Decision
2019, Vol. 6, No. 1, 17– 60© 2017 American Psychological Association
2325-9965/19/$12.00 http://dx.doi.org/10.1037/dec0000088

Reasoning About Social Sources to Learn From Actions
and Outcomes
Daniel Hawthorne-Madell and Noah D. Goodman

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Stanford University
Learning what others know, especially experts, is a crucial shortcut to understanding
the world. Other people’s actions and utterances are thus a powerful source of evidence.
However, people do not simply copy others’ choices or stated beliefs; rather, they infer
what others believe and integrate these beliefs with their own. In this paper, we present
a computational account of the inference and integration process that underpins
learning from a combination of social and direct evidence. This account formalizes thelearner’s intuitive understanding of psychology— or theory of mind (ToM)—including
attributes such as confidence, reliability, and knowledgeability. It then shows how ToM
is the lens used to interpret another person’s choices, weighing them against thelearner’s own direct evidence. To test this account, we develop an experimental
paradigm that allows for graded manipulation of social and direct evidence, and forquantitative measurement of the learner’s resulting beliefs. Four experiments test thepredictions of the model, manipulating knowledgeability, confidence, and reliability ofthe social source. Learners’ behavior is consistent with our quantitative and qualitativemodel predictions across all 4 experiments, demonstrating subtle interactions between
evidence and the attributes of those learned from.
Keywords: social cognition, theory of mind, information integration, Bayesiancomputational models

Which is the best stock, job candidate, or race
horse? When confronted with decisions likethese, people are information omnivores, flexibly learning from multiple information sources.
Stock investors, for example, look for direct
information about a company’s strength, scouring the stock’s fundamentals. They also look formore indirect, socially-mediated information.
What is Warren Buffet’s position on the stock?
How confident is he? What about a presumably
knowledgeable company insider? The choicesthese actors make represent a rich source ofsocial information, even when they are not intentionally trying to advise or influence others.1
At its core, social learning involves interpreting
intentional actions in the context of the learner’s
understanding of the situation, her prior beliefs,

and direct evidence. A theory of social learning
must both explain how a learner interprets anactor’s actions and how the results are integrated with nonsocial evidence and prior beliefs. This suggests two core questions of social
learning. First, what is the relative impact ofsocial sources? How do people integrate social
information with other evidence like direct observations? Do they heavily discount advice as
has been previously found (Harvey & Fischer,1997; Yaniv, 1997; Yaniv & Kleinberger,
2000)? Second, how do learners evaluate andunderstand a social source? What is the role ofeach trait attributed to the actor, how are they
represented, and how do they interact to determine an action’s informativity?
To understand the actions of an actor, people
must have a framework for relating people’s

This article was published Online First September 14,
2017.
Daniel Hawthorne-Madell and Noah D. Goodman, Department of Psychology, Stanford University.
Correspondence concerning this article should be addressed
to Daniel Hawthorne-Madell. E-mail: d.j.hawthorne@alumni.stanford.edu

1
Although the computational framework we propose isgeneral to multiple kinds of social cues, we focus our
discussion on reasoning about intentional actions and referto the social source with the generic actor. For clarity, we
subsequently refer to the actor with a male pronoun and thelearner with a female pronoun.

17

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18

HAWTHORNE-MADELL AND GOODMAN

knowledge, traits, thoughts and behavior—atheory of mind (ToM; Wellman, 1990; Perner,
1991; Premack & Woodruff, 1978). A learner
can use her ToM to reason about the informativeness of an actor’s intentional actions, inferring the actor’s beliefs and the traits that willaffect learning. Previous research shows thatlearners prefer to learn from people who they
think acted intentionally (e.g., Carpenter et al.,
1998; Olineck & Poulin-Dubois, 2005), confidently (e.g., Birch et al., 2010; Moore et al.,
1989; Sabbagh & Baldwin, 2001), and areknowledgeable (e.g., Pratt & Bryant, 1990) andreliable (e.g., Koenig et al., 2004; Birch et al.,2008). We refer collectively to these latent
properties as actor attributes. The ToM required to account for social learning thus relatesbeliefs, actions, and actor attributes.
In this article, we address the two core questions of social learning with a computationalmodel that formalizes how learners interpretsocial information and integrate it with direct
evidence, which we tested with a series of empirical studies. First, we review the factors thatlearners use to identify the informativeness ofsocial information. We then discuss the existing
empirical work in which learners have beenshown to integrate social information with othersources. Building off these experimental paradigms, we introduce one capable of both quantitatively manipulating an actor’s attributes andquantifying the information content of a learner’s direct and social evidence. We then introduce our model, which provides a unified account of how people interpret and learn fromsocial information and integrate it with othersources of knowledge; crucially, this model relies on inferences about the actor— his reliability, knowledge, and other attributes. We perform four experiments and show their close fit
to the predictions of our model. In the firstexperiment, we testeded how learners’ integration of an actor’s choices with their own evidence is affected by the actor’s knowledgeability and perceived reliability. The second
experiment tested the impact of an actor who
also reports his degree of confidence in his
action, providing metacognitive information
about his choice. The third experiment looks for
signatures of spontaneously attributing actors’
confidence when it is not explicitly provided. In
the final experiment, we manipulated the apparent reliability of the actor directly. Following

the experiments, we discuss alternative models
and then model two previous experiments toshow how our model captures how people learn
from and about others. We conclude with a
discussion of the model’s role in understandingsocial cognition, learning, and the future empirical and theoretical directions it inspires.
Actor Attributes Guide Social Learning
Learners make use of information from avariety of social cues like a person’s testimony,
pedagogical actions, advice, and goal-directedactions. Goal-directed actions are the simplestof these cues and therefore provide the clearest
domain to study how people infer an actor’s
attributes from social cues and how these attributes are used to determine from whom to learn
fand how to weigh their advice.2 Exploration ofthese complementary questions has been undertaken in two separate intellectual traditions. The
epistemic trust literature has focused on theearly developmental emergence of sensitivity toactor attributes when selecting an actor to learn
from (for a review see Harris, 2007; Koenig &Harris, 2005). The judge–adviser system (JAS)literature has explored how social informationfrom actors with different attributes are integrated with a person’s own opinions (for a review see Bonaccio & Dalal, 2006).3 This section provides an overview of the key social
factors these independent lines of inquiry have
converged upon— knowledgeability, reliability,
and intentionality.
Knowledgeability
An actor’s knowledgeability is the accuracyof his beliefs about the world. His knowledge isa valuable source of information to the extent
that it is independent from, or more extensive
than, what the learner knows. His value as an
information source could be the result of superior conceptual understanding of the domain, as
is the case with Warren Buffet, or he could
simply have more information (or unique infor2

Communicative actions, for example, require recursive
reasoning to interpret Shafto et al., 2012; Goodman &Stuhlmüller, 2013.
3
The classic adviser in the JAS literature simply states
what they would do akin to the actors in the developmental
literature.

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REASONING ABOUT SOCIAL SOURCES

mation).4 Learners therefore need to know theactor’s knowledgeability to appropriately learn
from him. Previous research has shown thatlearners are sensitive to a range of cues of anactor’s knowledgeability, such as what evidence
he has, his past actions, or his claimed knowledge.
An actor’s perceptual access is a very clear
cue explored extensively in the developmental
literature. A particularly clear case of an independently knowledgeable actor is when he has
seen something that the learner has not; that is,when the actor has privileged perceptual access.
Sensitivity to this kind of knowledge requires
understanding that other people have their own
perceptual access, which is evident in infants as
early as 14 months (Brooks & Meltzoff, 2002,
2005). Understanding that this independent perceptual access can allow other people to form
their own beliefs emerges later with the development of a full-fledged ToM (Flavell, 1999;
Wellman et al., 2001). Children with a ToM
understand that “looking leads to knowing” andcorrectly identify actors who have had perceptual access to the item in question (Pillow,
1989; Pillow & Weed, 1997; Pratt & Bryant,1990). Children are also sensitive to the relevance of the perceptual access to the question.
For example, when asked about the color of ahidden toy, children preferentially learn fromactors who saw the toy over those who felt it(Nurmsoo & Robinson, 2009; Robinson et al.,2011; Robinson & Whitcombe, 2003; Whitcombe & Robinson, 2000).5
Claimed knowledge or expertise is a moreindirect source of evidence of an actor’s knowledgeability. Children prefer to heed actors who
claim to be knowledgeable (Birch et al., 2010;
Jaswal & Malone, 2007; Koenig & Harris,
2005; Moore et al., 1989; Moore & Davidge,
2009; Sabbagh & Baldwin, 2001), and adults
have been shown to be more influenced byactors who claim to be knowledgeable experts(Jungermann & Fischer, 2005). Another indirect source of evidence of an actor’s knowledgeability is their past actions. Young children
are naturally trusting of older actors, but oncean actor makes a mistake they will preferentially learn from others (Birch et al., 2008; Harris & Corriveau, 2011; Jaswal & Neely, 2006;
Koenig et al., 2004; Koenig & Harris, 2005).
Similarly, adult learners use an actor’s past performances to infer their knowledgeability and19

are subsequently less influenced by poorly performing actors (Yaniv, 2004).
In the absence of information about perceptual access, past actions, or claimed knowledge,
learners fall back on more indirect cues. Infants
prefer actors that exhibit nonverbal confidence
cues (Birch et al., 2010) or accompany theiradvice with words like “know” instead of“think” (Jaswal & Malone, 2007; Matsui et al.,2006; Sabbagh & Baldwin, 2001; Stock et al.,2009). Adults are also more influenced by confident actors (Phillips, 1999; Sniezek & Buckley, 1995; Sniezek & Van Swol, 2001; Swol etal., 2005; Yaniv, 1997). Children use additional
heuristics like familiarity (Corriveau & Harris,2009), cultural similarity (Hu et al., 2013; Kinzler et al., 2010), expertise (Lutz & Keil, 2002;
Sobel & Corriveau, 2010), and age (Jaswal &Neely, 2006), whereas adults defer to actors ofgreater age, education, life experience, and wisdom (Feng & MacGeorge, 2006).
Reliability
Although reliability is often used interchangeably with knowledgeability in the epistemic trust literature, we argue for a distinct
meaning—a reliable actor chooses reasonableactions to achieve their goals, informed by theirbeliefs. Reliable actors’ actions are diagnosticof their knowledge, unlike actors that are noisy
decision makers or purposefully deceitful. As
discussed previously, preschoolers track actors’
past mistakes and prefer accurate, “reliable”
actors. Although these actors’ previous mistakes could be interpreted as stemming fromtheir lack of knowledge, the true cause is underspecified in many experiments (e.g., Clément et al., 2004; Jaswal & Neely, 2006; Koenig et al., 2004).
There is a fundamental distinction between
the possession of knowledge and the reliable
use of it, either of which could be the cause of
an informant’s past mistakes. Although difficult, some experimental paradigms have begun
4
If there is independent noise in the belief formation
process, then the actor’s beliefs are informative even is hehas observed the exact same information.
5
Children also appear to be sensitive to the independenceof actor’s knowledge, which is consonant with Whalen,
Buchsbaum, and Griffiths’ (2013) recently finding that
adults weigh the evidence of actor’s with independent information sources.

