The high availability of extreme events serves resource-rational decision-making

The high availability of extreme events serves resource-rational decision-making Falk Lieder (falk.lieder@berkeley.edu) Helen Wills Neuroscience Institute, University of California at Berkeley, CA, USA Ming Hsu (mhsu@haas.berkeley.edu) Haas School of Business, University of California at Berkeley, CA, USA Thomas L. Griffiths (tom griffiths@berkeley.edu) Department of Psychology, University of California at Berkeley, CA, USA Sampling as a Decision Strategy Abstract To evaluate a potential action a, decision makers should in- tegrate the probabilities P(o|A = a) of possible outcomes o with their utilities u(o) into the action’s expected utility E p(O|A=a) [u(O)] (Von Neumann & Morgenstern, 1944). In the real-world–unlike in the laboratory–each action has in- finitely many possible outcomes. As a consequence, the ex- pected utility of action a becomes an integral: Extreme events come to mind very easily and people over- estimate their probability and overweight them in decision- making. In this paper we show that rational use of limited cog- nitive resources can generate these ’availability biases.’ We hy- pothesize that availability helps people to quickly make good decisions in very risky situations. Our analysis shows that agents who decide by simulating a finite number of possible outcomes (sampling) should over-sample outcomes with ex- treme utility. We derive a cognitive strategy with connections to fast-and-frugal heuristics, and we experimentally confirm its prediction that an event’s extremity increases the factor by which people overestimate its frequency. Our model also explains three context effects in decision-making under risk: framing effects, the Allais paradox, and preference reversals. Z p(o|a) · u(o) do. E p(O|A=a) [u(O)] = Keywords: availability; Bayesian; bounded rationality; cog- nitive biases; heuristics; judgment and decision-making Introduction People overestimate the probability of extreme events such as terrorism (Sunstein & Zeckhauser, 2011) and other threats (Lichtenstein, Slovic, Fischhoff, Layman, & Combs, 1978; Rothman, Klein, & Weinstein, 1996) and overreact accord- ingly (Sunstein & Zeckhauser, 2011). This phenomenon has been explained by the availability bias (Tversky & Kahne- man, 1973) according to which people overestimate the prob- ability of events that come to mind very easily. The availability bias appears irrational, but here we ar- gue that it reflects the rational use of finite time (resource- rationality; Lieder, Griffiths, & Goodman, 2013, Griffiths, Lieder, & Goodman, in revision). In brief, we hypothesize that the high availability of extreme events helps decision- makers to allocate their finite time towards considering the most important consequences their decision might have. We model the strategy that might determine which events come to mind first and how they influence judgment and decision- making. Our model explains not only why people overesti- mate the probability of extreme events, but it also explains three context effects in decision-making under risk. The plan of this paper is as follows: The first section intro- duces the theoretical and empirical background. The second section derives a rational model of decision-making in high- risk situations. The third section presents an experiment test- ing the model’s predictions for frequency judgment, and the fourth section applies the model to explain context-effects in decision-making under risk. The final section discusses our results and their implications. In the general case, this integral is intractable to compute, but it can be approximated by sampling methods (Hammersley & Handscomb, 1964). Mental simulations of a decision’s poten- tial consequences can be thought of as samples. In fact, there is neural evidence (Fiser, Berkes, Orb´an, & Lengyel, 2010) and behavioral evidence (Vul, Goodman, Griffiths, & Tenen- baum, 2014; Denison, Bonawitz, Gopnik, & Griffiths, 2013; Griffiths & Tenenbaum, 2006) that the brain handles uncer- tainty by sampling. For instance, people’s predictions of an uncertain quantity X given partial information y are roughly distributed according to the posterior distribution p(X|y) as if they were sampled from it (Griffiths & Tenenbaum, 2006; Vul et al., 2014). These results suggest that people often use only one or very few samples, and this is what rational agents with finite computational resources (bounded rational agents) should do (Vul et al., 2014). The evidence for sam- pling in human cognition raises the question which sampling algorithm(s) are implemented in the brain. Importance sampling is a popular sampling algorithm in computer science and statistics (Hammersley & Handscomb, 1964; Geweke, 1989), and it has been shown to have connec- tions to both neural network (Shi & Griffiths, 2009) and psy- chological process models (Shi, Griffiths, Feldman, & San- born, 2010). Self-normalizing importance sampling estimates the expected value of a function by the weighted average of the function’s values at points drawn from a distribution q: X 1 , · · · , X s ∼ q, w j = IS E p [ f (X)] ≈ E ˆ q,s p(x j ) q(x j ) ∑ sj=1 w j s · ∑ w j · f (x j ), j=1 where w i is the weight of the i th sample. With finitely many samples, this estimate is generally biased. Following Zabaras