The anchoring bias reflects rational use of cognitive resources.

Psychon Bull Rev (2018) 25:322–349
DOI 10.3758/s13423-017-1286-8
THEORETICAL REVIEW
The anchoring bias reflects rational use of cognitive
resources
Falk Lieder1,2·Thomas L. Griffiths1,5·Q u e n t i nJ .M .H u y s2,4·Noah D. Goodman3
Published online: 8 May 2017
© Psychonomic Society, Inc. 2017
Abstract Cognitive biases, such as the anchoring bias, pose
a serious challenge to rational accounts of human cog-
nition. We investigate whether rational theories can meetthis challenge by taking into account the mind’s bounded
cognitive resources. We asked what reasoning under uncer-tainty would look like if people made rational use of theirfinite time and limited cognitive resources. To answer this
question, we applied a mathematical theory of bounded
rationality to the problem of numerical estimation. Our anal-ysis led to a rational process model that can be interpreted
in terms of anchoring-and-adjustment. This model provided
/envelopebackFalk Lieder
falk.lieder@berkeley.edu
Thomas L. Griffiths
tom griffiths@berkeley.edu
Quentin J. M. Huysqhuys@biomed.ee.ethz.ch
Noah D. Goodman
ngoodman@stanford.edu
1Helen Wills Neuroscience Institute, University of California,Berkeley, USA
2Translational Neuromodeling Unit, Institute for BiomedicalEngineering, University of Z ¨urich and Swiss Federal Institute
of Technology (ETH), Z ¨urich, Switzerland
3Department of Psychology, Stanford University, Stanford,USA
4Department of Psychiatry, Psychotherapy and Psychosomatics,Hospital of Psychiatry, University of Z ¨urich, Z ¨urich,
Switzerland
5Department of Psychology, University of California, Berkeley,USAa unifying explanation for ten anchoring phenomena includ-
ing the differential effect of accuracy motivation on the biastowards provided versus self-generated anchors. Our resultsillustrate the potential of resource-rational analysis to pro-vide formal theories that can unify a wide range of empirical
results and reconcile the impressive capacities of the human
mind with its apparently irrational cognitive biases.
Keywords Bounded rationality ·Heuristics ·Cognitive
biases ·Probabilistic reasoning ·
Anchoring-and-adjustment ·Rational process models
Many classic theories in economics, philosophy, linguistics,
social science, and psychology are built on the assump-
tion that humans are rational (Friedman & Savage, 1948;
Lohmann, 2008;H e d s t r ¨om & Stern, 2008;H a r m a n , 2013;
Frank & Goodman, 2012) and therefore act according to
the maxims of expected utility theory (V on Neumann &
Morgenstern, 1944) and reason according to the laws of
logic (Mill, 1882;N e w e l le ta l . ,1958;B r a i n e , 1978; Fodor,
1975) or probability theory (Oaksford & Chater, 2007). The
assumption that people are rational was challenged whena series of experiments suggested that people’s judgments
systematically violate the laws of logic (Wason, 1968)
and probability theory (Tversky & Kahneman, 1974). For
instance, Tversky and Kahneman ( 1974) showed that peo-
ple’s probability judgments appear to be insensitive to prior
probability and sample size but are influenced by irrel-
evant factors such as the ease of imagining an event or
the provision of an unrelated random number. These sys-tematic deviations from the tenets of logic and probability
are known as cognitive biases. According to Tversky and
Kahneman (1974), cognitive biases result from people’s use
of fast but fallible cognitive strategies known as heuristics.
Psychon Bull Rev (2018) 25:322–349 323
The discovery of cognitive biases was influential because
following the rules of logic and probability was assumed
to be the essence of rational thinking. Evidence that people
deviate from these rules brings human rationality into ques-
tion. This doubt is shaking the foundations of economics,the social sciences, and rational models of cognition. If the
human mind does not follow rational principles, then there
is little hope that we will be able to able derive unifying lawsof cognition from a basic set of axioms. Without the principles
of rationality, there is little guidance for how to translate as-
sumptions about cognitive processes into predictions aboutbehavior and how to generalize from our data. But if peoplewere systematically rational in some sense then all of this
would be possible, and creating artificial intelligence could
go hand in hand with understanding how the mind works.Therefore, the question whether people are rational is fun-
damental to how we study the mind, to how we model it,
and the implications of our theories for science and society.
Despite their cognitive biases, humans still outperform
intelligent systems built on the laws of logic and probabilityon many real-world problems. This poses a paradox: howcan we be so smart, if we appear so irrational? The argumentthat people are irrational rests on two premises: First, to be
rational is to follow the rules of logic and probability theory.
Second, human thought violates the rules of logic and prob-ability. Previous work supports the second premise (Shafir
& LeBoeuf, 2002), but in this article we question the first by
suggesting that notions of human rationality should take into
account that reasoning costs time. The number of compu-
tations required for exact logical or probabilistic reasoning
grows exponentially with the number of facts and variablesto be considered. As a result, to exactly and completelyreason through just a single complex everyday situation
involving hundreds of variables could require more com-
putations than can be performed in a human lifetime (VanRooij, 2008). Thus, if a person were to reason out every-
thing strictly according to the laws of logic and probabilitytheory she might die before she reached her first conclusion.
The laws of logic and probability theory are thus insuf-
ficient to give a definition of rationality relevant to anyreal intelligent agent, because the cost of computation hasto be taken into account. To be successful in the worldwe live in, we have to solve complex problems in finite
time despite bounded cognitive resources. In this paper,
we explore the implications of a different framework forcharacterizing rationality that captures this idea: resource-
rationality (Lieder et al., 2012; Griffiths et al., 2015), which
builds on the notion of bounded optimality proposed in the
artificial intelligence literature by Russell and colleagues
(Russell & Wefald, 1991; Russell & Subramanian, 1995;
Russell, 1997). We use this alternative characterization of
rationality to re-evaluate human performance in tasks used
to demonstrate that people’s judgments are biased becausethey are cognitive misers. Achieving demanding goals inlimited time requires balancing being quick and being accu-rate. We regret the opportunities we miss when we fail to
make up our mind on time, but we also regret the errors
we commit by jumping to conclusions. When we think toolittle our judgments can be skewed by irrelevant informa-
tion that we happened to see, hear, or think about a moment
ago. This phenomenon is known as anchoring. Anchoring
is one of the cognitive biases discovered by Tversky and
Kahneman (1974). It impacts many important aspects of our
lives including the outcome of salary negotiations (Galinsky& Mussweiler, 2001), economic decisions (e.g., Simonson &
Drolet, 2004), criminal sentences (Englich et al., 2006), and
even our ability to understand other people (Epley et al., 2004).
In their classic paper, Tversky and Kahneman ( 1974)
showed that people’s judgments could be systematically
skewed by providing them with an arbitrary number before
their judgment: The experimenter generated a random num-ber by spinning a wheel of fortune, and then asked partici-
pants to judge whether the percentage of African countries
in the United Nations was smaller or larger than that num-ber. Participants were then asked to estimate this unknownquantity. Strikingly, the participants’ estimates were biased
towards the random number: their median estimate was
larger when the random number was high than when itwas low. This appears to be a clear violation of ratio-
nality. According to Tversky and Kahneman (1974)t h i s
violation occurs because people use a two-stage processcalled anchoring-and-adjustment (see also Nisbett & Ross,
1980). In the first stage, people generate a preliminaryjudgment called their anchor . In the second stage, they
adjust that judgment to incorporate additional information,but the adjustment is usually insufficient. In Tversky and
Kahneman’s experiment people appear to have anchored
on the random number provided by the experimenter and
adjusted it insufficiently. Consequently, when the anchor
w
as low people’s judgments were too low, and when the
anchor was high their judgments were too high.
At first sight, anchoring appears to be irrational, because
it deviates from the standards of logic and probability which
are typically used to assess rationality. But it could also bea reasonable compromise between error in judgment andthe cost of computation, and hence be resource-rational.
Anchoring-and-adjustment has two components that could
be irrational: the generation of the anchor and the processby which it is adjusted. Previous research found that when
no anchor is provided, the anchors that people generate for
themselves are relevant quantities that are reasonably closeto the correct value and can be generated quickly (Epley
& Gilovich, 2006). Furthermore, research on human com-
munication suggests that in everyday life it is reasonable to
assume that other people are cooperative and provide rel-evant information (Schwarz, 2014). Applied to anchoring,
324 Psychon Bull Rev (2018) 25:322–349
this means that if somebody asks you in real life whether a
quantity you know very little about is larger or smaller thana certain value, it would be rational to treat that question as
a clue to its value (Zhang & Schwarz, 2013). Thus, having
the queried value in mind might make it rational to reuse
it as your anchor for estimating the unknown quantity. This
suggests that the mechanism by which people generate their
anchors could be rational in the real world.
1
Assuming that people generate or select anchors in a rea-
sonable way, the rationality of anchoring-and-adjustmenthinges on the question whether adjustment is a rationalprocess. To answer this question, we investigate whetherinsufficient adjustment can be understood as a rational
tradeoff between time and accuracy. If so, then how much
people adjust their initial estimate should adapt rationallyto the relative utility of being fast versus being accurate.
To formalize this hypothesis, we present a resource-rational
analysis of numerical estimation. Our analysis suggests thatthe rational use of finite resources correctly predicts the
anchoring bias and how it changes with various experimen-
tal manipulations (see Table 1). These results support the
conclusion that adjustment is resource-rational.
The remainder of this article begins with a brief survey
of empirical findings on anchoring and discusses the chal-lenges that they pose to existing accounts of anchoring-and-adjustment. We then present our resource-rational analysis
of numerical estimation, derive a rational process model that
can be interpreted in terms of anchoring-and-adjustment,and show it is sufficient to explain the reviewed phenomena.
We close by discussing our findings and their implications
for the debate about human rationality.
Empirical findings on the anchoring bias
Anchoring is typically studied in numerical estimationtasks. Numerical estimation involves making an informed
guess of the value of an unknown numerical quantity. Since
the first anchoring experiment by Tversky and Kahneman(1974) a substantial number of studies have investigated
when anchoring occurs and what determines the magnitude
of the anchoring bias (see Table 1).
The anchors that people use when forming estimates can
be relevant to the quantity they are estimating. For instance,Tversky and Kahneman ( 1974) found that people sometimes
anchor on the result of calculating 1 ×2×3×4 when the
task is estimating 1 ×2×3×4×···× 8. However, people
can also be misled, anchoring on numbers that are irrelevantto the subsequent judgment. For instance, many anchoring
1We will revisit this issue in more depth in the general discussion.experiments first ask their participants whether an unknownquantity is larger or smaller than a given value and then pro-ceed to have them estimate that quantity. Having compared
the unknown quantity to the value provided by the exper-
imenter makes people re-use that value as their anchor inthe subsequent estimation task. Those numbers are there-
fore known as provided anchors . Importantly this procedure
works with irrelevant numbers such as the random number
that Tversky and Kahneman ( 1974) generated for their par-
ticipants or one’s own social security number (Ariely et al.,2003).
Although asking people to compare the quantity to
a given number is particularly effective, the anchoring
bias also occurs when anchors are presented incidentally
(Wilson et al., 1996), although this effect is smaller and
depends on particulars of the anchor and its presentation
(Brewer & Chapman, 2002). Furthermore, anchoring-and-
adjustment can also occur without an externally provided
anchor: At least in some cases people appear to generate
their own anchor and adjust from it Epley and Gilovich
(2004). For instance, when Americans are asked to estimatethe boiling point of water on Mount Everest they often recall212
◦F (100◦C) and adjust downwards to accommodate the
lower air pressure in higher altitudes.
Although people’s adjustments are usually insufficient,
various factors influence their size and consequently the
magnitude of the anchoring bias. For instance, the anchoring
bias is larger the more uncertain people are about the quan-tity to be estimated (Jacowitz & Kahneman, 1995). Indeed,
Wilson et al. (1996) found that people knowledgeable aboutthe quantity to be estimated were immune to the anchoringbias whereas less knowledgeable people were susceptibleto it. While familiarity (Wright and Anderson, 1989)a n d
expertise (Northcraft & Neale, 1987) do not abolish anchor-
ing, expertise appears to at least reduce it Northcraft andNeale (1987). Other experiments have systematically va-
ried the distance from the anchor to the correct value. Their
results suggested that the magnitude of the anchoring biasinitially increases with the distance from the anchor to the cor-
rect value (Russo & Schoemaker, 1989). Yet this linear in
crease of the anchoring bias does not continue indefinitely.
Chapman and Johnson (1994) found that increasing analready unrealistically large anchor increases the anchor-
ing bias less than increasing a realistic anchor by the same
amount.
Critically for the resource-rational account proposed
here, the computational resources available to people alsoseem to influence their answers. Time pressure, cognitiveload, and alcohol decrease the size of people’s adjust-
ments and inter-individual differences in how much people
adjust their initial estimate correlate with relevant personal-ity traits such as the need for cognition (Epley and Gilovich,2006). In addition to effects related to cognitive resources,
Psychon Bull Rev (2018) 25:322–349 325
Ta b l e 1 Anchoring phenomena and resource-rational explanations
Anchoring effect Simulated results Resource-rational explanation
Insufficient adjustment from pro-
vided anchorsJacowitz and Kahneman ( 1974),
Tversky and Kahneman ( 1995)Rational speed-accuracy tradeoff.