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HAWTHORNE-MADELL AND GOODMAN

to dissociate these two potential causes by creating situations where the learner knows theactor is knowledgeable and yet does not act
reliably on their understanding (Chow et al.,2008; Poulin-Dubois et al., 2011). They findthat learners are sensitive to an actor’s reliability, preferentially learning from actors who
have reliably used their knowledge in the past.
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Intentionality
A final necessary condition for learning fromthe actions of a knowledgeable and reliable
actor is that the action be intentional. Unintended actions are effectively independent of
the actor’s knowledge. Eighteen-month-olds already display sensitivity to intentionality, selectively learning from purposeful actions overaccidental actions (Carpenter et al., 1998; Olineck & Poulin-Dubois, 2005). Infants preferentially reproduced the actions of adults playing
with a novel toy when the action was intentional(e.g., adult said “There!” while completing theaction) than when it was unintentional (e.g., theadult said “Whoops!” while completing the action). The sensitivity to the intentionality of anaction is also observed in adults who makestrong inferences about the causal structure of adevice when they are presented with evidence
from an intentional action but not from identicalphysical evidence or accidental action (Goodman et al., 2009).
Toward a Cohesive Account of Actor
Attributes
We have seen that learners spontaneously
reason about an actor’s attributes and preferentially learn from the actions of knowledgeable
and reliable actors who act intentionally andwith confidence. However, we have neither acoherent account of how learners combine these
attributes to interpret social evidence nor of how
this social evidence is integrated with their otherevidence, like direct observations. In the next
section, we propose such an account. Our account formally defines knowledgeability, reliability, and confidence in terms of a learner’s
ToM. It makes specific predictions about therelative importance of these factors and theirinteractions. We embed this ToM into an integrative framework to describe how learners
combine social and direct evidence. The result-ing model predicts how a Bayesian learnerwould (a) use available evidence to characterize
an actor, that is, infer how knowledgeable, confident, and reliable he is and (b) integrate social
evidence from the actor given the learner’s
characterization of him. That is to say that themodel provides a standard to evaluate whether
learners learn from social evidence as much as
they “should,” given their model of the actor
and their characterization of his attributes.
To test our model’s predictions requires anexperimental paradigm in which the influenceof social evidence can be quantitatively measured relative to a well understood reference
point (be it other social sources, direct evidence,or prior knowledge). Additionally, the learner’s
characterization of the actor’s attributes must be
measured or manipulated. However, previous
experimental paradigms have not been clear on
one or both points, limiting their ability to rigorously test our models predictions. We briefly
review some key experiments in the advice integration literature and describe the predictionsof our integrative framework and how carefullymeasuring the learner’s characterization of theactor and measuring his influence relative to aclear reference point could test them.
A number of experiments in the JAS literature explored advice utilization by comparing alearner’s initial estimate to her estimate after
receiving advice. The learner’s initial estimate
is effectively a second source of social evidence(a decision made by a person) and is used to
calculate how influential the adviser was. A
robust finding using this paradigm is that learners exhibit an “egocentric bias,” weighing theirinitial opinions more than they “should” (e.g.,
Gardner & Berry, 1995; Harvey & Fischer,1997; Yaniv, 1997; Yaniv & Kleinberger,
2000). However, from the perspective of alearner in these experiments, there is insufficient evidence to decide the extent to which
they should rely on the advice. For example,
there is considerable uncertainty about both theactor’s knowledgeability and their own. Our
framework can formally describe the learner’s
uncertainty about the adviser and herself
and represents how a Bayesian learner would
integrate the information provided in these experiments (this is further explored in the applied
section Learning from and about advisers in
JAS experiments). It suggests that the seeming
underutilization could be due to the substantial

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REASONING ABOUT SOCIAL SOURCES

uncertainty the learner has about the adviser,
which is compounded by the uncertainty she has
about herself. Testing this prediction requires
measuring the learner’s characterization of boththe adviser’s and her own attributes. Additionally, using a less complicated reference point
would allow for a clear test of whether people
truly exhibit an egocentric bias.
Direct evidence, like observations or baserate information, can provide a clearer reference point for social evidence. For example,
Birnbaum and Mellers (1983) used base rate
information to measure the impact of social
evidence while manipulating the adviser’s accuracy and bias. They found that learners
were sensitive to both. Learners listened toaccurate advisers more and discounted advice
from biased sources. Birnbaum and Meller’sparadigm enabled a clean test of how participants integrated social and direct evidence byparametrically manipulating the adviser’s accuracy and the learner’s direct evidence. Do
they weigh advice as much as an ideal Bayesian integrator would? Although the magnitudeof advice integration was well-captured by
the Bayesian model, a clear violation ofBayesian reasoning led Birnbaum and Mellers
to reject his formulation of an ideal Bayesian
integrator. When the adviser endorsed an option that was highly likely given the base-rate,
his advice paradoxically reduced the learner’s
estimate of its likelihood.
An alternative explanation to this surprising
result lies in learners’ characterization of theadviser in Birnbaum and Mellers’ experiment.
Although they carefully controlled the adviser’s
attributes in his experiment, they did not manipulate or measure the adviser’s confidence. In
the subsequent experiments, we find evidence
that learners spontaneously attribute confidence
to advisers, and we describe how spontaneously
attributing low confidence would result in paradoxical integration of advice. Again, our“learnercentric” perspective that focuses on thelearner’s characterization highlights a new interpretation of advice integration findings andpoints to the importance of pinning down thelearner’s characterization of advisers.
Taking stock of the strengths and limitations of these previous investigations, we candescribe the desiderata for an empirical paradigm to test our framework. Social evidence
should have a clear reference point to gauge

21

its relative influence— experimentally controlled direct observations. Having both thelearner and the adviser’s knowledge clearly
quantified in terms of direct observations ensures that they are both commensurable andamenable to parametric manipulation. Similarly, other actor attributes should be definedin a graded way and either explicitly manipulated or the learner’s unmanipulated estimate should be measured. We now introduce
a betting scenario built on these insights, capable of rigorously testing our framework andformalization of ToM.
Social Learning at the Racetrack
Testing how learners combine social and direct information requires an experimental design that quantifies both knowledge sources inthe same way. To do this, we use a horse racing
scenario in which both the learner and the actor
independently observe some number of races.
Hence, the knowledgeability of each is simply
the number of races seen, which can be individually manipulated. Instead of receiving a direct
report of the race results from the actor, thelearner only observes the bet placed by theactor. This requires the learner to infer the actor’s beliefs from his actions, and combine
those beliefs appropriately with the learner’s
own evidence.
We consider a restricted class of races inwhich the actor, who we refer to as Zach, has
followed two racehorses, x and y. Zach observed some number, nz, of their previous headto-head races. The learner has seen some number, nl, of other races from this same rivalry inwhich horse x has won some number of times,
kl. Although the learner knows the outcomesfrom his nl observations, she doesn’t know theoutcomes of the nz races that Zach saw. The
learner just sees Zach say which horse he bet
on, bz, and how many races he saw, nz.
Suppose that the learner sees horse x win two
out of five races and is told that Zach saw 10
races and bet on horse x in the next race— given
the available information (bz, kl, nz, nl), howshould they bet on subsequent races? This isprecisely the question of how learners integratesocial information with their own observations.
Given this experimental formulation of thequestion, we can construct a computationalmodel that explicitly formalizes how learners’

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HAWTHORNE-MADELL AND GOODMAN

ToM is used to interpret social evidence andintegrate it with other sources of evidence. The
model makes quantitative predictions for manipulations of bz, kl, and nz that suggest systematic tests of the model.

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A Computational Account of Learning
From Others
This section introduces a model of howlearners reason about social information andintegrate it with other sources of evidence.
Specifically, we outline a model of an integrative learner who infers a horse’s chance of
winning future races by integrating observations of its previous performance with theway Zach bets given his observations. We
describe the learner’s mental model of both
sources in terms of probabilistic generativemodels (Griffiths et al., 2008; Tenenbaum etal., 2011). These models describe the learner’s theory about the causal processes in theworld giving rise to her observations. For
direct observations, the generative model represents the learner’s theory of the domain, forexample, what latent properties dictate ahorse’s performance (Figure 1a.). The learner’s model of an actor’s actions represents her
theory of mind, that is, the learner’s prediction of how the actor uses his observations toform beliefs and decide which actions willfurther his goals (Figure 1d.). Combining
these two mental models results in the unified
representation used by the integrative learnerto learn from social and direct evidence (Figure 1c). Using Bayesian inference to learn
about the common latent properties of these
two information sources, the integrative
learner model represents a coherent reference
point for social learning. The integrative

Figure 1. Modeling learning from direct and social evidence in causal graph notation.
Random variables are represented as oval nodes, and dotted boxes specify the causal relation
between variables. Solid nodes are observed variables; empty nodes represent latent variables
whose values must be inferred. Panel a: The “direct learner” who learns about the environment from observations. Panel b: The “pure-social learner” who learns about the environment
from an agent’s actions. Panel c: The “integrative learner” who combines information fromboth observations and an agent’s actions to learn about the shared environmental cause. Paneld: The learner’s theory of mind (ToM). It represents the learner’s model of how agents form
beliefs and make decisions given observations, beliefs, and degree of rationality.

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REASONING ABOUT SOCIAL SOURCES

learner described subsequently is an instanceof a general framework for formalizing how
social information is interpreted and integrated with other evidence. The framework
can be used to describe learning in complicated situations requiring elaborate theories
of domain and mind, for example, why yourcar whines only when turning slightly left andthe implication of your mechanic’s knowing
smile. The simple racetrack scenario elides
very little of the general structure and allows
for a concise treatment of the integrative
learner with clear formalizations of the actor’s knowledgeability and reliability. The
full integrative learner model can be built
incrementally. We first describe the puredirect learner, who updates her beliefs about
the world from direct observations. We next
introduce the learner’s ToM, which combines
a belief update model with a decision rule tomodel how Zach forms beliefs from evidence
and then acts on those beliefs. We then show
how a pure-social learner uses this ToM tolearn from Zach’s actions. Combining thesecomponents, we introduce the full integrativemodel. Finally, we conclude with a discussionof the predictions of the integrative learnermodel.
Direct Learner
In the racetrack scenario, two horses, x and y,
repeatedly race against one another. By observing a number of races, the learner can gain
information about the probability horse x wins
the next race. The learning process is guided by
the learner’s theory of the domain—what causesa horse to win. Learners likely entertain a number of complicated theories about the factors
contributing to a horse’s performance (jockey,
pedigree, betting odds, etc.). However, the restricted information the learner and Zach receive admits of a simple domain theory inwhich the probability that a horse wins a race wn
is determined solely by their relative “skill,”
which remains constant. That is to say that thelearner’s estimate of horse x’s skill (sl) is theirestimate of the probability that horse x wins thenext race wn⫹1. Because the evidence is restricted to the results of repeated races between
the same two horses with no ordering information provided, sl is sufficient to describe thelearner’s beliefs about the domain (with 1 ⫺ sl

23

representing horse y’s relative skill).6 Given thisframing, races between the two horses are independent and their outcomes can be predictedby flipping a coin weighted by their skill foreach of the nl races:p(kl ⱍ sl, nl) ⬃ binomial(sl, nl).

(1)

Assuming learners use this simple domain theory and Bayesian inference to update their beliefs, their posterior beliefs after seeing somenumber nl of observations where horse x wins kl
times isp(sl ⱍ kl, nl) ⬀ p(kl ⱍ sl, nl)p(sl).

(2)

where p(sl) is the learner’s prior belief in ahorse’s chance of winning. Equation 2 represents the direct learner (Figure 1a) who relies
solely on direct observations to update her beliefs.
ToM
To interpret Zach’s actions, a learner musthave a theory of the relationship between Zach’s
observations, beliefs, goals, and choices—a ToM(see Figure 1d). The first component of the learner’s ToM is how Zach learns from evidence. We
assume that the learner thinks Zach shares thelearner’s domain theory (p(wn ⫽ x | sz)) andforms beliefs similarly, by Bayesian inference(Equation 2).
The second component of the learner’s ToM
captures how Zach makes decisions given his
beliefs. We understand Zach as an (approximately) rational decision maker, who (softly)
maximizes his utility (in the tradition of Goodman et al. (2009) and Baker et al. (2009)). In thehorserace scenario there are no “odds” or differential payoffs for the two horses, making
Zach’s utility function simply “1” if he bets on
the winning horse and “0” otherwise. Whichhorse Zach thinks will win is specified by his
beliefs p(wn⫹1 | sz). Given this belief distribu6
Indeed, under the assumption that the order of races is
unimportant (i.e., that the chance horse x will win m of nraces is the same independent of the order of the wins andloses), Finetti’s theorem on exchangeable distributions
Finetti (1974) implies that the complex theory of thehorses’ skill can be simplified to a single probability,
without losing any information.

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HAWTHORNE-MADELL AND GOODMAN

tion, an optimal decision theoretic Zach would
choose the action, bz, that maximizes his expected utility ⺕p共wn⫹1 ⱍ sz兲U共bz; wn⫹1兲, always betting on the horse he thinks is most likely to win.
If the learner thinks that Zach may not be entirely optimal, she may assume that Zach sometimes chooses other actions but does so in a waythat higher expected-utility actions will be more
likely; we model this as soft maximization, sothat the probability of a bet isp(bz ⱍ sz, ␣) ⬀ e␣·⺕p(wn⫹1 ⱍ sz)U(bz;wn⫹1) .

(3)

Or, given the simple structure of the racetrack
scenario, (p(wn⫹1 ⫽ x | sz) ⫽ sz), it is simplyp(bz ⫽ x ⱍ sz, ␣) ⬀ e␣·sz ,(4)where alpha is a “rationality” parameter that corresponds to the amount of noise in Zach’s decision
making. As alpha goes to infinity, the decision
rule approaches a strict max-rule where the action
with the highest expected utility is always taken.
High values correspond to “reasonable” agents
whose actions are predictable given their beliefs.7
When alpha approaches zero, actions are chosenrandomly, without regard to their expected utility.
Agents with low alpha values are “unreasonable”:
Their actions are erratic, permitting only weakinferences about the latent causes of the choice.
The alpha parameter therefore corresponds to theagent’s reliability. We simplify our model by positing that learners either think Zach is “reasonable” with a high alpha or “unreasonable” with a
low alpha.
Combining these two components— belief
formation (Equation 2) and action selection(Equation 3)—we have the learner’s model of
how Zach bets given his observations:8p(bz ⱍ kz, nz, ␣) ⫽兰 p(b ⱍ s , ␣)p(s ⱍ n , k ) ds . (5)
sz

z

z

z

z

z

z

Pure-Social Learner
The pure-social learner does not have any
direct observations and must therefore rely
solely on social information. In our racetrack
scenario, the pure-social learner is trying toinfer the horse skill most likely to make Zach
bet as he did. She uses her ToM (Figure 1d) toreason about Zach’s bet. By Bayes’ rule,p(sl ⱍ bz, nz, ␣) ⬀ p(bz ⱍ sl, nz, ␣)p(sl).

