The high availability of extreme events serves resource-rational decision-making Falk Lieder ([email protected]) Helen Wills Neuroscience Institute, University of California at Berkeley, CA, USA Ming Hsu ([email protected]) Haas School of Business, University of California at Berkeley, CA, USA Thomas L. Griffiths (tom griffi[email protected]) Department of Psychology, University of California at Berkeley, CA, USA

Abstract Sampling as a Decision Strategy To evaluate a potential action a, decision makers should in- Extreme events come to mind very easily and people over- tegrate the probabilities P(o|A = a) of possible outcomes estimate their probability and overweight them in decision- making. In this paper we show that rational use of limited cog- o with their utilities u(o) into the action’s expected utility nitive resources can generate these ’availability biases.’ We hy- Ep(O|A=a)[u(O)] (Von Neumann & Morgenstern, 1944). In pothesize that availability helps people to quickly make good the real-world–unlike in the laboratory–each action has in- decisions in very risky situations. Our analysis shows that agents who decide by simulating a finite number of possible finitely many possible outcomes. As a consequence, the ex- outcomes (sampling) should over-sample outcomes with ex- pected utility of action a becomes an integral: treme utility. We derive a cognitive strategy with connections to fast-and-frugal heuristics, and we experimentally confirm Z its prediction that an event’s extremity increases the factor Ep(O|A=a)[u(O)] = p(o|a) · u(o)do. (1) 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. In the general case, this integral is intractable to compute, but it can be approximated by sampling methods (Hammersley & Keywords: availability; Bayesian; bounded rationality; cog- nitive biases; heuristics; judgment and decision-making 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, Orban,´ & Lengyel, 2010) Introduction and behavioral evidence (Vul, Goodman, Griffiths, & Tenen- People overestimate the probability of extreme events such baum, 2014; Denison, Bonawitz, Gopnik, & Griffiths, 2013; as terrorism (Sunstein & Zeckhauser, 2011) and other threats Griffiths & Tenenbaum, 2006) that the brain handles uncer- (Lichtenstein, Slovic, Fischhoff, Layman, & Combs, 1978; tainty by sampling. For instance, people’s predictions of an Rothman, Klein, & Weinstein, 1996) and overreact accord- uncertain quantity X given partial information y are roughly ingly (Sunstein & Zeckhauser, 2011). This phenomenon has distributed according to the posterior distribution p(X|y) as been explained by the availability bias (Tversky & Kahne- if they were sampled from it (Griffiths & Tenenbaum, 2006; man, 1973) according to which people overestimate the prob- Vul et al., 2014). These results suggest that people often ability of events that come to mind very easily. use only one or very few samples, and this is what rational agents with finite computational resources (bounded rational The availability bias appears irrational, but here we ar- agents) should do (Vul et al., 2014). The evidence for sam- gue that it reflects the rational use of finite time (resource- pling in human cognition raises the question which sampling rationality; Lieder, Griffiths, & Goodman, 2013, Griffiths, algorithm(s) are implemented in the brain. Lieder, & Goodman, in revision). In brief, we hypothesize that the high availability of extreme events helps decision- Importance sampling is a popular sampling algorithm in makers to allocate their finite time towards considering the computer science and statistics (Hammersley & Handscomb, most important consequences their decision might have. We 1964; Geweke, 1989), and it has been shown to have connec- model the strategy that might determine which events come tions to both neural network (Shi & Griffiths, 2009) and psy- to mind first and how they influence judgment and decision- chological process models (Shi, Griffiths, Feldman, & San- making. Our model explains not only why people overesti- born, 2010). Self-normalizing importance sampling estimates mate the probability of extreme events, but it also explains the expected value of a function by the weighted average of three context effects in decision-making under risk. the function’s values at points drawn from a distribution q: The plan of this paper is as follows: The first section intro- p(x j) duces the theoretical and empirical background. The second X1,··· ,Xs ∼ q, w j = (2) q(x j) section derives a rational model of decision-making in high- s risk situations. The third section presents an experiment test- E ˆ IS 1 