Insufficient adjustment from self-generatedanchorsEpley and Gilovich ( 2006), Study 1 Rational speed-accuracy tradeoff.
Cognitive load, time pressure, and alcoholreduce adjustment.Epley and Gilovich ( 2006), Study 2 Increased cost of adjustment reduces the resource-
rational number of adjustments.
Anchoring bias increases with anchorextremity.Russo and Schoemaker ( 1989) Each adjustment reduces the bias by a constant
factor (3). Since the resource-rational number ofadjustments is insufficient, the bias is proportionalto the distance from anchor to correct value.
Uncertainty increases anchoring. Jacowitz and Kahneman ( 1995) The expected change per adjustment is small when
nearby values have similar plausibility.
Knowledge can reduce the anchoring bias. Wilson et al. ( 1996), Study 1 High knowledge means low uncertainty. Low
uncertainty leads to high adjustment (see above).
Accuracy motivation reduces anchoring
bias when the anchor is self-generated butnot when it is provided.Tversky and Kahneman ( 1974),
Epley and Gilovich ( 2005)1. People are less uncertain about the quantities forwhich they generate their own anchors.
2. Accuracy motivation increases the number of
adjustments but change per adjustment is lowerwhen people are uncertain.
Telling people whether the correct value
is larger or smaller than the anchor makesfinancial incentives more effective.Simmons et al. ( 2010), Study 2 Being told the direction of adjustments makes
adjustments more effective, because adjustmentsin the wrong direction will almost always be
rejected.
Financial incentives are more effective
when the anchor is extreme.Simmons et al. ( 2010), Study 3 Values on the wrong side of an extreme anchor
are much less plausible than values on the correctside. Therefore proposed adjustments in the wrongdirection will almost always be rejected.
adjustment also depends on incentives. Intuitively, accuracy
motivation should increase the size of people’s adjustmentsand therefore decrease the anchoring bias. Interestingly,
experiments have found that accuracy motivation decreases
the anchoring bias only in some cases, but not in others(Epley & Gilovich, 2006; Simmons et al., 2010). On ques-
tions where people generated their own anchors, financial
incentives increased adjustment and reduced the anchor-
ing bias (Epley & Gilovich, 2006; Simmons et al., 2010).
But on questions with provided anchors, financial incentives
have typically failed to eliminate or reduce the anchor-
ing bias (Tversky & Kahneman, 1974; Ariely et al., 2003)
with some exceptions (Wright & Anderson, 1989). A recent
set of experiments by Simmons et al. ( 2010) suggested
that accuracy motivation increases adjustment from pro-vided and self-generated anchors if and only if peopleknow in which direction to adjust. Taken together, these
findings suggests that the anchoring bias depends on how
much cognitive resources people are able to and willing toinvest.
Before the experiments by Simmons et al. (2010) demon-
strated that accuracy motivation can increase adjustmentfrom provided anchors, the bias towards provided anchorsappeared immutable by financial incentives (Tversky andKahneman, 1974; Chapman & Johnson, 2002; Wilson et al.,
1996), forewarnings and time pressure (Mussweiler &Strack, 1999; but see Wright & Anderson, 1989). Since in-
centives were assumed to increase adjustment and increasedadjustment should reduce the anchoring bias, the ineffec-
tiveness of incentives led to the conclusion that the anchor-
ing bias results from a mechanism other than anchoring-and-adjustment, such as selective accessibility (Mussweiler
& Strack, 1999; Chapman & Johnson, 2002;E p l e y , 2004).
Later experiments found that when people generate the
anchor themselves accuracy motivation and time pressure
are effective (Epley & Gilovich, 2005; Epley et al., 2004;
Epley & Gilovich, 2006). This led (Epley & Gilovich, 2006)
to conclude that people use the anchoring-and-adjustment
strategy only when they generated the anchor themselves
whereas provided anchors bias judgments through a differ-
ent mechanism.
The wide range of empirical phenomena summarized
in Table 1have suggested a correspondingly wide range
of explanations, including the idea that anchoring and
326 Psychon Bull Rev (2018) 25:322–349
adjustment is not a simple, unitary process. In the remain-
der of the paper we explore an alternative account, showingthat these disparate and seemingly inconsistent phenomena
can all be explained by a unifying principle: the rational
use of finite time and cognitive resources. From this princi-ple we derive a resource-rational anchoring-and-adjustment
model and show that it is sufficient to explain the anchor-
ing bias regardless of whether the anchor was provided orself-generated.
Anchoring and adjustment as resource-rational
inference
In this section we formalize the problem people solve in
anchoring experiments – numerical estimation – and ana-lyze how it can be efficiently solved in finite time with
bounded cognitive resources. We thereby derive a resource-
rational model of anchoring-and-adjustment. We then usethis model to explain a wide range of anchoring phenomena.
Conceptually, our model assumes that adjustment pro-
ceeds by repeatedly considering small changes to the currentestimate. The proposed change is accepted or rejected prob-abilistically such that the change is more likely to be made
the more probable the new value is and the less probable the
current one is (see Fig. 1). After sufficiently many adjust-
ments the estimate becomes correct on average and inde-
pendent of the initial guess. However, each small adjustment
costs a certain amount of time. According to our model, thenumber of steps is chosen to minimize the expected value ofthe time cost of adjustment plus the error cost of the result-
ing estimate. In the remainder of this section, we derive our
model from first principles, specify it in detail, and showthat the optimal number of adjustments is very small. As
Fig.1illustrates, this causes the final estimates to be biased
towards their respective anchors.
In contrast to previous theories of anchoring (Epley &
Gilovich, 2006; Simmons et al., 2010), our model precisely
specifies the number, size, and direction of adjustments as a
function of the task’s incentives and the participant’s knowl-edge. In contrast, to the proposal by Epley and Gilovich
(2006) our model covers adjustments from provided anchors
andself-generated anchors. Furthermore, while Epley and
Gilovich (2006) assumed that the correct direction of adjust-
ment is known, our model does not make this assumption
and allows the direction of adjustment to change from onestep to the next. The model by Simmons et al. ( 2010)
also makes these conceptual assumptions. However, it doesnot specify precisely how the direction and size of eachadjustment are determined. While their model predicts adeterministic back-and-forth in the face of uncertainty, our
model assumes that adjustments that improve the estimate
are probabilistically preferred to adjustments that do not.This enables our model to capture streaks of adjustmentsin the correct direction interrupted by small steps in thewrong direction, whereas the model by Simmons et al.
(2010) appears to predict that the direction of adjustment
should constantly alternate. Finally, while both previousmodels assumed that adjustment stops as soon as the cur-
rent estimate is sufficiently plausible (Epley and Gilovich,
2006; Simmons et al., 2010), we propose that the number of
adjustments is pre-determined adaptively to achieve an opti-
mal speed-accuracy tradeoff on average. In the subsequent
section we apply the resulting model to explain the variousanchoring phenomena summarized in Table 1.
Resource-rational analysis
Resource-rational analysis is a new approach to answering
a classic question: how should we think and decide given
that our time and our minds are finite? In economics this
problem was first identified by Simon ( 1955, 1956, 1972).
Simon pointed out that our finite computational capacities
make it impossible for us to always find the best course
of action, because we cannot consider all possible conse-quences. He illustrated this using the game of chess, where
choosing the optimal move would require considering about
10
120possible continuations. Thus, Simon concluded, to
adequately model human behavior we need a theory ofrationality that takes our minds’ limits into account. Simon
called such an approach bounded rationality, emphasizing
that it depends on the structure of the environment (Simon,
1956) and entails satisficing, that is accepting suboptimal
solutions that are good enough, rather than optimizing.
Subsequent research has identified simple heuristics thatmake good, but not necessarily optimal, decisions and judg-
ments (Tversky 1972; Thorngate 1980; McKenzie 1994;
Gigerenzer & Goldstein 1996) very efficiently. Thinking is
assumed to be costly and alternative strategies differ in the
amount of thinking they entail (e.g., Shugan, 1980). Based
on this line of reasoning, it has been proposed that peopleadaptively select their cognitive strategies from a toolboxof simple heuristics (Gigerenzer & Selten, 2002) according
to a cost-benefit analysis (Beach & Mitchell, 1978; Payne
et al., 1993).
While Simon (1955) provided some formal examples
of satisficing strategies, he viewed bounded optimality asa principle rather than a formal framework. Subsequentresearchers have tried to formally capture the tradeoffbetween time and errors. Good (1983) formulated this idea
in terms of the maximization of expected utility taking into
account deliberation cost. Intuitively, this means that ratio-nal bounded agents optimally trade off the expected utility
of the action that will be chosen with the corresponding
deliberation cost. Yet, Good (1983) did not make this notionmathematically precise. Furthermore, his formulation does
Psychon Bull Rev (2018) 25:322–349 327
0250500Nr. Adjustmentsp(x|k)
pprop
High Anchor Medium Anchorx xProbability
x i*optimal #adjustments
Low Anchor
Fig. 1 The figure illustrates the resource-rational anchoring-and-
adjustment. The three jagged lines are examples of the stochastic
sequences of estimates the adjustment process might generate startingfrom a low, medium, and high anchor respectively. In each iterationa potential adjustment is sampled from a proposal distribution p
prop
illustrated by the bell curves. Each proposed adjustment is stochasti-
cally accepted or rejected such that over time the relative frequency
with which different estimates are considered q(ˆxt)becomes the tar-
get distribution p(x|k). The top of the figure compares the empirical
distribution of the samples collected over the second half of theadjustments with the target distribution p(x|k). Importantly, this dis-
tribution is the same for each of the three sequences. In fact, it isindependent of the anchor, because the influence of the anchor van-ishes as the number of adjustments increases. Yet, when the number ofadjustments (iterations) is low (e.g., 25), the estimates are still biasedtowards their initial values. The optimal number of iterations i
⋆is
very low as illustrated by the dotted line. Consequently, the resultingestimates indicated by the red, yellow, and red cross are still biasedtowards their respective anchors
not take into account the deliberation cost of determining the
optimal tradeoff between expected utility and deliberation
cost. These problems were solved by Russell and colleagues
(Russell and Wefald, 1991; Russell & Subramanian, 1995;
Russell, 1997) who provided a complete, formal, mathe-
matical theory of the rationality of bounded agents. In this
framework, agents are considered to be rational if they fol-
low the algorithm that makes the best possible use of their
computational architecture (e.g., hardware) and time.
Resource-rational analysis leverages this abstract theory
for understanding the human mind. To be resource-rationalis to make optimal use of one’s finite time and limited cog-
nitive resources. Resource-rational analysis (Griffiths et al.,
2015) derives rational process models of cognitive abili-ties from formal definitions of their function and abstractassumptions about the mind’s computational architecture.
This function-first approach starts at the computational level
of analysis (Marr, 1982). When the problem solved by
the cognitive capacity under study has been formalized,
resource-rational analysis postulates an abstract computa-
tional architecture, that is a set of elementary operationsand their costs, with which the mind might solve this prob-
lem. Next, resource-rational analysis derives the algorithm
that is optimal for solving the problem identified at thecomputational level with the abstract computational archi-tecture. The resulting process model can be used to simulate
people’s responses and reaction times in a given experi-
ment. The model’s predictions are tested against empiricaldata. Based on this evaluation, the assumptions about thecomputational architecture and the problem to be solved are
revised.
Resource-rational analysis of numerical estimation
Having introduced the basic concepts of resource rational-
ity, we now apply resource-rational analysis to numericalestimation: We start by formalizing the problem solved bynumerical estimation. Next, we specify an abstract compu-
tational architecture. We then derive the optimal solution to
the numerical estimation problem afforded by the computa-tional architecture. This resource-rational strategy will then
be evaluated against empirical data in the remainder of this
article.
Function In numerical estimation people have to make an
informed guess about an unknown quantity Xb a s e do nt h e i r
knowledge K. In general, people’s relevant knowledge Kis
incomplete and insufficient to determine the quantity Xwith
certainty. For instance, people asked to estimate the boiling
point of water on Mount Everest typically do not know its
exact value, but they do know related information, such as
the boiling point of water at normal altitude, the freezingpoint of water, the qualitative relationship between altitude,
air pressure, and boiling point, and so on. We formalize peo-
ple’s uncertain belief about Xby the probability distribution
P(X|K)which assigns a plausibility p(X=x|K)to each
328 Psychon Bull Rev (2018) 25:322–349
potential value x. According to Bayesian decision theory,
the goal is to report the estimate ˆxwith the highest expected
utilityEP(X|K)[u(ˆx,x)]. This is equivalent to finding the
estimate with the lowest expected error cost
x⋆=arg minˆxEP(X|K)[cost(ˆx,x)], (1)
where x⋆is the optimal estimate, and cost( ˆx,x) is the error
cost of the estimate ˆxw h e nt h et r u ev a l u ei sx . Here, we
assume that the error cost is the absolute deviation of the
estimate from the true value, that is cost (ˆx,x)=/vextendsingle/vextendsingleˆx−x/vextendsingle/vextendsingle.