(6)

We make the further simplification that thelearner reasons using a fixed assumption about
Zach’s alpha (rather than jointly updating her
beliefs about alpha and other unknown quantities), unless there is direct and salient information about Zach’s rationality. In our experimental setting this means that alpha is only updated
when Zach’s observations and actions are bothobserved. Different learners may have different
beliefs about Zach’s a priori reasonability, resulting in a population (i.e., mixture) of learners
who think Zach is reasonable with p(␣) andunreasonable with 1 ⫺ p(␣). Each learner has
incomplete knowledge of other latent factors
governing Zach’s bet. She knows how many
races Zach saw (nz), but she doesn’t know thecontent of his observations (kz). The learnermust therefore consider both what Zach is likely
to have seen and his rationality in order toevaluate how likely he is to have made this bet
bz. For a learner attributing Zach rationality ␣,
the probability Zach bets bz isp(bz ⱍ sl, nz, ␣)⫽兺k p(bz ⱍ sl, nz, kz, ␣)p(kz ⱍ sl, nz, ␣) (7)z⫽兺k p(bz ⱍ nz, kz, ␣)p(kz ⱍ sl, nz),(8)zwhere the second line follows by simplifying
using the independencies evident in Figure 1b(namely, bz does not depend on sl once kz is
known). We see by combining Equations 6 and8 that the pure-social learner can be defined interms of the learner’s theory of mind and domain (Equations 5 and 2) and prior expectations
about Zach’s rationality and the horse’s skill.
The pure-social learner model formalizes
how a learner can use Zach’s bet to infer information about the world. It includes a nested
model of how Zach updates his beliefs andmakes decisions (Equation 5), which is inverted
7

We use the terms reliable, reasonable, and rational
interchangeably in our discussion of the model and ourexperiments. However, in the General Discussion we introduce distinctions between reliable and reasonable or rational.
8
We assume joint update of beliefs and action plans.

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REASONING ABOUT SOCIAL SOURCES

to infer what Zach saw from how he bet. The
outer level of the model must again use Bayes
rule to infer horse x’s skill from what Zach was
likely to have seen. Equation 6 therefore requires two normalizations to compute— one forthe application of Bayes rule in the inner modelof Zach and one for the application of Bayes
rule in the outer level. Nested models with innernormalizations emerge from social situations inwhich people use their ToM to reason about theactions of others (Stuhlmüller & Goodman,2013).
Integrative Learner
When a social learner who observes Zach’s
bet also has her own direct observations, howshould she combine these two information
sources? It depends on her understanding of thecausal relation between the sources. As seen in
Figure 1c, the integrative learner models bothZach’s and her own observations as being generated from horse x’s skill (sl). This common(unobserved) cause allows information to flow
between the learner’s model of Zach and her
own observations. The learner’s belief about
horse x’s skill constrains what she infers Zach
saw. Zach’s bet, in turn, provides evidence
about sl. This is to say that an integrative social
learner forms her belief (sl) by jointly considering her observations (kl) and what she infers
about Zach’s observations (kz). This requires
elaborating the model of the pure-social learner(Equation 6) to jointly consider Zach’s bet andthe learner’s observations. Using Bayes rule andthe independence between the learner’s direct
and social evidence, given sl,
p(sl ⱍ bz, nz, kl, nl, ␣) ⬀ p(bz, kl ⱍ sl, nl, nz␣)p(sl)(9)⬀ p(bz ⱍ sl, nz, ␣)p(kl ⱍ sl, nl)p(sl).

(10)

Equation 10 combines elements from both thedirect and pure-social learners (Equations 2 and
8). This integrative learner model formalizes
how a learner should combine her direct evidence with Zach’s bet to estimate horse x’s trueskill. Like the pure-social learner, the integrative learner model includes a nested model of
how Zach makes decisions (Equation 5). However, the top level of the integrative model is25

more complex, using Bayes rule to infer sl fromboth bz and kl.
Predictions of the integrative learner.
The integrative learner model makes quantitative predictions about the relationship between
Zach’s attributes and the learner’s estimate ofhorse x’s skill. These predictions are made clear
by the concise formal definitions of Zach’s attributes within the horserace scenario. The number of races observed represents both Zach’sand the learner’s knowledgeability (i.e., horse
x’s record in the races) and the content of theseobservations. The learner’s estimate of Zach’s
rationality is simply the amount of decision
noise (alpha) she thinks he has. Finally, thelearner understands Zach’s confidence (discussed subsequently) simply as his estimate of
the likelihood that his chosen action will resultin the desired outcome. To explore the predicted
impact of each of these actor attributes, we
conducted a series of simulations leading to four
core testable predictions. All simulations wereconducted in the probabilistic programming
language, Church (Goodman et al., 2008).9
The first three predictions concern the effectof Zach’s knowledgeability and reliability on
the learner’s estimate of horse x’s skill. We
simulated the learner’s estimate of horse x’s
skill when Zach’s knowledgeability and the directly observed horse records were manipulated(see Figure 2).10 We simulated this for learners
who thought that Zach was reliable and for
those who thought he was unreliable. The
model predicts that all three variables—Zach’s
knowledgeability, the learner’s estimate ofZach’s reliability, and the content of the learner’s observations—should have an impact on
her estimate of horse x’s skill. First, the simulation makes the straightforward prediction that,
all else being equal, as horse x becomes more
dominant in the learner’s observations, sheshould increase her estimate of the horse’s skill.
Second, the simulation shows that the influenceof Zach’s bet is determined by his knowledge9
All code is freely available and executable in a modernbrowser https://probmods.org/v1/lfz.html
10
In all simulations discussed, we used the same reliability and horse skill parameters. When Zach was reliable, hemaximized his utility (␣ ⫽ ⬁); when he was unreliable, he
bet randomly (␣ ⫽ 0). The learner had a uniform prior overhorse strengths and she imagined Zach had the same uniform prior.

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26

HAWTHORNE-MADELL AND GOODMAN

Figure 2. Simulation of the integrative learner model’s estimate of horse x’s skill (Equation10), varying the content of her observations, her estimate of Zach’s reliability, and Zach’s
knowledgeability. The learner saw horse x win zero, one, or two of five races, and Zach saw
zero, five, or 30 races. Zach always bet on horse x and had an ␣ ⫽ ⬁ when reliable and an␣ ⫽ 0 when unreliable. As the figure shows, the more reliable and knowledgeable Zach is, themore influential his bet is predicted to be. See the online article for the color version of thisfigure.

ability and perceived reliability. When Zachhasn’t seen any races, and therefore isn’t knowledgeable about horse x’s skill, the learnershould ignore Zach and rely on her own evidence. The following is the first major prediction of the integrative learner model:
Prediction 1: When Zach hasn’t seen any
races, his bet should be ignored. As Zachsees more races (becomes knowledgeable)
his bet should increasingly influence thelearner’s estimate of x’s chances.
Zach’s reliability should have a similar effect; thus we predict the following:
Prediction 2: Learners who think Zach
does not bet reasonably, that is, has a low
rationality parameter, should be less influenced by his bet than those who think hebets reasonably. In the extreme case when
Zach is completely unreliable, his bet
should be ignored.
As Figure 2 shows, there is an interaction
between Zach’s reliability and knowledgeability on his influence. Zach must be both reliableand knowledgeable.

Prediction 3: There should be an interaction between knowledgeability and reliability such that knowledgeability is only
relevant to the extent that Zach’s bets are
diagnostic of his knowledge, that is, he hasa high rationality parameter. Conversely,
Zach’s reliability is only relevant when
Zach has some relevant knowledge.
These model predictions are consistent withthe finding that children and adult advice-takers
learn more from reliable and knowledgeable
actors (Harris, 2007; Bonaccio & Dalal, 2006).
Thus, we testeded the model’s quantitative predictions for these effects. Beyond these factors
identified in the epistemic trust and JAS literatures, the model also predicts that a less explored variable should also modulate howknowledgeable learners incorporate informationfrom other’s actions: the learner’s estimate of
the actor’s observations.
Learners infer Zach beliefs about horse x’s
skill from his bet, and understand this belief asa reflection of his observations through Equation 2. Zach’s observations, in turn, provide thelearner with information about horse x’s skill.
Indeed, Zach’s observations screen off his otherREASONING ABOUT SOCIAL SOURCES

properties from the learner’s beliefs. Zach’s observations therefore form the link between his
actions and horse x’s skill. This leads to thefollowing prediction:

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Prediction 4: The learner’s estimate ofZach’s observations should fully mediate
the effect of his knowledgeability.
We tested all four predictions within thehorse betting experimental paradigm in Experiment 1. By manipulating the number of observations Zach sees we can test Prediction 1. By
asking learners whether they think Zach was
betting reliably we can test Prediction 2 and theinteraction, Prediction 3. By asking learners toestimate what Zach saw while manipulating his
bet and the proportion of horse x wins in theirsample, we can test Prediction 4. The models
predict that Zach’s bet will have the largest
impact in inconsistent trials, where the evidence
learners receive is inconsistent with Zach’s bet.
In particular, these situations maximize the difference between the direct, pure-social, and integrative models (assuming for the moment that
Zach is reliable). Thus, in the following experiments we focus on inconsistent trials.
Elaborating the Integrative Learner
We next extend the integrative learner modelto account for additional sources of informationthat have been shown to affect social learning.
We first elaborate the integrative learner’s ToM
to allow her to interpret an actor’s confidence, a
cue shown to have a strong effect on learning(e.g., Birch et al., 2010; Sniezek & Van Swol,
2001). We then turn to describing how the integrative learner can use her ToM to infer
Zach’s reliability directly from Zach’s bet inobserved races. More realistically, the learner’s
mental model allows her to simultaneously
learn from and about an actor (as is evident
early in social reasoner’s development; see
Kushnir, 2013), across partially observed situations.
Learning from confidence cues. The degree of confidence an actor has in his action isa highly influential cue for social learners. We
model the reliable actor’s confidence, cz, as his
metacognitive judgment about the likelihoodthat his action will result in the desired outcome(where 0 ⱕ cz ⱕ 1). In the horserace scenario,
Zach’s confidence in his bet is his estimate of27

the chances the horse he bets on will win, whichis equivalent to the probability he bets on thehorse:p(cz ⫽ ␪ ⱍ bz, sl, nz, ␣) ⫽ ␦␪⫽p(bz ⱍ sl,nz,␣) . (11)
Confidence is often expressed with natural
language as a gradable adjective, for example,“I am [not/somewhat/very] confident horse x
will win.” Similarly, we bin the continuous confidence variable cz into a discrete confidence
variable with low, medium, and high levels. A
reliable person sends confidence signal ␴ when
his confidence, cz, falls within the intervalcmin(␴) ⱕ cz ⱕ cmax(␴). We understand Zach’s
confidence signal as a function of his belief cz
and his reliability alpha vis-à-vis the decision
rule described by Equation 4. The probability
that Zach with reliability ␣ would send confidence signal ␴ in their bet bz is thencmax(␴)p(␴ ⱍ bz, sl, nz, ␣) ⬀ e␣·兰cmin(␴) p(␴,cz ⱍ bz,sl,nz) dcz(12)cmax(␴)⬀ e␣·兰cmin(␴) p(␴ ⱍ cz)p(cz ⱍ bz,sl,nz) dcz(13)
The second line relies upon the fact that ␴ isconditionally independent of bz, sl, and nz whencz is known. Elaborating our integrative social
learner with this understanding of how Zach
determines his confidence level allows us tomodel learning from variably confident actors:p(sl ⱍ u, bz, nz, kl, nl, ␣)⬀ p(␴, bz, kl ⱍ sl, nl, nz, ␣)p(sl)(14)⬀ p(␴ ⱍ bz, sl, nz, ␣)p(bz ⱍ sl, nz, ␣)p(kl ⱍ sl, nl)p(sl).
(15)
Given this formulation, learners know that a
reliable and highly confident Zach is more
likely to have seen a sample strongly dominatedby his chosen horse. Learners are therefore
more influenced by Zach’s bet when he ishighly confident in it.
In Figure 3 we examined the model’s predictions for how Zach’s confidence in his bet affects his influence on learners. We considered
the situation where Zach bet on horse x and has

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28

HAWTHORNE-MADELL AND GOODMAN

Figure 3. Simulation of the integrated learner model’s estimate of horse x’s skill when
Zach’s confidence is varied (Equation 14). The simulation was conducted when Zach saw 10
races and said he had high confidence, low confidence, or did not provide his confidence in
his bet. When Zach was unreliable, his confidence did not impact the learner’s assessment ofhorse x’s skill in the simulation. However, as the figure shows, when Zach was reliable, themore confident he was, the more influential his bet is predicted to be. See the online article
for the color version of this figure.

seen 10 races, varying the amount of direct
evidence the learner has and the confidence
Zach expresses in his bet. Consistent with previous empirical results (e.g., Sniezek & Van
Swol, 2001) the model predicts the following:
Prediction 5: When Zach is highly confident his bet should be more influential thanwhen he has low confidence or doesn’t
provide his confidence.
Looking at the left half of Figure 3 where
Zach’s bet is inconsistent with the learner’s
evidence, we see that when Zach has high confidence in his bet, the learner is highly influenced by him. Even when the leaner saw horse
x win only one of five races, she thinks there isa moderate chance horse x will win the next one
if Zach bet with high confidence. When Zach
bet but did not convey confidence, the learner
thinks that horse x is likely to lose the next race.
This seemingly muted influence of Zach stems
from the nature of the information he provides,a binary bet. A binary bet only licenses a weakinference about Zach’s knowledge; even if Zach
were perfectly rational, he could have seen thehorse he bet on win every time, half the time,or anything in between.