p[ f (X)] ≈ Eq,s = s · ∑ w j · f (x j), (3) ing the model’s predictions for frequency judgment, and the ∑ j=1 w j j=1 fourth section applies the model to explain context-effects in th decision-making under risk. The final section discusses our where wi is the weight of the i sample. With finitely many results and their implications. samples, this estimate is generally biased. Following Zabaras (2010), we approximate its bias and variance by though the expected utility is about −1000. In conclusion, under high risk, representative sampling is insufficient for Z p(x)2 ˆ IS 1 E resource-bounded decision-making. Bias[Eq,s] ≈ · · ( p[ f (x)] − f (x))dx (4) s q(x) What is the problem with representative sampling and how 1 Z p(x)2 can it be overcome? Representative sampling fails when it Var[Eˆ IS ] ≈ · · ( f (x) − E [X])2 dx (5) q,s s q(x) p neglects crucial eventualities. Neglecting some eventuali- ties is necessary, but particular eventualities are more im- Importance sampling can be used to approximate the ex- portant than others. Intuitively, the importance of potential pected utility Ep(O|A=a)[u(O)] of taking action a and to es- outcome oi is determined by |p(oi) · u(oi)| because neglect- ? timate the optimal decision a = argmaxa Ep(O|A=a)[u(O)]: ing oi amounts to dropping the addend p(oi) · u(oi) from the expected-utility integral (Equation 1). Thus, intuitively, the ? IS IS aˆ = argmaxUˆ (a), Uˆ (a) ≈ Ep(O|a)[u(O)] (6) problem of representative sampling can be overcome by con- a q,s q,s sidering outcomes whose importance (|p(o ) · u(o )|) is high 1 s i i Uˆ IS (a) = w · u(o ), o ,··· ,o ∼ q. (7) and ignoring those whose importance is low. q,s s w ∑ j j 1 s ∑ j=1 j j=1 Formally, the agent’s goal is to maximize the expected util- ity of a decision made from only s samples. The utility fore- Note that importance sampling is a family of algorithms: each gone by choosing a sub-optimal action can be upper-bounded importance distribution q yields a different estimator, and two by the error in a rational agent’s utility estimate. Therefore estimators may recommend opposite decisions. We thus con- the agent should minimize the expected squared error of its sider which distribution q yields the best decisions. utility estimate, which is the sum of its squared bias and vari- E ˆ IS E 2 ˆ IS 2 ˆ IS  Which distribution should we sample from? ance, i.e. (Uq,s − [U]) ] = Bias[Uq,s] + Var[Uq,s . As the number of samples s increases, the estimate’s squared bias Intuitively, the rational way to make a decision is to simu- decays much faster (O(s−2)) than its variance (O(s−1)); see late consequences o according to one’s best knowledge of the Equations 4-5. Therefore, as the number of samples s in- probability p that they will occur, i.e. q = p: creases, minimizing the estimator’s variance becomes a good s approximation to minimizing its expected squared error. RS 1 Uˆ (a) = u(oi), o1,··· ,os ∼ p(O). (8) According to variational calculus the variance (Equation 5) s s ∑ i=1 of the utility estimate in Equation 7 is minimized by This importance sampling algorithm, which we call represen- qvar(o) ∝ p(o) · |u(o) − E [U]|. (10) tative sampling, computes unbiased utility estimates. Yet– p surprisingly– representative sampling is insufficient for mak- Interestingly, the optimal sampling distribution overrepre- ing good decision with very few samples. Consider, for in- sents outcomes with large absolute utility, Thus, biased sam- stance, the choice to play Russian roulette with the popu- pling can lead to better decisions. Unfortunately, importance lar six-round NGant M1895 revolver. Playing the game will sampling with qvar is intractable, because it presupposes the 5 most likely, i.e. with probability p1 = 6 , reward you with a expected utility Ep[U] that importance sampling is supposed thrill and a gain in status (u(o1) = 1) but kill you otherwise to approximate. A priori the expected utility of a prospect 1 9 (p2 = 6 , u(o2) = −10 ). Ensuring that representative sam- whose outcomes may be good or bad is equally likely to be pling declines a game of Russian roulette at least 99.99% of positive or negative. The same is true for choosing action a the time, would require 51 samples–potentially a very time- over action b or vice versa. Therefore, the most principled consuming computation. Many real-life decisions involve guess an agent can make for the expected utility Ep[U] in high risks and are even more challenging, because the prob- Equation 10–before computing it–is 0. Thus when the ex- 1 ability of disaster is orders of magnitude smaller than 6 but pected utility is not too far from zero, then the