Model of mental computation How the mind should solve
the problem of numerical estimation (see Eq. 1) depends
on its computational architecture. Thus, to derive predic-
tions from the assumption of resource-rationality we have
to specify the mind’s elementary operations and their cost.
To do so, we build on the resource-rational analysis by Vulet al. (2014) which assumed that the mind’s elementary
computation is sampling .S a m p l i n gi sw i d e l yu s e dt os o l v e
inference problems in statistics, machine learning, and arti-
ficial intelligence (Gilks et al., 1996). Several behavioral
and neuroscientific experiments suggest that the brain uses
computational mechanisms similar to sampling for a wide
range of inference problems ranging from vision to causallearning (Vul et al., 2014; Denison et al., 2013; Bonawitz
et al., 2014; Bonawitz et al., 2014; Griffiths and Tenenbaum,
2006; Stewart et al., 2006; Fiser et al., 2010). One piece of
evidence is that people’s estimates of everyday events are
highly variable even though the average of their predictions
tends to be very close to the optimal estimate prescribedby Bayesian decision theory (see Eq. 1, Griffiths &
Tenenbaum, 2006, 2011). Furthermore, Vul et al. ( 2014)
found that the relative frequency with which people report acertain value as their estimate is roughly equal to its poste-rior probability, as if the mind was drawing one sample from
the posterior distribution.
Sampling stochastically simulates the outcome of an
event or the value of a quantity such that, on average, therelative frequency with which each value occurs is equal to
its probability. According to Vul et al. (2014), people mayestimate the value of an unknown quantity Xusing only
a single sample from the subjective probability distribution
P(X|K) that expresses their beliefs. If the expected error
cost (1) is approximated using a single sample ˜x, then that
sample becomes the optimal estimate. Thus, the observationthat people report estimates with frequency proportional to
their probability is consistent with them approximating theoptimal estimate using only a single sample.
However, for the complex inference problems that people
face in everyday life generating even a single perfect samplecan be computationally intractable. Thus, while samplingis a first step from computational level theories based on
probabilistic inference towards cognitive mechanisms, amore detailed process model is needed to explain how sim-
ple
cognitive mechanisms can solve the complex inference
problems of everyday cognition. Here, we therefore explore
a more fine-grained model of mental computation whose
elementary operations serve to approximate sampling. Instatistics, machine learning, and artificial intelligence sam-
pling is often approximated by Markov chain Monte Carlo
(MCMC) methods (Gilks et al., 1996). MCMC algorithms
allow the drawing of samples from arbitrarily complex
distributions using a stochastic sequence of approximate
samples, each of which depends only on the previous one.Such stochastic sequences are called Markov chains; hencethe name Markov chain Monte Carlo.
In the remainder of the paper, we explore the conse-
quences of assuming that people answer numerical estima-tion questions by engaging in a thought process similar to
MCMC. We assume that the mind’s computational archi-
tecture supports MCMC by two basic operations: The firstoperation takes in the current estimate and stochastically
modifies it to generate a new one. The second operation
compares the posterior probability of the new estimate tothat of the old one and accepts or rejects the modifica-tion stochastically. Furthermore, we assume that the cost
of computation is proportional to how many such oper-
ations have been performed. These two basic operationsare sufficient to execute an effective MCMC strategy for
probabilistic inference known as the Metropolis-Hastings
algorithm (Hastings, 1970). This algorithm is the basis
for our anchoring-and-adjustment models as illustrated in
Fig.1.
To be concrete, given an initial guess ˆx
0, which we will
assume to be the anchor a(ˆx0=a), this algorithm performs
a series of adjustments. In each step a potential adjustment
δis proposed by sampling from a symmetric probability
distribution Pprop(δ∼Pprop,Pprop(−δ)=Pprop(δ)). The
adjustment will either be accepted, that is ˆxt+1=ˆxt+δ,
or rejected, that is xt+1=ˆxt. If a proposed adjustment
makes the estimate more probable (P(X =ˆxt+δ|K) >
P(X =ˆxt|K)), then it will always be accepted. Other-
wise the adjustment will be made with probability α=
P(X=ˆxt+δ|K)
P(X=ˆxt|K), that is according to the posterior probability of
the adjusted relative to the unadjusted estimate. This strat-egy ensures that regardless of which initial value you startfrom, the frequency with which each value xhas been con-
sidered will eventually equal to its subjective probability ofbeing correct, that is P(X=x|K). This is necessary to cap-
ture the finding that the distribution of people’s estimatesis very similar to the posterior distribution P(X=x|K)
(Vul et al., 2014; Griffiths and Tenenbaum, 2006). More
formally, we can say that as the number of adjustments t
increases, the distribution of estimates Q(ˆx
t)converges to
the posterior distribution P(X|K). This model of compu-
tation has the property that each adjustment decreases an
Psychon Bull Rev (2018) 25:322–349 329
upper bound on the expected error by a constant multiple
(Mengersen & Tweedie, 1996). This property is known as
geometric convergence and illustrated in Fig. 2.
There are several good reasons to consider this com-
putational architecture as a model of mental computationin the domain of numerical estimation: First, the success
of MCMC methods in statistics, machine learning, and
artificial intelligence suggests they are well suited for thecomplex inference problems people face in everyday life.
Second, MCMC can explain important aspects of cognitive
phenomena ranging from category learning (Sanborn et al.,2010) to the temporal dynamics of multistable perception(Moreno-Bote et al., 2011;G e r s h m a ne ta l . ,2012), causal
reasoning in children (Bonawitz et al., 2014), and develop-
mental changes in cognition (Bonawitz et al., 2014). Third,
MCMC is biologically plausible in that it can be efficiently
implemented in recurrent networks of biologically plausible
spiking neurons (Buesing et al., 2011). Last but not least,
process models based on MCMC might be able to explain
why people’s estimates are both highly variable (Vul et al.,
2014) and systematically biased (Tversky & Kahneman,1974).
Optimal resource-allocation Resource-rational anchoring-
and-adjustment makes three critical assumptions: First, the
estimation process is a sequence of adjustments such that
05 1 0 1 5
Nr. Adjustments00.511.522.533.544.55Bias/Anchor=5
Anchor=4
Anchor=3
Anchor=2
Anchor=1
Fig. 2 In resource-rational anchoring-and-adjustment the bias of the
estimate is bounded by a geometrically decaying function of the num-
ber of adjustments. The plots shows the bias of resource-rationalanchoring-and-adjustment as a function of the number of adjustmentsfor five different initial values located 1, ···,5 posterior standard
deviations (i.e., σ) away from the posterior mean. The standard normal
distribution was used as both the posterior P(X|K)and the proposal
distribution P
prop(δ)after sufficiently many steps the estimate will be a represen-
tative sample from the belief P(X|K)about the unknown
quantity Xgiven the knowledge K. Second, each adjust-
ment costs a fixed amount of time. Third, the number ofadjustments is chosen to achieve an optimal speed-accuracytradeoff. It follows, that people should perform the optimal
number of adjustments, that is
t
⋆=arg mint/bracketleftbig
EQ(ˆxt)/bracketleftbig
cost(x,ˆxt)+γ·t/bracketrightbig/bracketrightbig
, (2)
where Q(ˆxt)is the distribution of the estimate after tadjust-
ments, xis its unknown true value, ˆxtis the estimate after
performing tadjustments, cost(x, ˆxt)is its error cost, and γ
is the time cost per adjustment.
Figure 3illustrates this equation showing how the
expected error cost – which decays geometrically with the
number of adjustments–and the time cost – which increases
linearly – determine the optimal speed-accuracy tradeoff.
We inspected the solution to Eq. 2when the belief and the
proposal distribution are standard normal distributions (i.e.
P(X|K)=P(Xprop)=N(0,1)) for different anchors. We
found that for a wide range of realistic time costs the optimal
number of adjustments (see Fig. 4a) is much smaller than
the number of adjustments that would be required to elimi-nate the bias towards the anchor. Consequently, the estimateobtained after the optimal number of adjustments is stillbiased towards the anchor as shown in the Fig. 4b. This is a
consequence of the geometric convergence of the error (seeFig.2) which leads to quickly diminishing returns for addi-
tional adjustments. This is a general property of this rational
model of adjustment that can be derived mathematically
(Lieder et al., 2012).
The optimal speed-accuracy tradeoff weights the costs in
different estimation problems according to their prevalencein the agent’s environment; for more information please seeAppendix B.
Resource-rational explanations of anchoring
phenomena
Following the definition of the bias of an estimator in
mathematical statistics, we quantify the anchoring bias by
B
t(x, a) =E[ˆxt|x,a]−x,w h e r e ˆxtis a participant’s esti-
mate of a quantity xafteriadjustments, and adenotes the
anchor. Figure 5illustrates this definition and four basic
ideas: First, the average estimate generated by anchoring-and-adjustment equals the anchor plus the adjustment. Sec-ond, the adjustment equals the relative adjustment times the
total distance from the anchor to the posterior expectation.
Third, adjustments tend to be insufficient, because the rela-tive adjustment size is less than one. Therefore, the average
330 Psychon Bull Rev (2018) 25:322–349
Fig. 3 The expected value of the error cost cost(x, ˆxn)shown in green
decays nearly geometrically with the number of adjustments n. While
the decrease of the error cost diminishes with the number of adjust-
ments, the time cost γ·tshown in red continues to increase at the
same rate. Consequently, there is a point when further decreasing the
expected error cost by additional adjustments no longer offsets theirtime cost so that the total cost shown in blue starts to increase. Thatpoint is the optimal number of adjustments t
⋆
estimate usually lies between the anchor and the correct
value. Fourth, because the relative adjustment is less than
one, the anchoring bias increases linearly with the distancefrom the anchor to the correct value.
More formally, the bias of resource-rational anchoring-
and-adjustment cannot exceed a geometrically decayingfunction of the number of adjustments as illustrated inFig.2:
B
t(x, a) =E[ˆxt|x,a]−x≤B0(x, a)·rt=(a−x)·rt,(3)where ris the rate of convergence to the distribu-
tionP(X|K) that formalizes people’s beliefs. Conse-
quently, assuming that the bound is tight, resource-rational
anchoring-and-adjustment predicts that, on average, peo-
ple’s predictions ˆxare a linear function of the correct value
xand the anchor a:
E[ˆxt|x,a]≈a·rt+(1−rt)·x. (4)
Therefore the anchoring bias remaining after a fixed numberof adjustments increases linearly with the distance from the
anchor to the correct value as illustrated in Fig. 5.
The hypothesis that the mind performs probabilistic
inference by sequential adjustment makes the interesting,
empirically testable prediction that the less time and com-
putation a person invests into generating an estimate, the
more biased her estimate will be towards the anchor. Asillustrated in Fig. 6a, the relative adjustment (see Fig. 5)
increases with the number of adjustments. When the num-ber of adjustments is zero, then the relative adjustment iszero and the prediction is the anchor regardless of how far it
is away from the correct value. However, as the number of
adjustments increases, the relative adjustment increases andthe predictions become more informed by the correct value.As the number of adjustments tends to infinity, the average
guess generated by anchoring-and-adjustment converges to
the expected value of the posterior distribution.
Our analysis of optimal resource-allocation shows that,
for a wide range of plausible costs of computation, theresource-rational number of adjustments is much smallerthan the number of adjustments required for convergence
to the posterior distribution. This might explain why peo-
ple’s estimates of unknown quantities are biased towardstheir anchor across a wide range of circumstances. Yet,
optimal resource allocation also entails that the number of
Fig. 4 Optimal number of adjustments (a) and the bias after optimal number of adjustments ( b) as a function of relative time cost and distance
from the anchor
Psychon Bull Rev (2018) 25:322–349 331
Fig. 5 If the relative adjustment is less than 100%, then the adjustment is less than the distance from the anchor and the prediction is biased
(Panel a) and the magnitude of the anchoring bias increases with the distance of the correct value from the anchor (Panel b)
adjustments increases with the relative cost of error and
decreases with the relative cost of time. Hence, our theory
predicts that the anchoring bias is smaller when errors are
costly and larger when time is costly; Fig. 6b illustrates this
prediction.
Although we derived the implications of making rational
use of finite cognitive resources for a specific computa-tional mechanism based on sampling, the crucial property
of diminishing returns per additional computation is a uni-
versal feature of iterative inference mechanisms includingapproximate Bayesian computation (Sunn ˚aker et al., 2013;
Turner & Sederberg, 2012), (stochastic) gradient descent,
variational Bayes, predictive coding (Friston & Kiebel,2009; Friston, 2009), and probabilistic computation in cor-
tical microcircuits (Habenschuss et al., 2013). Therefore,
the qualitative predictions shown in Figs. 3–6are not spe-
cific to the abstract computational architecture that we choseto analyze but characterize bounded rationality for a more
general class of cognitive architectures. Hence, while wedo notclaim that the brain implements the sampling algo-
rithm we have analyzed, there are many biologically andpsychologically plausible mechanisms that share the same
characteristics. We will elaborate on this idea in the GeneralDiscussion. In the following sections, we assess these and
other predictions of our model.
Simulation of anchoring effects
Having derived a resource-rational model of anchoring-
and-adjustment we performed computer simulations to test
whether this model is sufficient to explain the plethora of
anchoring effects reviewed above. To capture our assump-tion that people make adjustments in discrete steps, we
model the size of adjustments using the Poisson distribution
P(δ) =Poisson (|δ|;μ
prop). The simulated effects cover
a wide range of different phenomena, and our goal is toaccount for all of these phenomena with a single model.