Looking at the right half of Figure 3 where
Zach’s bet is consistent with the learner’s evidence, we see that Zach’s low confidence bets
become more influential with the strength of thelearner’s evidence. When Zach says he has low
confidence in his bet, the model thinks it islikely that his bet isn’t strongly supported by hisevidence, that is, he didn’t see horse x dominate.
In these circumstances, Zach’s bet on horse x
paradoxically decreases the learner’s estimateof x’s strength, thus we predict the following:
Prediction 6: If Zach states or is interpreted
as having low confidence, the learner has
strong evidence for an outcome, and Zachbets on that outcome his bet can decrease
her belief in that outcome. That is to saythat when these conditions hold and he bets
consistently with her evidence, his bet has
a paradoxical effect.
If learners fill in Zach’s confidence when hedoesn’t state it and some non-negligible proportion attributing Zach’s low confidence, thenPrediction 6 predicts that “paradoxical integration” should be evident in Experiment 1 because
confidence cues were not provided. Birnbaumand Mellers (1983) similarly observed this par-This document is copyrighted by the American Psychological Association or one of its allied publishers.
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REASONING ABOUT SOCIAL SOURCES

adoxical integration in their 1983 experiment,
and used this effect to reject Bayesian models of
their data. However, if some of his participants
similarly attributed to their advisers low confidence, their findings would be predicted by our
Bayesian integrative learner model.
We testeded Prediction 5 in Experiment 2 byhaving Zach provide either high or low confidence
in his bets. In addition to testing Prediction 5,
having a condition in which Zach should be highly
influential helps distinguish between possible explanations for the predicted modest effect of
Zach’s bet in Experiment 1. For example, although both our computational account andaccounts that find “under-utilization” of advice predict the muted impact of Zach inExperiment 1, our account more naturally
predicts the large relative impact of Zach’s
confidence in Experiment 2.
We testeded Prediction 6 in Experiment 3
where we provide the learner with strong direct
evidence that horse x is stronger and then see how
Zach betting on horse x influences her estimate.
We testeded Zach’s influence when he has high or
low confidence and compare these results to thecorresponding trials in Experiment 1 where Zachdoesn’t provide confidence information and baseline trials where there is only direct evidence.
Inferring reliability. The generativemodel underlying the integrative social
learner allows reasoning about arbitrary latent
properties of the actor. For example, in previous experiments, learners inferred the reliability of an actor from the way they behaved
toward known items (e.g., Poulin-Dubois etal., 2011) or named known words (e.g., Clément et al., 2004; Jaswal & Neely, 2006;
Pasquini et al., 2007). Did they call a cow a
duck or seem pleased with the contents of anempty box? The equivalent situation in thehorserace scenario is seeing how Zach bets
when the races he saw are known. Did Zach
bet on horse x when she only saw him win one
out of 10 races? To infer Zach’s reliability,
learners consider which alpha value is mostconsistent with his behavior. The probability
that Zach has reliability level alpha isp(␣ ⱍ bz, kz, nz) ⬀ p(bz ⱍ kz, nz, ␣)p(␣). (16)Although learners could consider Zach having
any ␣ 僆 ⺢⫹, we simplify by considering only

29

whether Zach is reasonable or unreasonable,
each corresponding to a particular alpha value.
The model indicates that learners should infer
Zach’s reliability from the consistency of his
action and the information available to him,thus we predict the following:
Prediction 7: When Zach’s observations
are more consistent with his bet, learners
are more likely to infer he is reliable.
The strength of this reliability inference depends on the learner’s model of how Zach bets
when he is reliable (a reliable Zach’s alpha). If
Zach maximizes his expected utility (␣ ⫽ ⬁),his consistency is highly influential on the learner’s estimate of his reliability. However if Zach
probability matches (␣ ⫽ 1), his consistency isless diagnostic. There is still some chance hecould be reliable and bet on horse x when x’s
record is poor.
We tested Prediction 7 in Experiment 3 byextending the horserace scenario with a complete information trial in which the learner sees
Zach’s observations and his bet. By manipulating the consistency of Zach’s bet with his observations we can test the model’s quantitative
prediction about his reliability inference. We
can further test whether learners’ updated estimate of Zach’s reliability affects Zach’s influence in subsequent trials.
Applying the Integrative Learner Model
The use of the integrative model outlined in
the preceding text to make quantitative predictions of human behavior requires the calculationof unspecified parameters. These parametersmust be filled in by additional hypotheses, left
“free” and allowed to take a value that maximizes the fit of the model to the behavioral data
it is trying to account for, or on the basis ofindependent empirical data. Of the three options, using independent data as an empiricalprior distribution of the unspecified variable
provides the most stringent test of a theoreticalframework. We therefore strove to constrain
unspecified variables with empirical data wherever possible. We used data from prior experimentation when available and otherwise leveraged the ease of online experimentation to
perform additional “prior elicitation experiments” to determine empirical priors. Following
this method, no model predictions were the re-This document is copyrighted by the American Psychological Association or one of its allied publishers.
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30

HAWTHORNE-MADELL AND GOODMAN

sult of “fitting” to the data they are trying toaccount for.
In each of the four experimental sections is amodel specification subsection that describes
the methods used to define the unspecified variables required to make quantitative predictions.
The resulting completely specified models were
implemented in the probabilistic programming
language, Church.11,12 We used exact inference
methods to calculate the posterior estimates.
The variability in the resulting model predictions is therefore a reflection of the variability ofthe empirical priors used, for example, p(sl).13
Experiment 1: Effect of Knowledgeability
and Reliability
We set out to test the first four predictions of
the integrative learner model by manipulating
the direct and social evidence available to participants in our simple racetrack scenario.
MethodFor an overview of the flow of the experiment, see Figure 4, and for the details about
how the evidence was provided, see Figure 5.
The primary stimuli consisted of a table summarizing the results of a series of races between
two horses. For each trial, the horses’ names
were randomly selected from a list of fictional
horse names. The questions in the trial werephrased in terms of one of the two horses,
henceforth horse x. Horse x’s position on thetable was randomized, and participants had noindication of which horse was going to be thesubject of the eventual query.
The experiment began with an introduction to
the betting game. Participants watched an animation of a series of head-to-head races
between two horses. As the winning horse
passed the finish line, its results were added to a
results table summarizing the series. Participants were then told that they were going to see
the results of a series of different head-to-headmatch-ups summarized like the results table
they had just seen. They were then given six
baseline trials. Each baseline trial consisted of asingle results table from a different match-up(as pictured in Figure 5c). In the baseline trials,
participants saw horse x win each of the possible number of times—zero through five. The
order of the baseline trials were randomized.

The baseline trials measured the participants’estimate of horse x’s skill based solely on theirown observations, corresponding to the direct
learner model. It was used to calculate theirprior beliefs about horse skills.
Following the baseline trials, participants
were introduced to the (fictional) last participant
to complete the experiment. The last participant
was given a random male name (i.e., Zach).
Participants were told that Zach “saw different
races from the same match-ups,” giving him
independent information about them. After
Zach was introduced, the test trials were presented. The six test trials consisted of a resulttable (Figure 5c), the number of races Zach saw(Figure 5b), and which horse he bet on (Figure5a). Each test trial was specified by four variables—the number of races the participant saw(nl), the proportion of times horse x won in
those races (kl), the number of races Zach saw(nz), and which horse Zach bet on (bz). The
independent variables kl and bz were manipulated within participants, whereas nz was manipulated between participants, and nl was held
constant at 5.
Participants always saw the results of fiveraces (each with five matches). Zach’s knowledgeability (nz) was manipulated by randomly
placing participants in one of five conditions inwhich Zach saw zero, five, 10, 20, or 30 races.
In the test trials, participants again saw resulttables where horse x won each of the possible
number of times—zero through five. In half ofthese combinations Zach’s bet was consistent
with the participants’ observations, for example, “I saw [horse x] win four out of fiveraces, and Zach bet on him,” whereas in theother half, his bet was inconsistent, for example, “I saw [horse x] win two out of five races,
and Zach bet on him.” Ignoring ordering,
there are three unique horse records (zero offive, one of four, and two of three). Participants saw each of the three unique horse
records twice for a total of six trials: one trial
in which Zach bets consistently with the record and one in which he bets inconsistently.
11

See https://probmods.org/v1/lfz.html.
The model can be edited interactively to see its sensitivity to the unspecified parameters.
13
However, the bootstrapped confidence intervals calculated were sufficiently small to be invisible on our plots.
This holds true for all subsequent model results.
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REASONING ABOUT SOCIAL SOURCES

31

Figure 4. Outline of Experiment 1. The number of races Zach saw was manipulated between
participants, from zero to 30. One trial sequence is pictured. Each participant saw six test trials
varying horse x’s record and Zach’s bet. See the online article for the color version of thisfigure.

The target horse (whose skill was queried)
was counterbalanced such that it was not always the weaker horse, and it was not always
the horse that Zach bet on. To maintain theplausibility of Zach’s inconsistent bets, thesetrials were not allowed to occur in a row.
Three pseudorandom trial orders maintaining
this constraint were used.
Following the presentation of all the evidence(see Figure 5), participants were asked to fill in
Zach’s empty results table with what they
thought Zach saw. They clicked on the question
marks and freely typed their response (Figure5b). Then the participants were asked “What do
you think is the chance that [horse x] wins thenext race?” They indicated their response byclicking on a continuum, ranging from certainly
will to certainly won’t, which was coded on ascale from 0 to 100, respectively. The evidence—participant s’ observed races table andZach’s bet—were presented when making thisassessment to ensure that participants’ judgments were not limited by memory.
After completing all the test trials, the participants answered a debriefing question(“Which best described Zach?”) on a fourpoint scale, ranging from very unreliable tovery reliable.14 For an overview of the parameters used in each trial of all the experiments,
see Table 1.

Participants
Two hundred participants (mean age ⫽ 29
years, 57% female) were recruited through Amazon’s Mechanical Turk crowd-sourcing service.
They completed the experiment in 6 min to 9 minfor a small payment. Forty participants were randomly assigned to each of five conditions in which
Zach saw zero, five, 10, 20, or 30 races.
Data Processing and Counterbalance
Testing
The data were put into a standard form foranalysis. Counterbalanced trials were transformed such that Zach always bet on horse x(the horse the participant was asked about). This
made all Zach-inconsistent trials occur whenhorse x was dominated in the participants’ observed races. Consider a Zach-inconsistent trial
in which the participant saw horse x win fourout of five races, saw Zach bet on horse y, andthen estimated that horse x had an 80% chanceof winning the next race. To transform this trial
into the standard form, all quantities were inverted such that the participant now observed x
win one of five, saw Zach bet on horse x, andestimated that x had a 20% chance of winning.
14

Participants in the condition in which Zach saw zeroraces did not answer this question.

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32

HAWTHORNE-MADELL AND GOODMAN

Figure 5. Panel a: Zach telling participant their bet and how many races he saw. Panel b:
Zach’s obscured result table where participants would click to fill in their estimates. Panel c:Participants’ results table. See the online article for the color version of this figure.

