Demons of Ecological Rationality

How can resource-bounded minds like our own make rational or otherwise “good” decisions in an uncertain and complex world (Oaksford & Chater, 1998; Simon, 1957, 1990)? The Adaptive Toolbox theory answers this question by defining human rationality in terms of a degree of adaptation of decision strategies (heuristics) to different environments (Gigerenzer & Todd, 1999; Todd & Gigerenzer, 2012). When heuristics are adapted to the environment and lead to “good enough” (or even high-quality) decisions, they are said to be ecologically rational. For almost two decades, this theory has been considered a tractable alternative to classical theories of human rationality based on logic or probability theory (Gigerenzer, 2015; Gigerenzer & Todd, 1999). These classical theories have been criticized for postulating intractable (e.g., NP-hard)1 computations (Arkes, Gigerenzer, & Hertwig, 2016; Gigerenzer, 2008; Oaksford & Chater, 1998), which suggests that humans must possess demonic computational powers in order to make rational decisions (so-called demons of rationality; Gigerenzer & Todd, 1999; Goldstein & Gigerenzer, 1999). It is widely assumed that the Adaptive Toolbox theory circumvents the intractability problem that plagues classical accounts of human rationality, because heuristics are by definition tractable. Yet the notion of ecological rationality hinges on the existence of tractable adaptation processes. Here, we present an argument that, contrary to common belief, the Adaptive Toolbox theory has not yet tamed the intractability demon. Rather, the demon is hiding in the theory's cornerstone assumption that ecological rationality is achieved by processes of adaptation, such as evolution, development, or learning. The Adaptive Toolbox theory provides an influential account with many empirical successes (Brighton & Gigerenzer, 2012a,b; Bröder, 2000; Gigerenzer & Goldstein, 1996; Pohl, 2006; Schooler & Hertwig, 2005; Todd, 2001; Todd & Gigerenzer, 1999; 2012), which has led to its adoption in cognitive science, psychology, business, economics, law, philosophy, cultural studies, and medicine (Marewski & Gigerenzer, 2012; Todd & Gigerenzer, 2012). Despite its empirical successes, the theory remains incomplete to date (Todd & Gigerenzer, 2012). So far, research has focused on hypothesizing and testing the various heuristics in the toolbox, while two key aspects of the theory so far remained unresolved: (a) the meta-decision process of selecting the right heuristic for a given environment (the selector) (Hafenbrädl, Waeger, Marewski, & Gigerenzer, 2016; Todd & Brighton, 2016) and (b) the adaptation process by which the adaptive toolbox of heuristics evolves, develops, or is learned (Schulz, 2011). First, several proposals about the nature of the selector have been suggested, but none so far is considered satisfactory (Marewski & Link, 2014). Be that as it may, it seems that to ensure tractability of the whole toolbox, minimally the selector must be fast and frugal like the heuristics that it selects (Gigerenzer & Todd, 1999, p. 32). Therefore—and to safeguard that our argument is not an artifact of a potentially intractable selector (cf. van Rooij, Wright, & Wareham, 2012)—we will work with the assumption that the selector itself is a heuristic as well. Second, the adaptation process involves creating and adapting the heuristics and the selector to be ecologically rational. It is assumed that both ontogenetic and phylogenetic adaptation processes can play a role (Todd & Brighton, 2016), but no explicit account of how this works has been put forth yet. To ensure generality of our result, we will make no assumption about the nature of the adaptation process other than that it yields toolboxes that are ecologically rational (cf. Otworowska et al., 2015). Earlier work (Schmitt & Martignon, 2006) has already shown that optimal toolbox adaptation (defined as a problem of cue ordering in the toolbox) is intractable. These results were used as a supporting argument for the idea that ecological rationality is not defined in terms of optimality but in terms of “good enough” cue orders (Gigerenzer, 2008). This presupposes that “good enough” toolbox adaptation would be tractable. Here we show, however, that even “good enough” toolbox adaptation is intractable. Importantly, intractability is not a property that can be derived from simulations, but given a proper formalization, it can be mathematically proven (van Rooij, 2008; van Rooij, Evans, Müller, Gedge, & Wareham, 2008). In the online supplementary materials,2 we prove the intractability of toolbox adaptation. We first formalize the notions of a toolbox (heuristics + selector), ecological rationality, and the environment (Box 1). Then, using these notions, we formally define the Toolbox Adaptation problem (i.e., given an environment, create an ecologically rational [good enough] toolbox for that environment) (Box 1). Lastly, we construct a mathematical proof that Toolbox Adaptation, so defined, is intractable (NP-hard) (Box 2). Boxes 1 and 2 sketch properties of the formalization and give the intuition behind the proof. Input: An environment, that is, a set of actions, and a set of situations (formalized as truth assignments for possible events). An upperbound on the number of heuristics () and the size of a heuristic (). A lowerbound for the level of adaptation that counts as ecologically rational (). Output: A toolbox , of bounded size, that is ecologically rational. Fig. 1 illustrates a possible (toy example) input and output for Toolbox Adaptation. Note that in this computational-level model, the toolbox consists only of fast and frugal trees (i.e., both the selector and heuristics are fast and frugal trees). A fast and frugal tree is a chain of cues with associated actions (in case of a heuristic) or a chain of cues with associated heuristics (in case of the selector). Each cue is a boolean function, evaluating whether an event () is true in a given situation. If the cue evaluates to true, then the heuristic associated with that selector cue is used (in case of the selector) or an action associated