Fig. 6 The number of adjustments increases the relative size of adjustments ( leftpanel). As the relative cost of time increases, the number of
adjustments decreases and so does the relative size of the adjustment ( right panel)
332 Psychon Bull Rev (2018) 25:322–349
Simulation methodology
We simulated the anchoring experiments listed in Table 1
with the resource-rational anchoring-and-adjustment model
described above. The participants in each of these exper-iments were asked to estimate the value of one or more
quantities X; for instance (Tversky & Kahneman, 1974)
asked their participant to estimate the percentage of African
countries in the United Nations. Our model’s prediction
of people’s estimates of a quantity Xdepends on their
probabilistic belief P(X|K) based on their knowledge K,
the number of adjustments, the anchor, and the adjust-
ment step-size. Thus, before we could apply our model to
simulate anchoring experiments, we had to measure peo-
ple’s probabilistic beliefs P(X|K)about the quantities used
on the simulated experiments. Appendix Cdescribes our
methodology and reports the estimates with obtained.
To accommodate differences in the order of magnitude of
the quantities to be estimated and the effect of incentives for
accuracy, we estimated two parameters for each experiment:
the expected step-size μ
prop of the proposal distribution
P(δ) =Poisson (|δ|;μprop)and the relative iteration cost
γ. These parameters were estimated by the ordinary least-
squares method applied to the summary statistics reported inthe literature. For experiments comprising multiple condi-tions using the same questions with different incentives for
accuracy we estimated a single step-size parameter that is
expected to apply across all conditions and a distinct relativetime cost parameter for each incentive condition.
Insufficient adjustment from provided
and self-generated anchors
Resource-rational anchoring-and-adjustment provides a the-
oretical explanation for insufficient adjustment from pro-
vided and self-generated anchors in terms of a ratio-nal speed-accuracy tradeoff, but how accurately does this
describe empirical data? To answer this question, we fit our
model to two well-known anchoring experiments: one withprovided and one with self-generated anchors.
Provided anchors As an example of adjustment from pro-
vided anchors, we chose the study by Jacowitz and Kahne-
man (1995), because it rigorously quantifies the anchoringbias. Jacowitz and Kahneman ( 1995) asked their partici-
pants two questions about each of several unknown quan-tities: First they asked whether the quantity is larger orsmaller than a certain value–the provided anchor .N e x tt h e y
asked the participant to estimate that quantity. For the firsthalf of the participants the anchor was a low value (i.e. the15
thpercentile of estimates people make when no anchoris provided), and for the second half of the participants theanchor was a high value (i.e. the 85
thpercentile). People’s
estimates were significantly higher when the anchor was
high than when it was low. Jacowitz and Kahneman (1995)
quantified this effect by the anchoring index (AI), whichis the percentage of the distance from the low to the high
anchor that is retained in people’s estimates:
AI=Median( ˆX
high anchor )−Median( ˆXlow anchor )
high anchor −low anchor·100%
(5)
Jacowitz and Kahneman (1995) found that the average
anchoring index was about 50%. This means that the differ-
ence between people’s estimates in the high versus the lowanchor condition retained about half of the distance between
the two anchors.
We determined the uncertainty σfor each of the 15
quantities by the elicitation method described above. Since
Jacowitz and Kahneman (1995) measured people’s median
estimates in the absence of any anchor, we used those val-
ues as our estimates of the expected values μ, because their
sample and its median estimates were significantly different
from ours.
Next, we estimated the adjustment step-size parameter
and the relative time cost parameter by minimizing the sumof squared errors between the predicted and the observed
anchoring indices. According to the estimated parameters,people performed 29 adjustments with an average step-size of 22.4 units. With these two estimated parameters the
model accurately captures the insufficient adjustment from
provided anchors reported by Jacowitz and Kahne ( 1995):
The model’s adjustments are insufficient (i.e. anchoring
index >0; see Eq. 5) on all questions for which this had
been observed empirically but not for the question on which
it had not been observed; see Fig. 7. Our model also cap-
tured the magnitude of the anchoring bias: the model’saverage anchoring index of 53 .22% was very close to its
empirical counterpart of 48 .48%. Furthermore, our model
also captured for which questions the anchoring bias was
high and for which it was low: the correlation between the
predicted and the empirical anchoring indices ( r(13)=
0.62,p=0.0135). The simulated and empirical anchoring
effects are shown in Fig. 7.
Self-generated anchors As an example of adjustment from
self-generated anchors we chose the studies reported inEpley and Gilovich (2006). In each of these studies par-
ticipants were asked to estimate one or more unknown
quantities such as the boiling point of water on MountEverest for which many participants readily retrieved a
Psychon Bull Rev (2018) 25:322–349 333
05 0 1 00 1 50
Predicted Anchoring Index in % 020406080100120Anchoring Index in %
Fig. 7 Simulation of the provided anchor experiment by Jacowitz and
Kahneman ( 1995)
well-known related quantity such as 272◦F (100◦C). After-
wards participants were asked whether they knew and had
thought of each intended anchor while answering the corre-sponding question. For each question, Epley and Gilovich
(2006) computed the mean estimate of those participants
who had thought of the intended anchor while answeringit. We combined the data from all self-generated anchorquestions without additional experimental manipulations
for which Epley and Gilovich (2006) reported people’s
mean estimate, i.e. the first five question from Study 1a,the first five questions from Study 1b, and the control
conditions of Study 2b (2 questions) and the first seven
questions from Study 2c.
2We determined the means and
uncertainties of the model’s beliefs about all quantities used
in Epley and Gilovich’s studies by the elicitation method
described above. The anchors were set to the intended self-generated anchors reported by Epley and Gilovich ( 2006).
We estimated the model’s time cost and adjustment step-
size parameters by fitting the relative adjustments reported
for these studies using the ordinary least-squares method.
The estimated parameters suggest that people performed
8 adjustments with an average step-size of 10.06 units. Withthese parameters the model adjusts its initial estimate by80.62% of the distance to the correct value; this is very close
to the 80.95% relative adjustment that Epley and Gilovich
(2006) observed on average across the simulated studies.Our model captures that for the majority of quantities (13out of 19) people’s adjustments were insufficient. It also
captures for which questions people adjust more and for
2The quantities were the year in which Washington was elected presi-
dent, the boiling point on Mt. Everest, the freezing point of vodka, the
lowest body temperature, the highest body temperature, and the dura-
tion of pregnancy in elephants. Some of these quantities were used in
multiple studies.which questions they adjust less from their uncertainties
a n da n c h o r s :a ss h o w ni nF i g .8 our model’s predictions of
the relative adjustments were significantly correlated with
the relative adjustments that Epley and Gilovich ( 2006)
observed across different questions ( r(17)=0.61,p =
0.0056). Comparing the parameter estimates between theexperiments with provided versus self-generated anchors
suggests that people adjusted less when they had generatedthe anchor themselves. This makes sense because self-
generated anchors are typically much closer to the correct
value than provided anchors.
Effect of cognitive load
In an experiment with self-generated anchors (Epley &
Gilovich, 2006) found that people adjust their estimate less
when required to simultaneously memorize an eight-letter
string. To investigate whether resource-rational anchoring-
and-adjustment can capture this effect, we fit our modelsimultaneously to participants’ relative adjustment with ver-
sus without cognitive load. Concretely, we estimated a
common step-size parameter and separate time cost param-eters for each condition by the least squares method. We
included all items for which Epley and Gilovich ( 2006)
reported people’s estimates. The resulting parameter esti-
mates captured the effect of cognitive load: when peoplewere cognitively busy, the estimated cost per adjustment
was 4.58% of the error cost, but when people were not cog-
nitively busy then it was only 0.003% of the error cost. The
estimated average step-size per adjustment was μ=11.69.
According to these parameters participants performed only
14 adjustments when they were under cognitive load but60 adjustments when they are not. With these parameters
05 0 1 00 1 50
Predicted Relative Adjustments in %020406080100120140160180Relative Adjustments in %
Fig. 8 Simulation of self-generated anchors experiment by Epley and
Gilovich ( 2006)
334 Psychon Bull Rev (2018) 25:322–349
our model captures the effect of cognitive load on relative
adjustment: cognitive load reduced the simulated adjust-ments by 18 .61% (83.45% under load and 102.06% without
load). These simulated effects are close to their empiricalcounterparts: people adjusted their estimate 72 .2% when
under load and 101 .4% without cognitive load (Epley &
Gilovich, 2006). Furthermore, the model accurately cap-
tured for which questions the effect of cognitive load washigh and for which it was low; see Fig. 9. Concretely, our
model explained 93.03% of the variance in the effect of cog-nitive load on relative adjustments ( r(5)=0.9645,p <
0.001).
The anchoring bias increases with anchor extremity
Next we simulated the anchoring experiment by Russo and
Schoemaker (1989). In this experiment business students
were first asked about the last three digits of their tele-
phone number. Upon hearing the number the experimenter
No Load High Load020406080100120Relative Adjustment in %Model
People
0 0.2 0.4 0.6
Predicted Decrease in Relative Adjustment00.10.20.30.40.50.60.7Decrease in Relative Adjustment
Fig. 9 Simulated versus observed effect of cognitive load on the size
of people’s adjustmentsannounced he would add 400 to this number (providing
an anchor) and proceeded to ask the participant whetherthe year in which Attila the Hun was defeated in Europe
was smaller or larger than that sum. When the participant
indicated her judgment, she was prompted to estimate theyear in which Attila had actually been defeated. Russo
and Schoemaker ( 1989) then compared the mean estimate
between participants whose anchor had been 500 ±100,
700±100,···, 1300 ±100. They found that their partici-
pants’ mean estimates increased linearly with the provided
anchor even though the correct value was A.D. 451.
To simulate this experiment, we determined the values
ofμandσby the elicitation method described above.
Since the variability of people’s estimates and confidenceintervals was very high, we increased the sample size ofthis one experiment to 200. We set the model parameters
to the values estimated from the provided anchor experi-
ments by Jacowitz and Kahneman ( 1995) (see above). As
Fig. 10shows, our model correctly predicted that people’s
estimates increase linearly with the provided anchor (Russo& Schoemaker, 1989). To determine whether the quanti-
tative differences between the model predictions and thedata reported by Russo and Schoemaker (1989) were due
to differences between business students in 1989 and peo-
ple working on Mechanical Turk in 2014, we ran an onlinereplication of their experiment on Mechanical Turk with 300
participants. There appeared to be no significant difference
between the estimates of the two populations. However,people’s estimates were highly variable. Consequently, the
error bars on the mean estimates are very large.
Taking into account the high variance in people’s judg-
ments, our simulation results are largely consistent with the
empirical data. In particular, both Russo and Shoemaker’s
400 500 600 700 800 900 1000 1100 1200 1300 14004005006007008009001000110012001300
AnchorMean Estimate (Year of Attila’s defeat)Model Fit
Russo, & Shoemaker (1989)
Replication with 95% CI
Fig. 10 Simulated effect of the anchor on people’s estimates of the
year of Atilla’s defeat and empirical data from Russo & Shoemaker
(1989)
Psychon Bull Rev (2018) 25:322–349 335
data and our replication confirm our model’s qualitative pre-
diction that the magnitude of the anchoring bias increaseslinearly with the anchor, although our model’s prediction
for the highest anchor was more extreme than the average
judgment.
The effects of uncertainty and knowledge
Several experiments have found that the anchoring bias is
larger the more uncertain people are about the quantity to be
estimated (Wilson et al. 1996; Jacowitz & Kahneman 1995).
To assess whether and how well our theory can explain
this effect, we re-analyzed our simulation of the experi-
ment by Jacowitz and Kahneman ( 1995) reported above.
Concretely, we computed the correlation between the uncer-
tainties σof the modeled beliefs about the 15 quantities and
the predicted anchoring indices. We found that resource-
rational anchoring-and-adjustment predicted that adjust-
ments decrease with uncertainty. Concretely, the anchoringindex that our model predicted for each quantity Xwas
significantly correlated with the assumed uncertainty (stan-dard deviation σ) about it (Spearman’s ρ=0.5857,p =
0.0243). This is a direct consequence of our model’s prob-
abilistic acceptance or rejection of proposed adjustments
on a flat (high uncertainty) versus sloped (low uncertainty)belief distribution P(X|K)=
N(μ, σ). Our model thereby
explains the negative correlation ( r(13)=− 0.68) that
Jacowitz and Kahneman (1995) observed between confi-dence ratings and anchoring indices.
Uncertainty reflects the lack of relevant knowledge. Thus
people who are knowledgeable about a quantity should beless uncertain and consequently less susceptible to anchor-ing. Wilson et al. (1996) conducted an anchoring experiment
in which people first compared the number of countries
in the United Nations (UN) to an anchor, then estimatedhow many countries there are in the UN, and finally ratedhow much they know about this quantity. They found that
people who perceived themselves as more knowledgeable
were resistant to the anchoring bias whereas people whoperceived themselves as less knowledgeable were suscep-
tible to it. Here, we asked whether our model can explain
this effect by smaller adjustments due to higher uncertainty.To answer this question, we recruited 60 participants on
Mechanical Turk, asked them how much they knew about
the number of nations in the UN on a scale from 0 (“noth-ing”) to 9 (“everything”) and elicited their beliefs by themethod described in Appendix C. We then partitioned our
participants into a more knowledgeable and a less knowl-edgeable group by a median split as in Wilson et al. ( 1996).