importance may or may not be large enough to warrant caution. Exam- distribution qvar is efficiently approximated by ples include risky driving, the stock market, and air travel. For some of these choices (e.g. texting while driving) there q˜(o) ∝ p(o) · |u(o)|. (11) may be a one in a million chance of disaster while all other outcomes have negligible utilities: This confirms our intuition and leads to an importance sam- pling scheme that we call utility-weighted sampling: u o 9 p o −6 i d u o ( d) = −10 , ( d) = 10 , ∀ 6= : | ( i)| ≤ 1 (9) s ˆ IS 1 Uq˜,s = s ∑ sgn(u(o j)), o j ∼ q˜, (12) If people decided based on n representative samples, they ∑ j=1 1/|u(o j)| j=1 would completely ignore the potential disaster with proba- bility 1 − (1 − 10−6)n. Thus to have at least a 50% chance where sgn(x) is +1 for positive x and −1 for negative x. of taking the potential disaster into account the agent would Utility-weighted sampling is a simple and psychologically have to generate almost 700 million samples. This is clearly plausible strategy for decision-making under uncertainty: It infeasible; thus one would almost always take this risk even generates examples of possible outcomes by an appropriately biased simulation and tallies how many of them are positive. 0.8 avg. estimate (95% CI) If more than half of the examples for the utility of choosing 0.6 natural statistics action a over action b in a two-alternative forced-choice are 0.4 positive, then the agent will choose action a; if more than probability 0.2 0 half of them are negative, then the agent will choose action b; mundane stressful death (all causes) otherwise it it will be indifferent. 100 Utility-weighted sampling succeeds in the two decision 50 problems in which representative sampling failed. In Rus-

sian roulette utility-weighted sampling requires only 1 rather 0 mundane stressful lethal than 51 samples to recommend the correct decision at least judged extremity (95% CI) event type 99.99% of the time, because the first sample is almost always Figure 1: Judged frequency and extremity by event type. the most important potential outcome, i.e. death. Likewise one single utility-weighted sample suffices to consider the potential disaster (Equation 9) at least 99.85% of the time, movies, headache, and food-poisoning), and two attention- whereas even 700 million representative samples would miss checks. Overestimation was measured by the ratio of a par- the disaster almost half of the time. Thus, utility-weighted ticipant’s estimate over the event’s frequency according to of- sampling would allow people to avoid both disasters even un- ficial statistics.1 The complete experiment can be inspected der extreme time pressure. The hypothesis that people decide online.2 Out of 100 participants 17 failed the attention check by utility-weighted sampling makes two predictions that we (wrong answer on attention-check questions, or mean judged test in the next two sections: extremity of lethal events less than 75%) and were excluded. 1. People overestimate an event’s probability more, the more Results and Discussion extreme the event is. Consistent with our theory, participants overestimated the fre- 2. Extreme potential outcomes are over-weighted in decision- quencies of all lethal events (all p < 0.0005) but none of making, and extremity is context-dependent. the mundane events (all p > 0.048 > αSidak = 0.0014, Sidak- correction for multiple comparisons). Participants also over- Overestimation of extreme events’ frequencies estimated 23 of the 30 stressful life events. Figure 1 shows that across event types overestimation gradually increase with We hypothesize that the mind re-uses utility-weighted sam- extremity: Absolute overestimation (p ˆ − p) was significantly pling (Equation 12) to estimate event frequencies, because larger for stressful life events than for mundane events (p < evolutionary fitness depended on good decisions rather than 0.01) and even larger for lethal events (p = 0.02). While accurate statements of probability. We therefore predict that our participants’ probability estimates were very accurate for people’s estimatep ˆk of the probability pk = p(ok) is mundane events, their (implicit) estimate of the annual death s 1 rate was 25-times higher than its true value. For stressful life ∑i=1 wi · (oi = ok) 1 pˆk = s , wi = , oi ∼ q˜. (13) events overestimation and judged extremity are both interme- ∑i=1 wi |u(oi)| diate. This pattern answers the open question whether avail- ability or regressed frequency causes the overestimation of Since the importance densityq ˜(o) ∝ p(o) · |u(o)| over- extreme events (Hertwig, Pachur, & Kurzenhauser,¨ 2005): if represents events proportionally to their extremity |u|, i.e. estimates were merely regressive towards a mean frequency, q˜(o) ∝ |u(o)|, we predict that people’s relative over-estimation p(o) as proposed by Gigerenzer (2008a), then people should un- pˆk is a monotonically increasing function of the event’s ex- derestimate frequent mundane events, but our participants did pk tremity |uk|. Formally, the bias (Equation 4) of utility- not. Figure 2 shows that the average relative overestimation weighted probability estimation (Equation 13) implies that of the 37 events increases with their judged extremity (Spear-   pˆk−pk = 1 − 1 +C , for some constant C. We tested this man’s rank correlation ρ = 0.53, p < 0.001). Furthermore, we pk s |ui| observed the same effect at the level of individual judgments prediction with a simple experiment. (Spearman’s rank correlation ρ = 0.28, p < 10−15). On aver- Methods age, the extremest event’s frequency (murder, 98% extreme) was overestimated by a factor of 972, whereas the frequency We recruited 100 participants on Amazon Mechanical Turk. of the least extreme event (headache, 20% extreme) was over- Each participant estimated how many of 1000 randomly se- estimated by merely 6% (n.s.). Consistent with this result, lected American adults had experienced each of 39 events in Rothman et al. (1996) found that prevalence is more heavily 2013 and judged the events’ valence (good or bad) and ex- overestimated for suicide than for divorce. tremity (0: neutral – 100: extreme). The 39 events comprised 30 stressful life events from Hobson et al. (1998), four lethal 1Hobson & Delunas (2001),www.cdc.gov/nchs/fastats/deaths.htm,www .mpaa.org/resources/3037b7a4-58a2-4109-8012-58fca3abdf1b.pdf,www.cdc events (suicide, homicide, lethal accidents, and dying from .gov/foodborneburden/, Rasmussen, Jensen, Schroll, & Olesen (1991). disease/old age), three more mundane events (going to the 2http://sites.google.com/site/falklieder/freq estimation.html ρ = 0.53, p < 0.001 40 on the expected value of the utility difference ∆U:

35 ( 30 u(o) − u(p · o) with probability p 25 ∆U = (15) 20 −u(p · o) with probability 1 − p

15 10 According to our theory, people simulate ∆U by sampling 5 IS relative overestimation rank fromq ˜(∆u) ∝ |∆u| · p(∆u) and estimate E[∆U] by ∆Uˆ ac- 0 q˜,s 0 5 10 15 20 25 30 35 40 judged extremity rank cording to Equation 12. When the bias of this estimate (Bias(∆Uˆ IS )) is positive, the agent will fancy the gamble pˆ−p q˜,s Figure 2: Overestimation p increases with perceived ex- (risk-seeking), but when it is negative, the agent will be risk- tremity (|u|). A cross represents one event’s average ratings. averse. The importance distributionq ˜ is proportional to |∆u|. Therefore the agent will overweight the gain/loss o of the lot- tery if p(o) is small, because then |u(o)−u(p·o)| > |u(p·o|). In conclusion, the experiment confirmed our theory’s pre- Conversely, the agent will underweight outcome o if p(o) is diction that an event’s extremity increase the relative overesti- large, because then |u(o)−u(p·o)| < |u(p·o|)). This renders mation of its frequency. However, additional experiments are the agent’s bias positive (risk-seeking) for improbable gains required to disentangle the effects of extremity and low prob- and probable losses but negative (risk-aversion) for probable ability, because these two factors were anti-correlated. Future gains and improbable losses (see Figure 3). Thus our model work will formally test our model against competing theories predicts the fourfold pattern of risk preferences which is used and investigate whether the effect of an event’s extremity is to explain how a person who is so risk-averse that he buys mediated by how many instances of this event people imagine insurance can also be so risk-seeking that he plays the lottery. or retrieve from memory (cf. Hertwig et al., 2005). Next we show by simulation that a bounded rational agent might indeed make both choices. First, we simulated the de- Context effects in decision-making cision whether or not to play the Powerball lottery.3 The jack- pot is at least $40 million, but the odds of winning it are less While our goal was to explain why people overestimate the than 1:175 million. In brief, people pay $2 to play a gam- probability of extreme events, utility-weighted sampling also ble whose expected value is only $1. We found that utility- predicts that people over-weight certain outcomes in eco- weighted sampling does, in expectation, fancy the lottery nomic decisions. To simulate such decisions we model the when the number of