The transformed trials were not distinguishable
from the untransformed counterbalanced trails2(␹df⫽1⫽ 7.4 ⫻ 10⫺2, p ⬎ .05).15 Therefore theresults neither significantly depended on which
horse was the subject of the query nor onwhether the horse that participants were asked
about was the same as the one that Zach had bet
on. Additionally, the different trial orderings
did not significantly influence participants’ re2
sponses (␹df⫽3⫽ 1.8, p ⬎ .05, comparing thebest-fitting linear model with and without thecategorically coded ordering variable).
Model Specification
The integrative model outlined previously(Equation 10) has two unspecified components:
The learner’s prior over the relative skill of thehorses, p(sl), and the probability that a learner
will think that Zach is reasonable, p(␣). We
discuss the empirical bases for specifying eachof these pieces in turn.
We used the nonsocial baseline trials toestimate p(sl). We bootstrapped an estimate of
the participant’s mean baseline scores with1,000 samples. We discretized p(sl) into a
multinomial distribution on 10 evenly spaced
points in [0, 1], and for each sampled mean
we assigned the points probability such thatthe mean absolute error between the direct
learner model and the sampled mean was
minimized (see Figure 6 for fit prior).16 To
lower the dimensionality of this optimization,
we added the constraint that the resulting
distribution be symmetrical around p(sl) ⫽ .5
(a simplification supported by the symmetryof responses seen in Figure 7). We fit ourmodel to the resulting distribution of fittedp(sl), enabling our model predictions to reflect
the variability in this empirically derived prior.
We set the probability of a learner thinking
Zach is reasonable, p(␣), to the proportion
measured in the debriefing question (.74). As
described in the model section, we are assuming that participants’ estimate of Zach’s reliability is constant throughout their trials, not
updated within or between trials. This is consistent with the ordering of the trials having
no effect (discussed in the previous section—
Data Processing and Counterbalance Testing). This is not to say that participants areunable to update their beliefs about Zach’s
reliability, only that they fail to do so from theweak evidence provided in this experiment(i.e., relatively few observed races and noinformation about what Zach saw). In Experiment 4, we focus on participants’ inferences

15
Chi-square statistic calculated from the difference indeviance of the full model and the (nested) model without
the predictor (Pinheiro & Bates, 2009; Barr et al., 2013).
16
It would have been natural to assume a beta distribution
prior on sl, resulting in a beta-binomial model for the baseline(nonsocial) trials. However, participants’ baseline scores deviated from a beta-binomial model, motivating our use of thebaseline data to directly estimate a less parametric prior. Participants’ baseline scores also deviated from canonical probability weighting functions (Tversky & Kahneman, 1992). The
empirically estimated prior exhibits the classic overweightingof extreme probabilities but has high fidelity for moderate
probabilities, contradicting both the central tendency predictedby the beta-binomial model and the shallower slope of standard probability weighting functions.

REASONING ABOUT SOCIAL SOURCES

33

Table 1
Overview of the Manipulated Parameters in the Four Experiments
Parameters

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Participant

Zach

Experiment

Numberparticipant
observations

Observed
proportion ofhorse x wins

Number of
Zach’sobservations

Observed
proportion ofhorse x wins

Baseline
1
2
3
4

5
5
5
5
5

a

—
(5, 10, 15, 20, 30)
10
10
10

–
—
—
—
8/10 ¡ —aa

5/5
4/5 ¡ 1/5

Chosen
horse
—
a
a

x
(x, y) ¡ —Confidence

Total
numberof trials——(low, high)(low, high)
—

6
6
6
6
2

Note. Identical baseline trials were conducted before the test trials of each condition. x ¡ y indicates that a trial withparameter x is Followed by parameter y. (x, y) indicates that the parameter was manipulated between participants with one
half seeing x and the other half seeing y. — Indicates that participants did not know the value of this parameter.
a
Indicates that each participant saw Zach betting consistently and inconsistently for each possible horse record.

about Zach’s alpha by providing stronger andmore salient evidence.
Finally, we fixed the alpha values to describe a “reasonable” and “unreasonable”
Zach to empirical values measured in a separate experiment. We elicited 40 participants’
intuitive theories regarding how Zach would
bet, given his observations and reliability.
Participants were randomly split into two
conditions. In one, they were told Zach was“reasonable,” and in the other, they were told
he was “unreasonable.” Zach always saw 10
races, and participants were asked how Zach
would bet when he saw horse x win five, six,seven, eight, nine, or 10 of them. When Zach
was reliable, all 20 participants said he would
bet on horse x as soon as he won a majority of
the races (six to ten). This corresponds toimagining that Zach used a strict maximizing
rule (a very high ␣). When Zach was unreliable, participants said Zach would bet on
horse x exactly half of the time. A logistic
regression found no effect of horse x’s recordon how participants’ thought Zach would bet
2(␹df⫽1⫽ .82, p ⬎ .05). This corresponds toimagining that an unreliable person just flips
a coin to decide who to bet on (␣ ⫽ 0),
effectively ignoring their observations.
All unspecified parameters of the integrative model are therefore determined by experimental measurements that were independentof the target judgments in integrative trials—
thus all parameters were measured rather than
fit in the usual sense. We use the same parameters where they are required to model
later experiments.

Figure 6. Empirically derived prior distribution over ahorse’s skill.

Figure 7. Mean of participants’ estimates of horse’s
chances on the basis of a summary table (i.e., the nonsocial“baseline trials”).

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34

HAWTHORNE-MADELL AND GOODMAN

Figure 8. Model’s and participants’ mean estimate of horse x winning the next race when
Zach bet on him, for each directly observed record and level of Zach’s knowledgeability.
Participants’ responses show a clear sensitivity to both Zach’s knowledgeability and theirdirect evidence captured by the integrative model (R2 ⫽ .96, MSE ⫽ 6.3 ⫻ 10⫺4). See theonline article for the color version of this figure.

Results and Discussion
The four predictions from the integrative
learner model guide our analysis. All predictions
were tested with linear mixed-effects regression
models to account for subject-level effects andavoid issues of multiple independent comparisons(Bates et al., 2014; Pinheiro & Bates, 2009). The
effect of Zach’s bet was predicted to get larger as
his bet became more inconsistent with the participants’ observations, so we focus our analysis onthese Zach-inconsistent trials.17
We first verified that the influence of Zach’s
bet was dependent on the races he observed. We
used a planned comparison of when Zach saw
no races, when Zach saw 10 races, and thebaseline scores. Looking at the inconsistent trials, we would expect participants who werelearning from Zach’s bet to move their estimateof horse x’s chances toward Zach’s bet, relative
to the baseline scores, which are solely based on
their own observed races. If Zach’s influence isbased on the fact that he saw some races theparticipant didn’t, then we expect him to be
influential when he saw 10 races but not when
he saw none. We used a mixed-effects model totest the planned comparison. The model also
included the participants’ observed race results
and a (random) term for each participant. In the
resulting model, participants’ estimates of horse

x’s chances were more consistent with Zach’s
bet when he was knowledgeable (saw races)than when he was not (saw no races; ␤ ⫽ .13,t ⫽ 5.0, p ⬍ .001).18 In fact, when Zach saw noraces, participants ignored him, responding indistinguishably from those in the baseline trials
where Zach was not yet introduced (␤ ⫽ 6.7 ⫻
10⫺4, t ⫽ 2.6 ⫻ 10⫺2, p ⬎ .05). Consistent withthe first prediction, participants listened to Zach
when he had independent information and ignored him when he didn’t.19
Figure 8 shows how the integrative learnermodel and participants respond to the full quantitative manipulation of Zach’s knowledgeability. The integrative learner model captures howlearners combine their own observations with
Zach’s bet to infer horse x’s chances at specificlevels of Zach knowledgeability (R2 ⫽ .96,
MSE ⫽ 6.3 ⫻ 10⫺4). It infers horse x’s chances
17
This is with the exception of the paradoxical integration
effect described in Prediction 6 and tested in Experiment 3,
which focuses on the maximally Zach-consistent trial.
18
Denominator degrees of freedom used to calculate pvalues were approximated using the Satterthwaite method(Kuznetsova, Bruun Brockhoff, & Haubo Bojesen Christensen, 2014).
19
Subsequent analyses focused on exploring the factors
modulating Zach’s influence will therefore omit the conditionin which Zach saw no races, as it was indistinguishable fromthe baseline trials.

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REASONING ABOUT SOCIAL SOURCES

from the number of races Zach observed, thenumber of times horse x won in his direct observations, its estimate of how many times Zach
saw horse x win, and its assessment of Zach’s
reliability. The best mixed-effects model of thedata had precisely these predictors with the addition of a (random) participant term.
Analyzing the mixed-effects model we first
see that, like the integrative model, participants
used both Zach’s bet and their observed races toinfer horse x’s chances. When x was more dominant in the participants’ sample, participants2
rated him more likely to win (␤ ⫽ .42, ␹df⫽1⫽
7.5, p ⬍ .001).20 The number of races Zach saw
was also predictive; as Zach saw more races,
participants estimated horse x’s chances more in
2line with his bet (␤ ⫽ 4.9 ⫻ 10⫺3, ␹df⫽1⫽ 19,
p ⬍ .001). In line with the first prediction,
Zach’s influence increases with his knowledgeability (see Prediction 1 in Table 2). Whether
Zach was thought to be reliable was also predictive (see Prediction 2 in Table 2). The 74%
of participants who thought Zach was reliablegave responses more consistent with his bet
than did the 26% who did not think he was2reliable (␤ ⫽ .12, ␹df⫽1⫽ 25, p ⬍ .001).21
Finally, participants’ estimate of the percentageof the races horse x won in Zach’s observations
was also predictive of their estimate of horse x’s
2chances (␤ ⫽ 9.8 ⫻ 10⫺2, ␹df⫽1⫽ 7.5, p ⬍.001). The better they thought horse x’s chanceswere, the more they thought x dominated Zach’s
observations.
As the model predicted, participants’ estimate of Zach’s observations mediated the effectof his knowledgeability (see Prediction 4 inTable 2). Including both the number of races
Zach saw and the participants’ estimate ofZach’s observations resulted in a significant effect for his observations (␤ ⫽ 7 ⫻ 10⫺3, t477 ⫽
3.5, p ⬍ .05), and the previous effect of Zach’s
knowledgeability is absent (␤ ⫽ 1 ⫻ 10⫺3,t477 ⫽ .20, p ⬎ .05). The Sobel test showed asignificant reduction in the effect of Zach’s
knowledgeability on participants’ estimate ofhorse x’s chances (z ⫽ 3.5, p ⬍ .05).
Finally, Figure 9 shows that the effect of
Zach’s knowledgeability is dependent on him
betting reasonably in both the data and the integrative learner model (see Prediction 3 in Table 2). The model captures this interaction(R2 ⫽ .88, MSE ⫽ 2.5 ⫻ 10⫺3). Adding theinteraction of Zach’s reliability with his knowl-

35

edgeability to our previous mixed-effects modelyields a significant predictor and a better model2
fit (␤ ⫽ 5.0 ⫻ 10⫺3, ␹df⫽1⫽ 4.6, p ⬍ .05). As
is evident in Figure 9, participants who said
Zach bet unreasonably showed no effect ofZach seeing more races (␤ ⫽ 1.5 ⫻ 10⫺3,2
␹df⫽1⫽ .84, p ⬎ .05). For participants who said
Zach bet reasonably, the more he saw, the more2⫽
they listened to him (␤ ⫽ 6.4 ⫻ 10⫺3, ␹df⫽1
21, p ⬍ .001).
In the integrative learner model, Zach’s observations are the common currency between
his bet and what the learner already knows
about the world. They form the bridge between
the learner’s observations and her final estimateof horse x’s chances. To test the model’s predictions about the crucial role played by Zach’s
unseen observations, participants were asked to
predict what Zach saw. Explicitly posing thisquestion could have induced participants toconsider Zach’s observations, which they may
not have done otherwise. Investigations into thispossibility indicate that asking participants toestimate what Zach saw does not influence theirestimate of horse x’s chances.22 This implies
that learners spontaneously appreciate the importance of what Zach saw and form an estimateof this quantity in the course of reasoning about
horse x’s chances.
20
All ␤ values reported are taken from full model, andthe chi-square statistic is calculated from the difference indeviance of the full model and the (nested) model without
the predictor (Pinheiro & Bates, 2009; Barr et al., 2013).
21
The proportion of participants that thought Zach was
reliable was independent of the number of races he saw
2(␹df⫽3 ⫽ 1.2, p ⬎ .05).
22
Thirty participants were recruited through Amazon’s
Mechanical Turk crowd-sourcing service and completed amodified version of Experiment 1. The experiment was
identical to the Zach sees 10 races condition, except thatparticipants did not report what they thought Zach saw. The
30 participants were not distinguishable from those in thecorresponding Zach sees 10 races condition of Experiment
1. An exact Wilcoxon’s Mann–Whitney rank sum test was
performed that found that the estimate of horse x’s chances
was not affected by estimating what Zach saw (z ⫽ .95, p ⬎.05). An additional analysis was conducted in which the bestmodel with the addition of a dummy coded condition variable was applied to this control and the corresponding
Experiment 1 data. In the resulting data model, the dummy
coded condition was neither predictive of the results, nor
was its interaction with participants’ observed proportion or2
reliability attribution (respectively, ␹df⫽1⫽ 1.8, p ⬎ .05;
2
2
␹df⫽1
⫽ 2.3, p ⬎ .05; ␹df⫽1⫽ 2.0 ⫻ 10⫺3, p ⬎ .05).

36

HAWTHORNE-MADELL AND GOODMAN

Table 2
Results of the Major Predictions of the Integrative Learner Model Simulations
Prediction

Description

Result1
2
3

The more knowledgeable Zach is, the more influential he should be.
The more reliable Zach is, the more influential he should be.
Zach’s influence is dependent upon him being both reliable andknowledgeable (interaction).
The learner’s estimate of Zach’s observations should mediate the effectof his knowledgeability.
When Zach is highly confident his bet should be more influential thanwhen he has little confidence or doesn’t provide his confidence.
Zach’s bet can have a paradoxical effect when he has low confidence
and his bet is consist with strong direct evidence.
When Zach’s bet is inconsistent with his observations, learner’s should
infer he is unreliable.

