Modern theories of decision-making typically model uncertainty about decision options using the tools of probability theory. This is exemplified by the Savage framework, the most popular framework in decision-making research. There, decision-makers are assumed to choose from among available decision options as if they maximized subjective expected utility, which is given by the utilities of outcomes in different states weighted with subjective beliefs about the occurrence of those states. Beliefs are captured by probabilities and new information is incorporated using Bayes’ Law. The primary concern of the Savage framework is to ensure that decision-makers’ choices are rational. Here, we use concepts from computational complexity theory to expose two major weaknesses of the framework. Firstly, we argue that in most situations, subjective utility maximization is computationally intractable, which means that the Savage axioms are implausible. We discuss empirical evidence supporting this claim. Secondly, we argue that there exist many decision situations in which the nature of uncertainty is such that (random) sampling in combination with Bayes’ Law is an ineffective strategy to reduce uncertainty. We discuss several implications of these weaknesses from both an empirical and a normative perspective.This article is part of the theme issue ‘Risk taking and impulsive behaviour: fundamental discoveries, theoretical perspectives and clinical implications’.

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Current models of decision-making more often than not ignore the level of difficulty of choices or treat it only informally. Yet, difficulty has been shown to affect human decision quality. We propose instance complexity (IC), a measure of computational resource requirements, as a generalisable framework to quantify difficulty of a choice based on a small number of properties of the choice. The main advantage of IC compared to other measures of difficulty is fourfold. Firstly, it is based on the theory of computation, a rigorous mathematical framework. Secondly, our measure captures complexity that is intrinsic to a decision task, that is, it does not depend on a particular solution strategy or algorithm. Thirdly, it does not require knowledge of a decision-maker's attitudes or preferences. And lastly, it allows computation of difficulty of a decision task ex-ante, that is, without solving the decision task. We tested the relation between IC and (i) decision quality and (ii) effort exerted in a decision using two variants of the 0-1 knapsack problem, a canonical and ubiquitous computational problem. We show that participants exerted more effort on instances with higher IC but that decision quality was lower in those instances. Together, our results suggest that IC can be used as a general framework to measure the inherent complexity of decision tasks and to quantify computational resource requirements of choices. The latter is particularly relevant for models of resource allocation in the brain (meta-decision-making/cognitive control). Our results also suggest that existing models of decision-making that are based on optimisation (rationality) as well as models such as the Bayesian Brain Hypothesis, are computationally implausible.Most theories of decision-making ignore the difficulty of making a decision [1][2][3]. They 2 assume that the decision-maker is always able to identify the best option-whether 3 it is a choice between two flavours of ice cream or a choice of investment option for 4 a retirement portfolio from thousands of available options. This is the case not only 5 for rational choice theories of decision-making [4-6], but also for theories of bounded 6 rationality [7-9] and theories of computational rationality [10,11]. All of those theories 7 assume, implicitly or explicitly, that an observed choice is the outcome of a (possibly 8 constrained) optimisation problem. 9Where decision difficulty has been taken into account, it has been done either 10 informally or in a highly domain-specific way. An example of the former are approaches 11 based on heuristics [12,13]. In this line of research, it is proposed that decision-makers 12 use simple rules or procedures as 'short cuts' to overcome various forms of cognitive 13 limitations. These approaches do not usually demonstrate, however, if and in what ways 14 the proposed heuristics overcome various cognitive limits. 15Other work on decision difficulty is domain-specific and cannot necessarily be gen-16 eralised. For example, it has been shown that the ab...