samples is less than 4 (E[Uˆ IS ] ≥ 59.8 utility of winning or losing money by the following function q˜,s for s ≤ 3). If the agent chose based on one sample (cf. Vul based on prospect theory (Tversky & Kahneman, 1992): et al., in press) and was given the choice twice a week, it ( would buy about seven lottery tickets per year. This value o0.85 , if o ≥ 0 u(o) = . (14) is a lower bound, because it ignores debt and other factors. −|o|0.95 , if o < 0 Nevertheless, it demonstrates that playing the lottery is com- patible with resource-rational decision-making. Second, we When choosing between receiving a high payoff with low simulated the decision whether or not to buy insurance. The probability (risky gamble) or a low payoff with high proba- magnitude of an insured loss l can be modeled by the Pareto α −α−1 9 bility (safe gamble), people often prefer the risky gamble in distribution (e.g., p(l) = α · xmin · x ,xmin < x < 10 with one context but the safe gamble in another. This suggests α = xmin = 1). We found that the bias of utility-weighted that people’s risk preferences are inconsistent and context- sampling (Equation 4) would make people overestimate the 244 dependent (Tversky & Kahneman, 1992). Utility-weighted value of insurance against such a loss by s %, where s is sampling can explain several such inconsistencies: (i) fram- the number of samples. This resolves the apparent paradox ing effects, (ii) the Allais paradox, and (iii) preference rever- of being willing to buy both lottery tickets and insurance. sals. We now consider these in turn. The Allais paradox

Framing effects on risk attitudes In the two gambles L1(z) and L2(z) defined in Table 1 the chance of winning z dollars is exactly the same. Yet, when z = Framing outcomes as losses rather than gains can reverse peo- 2400 most people prefer L2 over L1, but when z = 0 the same ple’s risk preferences (Tversky & Kahneman, 1992): In the people prefer L over L . This inconsistency is known as the o 1 2 domain of gains people prefer a lottery ( dollars with proba- Allais paradox. Table 2 shows how our theory resolves this p risk seeking p < . bility ) to its expected value ( ) when 0 5, but paradox: According to the importance distributionq ˜ (Equa- p > . risk-aversion when 0 5 they prefer the expected value ( ). tion 11) people overweight the event for which the utility dif- To the contrary, in the domain of losses people are risk seek- ference between the two gambles’ outcomes (O and O ) is ing for p < 0.5 but risk averse for p > 0.5. This phenomenon 1 2 largest (∆U = u(O )−u(O )). Thus when z = 2400, the most is known as the fourfold pattern of risk preferences. 1 2 Whether or not people should choose the gamble depends 3www.calottery.com/play/draw-games/powerball Table 3: Utility-weighted sampling explains preference reversals. E E ˆ IS utility of [U] [Uq˜,2] safer gamble (Ls : $1, p = 0.8) 0.80 1 riskier gamble (Lr : $2, p = 0.4) 0.72 1.8 choosing Ls over Lr 0.079 0.075

weighted sampling’s bias in favor of the riskier option is weaker in choice than in pricing: so weak that it is overwrit- ten by the risk-aversion due to concavity (flattening) of the utility function (cf. Equation 14); see Table 3. This illustrates Figure 3: Risk preferences (E[∆Uˆ ]) predicted by utility- weighted sampling with s = 2. that utility-weighted sampling weights events differently de- pending on the problem to be solved. Table 1: Allais Gambles General Discussion (o1, p1)(o2, p2)(o3, p3) Our resource-rational analysis of decision-making in high- L (z) : (z,0.66)(2500,0.33)(0,0.01) 1 risk situations suggested that people should decide by utility- L2(z) : (z,0.66)(2400,0.34) weighted sampling (Equation 12). Utility-weighted sampling explains not only how we are able to make sensible decisions over-weighted event is the possibility that gamble L1 yields under high risk but also why we overestimate the frequency of o1 = 0 and gamble L2 yields o2 = 2400 (∆U = −u(2400)); extreme events and have inconsistent risk preferences (fram- consequently the bias is negative and the first gamble ap- ing effects, the Allais paradox, and preference reversals). E ˆ IS pears inferior to the second ( [∆Uq˜,2] = −152). But when While overestimation was previously explained by ‘avail- z = 0, then L1 yielding o1 = 2500 and L2 yielding o2 = 0 ability’ (Tversky & Kahneman, 1973), our theory specifies (∆U = +u(2500)) becomes the most over-weighted event what exactly the availability of events should correspond to– E ˆ IS making the first gamble appear superior ( [∆Uq˜,2] = +4.1). namely their importance distributionq ˜ (Equation 11)–and This explains why people’s