2
␹df⫽1⫽ 16, p ⬍ .001
2
␹df⫽1
⫽ 22, p ⬍ .001
2
␹df⫽1
⫽ 4.6, p ⬍ .05

4

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5
6
7
Note.

z ⫽ 3.5, p ⬍ .05
2
␹df⫽1⫽ 16, p ⬍ .001

t238 ⫽ 2.74, p ⬍ .05
2
␹df⫽1⫽ 16, p ⬍ .001

All statistics are derived from relevant mixed-effects models.

Participants were influenced by Zach’s bet,
which reflected the knowledge he derived fromhis observations. However, the influence of his
bet, even when he saw six times as many races
as the participant, was modest. Interestingly,
although this small effect is in line with themodel’s prediction, it did not resonate with theauthors’ intuitions: If you saw horse x win zeroout of five races and learn that Zach would bet
on him, how many observations would Zachneed to have seen before you would believe him
and bet on horse x (i.e., estimate horse x’s
chances greater than .5)? Our initial intuition
was that Zach would have to have seen twice as
many. This was corroborated by the free responses of 34 Amazon Mechanical Turk responders. The mean number of observations
Zach would have to have seen to persuade themwas 12.3 when they saw zero of five, with theamount decreasing as their observations became
more consistent with his bet. Hence, thoughparticipants’ behavior conformed closely to thatof the normative integrative learner model inExperiment 1, explicit intuitions were highly
inaccurate. To put this another way, participants’ online behavior was consistent with anagent reasoning rationally over the causal structure of the integrative learner model (see Figure
8, Panels c and d), but the same reasoning was
not used to explicitly predict the outcome of theexperiment. This décollage is an interesting
subject for future exploration.
The full integrative model predicts the modest
observed effect of Zach’s bet because a single bet
only licenses a limited inference about Zach’s

observations. Although Zach betting multiple
times would license a slightly stronger inference,
the model predicts that the real bottleneck is theamount of information a binary choice confers(when made at a 50% cutoff point). However,with the addition of simple, low or high confidence ratings, the model predicts that participants
should be able to infer what Zach saw with greater
precision and, subsequently, weigh his high confidence bets more heavily. Experiment 2 tests thisprediction.
In Experiment 3 we examine a surprising
result observed in the Zach-consistent trials—we find that Zach’s bet on horse x causes
participants to think that horse x winning is lesslikely than if they had not seen his bet andsimply had the same direct evidence. This paradoxical effect of Zach’s bet seems to clash
with the measured way in which participants
integrated his bet on the inconsistent trials,
which was well-captured by our Bayesian integrative model. However, this effect is consistent
with our model if some learners spontaneously
attributed Zach low confidence, similar to howsome learners seemed to spontaneously attribute him low reliability. We explore this possibility in Experiment 3.
The 24% of participants who spontaneously
thought that Zach was betting unreasonably also
contributed to the relatively muted effect. When
Zach does not bet reasonably, his action is notdiagnostic of his observations. Although thepredicted effect is evident in the results (seeFigure 9), the data admits an alternative explanation: Participants with low estimates of horseThis document is copyrighted by the American Psychological Association or one of its allied publishers.
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REASONING ABOUT SOCIAL SOURCES

37

Figure 9. Model’s and participants’ mean estimate of horse x winning the next race when
Zach bet on him, for each level of Zach’s knowledgeability and split by participants’assessment of Zach’s reasonableness. By marginalizing across the participants’ observations,
we can clearly see that Zach’s knowledgeability and reliability interact—Zach’s knowledge isonly influential when he is thought to be reliable, as predicted by the model (R2 ⫽ .88, MSE ⫽2.5 ⫻ 10⫺3). See the online article for the color version of this figure.

x’s chances could simply be more likely to saythat Zach is unreliable, as his bet is dissonantwith their estimate. Conversely, those who
think horse x’s chances are good would be more
likely to say Zach is reliable. This alternative
explanation would not explain the observed interaction between reliability and knowledgeability, but the nature of the reliability inference
merits further study. In Experiment 4, we explicitly manipulate the evidence of Zach’s reliability to explore both the source and flexibilityof participants’ reliability estimates.
Experiment 2: Effect of Confidence
MethodAll procedures were identical to those in Experiment 1, with one addition to the stimuli. In
Zach’s speech bubble, where he stated which
horse he bet on, he now also gave his level ofconfidence—“I bet on horse x with [high/low]
confidence.” Zach had high confidence for all bets
consistent with the participants’ information andhad moderate confidence on the first trial, which
had ambiguous direct evidence (horse x wining
two of four races). When Zach’s bet was inconsistent with the participant’s information, hewould either have low confidence in the low con-fidence condition or have high confidence in thehigh confidence condition.
Participants
Eighty participants (mean age ⫽ 26 years,
59% female) were recruited through Amazon’s
Mechanical Turk crowd-sourcing service. They
completed the experiment in 6 to 9 min for asmall payment. Forty participants were randomly assigned to each of the high and low
confidence conditions.
Model Specification
For this experiment, the integrative learnermodel was extended with a straightforwarddefinition of confidence—what Zach thinks
his chances of success are, as described by
Equation 14. Given this definition we mustassess how participants interpreted the utterances, “I bet on horse x with [high/low] confidence,” that is, what learners think Zach
thinks about the chance that his bet is correct
given his stated confidence level. This corresponds to the utterance thresholds used in
Equation 14. We conducted a separate experiment with 40 participants to measure thismapping. Participants were shown a (fic-This document is copyrighted by the American Psychological Association or one of its allied publishers.
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38

HAWTHORNE-MADELL AND GOODMAN

tional) previous “Zach,” who made a bet on ahorse with either high, moderate, or low confidence. Participants saw one trial at eachlevel of confidence. No information beyond
Zach’s confidence was provided. Participants
responded by providing the lower and upperbounds of an interval, via two free responseboxes, which indicated their estimate ofZach’s beliefs about the likelihood of his bet
being correct. We calculated the median point
at which participants thought Zach would say
he had moderate and high confidence, whichwere 51% and 73%, respectively.23
The model used the same horse skill prior andalpha values used in Experiment 1. The probability that Zach was reliable was fixed to the
proportion measured in this experiment (.83).
Results and Discussion
Did Zach’s high confidence in his bet increase its influence on participants’ subsequentestimate of horse x’s chances (vis-à-vis Prediction 5)? We tested for this effect by expanding
the additive mixed-effects model described inExperiment 1 with a dummy coded confidence
variable (and removing the knowledgeability
variable because that was held constant in thisexperiment). As seen in Figure 10, Zach’s confidence did have a significant effect (␤ ⫽ .13,
2
⫽ 11, p ⬍ .05), which the integrative␹df⫽1
learner model captured (R2 ⫽ .99, MSE ⫽1.6 ⫻ 10⫺3). When Zach had high confidence in
his bet, it was 2.6 times as influential as when he
had low confidence in it (comparing the impactof Zach’s bet in each condition to the baseline
scores). Compared with the corresponding trials
in Experiment 1 in which Zach saw 10 races andgave no confidence information, Zach’s highconfidence bet was 2.1 times as influential andhis low confidence bet was.9 times as influential.
All the significant predictor variables from
Experiment 1 were again predictive of participants’ estimate of horse x’s chances. Participants were receptive to the proportion of times
horse x won in their five observed races (␤ ⫽2.37, ␹df⫽1⫽ 34, p ⬍ .001). Their assessment of
Zach’s reliability influenced how much they2
incorporated his bet (␤ ⫽ .15, ␹df⫽1⫽ 8.7, p ⬍.05), and participants’ estimate of Zach’s observed proportion of x wins was also a signifi-

2cant predictor of his bet (␤ ⫽ .17, ␹df⫽1⫽ 4.7,
p ⬍ .05).
Participants’ assessment of Zach’s reliability
was not driving the effect of confidence: The
proportion of participants who thought Zach
was reliable did not differ between confidence
2
conditions (␹df⫽1⫽ 0, p ⬎ .05). However,
Zach’s perceived reliability did interact with theeffect of his confidence, just as his perceived
reliability interacted with the effect of his
knowledgeability in Experiment 1. Adding theinteraction of Zach’s reliability and his confidence to the additive mixed model yields a2
significant predictor (␤ ⫽ .15, ␹df⫽1⫽ 4.0, p ⬍.05). The integrative learner model also captures
this interaction (R2 ⫽ .99, MSE ⫽ 1.6 ⫻ 10⫺3
see Figure 11). Looking at the participants who
said Zach bet unreasonably, there was no effect2of Zach’s confidence (␤ ⫽ 8.7 ⫻ 10⫺3, ␹df⫽1⫽⫺2
1.5 ⫻ 10 , p ⬎ .05). For participants who said
Zach bet reasonably, his bet had a larger impact2
when he was more confident (␤ ⫽ .15, ␹df⫽1⫽
12, p ⬍ .001).
As the integrative model predicted, when
Zach had high confidence he was much more
influential, even more iso than if he had seen
three times as many races. This finding indicates that the modest effect of Zach’s bet inExperiment 1 was not due to a general discounting of social information. The integrative model
predicted the small effect of knowledgeability
in Experiment 1 from the limited inferences
about Zach’s beliefs that his binary bet permitted. In the model, the addition of confidence
enabled a stronger inference about Zach’s beliefs, resulting in his bet being more informative, consistent with participants’ responses.
Again, participants’ estimate of Zach’s reliability mediated his influence.
We next examine the impact of Zach’s confidence when his bet is consistent with strong
evidence. Do his low confidence bets have aparadoxical effect on participants as the model
predicts?

23
For each participant, the upper and lower estimate ofadjacent intervals were averaged. For example, the upperbound where Zach would say he had low confidence and thelower bound where he would say he had moderate confidence were averaged to yield the moderate confidencebound. The median of these within subject estimates wasused.

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REASONING ABOUT SOCIAL SOURCES

39

Figure 10. Outline of Experiment 2 and 3. Experiment 2 was identical to Experiment 1
except that Zach always saw 10 races and stated his confidence in his bet. Zach’s confidence
was manipulated between participants such that he had either “low” or “high” confidence in
the inconsistent trials. As in Experiment 1, each participant saw six test trials, varying horse
x’s record and Zach’s bet. See the online article for the color version of this figure.

Experiment 3: Paradoxical Social Evidence
Integration
As is shown in Figure 12, participants from
Experiment 1 thought that horse x had worse
chances when they heard Zach bet on horse x(with unspecified confidence) than in the corresponding baseline trial (where they had no social

evidence). Looking at the figure from left to right,
we observe that the “paradoxical” effect of social
evidence increased as the direct evidence became
stronger with the largest effect in the maximallyconsistent trials in which participants saw horse x
win all five races.
This paradoxical updating of beliefs is inconsistent with the integrative learner model’s predic-

Figure 11. Model’s and participants’ mean estimate of horse x winning the next race at eachlevel of participants’ observations when Zach bets on him with either high or low confidence.
Participants were influenced by Zach’s high confidence bets, as quantitatively captured by themodel (R2 ⫽ .99, MSE ⫽ 8.1 ⫻ 10⫺4). See the online article for the color version of this figure.

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40

HAWTHORNE-MADELL AND GOODMAN

Figure 12. Model’s and participants’ mean estimate of horse x winning the next race when
Zach bets on him with either high or low confidence and split by participants’ assessment of
Zach’s reasonableness. By marginalizing across the participants’ observations, we can clearly
see the interaction or confidence and reasonableness. Only when Zach is thought to bereasonable does his increase in confidence increase his influence, as predicted by the model(R2 ⫽ .99, MSE ⫽ 1.6 ⫻ 10⫺3). See the online article for the color version of this figure.

tion that seeing Zach’s bet would produce a (verymodest) increase in the learner’s estimate of horse
x’s chances (see the middle panel of Figure 12).
The only way our framework would capture thispattern of reasoning would be if people were automatically thinking about Zach’s confidence in
his bet and some people thought he bet with low
confidence.24 We therefore tested whether paradoxical integration can be replicated by explicitly
providing confidence cues (vis-à-vis Prediction 6)
in the maximally consistent trials in which theeffect expected to be the strongest. Crucially, themodel predicts that the effect should occur when
Zach bets with low confidence but not when hebets with high confidence, and only if Zach is also
thought to be reliable.
MethodThe procedure was similar to that of Experiment 2 in which Zach provides his confidence in his bets, but it focused on the maximally consistent trials in which Zach bet on
horse x and participants had strong direct
evidence supporting horse x winning (horse x
won five of the five the trials they saw). The
crucial comparison was between the maximally consistent trial in which Zach bets withhigh confidence and the maximally consistent
trial in which he bets with low confidence. It

is this comparison that should provide theclearest support for the model’s prediction.
However, it was important that participants
knew the alternative confidence levels Zach
could have stated prior to these two trials.
Therefore, after the baseline trials, participants first saw three trials in which Zach bet
with low, moderate, and high confidence. After these three trials, the two trials of interest
were interspersed with other consistent trials
to lessen demand characteristics. Thus, participants saw eight pseudorandom trials after
the baseline trials. First, they saw the three“alternatives” trials in which Zach bet on
horse x with each of the confidence levels. For
these trials, the direct evidence was randomly
selected among horse x winning two, three, orfour out of five races. Then participants had
one of the two trials of interest (randomly
selected), which were followed by three randomly selected consistent trials in which Zach
bet on horse x with randomly selected confidence and direct evidence randomly selectedfrom horse x wins three or four out of five.
The final trial was the remaining trial of interest.
24

They automatically reasoned about Zach’s reliability inExperiment 1.