We model the beliefs elicited from the two groups by two
separate normal distributions (Appendix C).
We found that the high-knowledge participants were less
uncertain than the low-knowledgeable participants ( σ
high=35.1 vs. σlow=45.18). Furthermore, their median estimate
was much closer to the true value of 193 ( μhigh=185
vs.μlow=46.25). We fit the relative adjustments from
the anchor provided in Wilson et al.’s experiment (1930)
by the least-squares method as above. With the estimatedparameters (17 adjustments, step-size 488.2) the model’s
predictions captured the effect of knowledge: For the low-
knowledge group the model predicted that providing thehigh anchor would raise their average estimate from 45.18
to 252.1. By contrast, for the high-knowledgeable group our
model predicted that providing a high anchor would fail toincrease people’s estimates (185 without anchor, 163 withhigh anchor).
Differential effects of accuracy motivation
People tend to invest more mental effort when they are
motivated to be accurate. To motivate participants to be
accurate some experiments employ financial incentives foraccuracy, while others warn their participants about poten-
tial errors that should be avoided (forewarnings). Consistent
with the effect of motivation, resource-rational anchoring-and-adjustment predicts that the number of adjustments
increases with the relative cost of error. Yet, financial incen-
tives for accuracy reduce the anchoring bias only in somecircumstances but not in others: First, the effect of incen-tives appeared to be absent when anchors were provided
but present when they were self-generated (Tversky and
Kahneman, 1974; Epley & Gilovich, 2005). Second, the
effect of incentives was found to be larger when people
were told rather than asked whether the correct value is
smaller or larger than the anchor (Simmons et al., 2010).
Here, we explore whether and how these interaction effects
can be reconciled with resource-rational anchoring-and-
adjustment.
Smaller incentive effects for provided anchors than for
self-generated anchors Epley and Gilovich (2005) found
that financial incentives and forewarnings decreased the
anchoring bias when the anchor was self-generated but not
when it was provided by the experimenter. From this find-ing Epley and Gilovich (2005) concluded that people use
anchoring-and-adjustment only when the anchor is self-
generated but not when it is provided. By contrast, Simmonset al. (2010) suggested that this difference may be mediatedby people’s uncertainty about whether the correct answer
is larger or smaller than the anchor. They found that peo-
ple are often uncertain in which direction they should adjustin questions used in experiments with provided anchors; so
this may be why incentives for accuracy failed to reduce
the anchoring bias in those experiments. Here we showthat resource-rational anchoring-and-adjustment can cap-
ture the differential effectiveness of financial incentives in
336 Psychon Bull Rev (2018) 25:322–349
experiments with provided versus self-generated anchors.
First, we show through simulation that given the amountof uncertainty that people have about the quantities to be
estimated our model predicts a larger effect of accuracy
motivation for the self-generated anchor experiments byEpley and Gilovich (2005) than for the provided anchor
experiments by Tversky and Kahneman (1974) and Epley
and Gilovich (2005).
First, we analyze people’s beliefs about the quantities
used in experiments with provided versus self-generatedanchors with respect to their uncertainty. We estimated themeanμand standard deviation σof people’s beliefs about
each quantity Xby the elicitation method described above.
Because the quantities’ values differ by several orders ofmagnitude, it would be misleading to compare the stan-dard deviations directly. For example, for the population
of Chicago (about 2,700,000 people) a standard devia-
tion of 1,000 would express near-certainty, whereas forthe percentage of countries in the UN the same standard
deviation would express complete ignorance. To overcome
this problem, the standard deviation has to be evaluatedrelative to the mean. We therefore compare uncertain-ties in terms of the signal-to-noise ratio (SNR). We esti-
mated the SNR by the median of the signal-to-noise ratios
of our participants’ beliefs (SNR
s=μ2
s/σ2
s). We found
that people tended to be much more certain about the
quantities Epley and Gilovich (2005)u s e di nt h e i rs e l f -
generated anchors experiments (median SNR: 21.03) thanabout those for which they provided anchors (median SNR:
4.58). A Mann-Whitney U-test confirmed that the SNR was
significantly higher for self-generated anchoring questionsthan for questions with provided anchors (U( 18)=74.0,
p=0.0341).
Given that people were more uncertain about the quan-
tities used in the experiments with provided anchors, weinvestigated how this difference in uncertainty affects the
effect of financial incentives on the anchoring bias predicted
by our resource-rational model. To do so, we simulatedStudy 1 from Epley and Gilovich ( 2005), in which they
compared the effects of financial incentives between ques-tions with self-generated versus provided anchors, and theprovided anchors experiment by Tversky and Kahneman(1974). To assess whether our model can explain why the
effect of motivation differs between questions with provided
versus self-generated anchors, we evaluated the effects ofmotivation as follows: First, we fit our model to the data
from the condition with self-generated anchors. Second,
we use the estimated numbers of adjustments to simulateresponses in the condition with provided anchors. Third,
for each question, we measured the effect of motivation
by the relative adjustment with incentives minus the rel-ative adjustment without incentives. Fourth, we averagedthe effects of motivation separately for all questions withself-generated versus provided anchors and compared theresults.
We fit the relative adjustments on the questions with
self-generated anchors with one step-size parameter andtwo relative time-cost parameters: The estimated step-sizewas 17.97. The estimated number of adjustments was 5 for
the condition without incentives and 9 for the condition
with incentives. According to these parameters, motiva-tion increased the relative adjustment from self-generated
anchors by 12.74% from 65.62% to 78.35%. This is con-
sistent with the significant effect of 33 .01% more adjust-
ment that Epley and Gilovich ( 2005) observed for ques-
tions with self-generated anchors. For the condition withprovided anchors Epley and Gilovich (2005) used four
questions from the experiment by Jacowitz and Kahneman(1995) simulated above and the same incentives as in the
questions with self-generated anchors. We therefore simu-
lated people’s responses to questions with provided anchorsusing the step-size estimated from the data by Jacowitz and
Kahneman (1995) and the number of adjustments estimated
from questions with self-generated anchors. Our simula-tion correctly predicted that incentives for accuracy failto increase adjustment from provided anchors. Concretely,
our simulation predicted 44 .09% adjustment with incentives
and 44.48% without. Thus, as illustrated in Fig. 11, our
model captures that financial incentives increased adjust-ment from self-generated anchors but not from provided
anchors. According to our model, this difference is just anartifact of the confound that people know more about the
quantities used in experiments with self-generated anchors
than about the quantities used in experiments with providedanchors.
Finally, we simulated Study 2 from Epley and Gilovich
(2005) in which they compared the effect of warningparticipants about the anchoring bias between questionswith provided versus self-generated anchors. This study
had 2 (self-generated anchors vs. provided anchors) ×2
(forewarnings vs. no forewarnings) conditions. Epley and
Gilovich (2005) found that in the conditions with self-
generated anchors forewarnings increased adjustment, but
in the conditions with provided anchors they did not. Asbefore, we set the model’s beliefs about the quantities usedin this experiment using the elicitation method described
above. We fit our model to the relative adjustments in the
conditions with self-generated anchors. Concretely, we usedthe least-squares method to fit one step-size parameter and
two time cost parameters: one for the condition with fore-
warnings and one for the condition without forewarnings.With these parameters, we simulated people’s estimates in
the conditions with self-generated anchors (to which the
parameters were fit) and predicted the responses in the pro-
vided
anchor conditions that we had notused for parameter
estimation.
Psychon Bull Rev (2018) 25:322–349 337
Selfgenerated Anchors Provided AnchorsEffect of Motivation on Rel. Adjustment %
01020304050607080Without Incentives
With Incentives
Fig. 11 Simulation of Study 1 from Epley and Gilovich (2005): Pre-
dicted effects of financial incentives on the adjustment from provided
versus self-generated anchors
According to the estimated parameters, forewarnings
increased the number of adjustments from 8 to 28. We there-
fore simulated the responses in both conditions with fore-
warnings (provided and self-generated anchor questions)with 8 adjustments and all responses in the two conditions
without forewarnings (provided and self-generated anchor
questions) with 28 adjustments. For the questions with self-generated anchors, forewarnings increased the simulated
adjustments by 30% from insufficient 81% to overshooting
111% of the total distance from the anchor to the correctvalue.
3By contrast, for questions with provided anchors
forewarnings increased the simulated adjustments by only
12.5% from 6.9% to 19.4%. Thus, assuming that forewarn-
ings increase the number of adjustments from provided
anchors by the same number as they increase adjustments
from self-generated anchors our model predicts that their
effect on people’s estimates would be less than one third ofthe effect for self-generated anchors; see Fig. 12. Accord-
ing to our model, the reason is that people’s uncertaintyabout the quantities for which anchors were provided isso high that the effect of additional adjustments is muchsmaller than in the questions for which people can read-
ily generate their own anchors. Our results are consistent
with the interpretation that the absence of a statisticallysignificant effect of forewarnings on the bias towards the
provided anchors in the small sample of Epley and Gilovich
(2005) does not imply that the number of adjustments didnot increase. Therefore adjustment from provided anchors
cannot be ruled out.
3Overshooting is possible, because the expected value of the estimated
belief P(X|K)=N(μ, σ) can be farther away from the anchor than
the correct value.Self-Generated Anchors Provided Anchors020406080100120Relative Adjustment in %Model
People
Fig. 12 Simulation of Study 2 from Epley and Gilovich (2005): Pre-
dicted effects of forewarnings for questions from experiments with
provided versus self-generated anchors
Direction uncertainty masks the effect of incentives
Simmons et al. (2010) found that accuracy motivation
decreases anchoring if people are confident about whether
the quantity is larger or smaller than the anchor but not whenthey are very uncertain. Simmons et al. (2010) showed thateven when the anchor is provided, incentives for accuracy
can reduce the anchoring bias provided that people are con-
fident about the correct direction of adjustment. Concretely,Simmons et al.’s second study unmasked the effect of incen-
tives on adjustment from provided anchors by telling instead
of asking their participants whether the true value is largeror smaller than the anchor. Similarly, in their third study,
Simmons et al. (2010) found that the effect of incentives is
larger when the provided anchor is implausibly extreme thanwhen it is plausible. Here we report simulations of both ofthese effects.
First, we show that our model can capture that the effect
of incentives increases when people are told the correctdirection of adjustment. Simmons et al.’s second study mea-
sured the effect of accuracy motivation on the anchoring
index as a function of whether people were asked or toldif the correct value is larger or smaller than the anchor.
We modeled the effect of being told that the quantity X
is smaller or larger than the anchor aby Bayesian updat-
ing of the model’s belief about XfromP(X|K)toP(X|K,
X<a )a n d P(X|K,X>a ) respectively. The original
beliefs P(X|K)were determined by the elicitation method
described in Appendix C. We fit the model simultaneously
to all anchoring indices by ordinary least squares to estimateone step-size parameter and one number of adjustments
for each incentive condition. According to the estimatedparameters, incentives increased the number of adjustments
from 5 to 1000 and the average adjustment step-size was
338 Psychon Bull Rev (2018) 25:322–349
11.6 units. For both incentive conditions, our model cap-
tured the variability of adjustments across trials: For trialswith incentives for accuracy the correlation between sim-
ulated and measured anchoring indices was r(18)=0.77
(p=0.0001), and for trials without incentives this cor-
relation was r(18)=0.61 (p =0.004). Our model also
captured the overall reduction of anchoring with incentives
for accuracy observed by Simmons et al. ( 2010), although
the predicted 42% reduction of anchoring with incentives
for accuracy was quantitatively larger than the empirical
effect of 8%. Most importantly, our model predicted theeffects of direction uncertainty on adjustment and its inter-action with accuracy motivation: First, our model predicted
that adjustments are larger if people are told whether the
correct value is larger or smaller than the anchor. The pre-dicted 13.7% reduction in the anchoring index was close
to the empirically observed reduction by 18 .8%. Second,
our model predicted that the effect of accuracy motivation
will be 6.3% larger when people are told the direction of
adjustment. The predicted effect of direction uncertainty is
smaller than the 21% increase reported by Simmons et al.(2010) but qualitatively consistent. Therefore, our modelcan explain why telling people whether the correct value
is larger or smaller than the anchor increases the effect
of accuracy motivation. According to our model, financialincentives increase the number adjustments in both cases,
but knowing the correct direction makes adjustment more
effective by eliminating adjustments in the wrong direction.
Second, we simulated Study 3b of Simmons et al. ( 2010)
in which they showed that financial incentives increaseadjustments away from implausible anchors. Concretely,this study compared the effect of accuracy motivation onadjustments between plausible versus implausible provided
anchors. As before, we determined the model’s beliefs by
the procedure described above and estimated the number ofadjustments with and without incentives (781 and 188) and
the adjustment step-size (0.01) by fitting the reported rela-
tive adjustments by ordinary-least squares.