preferences switch from the sec- why this is useful. Inconsistent risk preferences were pre- ond to the first gamble as z switches from 2400 to 0. viously explained by regret (Loomes & Sugden, 1982) or salience (Bordalo, Gennaioli, & Shleifer, 2012). Like these Preference Reversals explanations our theory assumes an amplified impact of large When people first price a risky gamble and a safe gamble with utility differences. While regret theory explains this ampli- similar expected value and then choose between them, their fication by altered subjective utilities, utility-weighted sam- preferences are inconsistent almost 50% of the time: most pling and salience theory explain the amplification by al- people price the risky gamble higher than the safe one, but tered probability-weighting. Despite this commonality, our many of them nevertheless choose the safer one (Lichtenstein account offers three advances over salience theory. First, & Slovic, 1971). Utility-weighted sampling over-estimates we do not only describe the effect of utility on probability- E the expected utility prisky [u(Orisky)] of high-payoff (risky) weighting, but we also model the cognitive strategy that gen- lotteries more than for low-payoff (safe) lotteries, because erates it. Second, our theory reconciles this seemingly ir- q˜(o) ∝ |u(o)|. This explains why people price the risky gam- rational effect with rational information processing. Con- ble higher than the safe one. To choose between two lot- cretely, the resource-rational basis of the salience of a util- teries our model estimates the expected utility difference, ity difference ∆U = u(O1) − u(O2) is the relative frequency E i.e. psafe,risky [u(Osafe) − u(Orisky)]. In this estimation prob- with which it should be simulated, i.e. the importance dis- lem there are 2 × 2 rather than just 2 possible outcomes, and tributionq ˜(∆u) ∝ p(∆u) · |∆u|.4 Third, since our explanation the importance of positive outcomes is counterbalanced by instantiates a more general theoretical framework–resource- the importance of the negative outcome. As a result, utility- rationality–it can be extended to multi-alternative decisions, decisions from experience, and many other problems. Table 2: Utility-weighted sampling explains the Allais paradox. We have proposed a psychologically plausible cognitive strategy that approximates the normative–but intractable– ∆U p q˜ q˜/p 0 0.66 0 0 decision-rule of expected utility theory in two simple steps: z = 2400: u(2500) − u(2400) 0.33 0.54 1.6 (i) generate example outcomes by biased mental simula- −u(2400) 0.01 0.46 46 tion, and (ii) count how many of them are positive. Inter- 0 0.66 · 0.67 0 0 estingly, these steps resemble two fast-and-frugal heuristics −u(2400) 0.67 · 0.34 0.5 2.19 z = 0 : u(2500) − u(2400) 0.33 · 0.34 0.01 0.08 (Gigerenzer, 2008b): Biased mental simulation (stochasti- u(2500) 0.33 · 0.66 0.49 2.26 cally) considers the most important consequence first–like Note: The agent’s simulation yields ∆U = ∆u with probability take-the-best–and binary choices are made by tallying if there q˜(∆u) ∝ p(∆u) · |∆u| where p is ∆u’s objective probability. 4This definition satisfies two of Bordalo et al.’s (2012) three axioms of salience. are more positive than negative simulated outcomes–as in Gigerenzer, G. (2008b). Why heuristics work. Perspect. Pschol. the tallying heuristic. The fact that we derived this strategy Sci., 3(1), 20–29. Gigerenzer, G., & Brighton, H. (2009). Homo heuristicus: Why as a resource-efficient approximation to normative decision- biased minds make better inferences. Top. Cogn. 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(1996). Absolute ysis suggested that the availability bias is a manifestation and relative biases in estimations of personal risk. J. Appl. Soc. of resource-rational decision-making. This conclusion sup- Psychol., 26(14), 1213–1236. ports the emerging view that cognitive biases are a window on Shi, L., & Griffiths, T. (2009). Neural implementation of hierar- chical Bayesian inference by importance sampling. In Y. Bengio, resource-rational computation rather than a sign of irrational- D. Schuurmans, J. Lafferty, C. K. I. Williams, & A. Culotta (Eds.), ity (Lieder, Griffiths, & Goodman, 2013; Lieder, Goodman, Adv. neural inf. process. syst. 22 (pp. 1669–1677). Red Hook: Cur- & Griffiths, 2013). Being biased can be resource-rational. ran Associates, Inc. Shi, L., Griffiths, T., Feldman, N., & Sanborn, A. (2010). Exemplar Acknowledgments. This work was supported by grant number models as a mechanism for performing Bayesian inference. 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