REASONING ABOUT SOCIAL SOURCES

Participants
Forty participants (mean age ⫽ 26 years,
59% female) were recruited through Amazon’s
Mechanical Turk crowd-sourcing service. They
completed the experiment in 6 min to 9 min fora small payment.

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Model Specification
The same model described in Experiment 2,
and Equation 14 also applies to this experiment
without modification. We used the same empirically derived priors described earlier: the horseskill prior and alpha values used in Experiment
1, and the meaning of the confidence levels
from Experiment 2. The probability that Zach
was reliable was fixed to the proportion measured in this experiment (.85).
To apply our model to Experiment 1, we
similarly needed to determine the probability
that Zach was confident p(␴). We fixed theprobability that Zach was confident to the proportion measured in a separate experiment. This
experiment was a modified version of Experiment 1 in which we elicited 40 participants’confidence attributions. The experiment was
identical to Experiment 1 except that after thethird test trial participants were asked whether
Zach had high or low confidence in his bet (atwo alternative forced choice). On this random
trial, 21 of the 40 participants (53%) said they
though Zach had high confidence in his bet.
The resulting model deals with a fully specified Zach with known knowledgeability andwith confidence and reliability spontaneously
attributed to him. This fully specified integrative
learner model, like the integrative learnermodel, has all parameters measured independently.
Results and Discussion
Participants who saw Zach bet on horse x
with low confidence exhibited paradoxical integration, believing that horse x had worse
chances than when they had not seen Zach’s bet(t39 ⫽ 2.5, p ⬍ .05).25 Conversely, when Zach
bet on horse x with high confidence, there wasno significant difference between his bet and thebaseline (t39 ⫽ ⫺.83, p ⬎ .05). We therefore
see paradoxical updating from low confidence
bets but not from high confidence bets (as expected from Prediction 6). We can see this41

effect in Figure 13 as the low-confidence point
being deflected significantly below the blackline representing the baseline score and thehigh-confidence point.26
We further expect that the paradoxical integration should only be evident when participants attribute low confidence to Zach and thinkthat he is reasonable (as seen in the interaction
in Experiment 2, Figure 11). Indeed, Zach’s
low-confidence bet on horse x only led to paradoxical integration in the reasonable condition(reasonable, t33 ⫽ 3.5, p ⬍ .001; unreasonable,t5 ⫽ ⫺.42, p ⬎ .05, as seen in the differential
effect of confidence between the columns of
Figure 13). The paradoxical integration from
Experiment 1 was consistent with this pattern.
Only participants who thought Zach was reasonable showed the discounting effect (for all
conditions where Zach saw some number ofraces: reasonable, t118 ⫽ 3.8, p ⬍ .001; unreasonable, t40 ⫽ 1.0, p ⬎ .05).27
The observed evidence is therefore consistent
with paradoxical integration being driven by
low confidence attribution. If participants attributed either high or low confidence to Zach in theabsence of explicit confidence cues (Experiment 1), then averaging this mixture should
yield an intermediate estimate between conditions in which all participants knew Zach had
high or low confidence.28 Indeed, as seen in the“reasonable” column of Figure 13, participants’estimate when Zach’s confidence was not stated(the point labeled Unspecified representing Experiment 1 data) fell between their estimateswhen Zach bet with high and low confidence.
In conclusion, the paradoxical discounting
observed in Experiment 1 and its interaction
25
All t tests conducted in this section were testing onesided predictions derived from our model.
26
Additionally, we replicate the main effect from Experiment 2 finding that when Zach bet on horse x with highconfidence, participants thought horse x was more likely to
win than when Zach had low confidence (t39 ⫽ 3.9, p ⬍.001).
27
Interestingly, Zach’s reasonableness had the opposite
effect than it had for the inconsistent trials in Experiment 1.
When Zach is unreasonable and bet on horse x, participants
had a higher estimate of x’s chances that when he wasreasonable.
28
People may, in fact, have a finely graded sense ofothers’ confidence and their initial attributions to them may
reflect this, however we simplify our discussion by assuming that participants only thought Zach had high or low
confidence.

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42

HAWTHORNE-MADELL AND GOODMAN

Figure 13. Participant’s and model’s mean estimate of horse x winning the next race when
he dominated their observations. The baseline estimates based solely on direct information arecompared with the knowledgeable Zach estimate, which are the estimates (averaged across allconditions where Zach saw some number of races) from Experiment 1 where Zach bet on
horse x, consonant with the participants’ observations. Participants’ “paradoxical” integrationof Zach’s bet is evident. They thought horse x was less likely after seeing Zach bet on him.
The integrative learner seen in the second panel does not predict this effect (R2 ⫽ .85, MSE ⫽3.7 ⫻ 10⫺3), but the effect is captured by the fully specified integrative learner model (R2 ⫽.97, MSE ⫽ 1.4 ⫻ 10⫺4). See the online article for the color version of this figure.

with perceived reasonableness is well describedby a model that posits that participants spontaneously fill in both Zach’s reliability and his
confidence (R2 ⫽ .90, MSE ⫽ 9.2 ⫻ 10⫺4).
Importantly, if we use this model to fit all thedata from Experiment 1, then we see that its
predictions only significantly diverge from theintegrative model (see Equation 10 and Figure8) when Zach’s bet is consonant with verystrong direct evidence, which is consistent with
our data and Birnbaum and Mellers’ (1983)
finding. The model that includes learners attributing confidence to Zach therefore fits the Experiment 1 data in Figure 8 when Zach’s bet
was inconsistent (R2 ⫽ .94) and the paradoxical
updating observed when Zach’s bet was consistent with direct observations.
In our last experiment, we tested whether participants update their initial attributions when provided with strong evidence, inferring Zach’s latentstates from his actions and the their own knowledge. Will evidence of Zach acting inconsistently
with his observations cause people to revise theirestimate of his reliability, consistent with our model’s predictions?
Experiment 4: Inferring Reliability From
Action
MethodStimuli in Experiment 4 were similar to thosein Experiment 1, with one major alteration (seeFigure 15). In the first test trial, the complete
information trial, the participants were able tosee Zach’s unobscured results table in addition
to his bet. Zach always saw horse x win eightout of 10 times in this trial, which was consistent with the four out of five wins the participants saw. Participants were randomly placed

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REASONING ABOUT SOCIAL SOURCES

into either the consistent condition in which
Zach’s bet was supported by his observations or
the inconsistent condition in which he bets
against his observations.
The trial following the complete information
trial, the transfer trial, was the same for bothconditions and identical to the standard test
trials in Experiment 1. The participants saw
horse x win one of five trials and Zach bet on
horse x. The transfer trial was followed by thesame debriefing question from Experiment 1
assessing the participants’ judgment of whether
Zach was reliable.
Participants
Eighty participants (mean age ⫽ 31 years,
51% female) were recruited through Amazon’s
Mechanical Turk crowd-sourcing service. They
completed the experiment for a small payment
in 4 min to 7 min. Forty participants wererandomly assigned to each of the Zach bets
consistently or inconsistently conditions.
Model Specification
The integrative model from Experiment 1 wasagain used with all the same parameters except forthe probability of Zach being reliable. In Experiment 1 and 2, the posterior probability of Zachbeing reliable was fixed to the measured proportion. For this experiment, we let the model infer
Zach’s reliability given his behavior in the complete information trial, like the participants (asdescribed in Equation 16). The prior of Zach being reliable was set to the average measured proportion from Experiments 1 and 2 (.77).
Results and Discussion
The manipulation of the consistency of
Zach’s bet led to the predicted assessment of
Zach’s reliability (see Prediction 7). In the consistent condition, participants said Zach was
reliable 88% of the time, compared with just
33% of the time in the inconsistent condition
2⫽ 23, p ⬍ .001). The inte(see Table 3; ␹df⫽1
grative learner model makes a similar inference,concluding that Zach is more likely to be reliable in the consistent trial (MSE ⫽ 1.6 ⫻ 10⫺3).
Does the assessment of Zach’s reliability,
inferred in the complete information trial, havean effect on how his bet is integrated in thetransfer trial? Using a linear regression model

43

with consistency condition as a predictor ofhorse x’s chances in the transfer trial, we canconclude that the consistency manipulation had
the predicted effect on the transfer trial (t78 ⫽2.1, p ⬍ .05). When Zach previously bet consistently, he was subsequently more influential
than when he bet inconsistently as the integrative learner model predicted (MSE ⫽ 3.8 ⫻
10⫺2; see Table 4).
In the integrative learner model, the complete
information trial influences the transfer trial
through the learner’s estimate of Zach’s reliability. Because Zach’s reliability is assumed to beconstant across his bets, forming a strong impression that he is reliable or unreliable should
influence the amount of information his bets
convey in subsequent trials. Indeed, the effect of
Zach’s (in)consistency in the complete information trial on judgments in the transfer trial
should therefore be completely mediatedthrough the manipulation-induced reliability attribution. A mediation analysis concluded thatthe effect of condition on the transfer trial was
mediated by the manipulation-induced reliability attribution (Baron & Kenny, 1986). Including both condition and reliability attribution in
the linear model yielded a highly significant
effect for reliability (␤ ⫽ .14, t77 ⫽ 3.0, p ⬍.05), and the previous effect of condition isabsent (␤ ⫽ .009, t77 ⫽ .20, p ⬎ .05). The Sobel
test showed significant reduction in the condition’s effect on participants’ estimate of horse
x’s chances (z ⫽ 2.6, p ⬍ .05).
Participants were more likely to infer that Zach
was reliable when his bet was consistent with his
information, and they subsequently use this inferred quality to determine how much they should
learn from Zach’s subsequent bets. It was theinferred reliability of Zach that drove them tolisten to him 2.9 times more in the consistentcondition than in the inconsistent condition.
Alternative Model Comparison
In this section we assess the empirical support
for different key components of the full integrative
learner model (Figure 1c; Equations 10, 15, and17) to simpler models with these components “lesioned” (see Table 5). The integrative learnermodel posits that learners reason about Zach’s bet
with a generative model of behavior that describes
how Zach’s observations influence his beliefs andhow his beliefs, desires, and reliability guide hisThis document is copyrighted by the American Psychological Association or one of its allied publishers.
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44

HAWTHORNE-MADELL AND GOODMAN

Figure 14. Participant’s and model’s mean estimate of horse x winning the next race when
they observe him win his last five and Zach bets on horse x (R2 ⫽ .90, MSE ⫽ 9.2 ⫻ 10⫺4).
Zach saw 10 races and either bet with high or low confidence (Experiment 3) or withunspecified confidence (data from Zach saw 10 races condition of Experiment 1). Resultsfrom participants who thought Zach was reasonable are displayed in the left column, whereas
those who thought he was unreasonable is displayed on the right. The horizontal black line
represents participant’s baseline estimate without seeing Zach’s bet. The significant decreasein estimates of horse x’s chances from participants who heard Zach bet with unspecified
confidence compared with the baseline in the first column is an example of “paradoxical”
social information integration (the downward divergence of the purple unspecified point fromthe black line). The model suggests that this is driven by participants spontaneously attributing
low confidence and should only occur when participants also attribute reasonability, which isobserved in the paradoxical integration of the low confidence estimate in the reasonable
column but not the unreasonable column. See the online article for the color version of thisfigure.

actions (see Figure 1d). It interprets Zach’s bet
with this intuitive ToM and integrates it withdirect observations to learn about the common
cause—the horse’s skill (see Figure 1c). Are all of
the components of the integrative learner model
necessary to explain the pattern of results acrossthe three experiments? Specifically, do the data
support a model that (a) integrates social anddirect evidence, (b) interprets social evidence witha nested inference about Zach and his attributes,(c) spontaneously fills in Zach’s unobserved attributes like reliability and confidence such that the
Zach is “fully specified”, (d) updates these spontaneous attributions in light of strong evidence.
We first describe the simplified models that missone or more of these components and then com-pare their ability to account for the data to theintegrative learner model.
Introduction to Alternative Models
Each model corresponds to a row in Table 5
with a model number (M1, M2, etc.) where its
ability to fit the data across our four experiments
was assessed. The integration of direct and social evidence of the integrative learner–M7 wascontrasted with the nonintegrative direct–M1
and pure-social–M2 models (Figure 1, Panels aand b, respectively). Each of these models exclusively relies on one source of evidence. The
direct model simply infers the horse’s skill fromthe direct observations, ignoring social cues,This document is copyrighted by the American Psychological Association or one of its allied publishers.
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REASONING ABOUT SOCIAL SOURCES