4With this single
set of parameters we simulated adjustments from plausible
versus implausible provided anchors. The predicted adjust-
ments captured a statistically significant proportion of theeffects of anchor type, motivation, and quantity on thesize of people’s adjustments: ρ(22)=0.72,p < 0.0001.
Most importantly, our simulations predicted no statisticallysignificant effect of accuracy motivation on absolute adjust-ment (mean effect: 0.76 units; 95% CI: [−0.42; 1.94])
when the anchor was plausible but a substantially largerand statistically significant effect when the anchor was
4The reason that the estimated step-size is so small appears to be that
all quantities and distances in this experiment are small compared to
those in other experiments such as Study 2 by the same authors. The
increase in the number of adjustments appears to compensate for the
reduced step-size.implausible (17.8 units; 95% CI: [9.76; 25.91]); see Fig. 13.
This prediction results from the fact that large adjustments
away from plausible anchors will often be rejected because
they decrease the estimate’s plausibility and small adjust-
ments in the wrong direction are almost as likely to beaccepted as adjustment in the correction direction because
values on either side of the plausible anchor are almost
equally plausible if the distribution is symmetric around itsmode. Thus the expected change per adjustment is rather
small.
In conclusion, resource-rational anchoring-and-adjust-
ment can explain why motivating participants to be accurate
reduces the anchoring bias in some circumstances but not
in others. In a nutshell, our model predicts that incentives
for accuracy have little effect when adjustments in eitherdirection hardly change the estimate’s plausibility. The sim-
ulations reported above demonstrate that this principle is
sufficient to explain the differential effect of accuracy moti-vation on adjustments from provided versus self-generated
anchors. Therefore, a single process – resource-rational
anchoring-and-adjustment – may be sufficient to explainanchoring on provided and self-generated anchors.
Summary
Our resource-rational analysis of numerical estimation
showed that under-adjusting an initial estimate can be a ratio-
nal use of computational resources. The resulting model
can explain ten different anchoring phenomena: insufficientadjustments from both provided and self-generated anchors,
the effects of cognitive load, anchor extremity, uncer-
tainty, and knowledge, as well as the differential effects offorewarnings and financial incentives depending on anchor
Fig. 13 Simulation of Experiment 3 from Simmons et al. ( 2010): Pre-
dicted effect of accuracy motivation on adjustments from plausible
versus implausible provided anchors
Psychon Bull Rev (2018) 25:322–349 339
type (provided vs. self-generated), anchor plausibility, and
being asked versus being told whether the quantity issmaller or larger than the anchor (see Table 1). None of
the previous models (Epley and Gilovich, 2006; Simmons
et al., 2010) was precise enough to make quantitative pre-
dictions about any of these phenomena let alone precisely
predict all of them simultaneously. The close match between
our simulation results and human behavior suggests thatresource-rational anchoring-and-adjustment provides a uni-
fying explanation for a wide range of disparate and appar-
ently incompatible phenomena in the anchoring literature.Our model was able to reconcile these effects by captur-ing how the effect of adjustment depends on the location
and shape of the posterior distribution describing the par-
ticipants’ belief about the quantity to be estimated. Forinstance, our model reconciles the apparent ineffectiveness
of financial incentives at reducing the bias towards provided
anchors (Tversky & Kahneman, 1974) with their apparent
effectiveness at reducing bias when the anchor is self-
generated (Epley & Gilovich, 2005). To resolve this appar-
ent contradiction, we did not have to postulate additional
processes that operate only when the anchor is provided–unlike Epley and Gilovich (2006). Instead, our computa-
tional model directly predicted this difference from people’s
higher uncertainty about the quantities used in experimentswith provided anchors, because when the uncertainty is high
then adjustments in the wrong direction are more likely to be
accepted. Our model thereby provides a more parsimoniousexplanation of these effects than the proposal by Epley and
Gilovich (2006). While Simmons et al. (2010) offered a con-
ceptual explanation along similar lines, our model predictedthe exact sizes of these effects a priori.
The parameter estimates we obtained differed signifi-
cantly across the simulated phenomena. This is partly duedifferences in the incentives and other experimental manip-ulations. Additional reasons for the variability in the param-
eter estimates are somewhat arbitrary differences in the
resolution of the hypothesis spaces across different quan-tities and the interdependence between the average change
per adjustment and the number of adjustments: the same
amount of adjustment can be explained either by a smallnumber of large steps or a large number of small steps. Forsome experiments maximum likelihood estimation chose
the former interpretation and for others it chose the latter.
But because a larger step size can compensate for a smallernumber of adjustments, it is quite possible that the model
could have explained all of the findings with a very similar
step size and number of adjustment parameters if we knewthe structure and resolution of people’s hypothesis spaces
for the quantities used in each experiment. Although the
model’s parameters were unknown and had to be estimatedto make quantitative predictions, all of the qualitative phe-nomena we simulated logically follow from the structure ofthe model itself. In this sense, our model did not just capturethe simulated phenomena but predicted them. Most impor-tantly, our theory reconciles the apparently irrational effects
of potentially irrelevant numbers with people’s impressive
capacity to efficiently handle a large number of complexproblems full of uncertainty in a short amount of time.
General discussion
Anchoring and adjustment is one of the classic heuris-tics reported by Tversky and Kahneman (1974)a n di tseems hard to reconcile with rational behavior. In this arti-
cle, we have argued that this heuristic can be understood
as a signature of resource-rational information process-ing rather than a sign of human irrationality. We havesupported this conclusion by a resource-rational analysis
of numerical estimation and simulations of ten anchor-
ing phenomena with a resource-rational process model.We showed that anchoring-and-adjustment can be inter-
preted as a Markov chain Monte Carlo algorithm–a rational
approximation to rational inference. We found that acrossmany problems the optimal speed-accuracy tradeoff of
this algorithm entails performing so few adjustments that
the resulting estimate is biased towards the anchor. Oursimulations demonstrated that resource-rational anchoring-and-adjustment, which adaptively chooses the number of
adjustments to maximize performance net the cost of com-
putation, provides a unifying explanation for ten differentanchoring phenomena (see Table 1).
Although we explored the implications of limited time
and finite cognitive resources assuming an abstract compu-tational architecture based on sampling, we do notclaim that
the brain implements the sampling algorithm we analyzedabove. Instead, our goal was to illustrate general prop-erties of resource-rational information processing. Manyother iterative inference mechanisms also have the prop-
erty of diminishing returns for additional computation that
our analysis is based on. Hence, the qualitative predictionss h o w ni nF i g s . 3–6characterize bounded rationality for a
more general class of cognitive architectures. Importantly,this class includes biologically plausible neural networkimplementations of Bayesian inference (Habenschuss et al.,
2013; Friston and Kiebel, 2009; Friston, 2009) and mech-
anisms that implement the general principles of our model
in a more psychologically plausible fashion. For instance,while our model’s assumption that people can evaluate the
exact likelihood of the observed data under each sampled
hypothesis is questionable, our analysis also applies to sam-pling methods that approximate the likelihood through sim-
ulation (Turner & Sederberg, 2012; Sunn ˚aker et al., 2013).
Likewise, while we do not propose a neural implementa-
tion of probabilistic inference, our analysis also applies to
340 Psychon Bull Rev (2018) 25:322–349
Markov chain Monte Carlo algorithms implemented in cor-
tical microcircuits (Habenschuss et al., 2013), stochastic
gradient descent, and the predictive coding implementa-
tion of variational inference postulated by the free-energy
principle (Friston & Kiebel, 2009; Friston, 2009). There-
fore, our results support the adaptive allocation of finite
computational resources and the resource-rationality of bias
regardless of the specific cognitive mechanism that peopleuse to draw inferences.
In the remainder of this paper we will discuss the impli-
cations of our results for general theoretical questions. Westart by discussing how our model is related to previous the-ories of anchoring and how they can be integrated into our
resource-rational framework. We then turn to two questions
about rationality: First, we discuss existing evidence for thehypothesis that anchors are chosen resource-rationally and
how it can be tested in future experiments. Second, we argue
that resource-rationality, the general theory we have appliedto explain the anchoring bias, provides a more adequate
normative framework for cognitive strategies than classical
notions of rationality. We close with directions for futureresearch.
Relation to previous theories of anchoring
and adjustment
Previous models of anchoring-and-adjustment (Epley &
Gilovich, 2006; Simmons et al., 2010) assumed that
adjustment terminates when the plausibility of the current
estimate exceeds a threshold. From an information pro-cessing perspective, the limitation of models postulating
that adjustment stops when plausibility exceeds a thresh-
old is that there is no single threshold that works wellacross all estimation problems. Depending on the level of
uncertainty successful estimation requires different thresh-
olds. A threshold that is appropriate for low uncertaintywill result in never-ending adjustment in a problem with
high uncertainty. Conversely, a threshold that is appro-
priate for a problem with high uncertainty would be tooliberal when the uncertainty is low. In addition, Sim-mons et al. (2010) postulate that people reason about
the direction of their adjustment whereas resource-rational
anchoring-and-adjustment does not. It would be interestingto see whether an extension of our model that incorporates
directional information would perform better in numeri-
cal estimation and better predict human behavior. We willreturn to this idea when we discuss directions for future
research.
According to the selective-accessibility theory of anchor-
ing (Strack and Mussweiler, 1997), comparing an unknown
quantity to the provided anchor increases the accessibilityof anchor-consistent knowledge and the heightened avail-
ability of anchor-consistent information biases people’sestimates. There is no quantitative mathematical modelof selective accessibility that could be tested against ourresource-rational anchoring-and-adjustment model using
the data we have collected. The evidence that some
anchoring biases result from selective accessibility (Strack& Mussweiler, 1997) does not undermine our analysis,
because the existence of selective accessibility would notrule out the existence of anchoring-and-adjustment andvice versa. In fact, from the perspective of resource-
rational probabilistic inference a mechanism similar to
selective accessibility is likely to coexist with anchoring-and-adjustment. Concretely, we have formalized the prob-lem of numerical estimation of some quantity Xas min-
imizing the expected error cost of the estimate ˆxwith
respect to the posterior distribution P(X|K)where Kis the
entirety of the person’s relevant knowledge. This problem
can be decomposed into two sub-problems: conditioning
on relevant knowledge to evaluate (relative) plausibilityand searching for an estimate with high plausibility. It
appears unlikely that the mind can solve the first problem
by simultaneously retrieving and instantly incorporatingeach and every piece of knowledge relevant to estimat-ingX. Instead, the mind might have to sequentially recall
and incorporate pieces K
(1),K(2),K(3),···of its knowl-
edge to refine P(X) toP(X|K(1))toP(X|K(1),K(2))to
P(X|K(1),K(2),K(3)), and so forth. This process could
be modeled as bounded using a sequential Monte Carloalgorithm (Doucet et al., 2001) and bounded conditioning
(Horvitz et al., 1989). Furthermore, it would be wasteful not
to consider the knowledge that has been retrieved to answerthe comparison question in the estimation task and impos-sible to retrieve all of the remaining knowledge. Selectiveaccessibility may therefore result from the first process.
Yet, regardless of how the first problem is solved, the mind
still needs to search for an estimate ˆxwith high posterior
probability, and this search process might be implemented
by something like anchoring-and-adjustment. Furthermore,
the knowledge retrieved in the first step might also guidethe generation of an anchor. Importantly, both processes
are required to generate an estimate. Therefore, we agree
with Simmons et al. (2010) that selective accessibility andanchoring-and-adjustment might coexist and both of themmight contribute to the anchoring bias.
Like the model by Simmons et al. (2010), our theory
deviates from Epley and Gilovich (2005) by suggestingthat anchoring and adjustment is a unifying mechanisms
for the anchoring biases observed for self-generated as
well as provided anchors. Our simulations show that thisassertion is compatible with the results reviewed by Epley
and Gilovich (2006) because the effect of financial incen-
tives declines with the uncertainty about the quantity to beestimated. This explanation is similar to the argument bySimmons et al. (2010), but our formal model does not need
Psychon Bull Rev (2018) 25:322–349 341
to assume that people reason about the direction of their
adjustments.
Our model is consistent with the recently proposed
anchor integration model (Turner and Schley, 2016). Both
models describe the effect of the anchor in terms ofBayesian inference, but while the anchor integration model
is agnostic about the mechanism by which the anchor affects
people’s judgments and whether or not this is rational, wehave developed a rational process model.
In summary, our resource-rational analysis of estimation
problems sheds new light on classic notions of anchoring-and-adjustment (Tversky & Kahneman, 1974;E p l e y&
Gilovich, 2006), explaining why they work and why peo-
ple use them. Furthermore, our framework is sufficientlygeneral to incorporate and evaluate the additional mecha-nisms proposed by Simmons et al. ( 2010) and Strack and
Mussweiler (1997) and many others. Exploring these exten-sions is an interesting direction for future work.
Are anchors chosen rationally?
Anchoring-and-adjustment has two components: generating
an anchor and adjusting from it. Our simulations supported
the conclusion that adjustment is resource-rational. Thus, a
natural next question is whether anchors are also generatedresource-rationally.
Self-generated anchors are usually close to the correct
value, but provided anchors can be far off. For instance, itappears irrational that people can be anchored on their socialsecurity number when they estimate how much they would
be willing to pay for a commodity (Ariely et al., 2003).