with that heuristic cue is executed (in case of a heuristic). If the cue is false, the next cue is evaluated until the last cue is reached. If this last cue is false, the first heuristic is used (in case of the selector) or the last action in the tree is performed (in case of a heuristic). The choice for fast and frugal trees is without loss of generality, because (a) many other heuristics proposed for the Adaptive Toolbox theory, such as for example, fluency heuristic, take the Best, satisficing, 1/N, default heuristic, tit-for-tat, imitate the majority, and imitate the successful (Gigerenzer, 2008), can be formally rewritten as fast and frugal trees (Sweers, 2015); and (b) if adaptation of toolboxes is intractable for some subset of heuristics, then it is also intractable for toolboxes for any superset of that. Given that our computational model of Toolbox Adaptation is an input-output mapping, it is neutral with respect to the nature of the adaptation process by which the output is reached. For instance, this process could be an ontogenetic or phylogenetic process, or a mixture of these. Furthermore, it does not make specific assumptions about how these processes are realized, for example, algorithmically. The results of (in)tractability analyses of a model like this will therefore hold for any type of algorithmic-level implementation, which could be either evolutionary, neural network, probabilistic, incremental, hill climbing, or any other type of algorithm. The reason is that computational intractability (i.e., NP-hardness) is a property of the input-output mapping, and not of a specific algorithm for computing it (Garey & Johnson, 1979). This is, in a nutshell, the strategy we used to prove that Toolbox Adaptation is NP-hard. Fig. 2 illustrates this strategy. The NP-hardness proof for Toolbox Adaptation establishes that there does not exist any general polynomial-time computable process (neither deterministic nor probabilistic3; see also van Rooij, 2008) that can adapt toolboxes to be ecologically rational (“good enough”), for all possible environments. This applies regardless of the nature of this process.4 More important, it demonstrates that Toolbox Adaptation is as difficult to compute as many other known NP-hard functions, including logic problems, such as deciding logical satisfiability of a set of logical clauses (Gary & Johnson, 1979; Oaksford & Chater, 1998), and probabilistic inference problems, such as exact or approximate inference in Bayesian networks (Abdelbar, Hedetniemi, & Hedetniemi, 2000; Kwisthout, Wareham, & van Rooij, 2011). This is an interesting observation given that one of the prime motivations for the Adaptive Toolbox theory was to move away from classical notions of rationality, based on logic or probability, in order to ensure tractability. Our proof that Toolbox Adaptation is intractable may be surprising, given that it is so widely believed that the Adaptive Toolbox theory is a tractable account of human rationality. We suspect that the belief could persist, however, because researchers have been focusing on Toolbox Application, while taking Toolbox Adaptation for granted. Here, Toolbox Application refers to the process of making ecologically rational decisions in a given environment, using a toolbox of heuristics that has already been adapted to that environment by some unspecified process. Even if Toolbox Application is free from computational demons, the demons are still hiding in Toolbox Adaptation. It is not uncommon for cognitive scientists to try to discredit theories in competing frameworks by pointing out that those frameworks run into intractability issues. But this is to no avail and is in no way our purpose here. We see intractability not as a problem for specific theories, or even for specific theoretical frameworks, but a ubiquitous feature of theoretical frameworks with high degrees of generality (van Rooij, 2008, 2015). For instance, Bayesians originally criticized logical accounts of rationality for their intractability (Chater & Oaksford, 1993; Oaksford & Chater, 1998), only to later discover that Bayesian theories themselves face intractability charges that are not easily fenced off by appeals to “approximation” or “as if” explanation (Kwisthout et al., 2011; van Rooij, Wright, Kwisthout, & Wareham, 2014). Similarly, Gigerenzer and colleagues have criticized both logical and Bayesian accounts of rationality for their intractability (Gigerenzer & Todd, 1999; Todd, 2001). By overlooking the question how adaptation of toolboxes of heuristics can itself be tractable, Gigerenzer and colleagues may not have realized that the Adaptive Toolbox theory faces exactly the same intractability charge, albeit in a different guise. From a complexity-theoretic perspective this is not surprising, but a natural consequence of the theory's high degrees of expressiveness (i.e., it has the degrees of freedom needed to encode NP-hard problems). Adopting this methodology not only has the benefit that it can potentially render a tractable version of the Adaptive Toolbox theory, but it may also sharpen the debate among logicists, Bayesians, and heuristicists. After all, classical approaches to rationality have the same methodology at their disposal (see also Kwisthout et al., 2011; van Rooij et al., 2014; van Rooij and Wareham, 2008). Applying the “tractable design cycle” to both ecological and classical accounts of rationality is a rigorous way to move forward on the question how rationality can be “tractable in the real world in which people live, not only in the small world of an experiment” (Gigerenzer et al., 2008, p. 236), as well as to assess whether or not the ecological account can really explain this better than classical accounts. The authors thank Richard Cooper and two anonymous reviewers for their useful comments. M.O. was supported by a Donders Centre for Cognition PhD grant awarded to I.v.R., and T.W. was supported by National Science and Engineering Research Council (NSERC) Discovery grants 228104-2010 and 228104-2015. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

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