Uncertainty and computational complexity

Abstract

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’.

Imprecise neural computations as source of human adaptive behavior in volatile environments

“…see Results). This finding confirms that as previous advocated 12,15 , resolving temporal structural uncertainty through higher-order inferences bearing upon the actual, highdimensional space of possible temporal structures is actually ineffective and even deleterious in view of the informative poverty of environment feedbacks.…”

“…Empirical studies show that consistently, humans adjust their behavior with respect to the environment volatility and in accordance with the computation of such higher-order inferences 1,11,12,13,14 . However, these inferences are complex and may rapidly yield intractable computations 13,15 . This computational complexity problem casts severe doubt upon the hypothesis that such higher-order inferential processes operate in the brain 13 .…”

The importance of constraints on constraints

“…In sharp contrast to the prescription of expected utility theory that actions should be chosen based on their expected consequences, people often act habitually without deliberating about consequences (Dolan & Dayan 2013). The contrast between the enormous computational complexity of expected utility maximization (Bossaerts & Murawski 2017;Bossaerts et al 2018) and people's limited computational resources and finite time suggests that habits may be necessary for bounded-optimal decisionmaking. Reusing previously successful action sequences allows people to save substantial amounts of time-consuming and errorprone computation; therefore, the principle of resource rationality in Equation 3 can be applied to determine under which circumstances it is rational to rely on habits.…”

Advancing rational analysis to the algorithmic level

“…They propose that instead of making optimal use of noisy representations, the brain uses approximations that entail systematic biases (Beck et al 2012). From the perspective of bounded optimality, approximations are necessary because the computational complexity of decision-making in the real world far exceeds cognitive capacity (Bossaerts & Murawski 2017;Bossaerts et al 2018). People cope with this computational complexity through efficient heuristics and habits.…”