Yet, the strategy failing people in this specific instance may
nevertheless be resource-rational overall for at least four
reasons: First, it is sensible to assume that the experimenter
is reasonable and cooperative. Therefore her utterancesshould follow the Gricean maxims. Specifically, accordingto Grice’s maxim of relation the stated anchor should be rel-
evant (Zhang & Schwarz, 2013). Furthermore, as a rational
information-seeking agent the experimenter should ask the
question whose answer will be most informative. The most
informative anchor to compare the true value to would be at
the center of the experimenter’s belief distribution. This toosuggests that it is reasonable to treat the provided anchor as
a starting point. A weaker version of this argument might
apply even to the experiment in which Tversky and Kah-neman (1974) asked participants to compare the number ofAfrican countries in the UN to a randomly generated num-
ber: It seems reasonably for participants to assume that the
experimenter would not be asking them whether the num-ber of African countries in the UN is larger or smaller than
the number on the wheel of fortune if the answer was obvi-
ous to him. Hence, assuming the logic of conversation, thefact that the experimenter did ask would suggest that thenumber was within the range of values he considered plau-sible. Under these assumptions, the question constitutes aninformative endorsement of the anchor regardless of how it
was generated. This makes it reasonable to use that value as
a starting point.
Second, subsequent thoughts and questions are usually
related. So it is reasonable to use the answer to a precedingquestion as the starting point for next thought. This holdsfor sequences of arithmetic operations such as 8 ×7×6×
5×4×3×2×1 for which people anchor on their interme-
diate results when they are forced to respond early (Tversky& Kahneman, 1974) and in many other cases too. Third,
when the provided anchor is the only number available in
working memory, then using it may be faster and require
less effort than generating a new one. This assumption isconsistent with evidence for spreading-activation theories
of semantic processing and memory retrieval (Collins &
Loftus, 1975; Neely, 1977; Anderson, 1983). For instance,
when primed with one word people are faster to recognize
words that are associated with the prime than words that are
not (Neely, 1977; Collins & Loftus, 1975). The spreading
of activation to associated mental representations appearsto be fast and automatic (Neely, 1977) and inhibiting it
would be slow and effortful (Diamond, 2013). Furthermore,
according to spreading-activation theories of memory recall(Anderson, 1983;
Neely, 1977) and rational process mod-
els of memory search (Bourgin et al., 2014; Abbott et al.,
2015), the generation of a new anchor from memory mightbe subject to the same limitations as the adjustment pro-
cess itself. Last but not least, one’s beliefs may be wrong
and the anchor may be more accurate. This was the case inRusso and Shoemaker’s experiment: People overestimatedthe year in which Attila the Hun was defeated in Europe so
much that the anchor was usually closer to the correct value
(A.D. 451) than the mean of their unbiased estimates (A.D.953.5). For these reasons, the observation that people anchor
on irrelevant values provided in psychological experiments
does not imply that anchors are selected irrationally. Anchorselection could be well adapted to the real-world. Conse-
quently, anchoring biases in everyday reasoning would be
much more benign than those observed in the laboratory.This is probably true, because most anchoring experimentsviolate people’s expectation that the experimenter will pro-
vide relevant information, provide negligible incentives for
accuracy, and ask people to estimate quantities about whichthey know very little.
There also is empirical evidence suggesting that peo-
ple do not always use the provided value as their anchor.For instance, in the experiment by Strack and Mussweiler
(1997) the provided anchor influenced the participants’ esti-
mates only when it was semantically related to the quantityto be estimated. Pohl ( 1998) found that the anchoring bias
was absent when the anchor was perceived as implausible,
342 Psychon Bull Rev (2018) 25:322–349
and Hardt and Pohl ( 2003) found that the bias was smaller
on trials where the anchor’s judged plausibility was below
the median plausibility judgment. Thus, at least under some
circumstances, people appear to discard the provided value
when it appears irrelevant or misleading.
However, realizing that the provided anchor is implau-
sible and generating a better anchor require knowledge,effort, and time. Therefore, when people are asked to esti-mate a quantity they know almost nothing about, it may be
resource-rational for them to anchor on whatever the exper-
imenter suggested. This seems applicable to most anchoringexperiments, because participants are usually so uncertainthat they do not even know in which direction to adjust from
the provided anchor (Simmons et al., 2010). If you can-
not even tell whether the correct value is larger or smaller
than the anchor, how could you generate a better one? The
effect of the anchor is largest in people with little knowl-
edge and high uncertainty about the quantity to be estimated(Jacowitz & Kahneman, 1995; Wilson et al., 1996). These
people would benefit from a better anchor, but they cannoteasily generate one, because they lack the relevant knowl-edge. Conversely, our simulation of the effect of knowledgesuggested that people knowledgeable enough to generate
good anchors, will perform well even if they start from a
highly implausible anchor. Although this argument is spec-ulative and has yet to be made precise it suggests that, at
least in some situations, self-generating an anchor might not
be worth the effort regardless of one’s knowledge.
In conclusion, existing data are not necessarily incon-
sistent with the idea that anchors are chosen resource-rationally. Thus, whether anchors are chosen rationallyis still an open question. Experimental and theoreticalapproaches to this question are an interesting avenue for
future research that we will discuss below.
Resource-rationality: A better normative standard
for human cognition?
When people estimate probabilities, the anchoring bias and
other cognitive biases can cause their judgments to vio-late the laws of probability. This could be interpreted as
a sign of human irrationality. However, adherence to the
laws of logic and probability is just one of many notionsof rationality. Existing definitions of rationality differ along
four dimensions: The first distinction is whether rational-
ity is defined in terms of beliefs (theoretical rationality)or actions (practical rationality, Harman, 2013;S o s i s&
Bishop, 2014). The second distinction is whether rationality
is judged by the reasoning process or its outcome (Simon,1976). Third, some notions of rationality take into accountthat the agent’s computational capacity is bounded whereas
others do not (Lewis et al., 2014; Russell, 1997). Fourth,
rationality may be defined either by the agent’s performanceon a specific task or by its average performance in its natu-
ral environment (ecological rationality, Chater & Oaksford,2014; Gigerenzer, 2008; Lewis et al., 2000).
In this taxonomy, Tversky and Kahneman’s notion of
rationality can be classified as theoretical, process-based,unbounded, and task-specific rationality. It is a notion of
theoretical rationality, because it evaluates beliefs rather
than actions. It is a form of process rationality, becauseit evaluates people by how they reason; specifically by
whether or not their thoughts follow the rules of logic andprobability theory. It is a notion of rationality for unboundedagents because it ignores the computational complexityof logical and probabilistic inference (Van Rooij, 2008).
It is task-specific because it evaluates human rationalityby people’s performance on laboratory tasks specificallydesigned to elicit errors rather than representative everyday
reasoning. We have argued that this is an unsuitable met-
ric of human rationality and proposed a concrete alternative:resource-rationality. Resource-rationality differs from clas-
sical rationality along three of the four dimensions: First,
it evaluates reasoning by its utility for subsequent deci-sions rather than by its formal correctness; this makes itan instance of practical rather than theoretical rationality.
For instance, we evaluated anchoring-and-adjustment not
by the correctness of the resulting estimates but by therewards that people earned by using those estimates. Sec-
ond, it agrees with Tversky and Kahneman’s approach in
that resource-rationality is an attribute of the process thatgenerates conclusions and decisions. Third, it takes into
account the cost of time and the boundedness of people’s
cognitive resources. Fourth, resource-rationality is definedwith respect to the distribution of problem’s in the agent’senvironment rather than a set of arbitrary laboratory tasks.
Arguably, all three of these changes are necessary to obtain
a normative–yet realistic–theory of human rationality. Thisnew metric of rationality allowed us to re-evaluate the
anchoring bias as a consequence of resource-rational com-
putation rather than irrationality. Heuristics and rationalmodels are often seen as opposites, but once the cost of com-
putation is taken into account heuristics can be resource-
rational. This illustrates the potential of resource-rationalanalysis to reconcile cognitive biases, such as the anchoringbias, with the fascinating capacities of human intelligence,
and to connect rational theories, such as Bayesian models
of cognition and rational analysis, to heuristics and otherpsychological process models (Griffiths et al., 2015).
Resource-rational analysis is closely related to other the-
oretical frameworks for analyzing cognition. The most clo-sely related one is the computational rationality approach
proposed by Lewis et al. (2014), which draws the same ins-piration from Russell’s work but focuses on finding optimalalgorithms within a fixed cognitive architecture. Anderson’sframework of rational analysis (1990, 1991),
 is also part
Psychon Bull Rev (2018) 25:322–349 343
of the inspiration of resource-rationality, although it pro-
vides only minimal treatment of the computational con-straints under which organisms operate. Finally, the idea
that human cognition is based on simple heuristics (Tversky
and Kahneman, 1974; Gigerenzer & Selten, 2002)i sc o m -
patible with resource-rationality – trading off errors with
the cost of computation is exactly what good heuristics do.
However, far from interpreting the cognitive biases result-ing from such heuristics as evidence for human irrationality
(Kahneman & Tversky, 1972; Nisbett & Borgida, 1975;
Slovic et al., 1977) resource-rational analysis assumes that
these biases are simply the consequence of rational use of
limited computational resources.
Even though resource-rationality is a very recent
approach, it has already shed some light on a wide rangeof cognitive abilities and provides a unifying framework
for the study of intelligence in psychology, neuroscience,
and artificial intelligence (Gershman et al., 2015). For
example, we have recently applied the resource-rational
framework to decision-making (Lieder et al., 2014), plan-
ning (Lieder et al., 2013), and strategy selection (Lieder
et al., 2014; Lieder & Griffiths, 2015). In conclusion,
resource-rationality appears to be a promising frameworkfor normative and descriptive theories of human cognition.
Directions for future research
In a companion paper (Lieder, Griffiths, Huys, & Goodman,
2017) , we empirically confirm our model’s prediction thatadjustment increases with error cost but decreases with time
cost. We show that this is true regardless of whether the
anchor was provided or self-generated. This confirms oursimulations’ assumption that participants in numerical esti-
mation experiments with provided anchors use the same
cognitive strategy as participants in numerical estimationexperiments with self-generated anchors.
The question to which extent anchors are chosen
resource-rationally is one interesting avenue for futureresearch. The hypothesis that anchors are chosen rationallypredicts that if everything else is equal, then people will
choose a relevant anchor over an irrelevant one. This could
be probed by providing people with two anchors rather thanjust one. Alternatively, one could manipulate the ease of
self-generating a good anchor and test whether this ease
decreases the bias towards an implausible provided anchor.To analyze such experiments, the models developed couldbe used to infer which anchor people were using from the
pattern of their responses.
An additional direction for future work is to extend
the resource-rational anchoring-and-adjustment model. Thiscould be done in several ways. First, the model could
be extended by mechanisms for choosing and generatinganchors. Second, the model could be extended by specifyinghow the mind approximates optimal resource allocation. A
third extension of our model might incorporate directionalinformation into the proposal distribution as in the Hamilto-
nian Monte Carlo algorithm (Neal, 2011) to better capture
the effects of direction uncertainty discovered by Simmons
et al. (2010). A fourth extension might capture the sequential
incorporation of relevant knowledge by iterative condition-
ing and explore its connection to the selective accessibilitytheory of the anchoring bias (Strack and Mussweiler, 1997).
A fifth frontier is to make resource-rational anchoring-and-adjustment more adaptive: How can the proposal dis-tribution and a mechanism for choosing the number ofadjustments be learned from experience? Can better per-
formance be achieved by adapting the proposal distribution
from one adjustment to the next? Finally, our resource-rational anchoring-and-adjustment model only uses a single
sample, but it can be generalized to using multiple samples.
Each of these extensions might improve the performanceof the estimation strategy and it is an interesting question
whether or not those improvements would bring its pre-
dictions closer to human behavior. Future studies mightalso evaluate additional alternatives to our model, such asan anchoring model with adaptive plausibility threshold
or algorithms that directly approximate the most probable
estimate rather than a sample from the posterior distribution.
Most previous models of heuristics are formulated for
the domain in which the corresponding bias was discovered.For instance, previous models of anchoring-and-adjustmentwere specific to numerical estimation (Epley & Gilovich,
2006; Simmons et al., 2010). Yet, everyday reasoning is not
restricted to numerical estimation and anchoring also occurs
in very different domains such as social cognition (Epleyet al., 2004). This highlights the challenge that models of
cognition should be able to explain not only what peopledo in the laboratory but also their performance in the real-world. Heuristics should therefore be able to operate on the
complex, high-dimensional semantic representations people
use in everyday reasoning. Resource-rational anchoring-and-adjustment lives up to this challenge, because Markov-
chain Monte Carlo methods are as applicable to semantic
networks (Bourgin et al., 2014 ; Abbott et al., 2015)a
 st h e y
are to single numbers. In fact, resource-rational anchoring-and-adjustment is a very general mechanism that can oper-
ate over arbitrarily complex representations and might be
deployed not only for numerical estimation but also in manyother cognitive faculties such as memory retrieval, lan-
guage understanding, social cognition, and creativity. For
instance, resource-rational anchoring-and-adjustment maybe able to explain the hindsight bias in memory recall
(Pohl, 1998; Hardt & Pohl, 2003), primacy effects in
sequential learning (Abbott & Griffiths, 2011), and the
dynamics of memory retrieval (Abbott et al., 2015; Bourgin
et al., 2014).
