Volz and Gigerenzer Decisions under risk versus uncertainty under risk does not necessarily generalize to what it should do under uncertainty. 2. Cognitive processes in decisions under risk are not the same as in decisions under uncertainty. We argue that cognitive processes observed under risk do not necessarily generalize to those the brain uses under uncertainty. Speciﬁcally, we argue: Risk: Value-based statistical thinking (e.g., Bayesian proba- bility updating plus utilities) is sufﬁcient for making good decisions, provided that the problem is computationally tractable. Uncertainty: Statistical thinking is no longer sufﬁcient; heuristic thinking is required. Much of cognitive neuroscience does not distinguish between risk and uncertainty. For instance, consider the claim, made in various forms, that the brain is Bayesian (e.g., Friston, 2010). Such a brain will likely provide optimal decisions only in small worlds, which are rare. Or consider the claim that there are two systems of reasoning: System 1, which is fast, heuristic, and prone to error, and System 2, which is slow, in keeping with the laws of probability, and rational (Sloman, 1996; Kahneman, 2011; for a critique, see Gigerenzer and Regier, 1996; Keren and Schul, 2009; Kruglanski and Gigerenzer, 2011). This two-system view does not consider that the laws of probability are sufﬁcient for rationality in a small world only. In uncertain worlds, however, heuristics are indis- pensable. That is, both logic and heuristics are tools for different classes of problems. For instance, the recent ﬁnancial crises illus- trate that statistical tools for estimating risk, Bayesian or otherwise, failed consistently in the real, uncertain world of ﬁnance (Taleb, 2010). They are optimal when risks are known, but not in a world of uncertainty. Applying normative theories of risk to uncertain worlds can in fact lead to disasters. With respect to the ﬁnancial crash of 2008, Stiglitz (2010) noted: “It simply wasn’t true that a world with almost perfect information was very similar to one in which there was perfect information” (p. 243). In sum, norms derived from assuming known risks do not simply generalize to norms under uncertainty. RATIONALITY OF RISK 6D RATIONALITY OF UNCERTAINTY The point that the calculus of probability can determine the best action under risk but not under uncertainty is not new; it has been made as often by statisticians as it has been forgotten by cogni- tive scientists. Savage (1954) devoted the ﬁrst half of his seminal book Foundations of Statistics to Bayesian decision theory, and the second half to heuristic decisions, such as minimax (choose the option that minimizes the maximal loss). Arrow (2004) similarly writes that in uncertain, ill-speciﬁed worlds, unbounded ratio- nality (i.e., expected utility optimization) “has no meaning at all” (p. 54). What is new are scientiﬁc demonstrations that show that applying an optimization model to an uncertain world can lead to decisions that are normatively inferior to simple heuristics (see Gigerenzer et al., 2011). Here is an illustration: MEAN-VARIANCE OPTIMIZATION LEADS TO INFERIOR RESULTS IN THE REAL WORLD Consider ﬁnancial investment A normative theory of how to allocate money to N assets is Markowitz’s Nobel prize-winning mean-variance model. Like all optimizing theories, it assumes a small world with perfect knowl- edge about the relevant parameters. Is this theory also optimal in the real, uncertain world of ﬁnancial investment, where parame- ter values are not known for certain but need to be estimated? De Miguel et al. (2009) compared the mean-variance model with a heuristic called 1/N, or equality heuristic. The heuristic sim- ply allocates money to N assets equally. The result was that 1/N consistently performed better in out-of-sample prediction (an ele- mentary form of uncertainty). Cross-validation is a prime example of out-of-sample prediction: the data is divided into two com- plementary subsets: the in-sample data set, which is used for ﬁtting the parameters of the competing models and an out-of- sample data set, which is used for testing how well the models predict (see also below). Note that in data ﬁtting, that is, when all data are known, the optimizing model always wins, but not in prediction. None of 12 other optimization models, Bayesian or otherwise, could consistently predict better than the simple heuristic. This result contradicts the widespread view that heuristics are always second best to logic and statistical optimization models. This view makes no distinction between risk and uncertainty. Researchers in this tradition have evaluated people’s reliance on 1/N negatively and attributed it to their cognitive limitations. However, ignoring part of the information is what makes heuris- tics robust for the unknown future, whereas by trying to integrate all information and estimate the weights, complex strategies such as the mean-variance portfolio suffer from overﬁtting the past. The mathematically sophisticated reader who wants to understand why and when simple heuristics can be more accurate than com- plex statistical methods will ﬁnd an answer in the bias-variance dilemma (Gigerenzer and Brighton, 2009). THE ECOLOGICAL RATIONALITY OF SIMPLE HEURISTICS The fact that simple heuristics often outperform “optimization” models in situations of uncertainty has been demonstrated many times over (see Czerlinski et al., 1999; Gigerenzer and Brighton, 2009; Gigerenzer et al., 2011). In order to deal with an uncertain world, the brain relies on an adaptive toolbox of heuristics. Accord- ingly, intelligence is deﬁned as the degree of knowing in which situation to use which heuristic. The scientiﬁc study of this nor- mative question is called the study of the ecological rationality of a heuristic. For instance, 1/N tends to outperform mean-variance optimization in situations where predictive uncertainty is high (stocks are hard to predict), the number of options N is large (the optimization models have to estimate more parameters which leads to more error), and the sample size is relatively small. In uncertain worlds with these features, 1/N can be expected to be both faster and more accurate than the mean-variance optimiza- tion. When would mean-variance outperform 1/N ? De Miguel et al. (2009) estimated that with 50 assets, one would need some 500 years of stock data before the optimization model is proﬁtable. Humans rely on the 1/N heuristic not only for ﬁnancial invest- ment. Many parents who have two or more children try to distrib- ute their time and love equally. For three or more children, this heuristic paradoxically predicts interesting inequalities in the long run because the ﬁrst and last-born get more time, dependent on the spacing between births. Tests have provided empirical evidence Frontiers in Neuroscience | Decision Neuroscience July 2012 | Volume 6 | Article 105 | 2