance imaging (fMRI) data (Volz et al. Reference Volz, Schooler and von Cramon2010), and the amount of information looked up on a computer screen (Johnson et al. Reference Johnson, Schulte-Mecklenbeck and Willemsen2008). Process models help to gain a deeper understanding of behavior, and their predictions – albeit not always free of scholarly disputes (e.g., Marewski et al. Reference Marewski, Pohl and Vitouch2010; 2011) – can eventually differentiate among competing theories that make identical outcome predictions.

Second, we posit that the process models we ask for should be informed by an analysis of the structure of the environment – after all, it takes two to tango. As has been stressed by many cognitive modelers (Anderson & Schooler Reference Anderson and Schooler1991; Dougherty et al. Reference Dougherty, Gettys and Ogden1999; Gigerenzer et al. Reference Gigerenzer, Hoffrage and Kleinbölting1991; Oaksford & Chater Reference Oaksford and Chater1998; Simon Reference Simon1956), the human cognitive system is adapted to its ecology. To use Brunswik's (Reference Brunswik, Bruner, Brunswik, Festinger, Heider, Muenzinger, Osgood and Rapaport1964) metaphor, mind and environment are like spouses who have co-evolved, who mutually shaped each other, and who cannot and should not be separated. The heuristics-and-biases program does not consider the situated nature of cognitive processes; rather, typically its heuristics are hypothesized to be invoked independent of context and there is no ecological analysis specifying how performance depends on environmental factors (Gigerenzer Reference Gigerenzer1996). P&B use the heuristics that have been proposed in the heuristics-and-bias tradition to motivate the QP program, pointing out that these heuristics reveal the insufficiencies of the classical probability program. While the QP program addresses some of these insufficiencies, others might be remedied by more explicitly considering (1) how environmental structures have shaped cognitive processes, (2) how such structures can select cognitive processes, and (3) how these processes, once selected, perform differentially in environments with different statistical structures. For instance, the shape of the distribution of environmental variables influences the performance of the QuickEst and social-circle heuristics, two fast-and-frugal heuristics for estimating quantities and frequencies, respectively (Hertwig et al. Reference Hertwig, Hoffrage, Martignon, Gigerenzer and Todd1999; Pachur et al. Reference Pachur, Hertwig, Rieskamp, Hertwig and Hoffrage2013). More examples of how the functioning of cognitive processes depends on environmental structures and how an analysis of the environment can assist researchers to develop – what one might term – ecological process models can be found in Ecological Rationality: Intelligence in the World (Todd et al. 2012) and in Simple Heuristics in a Social World (Hertwig et al. 2013).

Third, we posit that such ecological process models should be integrated into cognitive architectures. As P&B correctly point out, any theory requires constraints. One way to constrain psychological theories consists of introducing axioms and positing that cognition adheres to these axioms. Another way to constrain theories requires formulating psychologically grounded guiding principles that reflect empirical observations about behavior (cf. Anderson & Lebiere Reference Anderson and Lebiere2003). Consistent with the latter approach, we and others (Marewski & Mehlhorn Reference Marewski and Mehlhorn2011; Marewski & Schooler Reference Marewski and Schooler2011; Nellen Reference Nellen, Detje, Dörner and Schaub2003; Schooler & Hertwig Reference Schooler and Hertwig2005) have modeled a few of the fast-and-frugal heuristics – including their dependencies on environmental, mnemonic, and perceptual variables – with the ACT-R architecture (Anderson et al. Reference Anderson, Bothell, Byrne, Douglass, Lebiere and Qin2004). This architecture integrates models of memory, perception, and other components of cognition into one unified theory. In doing so, ACT-R incorporates the findings of decades of empirical work into computer code and mathematical equations, which, in turn, helps to constrain new models of cognition – including heuristics – when these are implemented in the architecture. We wonder to what extent the QP program's axiomatic approach provides added value for constraining ecological process models beyond what can be achieved by implementing them into psychologically plausible architectures. We hasten to add that we also wonder whether the QP program's axiomatic approach might aid, conversely, in further refining such architectures.

To conclude, we believe the QP program may do a lot of good. However, we ask ourselves whether it could do even better if (1) its presumed focus on outcome models were complemented by process models that (2) are, ideally, informed by ecological analyses and (3) integrated into cognitive architectures.