344 Psychon Bull Rev (2018) 25:322–349
Conclusion
Resource-rational anchoring-and-adjustment provides a
unifying, parsimonious, and principled explanation for aplethora of anchoring effects including some that were
previously assumed to be incompatible with anchoring-
and-adjustment. Interestingly, we discovered this cognitivestrategy purely by applying resource-rational analysis to the
problem of estimation under uncertainty. It is remarkable
that the resulting model is so similar to the anchoring-and-adjustment heuristic. Our simulations support the conclu-
sion that people rationally adapt the number of adjustments
to the environment’s incentives for speed and accuracy.Resource-rational anchoring and adjustment thereby recon-ciles the anchoring bias with people’s adaptive intelligence
and Bayesian models of reasoning under uncertainty. Con-
cretely, the anchoring bias may reflect the optimal speed-accuracy tradeoff when errors are benign, which is true
of most, if not all, laboratory tasks. Yet, when accuracy
is important and speed is not crucial, then people performmore adjustments and the anchoring bias decreases. Hence,
while people’s estimates are biased in the statistical sense
of the word ( E/bracketleftBig
ˆX|K/bracketrightBig
/negationslash=E[X|K]), our theory suggests
that this is consistent with how they ought to reason. In this
sense, the anchoring “bias” might not be a cognitive biasafter all. Instead, the anchoring bias may be a window on
resource-rational computation rather than a sign of human
irrationality. Being biased can be resource-rational, andheuristics can be discovered by resource-rational analysis.
Acknowledgements This research was supported by grant number
ONR MURI N00014-13-1-0341 from the Office of Naval Research
(TLG and NDG), grant number FA-9550-10-1-0232 from the AirForce Office of Scientific Research (TLG), and a John S. McDonnell
Scholar Award (NDG).
Appendix A
Notation
X: numerical quantity to be estimated
ˆx: people’s estimates of quantity X
t: number of adjustments
ˆxt: people’s estimates of quantity Xafter
tadjustments
Kory: knowledge or information about X
P(X|K),P(X|y): posterior belief about X
P(R|y): distribution of people’s responses to
observation y
cost(ˆx,x) : error cost of reporting estimate ˆxwhen
the true value is x
t⋆: resource-rational number of adjustments
γ: relative time cost per iterationce,ct: cost of time, cost of error
Q: approximate posterior belief
H: hypothesis space
μprop: average size of proposed adjustments
μ⋆
prop: resource-rational step-size of pro-
posed adjustments
a: anchor
Appendix B
Generalization of optimal speed-accuracy tradeoff
from problems to environments
Together, a person’s knowledge Kabout a quantity X,t h e
cost function cost( ˆx,x) , and the correct value xdefine an
estimation problem. However, in most environments people
are faced with many different estimation problems rather
than just a single one, and the true values are unknown.
We therefore define a task environment Eby the relative
frequency P(X,K, cost|E)with which different estimation
problems occur in it. Within each of the experiments thatwe are going to simulate, the utilities, and the participant’sknowledge are constant. Thus, those task environments arefully characterized by P(X,K |E)and cost( ˆx,x) .
The optimal speed-accuracy tradeoff weights the costs
in different estimation problems according to their preva-lence in the agent’s environment. Formally, the agent should
minimize the expected error cost in Eq. 2with respect to
the distribution of estimation problems P(X,K |E)in its
environment E:
t
⋆=arg max
tEP(X,K |E)/bracketleftbig
EQ(ˆxt|K)/bracketleftbig
u(x,ˆxt)−γ·t/bracketrightbig/bracketrightbig
.(6)
Thus, the number of adjustments is chosen to optimize the
agent’s average reward rate across the problem distributionof the task environment (cf. Lewis et al., 2014). If the task
environment is an experiment with multiple questions, thenthe expected value is the average across those questions.
Appendix C
Estimating beliefs
For each simulated experiment we conducted one short
online survey for each quantity Xthat its participants were
asked to estimate. For each survey we recruited 30 par-
ticipants on Amazon Mechanical Turk and asked the four
questions that Speirs-Bridge et al. ( 2010) advocate for the
elicitation of subjective confidence intervals: “Realistically,
what do you think is the lowest value that the ... could be?”,
“Realistically, what do you think is the highest value that
Psychon Bull Rev (2018) 25:322–349 345
Ta b l e 2 Estimated Beliefs: Insufficient adjustment from provided anchors
Study Quantity μσ Correct
Tversky and Kahneman ( 1974) African countries in UN (in %) 22.5 11.12 28
Jacowitz and Kahneman ( 1995) length of Mississippi River (in miles) 1,525 770 2,320
Jacowitz and Kahneman ( 1995) height of mount Everest (in feet) 27,500 3,902 29,029
Jacowitz and Kahneman ( 1995) amount of meet eaten by average American (in pounds) 238 210 220
Jacowitz and Kahneman ( 1995) distance from San Francisco to New York (in miles) 3000 718 2,900
Jacowitz and Kahneman ( 1995) height of tallest redwood tree (in feet) 325 278 379.3
Jacowitz and Kahneman ( 1995) number of United Nations members 111 46 193
Jacowitz and Kahneman ( 1995) number of female professors at the University of California, Berkeley 83 251 805
Jacowitzx and Kahneman ( 1995) population of Chicago (in millions) 5 3 2.715
Jacowitz and Kahneman ( 1995) year telephone was invented 1885 35 1876
Jacowitz and Kahneman ( 1995) average number of babies born per day in the United States 8,750 15,916 3,952,841
Jacowitz and Kahneman ( 1995) maximum speed of house cat (in mph) 17 10 29.8
Jacowitz and Kahneman ( 1995) amount of gas used per month by average American (in gallons) 55 84 35.2
Jacowitz and Kahneman ( 1995) number of bars in Berkeley, CA 43 55 101
Jacowitz and Kahneman ( 1995) number of state colleges and universities in California 57 112 248
Jacowitz and Kahneman ( 1995) number of Lincoln’s presidency 6 2 16
Ta b l e 3 Estimated beliefs: Insufficient Adjustment from self-generated anchors
Study by Epley, & Gilovich (2006) Quantity Mean SD Correct
Study 1a Washington’s election year 1786.5 7.69 1789
Study 1a Boiling Point on Mount Everest in◦F 158.8 36.82 160
Study 1a Freezing Point of vodka in◦F 3.7 17.052 −20
Study 1a lowest recorded human body temperature in◦F 86 14.83 55.4
Study 1a highest recorded human body temperature in◦F 108 3.39 115.7
Study 1b Washington’s election year 1786.5 7.69 1789
Study 1b Boiling point in Denver in◦F 201.3 9.93 203
Study 1b Number of US states in 1880 33.5 8.52 38
Study 1b year 2nd European explorer reached West Indies 1533.3 33.93 1501
Study 1b Freezing point of vodka in◦F 3.7 17.05 –20
Ta b l e 4 Estimated beliefs: Effect of cognitive load
Study by Epley, & Gilovich (2006) Quantity Mean SD Correct
Study 2b Washington’s election year 1786.5 7.69 1789
Study 2b second explorer 1533.3 33.93 1501
Study 2c Washington’s election year 1786.5 7.69 1789
Study 2c second explorer 1533.3 33.93 1501
Study 2c Highest body temperature 108 3.39 115.7
Study 2c boiling point on Mt. Everest 158.8 36.82 160
Study 2c Lowest body temperature 86 14.83 55.4
Study 2c freezing point of vodka 3.7 17.05 –20
Study 2c number of U.S. states in 1880 33.5 8.52 38
346 Psychon Bull Rev (2018) 25:322–349
Ta b l e 5 Estimated beliefs: effects of distance and knowledge
Study Quantity Mean SD Correct
Russo and Shoemaker ( 1989) year of Atilla’s defeat 953.5 398.42 451
Wilson et al. ( 1996); less knowledgeable group Number of countries in the world 46.25 45.18 196
Wilson et al. ( 1996); knowledgeable group Number of countries in the world 185 35.11 196
the ... could be?”, “Realistically, what is your best guess
(i.e. most likely estimate) of the ... ?”, and “How confidentare you that your interval from the lowest to the highestvalue could contain the true value o the ... ? Please enter
a number between 0 and 100%.”. These questions elicit a
lower bound ( l
s) and an upper bound (h s)o nt h ev a l u eo f
X, an estimate (m s), and the subjective probability psthat
Xlies between the lower and the upper bound ( P(X ∈
[ls,hs]|K) respectively, for each participant s. To estimate
people’s knowledge about each quantity from the reported
confidence intervals, we modeled their belief P(X|K)by a
normal distribution N(μs,σs). We used the empirical esti-
matemsasμs,a n ds e t σstohs−ls
/Phi1−1((1+p s)/2)−/Phi1−1(1−(p s+1)/2),
where /Phi1is the cumulative distribution function of the stan-
dard normal distribution. Finally, we took the medians of
these estimates as the values of μandσused in our sim-
ulations. We applied this procedure separately for each
quantity from each experiment that will be simulated below.The quantities and the estimated beliefs are summarized in
Appendix C.
The hypothesis space Hfor each quantity was assumed
to contain all evenly spaced values (interval =σ
20)i nt h e
range spanned by the 0.5th and the 99.5th percentile of thebelief distribution P(X|K)and the anchor(s) plus or minus
one standard deviation. We simulated the adjustments peo-
ple consider by samples from a Poisson distribution, that is
P(δ=h
k−hj)=Poisson (|k−j|;μprop),w h e r e hkandhj
are the kthand the jthvalue in the hypothesis space H,a n d
μpropis the expected step-size of the proposal distribution
P(δ) . This captures the intuition that people consider only
a finite number of discrete hypotheses and that the adjust-ments a person will consider have a characteristic size that
depends on the resolution of her hypothesis space.
The following tables summarize our estimates of peo-
ple’s beliefs about the quantities used in the simulatedanchoring experiments. Since the estimated probabilistic
Ta b l e 6 Estimated beliefs: Anchor type moderates effect of accuracy motivation; Abbreviations: EG– Epley & Gilovich ( 2005), TK– Tversky &
Kahneman ( 1974)
Study Quantity Mean SD Correct
EG, Study 1 population of Chicago 5,000,000 2,995,797.04 2,719,000
EG, Study 1 height of tallest redwood tree 200 76.58 379.3
EG, Study 1 length of Mississippi river (in miles) 1875 594.88 2,320
EG, Study 1 height of Mt. Everest (in feet) 15400 4657.90 29,029
EG, Study 1 Washington’s election year 1788 6.77 1789
EG, Study 1 year the 2nd explorer after Columbus reached the West Indies 1507.75 34.34 1501
EG, Study 1 boiling point on Everest (in◦F) 150.25 36.82 160
EG, Study 1 freezing point of vodka (in◦F) −1.25 14.73 −20
EG, Study 2 Washington election year 1788 6.77 1789
EG, Study 2 2nd explorer 1507.75 34.34 1501
EG, Study 2 boiling point on Mt. Everest (in◦F) 150.25 36.82 160
EG, Study 2 number of US states in 1880 33.5 8.52 38
EG, Study 2 freezing point of vodka (in◦F) −1.25 14.73 −20
EG, Study 2 population of Chicago 3000000 1257981.51 2,719,000
EG, Study 2 height of tallest redwood tree (in feet) 200 76.58 379.3
EG, Study 2 length of Mississippi river (in miles) 1875 594.88 2320
EG, Study 2 height of Mt. Everest 15400 4657.90 29,029
EG, Study 2 invention of telephone 1870 54.48 1876
EG, Study 2 babies born in US per day 7875 8118.58 3,952,841
TK African countries in UN 22.5 11.12 28
Psychon Bull Rev (2018) 25:322–349 347
Ta b l e 7 Estimated beliefs: Effects of direction uncertainty
Simmons et al. ( 2010) Quantity Mean SD Correct
Study 2 length of Mississippi river (in miles) 1625 752.3 2,320
Study 2 average annual rainfall in Philadelphia (in inches) 36.5 23.80 41
Study 2 Polk’s election year 1857.5 45.42 1845
Study 2 Maximum speed of a house cat (miles per hour) 16 9.40 30
Study 2 Avg. annual temperature in Phoenix (in◦F) 82.75 13.82 73
Study 2 Population of Chicago 2,700,000 1,560,608 2,719,000
Study 2 Height of Mount Everest (in feet) 23,750 7,519.70 29,032
Study 2 Avg. lifespan of a bullfrog (in years) 5.75 6.68 16
Study 2 Number of countries in the world 216.25 77.21 192
Study 2 Distance between San Francisco and Kansas city (in miles) 1,425 547.86 1,800
Study 3b Year Seinfeld first aired 1991 2.23 1989
Study 3b Average temperature in Boston in January 26.5 14.86 36
Study 3b Year JFK began his term as U.S. president 1961.25 2.26 1961
Study 3b Avg. temperature in Phoenix in Aug. 96 10.21 105
Study 3b Year Back to the Future appeared in theaters 1985 1.54 1985
Study 3b Avg. temperature in NY in Sept. 70 10.51 74
beliefs are normal distributions, we summarize each of them
by a mean μand a standard deviation σ.
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