45

Figure 15. Outline of Experiment 4. Participants first observed the “transparent trial” inwhich Zach’s observations are visible. They then see him bet (in)consistently with hisevidence, depending on which (between-participants) condition they are in. Participant’s thenhave a “transfer trial,” identical to those in Experiment 1, in which Zach sees 10 races and theysee horse x win one of five. See the online article for the color version of this figure.

whereas the pure-social model ignores direct
evidence and instead uses Zach’s bet to infer thehorse’s skill. The direct model uses the horseskill prior empirically fit to independent data, as
discussed in Experiment 1’s Model Specification section. The best pure-social model shares
all seven parameters of the integrative learnermodel, fit to independent data as described earlier.
The fully specified integrative learner–M7 posits that learners fill in Zach’s unobserved attributes
like his reliability and confidence. We contrast thiswith simple-reliability–M3 that assumes thatlearners do not attribute reliability (instead modeling all actors with a fixed degree of decision
noise, fit as a free parameter). M7 claims that, foreach attribute, there is heterogeneity in the population of learners, for example, some believe Zach
is a priori reasonable with p(␣), whereas others
believe that he is unreasonable with 1 ⫺ p(␣).
Similarly, some think a priori Zach bets with highconfidence with p(␴). The probability that learners
make these attributions—p(␣) and p(␴)—is fixed
to the observed proportion in the data. Similarly,
the interpretation of high and low confidence anda reliable and unreliable actor are fixed to independently collected data. The fully specified–M7
therefore has seven measured parameters, these
six and the horse-skill prior, and no free parameters.
When Zach’s confidence and reliability are
not observed, the model’s characterization ofthese attributes reflects its priors. The model
therefore has to specify when this belief is updated, particularly reliability, which is defined
as a stable attribute of Zach’s that impacts subsequent decisions. An ideal Bayesian reasoner
would always update his or her estimate ofZach’s reliability with new evidence as the update-reliability–M6 does. Departing from thisideal reasoner, the flexibly update-reliability–
M5/M7 posits that, perhaps because of resource

Table 3
Mean Probability Zach is Reliable With Standard
Error of the Mean

Table 4
Mean Estimate of Horse X’s Chances With
Standard Error of the Mean

Condition

Data

Simulation

Condition

Data

Simulation

Unreliable
Reliable.33 ⫾ .074
.88 ⫾ .052

.17
.80

Unreliable
Reliable.30 ⫾ .023
.38 ⫾ .034

.26
.37

46

HAWTHORNE-MADELL AND GOODMAN

Table 5Comparison Between Learner Models That Make Different Assumptions About the Structure and Content
of the Representations Employed by Participants Across the Three Experiments

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Simple model
Model

Kind

Figure812
7

DirectPure-social
Integrative

.56
.36
.96

Figure9

Figure11

Figure12

Average
fit

0

.38
.57
.99

0

.22
.46
.96

.74
.88

.17
.99

Integrative model
Reliability

3
4
5
6
7

Confidence

Attribute

UpdateAttribute

Figure8

Figure9

Figure11

Figure12

Figure13

Figure14

Tables 2
and 3

Average
fit

No
Yes
Yes
Yes
Yes

Never
Flexibly
Always
Flexibly

No
No
No
No
Yes

.96
.96
.96
.73
.96

.45
.88
.88
.17
.88

.99
.99
.99
.74
.99

.54
.99
.99
.17
.99

.85
.85
.85
.85
.97

.35
.86
.86
.86
.90

0
0
.98
.98
.98

.59
.79
.93
.64
.95

Note. Each column represents the R2 of the model’s predictions for the data represented in the named figure. The data in
Figure 8 show the effect of the manipulation of Zach’s knowledgeability. The data in Figure 9 Illustrate the interaction
between perceived reliability and the manipulation of Zach’s knowledgeability and reliability. Figure 11 demonstrates theeffect of Zach’s confidence, and Figure 12 shows that the dependence is on Zach being reliable. Finally, Tables 2 and 3 showthat participants update their estimate of Zach’s reliability when presented with clear, salient cues about it and use thisupdated estimate when learning from him in the future. All the models considered had no free parameters.

limitations, learners only update their beliefs
about Zach’s reliability when provided withstrong evidence.29 We contrast the ideal updatereliability–M6, the flexibly update-reliability–
M5/M7, and the fixed-reliability–M4 that never
updates its belief about Zach’s reliability. Thesemodels share the same representation of Zach’s
reliability but differ in when this representation
is updated.
Results and Discussion
Each model’s fit to the three experiments wastested (presented in Table 5). We focused ouranalysis on the results summarized in Figures 8,
9, 10, 11, 12, and 13 and Tables 3 and 4 forcontinuity with our discussion in the experimental sections. Across these summaries of thefour experiments, the fully specified integrative
learner–M7 provided the best average fit withno free parameters. Looking at the pattern offailures of the simpler models reveals the components of M7 that capture different aspects of
the learner’s social reasoning about Zach andthe world.
Participants’ responses across the experiments reflected both the races they directly ob-

served and Zach’s bet. This integration of social
and nonsocial cues is evident in the poor fit ofthe direct–M1 and the pure-social–M2. Participants were influenced by Zach’s bet and showedsensitivity to Zach’s knowledgeability (see Figure 8), confidence (see Figure 10), and reliability (see Figures 9 and 11 and Tables 3 and 4)
that direct–M1 couldn’t account for. Conversely, pure-social–M2 could not account forparticipants’ sensitivity to direct evidence (seeFigures 8 and 10). To account for the data wetherefore need a model that integrates direct andsocial evidence like M3 through M7.
Participants’ spontaneous assessment of
Zach’s reliability affected how much they were
persuaded by his bet. This effect of reliability
29This situation-dependent updating of the learner’s estimate of Zach’s reliability was motivated by the data inExperiment 1’s Data Processing and Counterbalance Testing section. The model itself does not specify what circumstances will lead learners to update their estimate of adviser
attributes; we make the auxiliary hypothesis that this happens in our setting only when explicit and strong evidence
is available. This is clearly an important direction for future
work, which is further discussed in the Reliability subsection of the Limitations and Future Work section.

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could not be captured by no-reliability–M3. The
model’s one free parameter allowed it to capture
the main effect of Zach’s knowledgeability (seeFigure 8) and confidence (see Figure 10), but it
couldn’t account for the effect of participants’
self-reported estimate of Zach’s reliability (as isevident in Figure 9 and Figure 11) or the effectof manipulating evidence of Zach’s reliability
in Experiment 4. All models that had two levelsof reliability that were spontaneously attributed
to Zach (M3 through M7) were able to capture
the influence of participants who have different
beliefs about Zach’s reliability. Similarly, themodels that did not spontaneously attribute confidence to Zach (M3 through M6) were unable
to explain the paradoxical discounting evident
in Figure 12 and 13. Only fully specified integrative learner–M7 that spontaneously attributed both reliability and confidence could fully
account for the data across the four experiments.
Comparing the predictions of M4 throughM6 —the integrative learner, constant-reliability, and update-reliability models— can reveal
when learners updated their estimate of Zach’s
reliability. Fixed-reliability:M4 captured learners’ social information integration across thefirst two experiments (see Figures 8, 9, 10, and11). In these experiments the participants’ behavior is consistent with the model’s assumption that learners do not update their belief
about Zach’s reliability but do use this fixed
belief to interpret Zach’s bet. However, themodel fails to account for the impact of thetransparent trial in Experiment 4 in which participants update their belief about Zach’s reliability and then use this updated belief whenlearning from Zach in subsequent trials (butfixed-reliability–M4 does account for how thisupdated estimate is used in subsequent trials).
On the other hand, update-reliability–M6 cannot account for the participant’s estimate in thefirst two experiments. On the trials in which
Zach’s bet is strongly inconsistent with the participants’ direct observations, update-reliability–M6 infers that Zach is likely unreliable anddiscounts his bet. Participants’ integration does
not exhibit this effect, and instead is substantially influenced by Zach on these incongruent
trials, as fixed-reliability model–M4 predicts.
The comparison of M4 –M6 suggests thatlearners are capable of updating their estimateof Zach’s reliability, but do so only in certain

47

circumstances, like flexibly updating M5. However, the data presented here are insufficient tobuild a formal understanding of the circumstance in which learners should update theirbeliefs about latent actor attributes like reliability. One major difference between Experiment 1
and 2, in which participants did not update theirbelief about Zach’s reliability, and Experiment
4, in which they did, was that the evidence inExperiment 4 was more diagnostic of Zach’s
reliability (according to update-reliability–M6).
Given this difference, one hypothesis is thatlearners are flexible “resource-rational reasoners” who conserve computational resources—
only reasoning about reliability when the added
inferential complexity results in substantially
improved accuracy (Lieder et al., 2012). Alternatively, learners could have a fixed cognitive
limitation such that they are unable to perform ajoint inference over the horse’s skill, Zach’s
observations, and his reliability. Under this alternative hypothesis, learners update their beliefin Zach’s reliability in Experiment 4 not because of the strong inference it licensed, but
because Zach’s observations were known,
which simplified the complexity of the inference to Equation 16.
In conclusion, the fully specified integrative
learner–M7 provides the best fit to participantssocial reasoning and information integration
across the four experiments (overall fit R2 ⫽.96). Simpler models that did not integrate social and direct evidence or fully specify Zach’s
attributes could not account for all the major
effects in the data. Neither could models that
did not selectively update Zach’s reliability.
The success of M7 is not likely due to overfitting the data. The full model’s predictions
across the four experiments had seven parameters fit to independent data (as described in eachof their Model Specification sections), and these
fit values were reused across the four experiments with no remaining free parameters(though the choice of which of the four experimental situations led to reliability updates
could be seen as a single free parameter).
Insights From a Computational Account of
Learning From Others
Learners’ flexible social reasoning is evident
in their ability to use what they know to infer
what they do not. We saw this clearly across ourThis document is copyrighted by the American Psychological Association or one of its allied publishers.
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48

HAWTHORNE-MADELL AND GOODMAN

four experiments; learners used their model of
Zach to interpret his actions to learn from him.
Similarly, we saw in Experiment 4 that learners
could learn about Zach’s latent attributes when
they had more certainty about what he saw (asKushnir [2013] stated, learners can learn bothfrom and about people; also see Landrum,
Eaves, and Shafto [2015] complimentary treatment of “learning to trust and trusting to learn”,
p. 1).30
The fit of the fully specified integrative
learner model across our experiments demonstrate that reasoning over a generative modelof action and outcomes naturally captures
these flexible patterns of inference. Further,by precisely defining the learner’s ToM andwhat she knows about the situation— her direct evidence, prior knowledge, and theory of
domain— our framework describes howlearners learn from and about people and thefactors that determine how much they learn
from and about them. It allows us to distilllearning from and about people into two interlocking patterns of social learning. First,
learners use evidence judiciously, that is, they
integrate different sources according to theirdiagnosticity. Because an actor’s attributes
determine their diagnosticity, learners use
available information about them to weigh therelative value of his actions. Second, learners
use available evidence to infer the unknown
attributes of an actor and in the absence ofevidence fill these attributes in. Our framework describes how these patterns inform one
another—learners use available evidence(e.g., the discrepancy between the learner’s
initial estimate and the actor’s estimate) toinfer the unknown attributes of an actor,
which, in turn, they use to weigh his actions.
In this section, we apply our framework to
two experiments that exhibit these patterns ofreasoning. The first experiment by Harvey andFischer (1997) demonstrated that learners use
all available information about an adviser toguide how they integrate that adviser’s actions
with their own beliefs. The second experiment
by Yaniv (2004) shows that when information
about an adviser’s attributes is not present,
learners flexibly use their understanding of thesituation to reason about the adviser and how
informative his actions are.

Learning From and About Advisers in JAS
Experiments
Both Yaniv and Harvey and Fischer’s experiments follow the same basic structure commonly used in the JAS literature. First theyelicit an initial estimate from the learner. They
then show the actor’s estimate, and while displaying both the initial and actor’s estimate,they elicit a final estimate.
The difference between the learner’s estimates before and after receiving advice is theiradvice utilization. Researchers have consistently found that learners under-utilize advice
across a range of situations and methods forcalculating learners’ advice utilization (as described previously in the section Toward a Cohesive Account of Actor Attributes). That is to
say that learners weigh their initial opinions
more than they “should.” Here the amount thatlearners “should” incorporate advice is theamount that would result in the correct answer(e.g., how much the learner should have moved
her estimate toward the actor’s estimate to arrive at the correct year the Suez Canal opened).
We can think of learners that approached thisstandard as ecologically normative.
However, from the perspective of a learner inthese experiments, there is insufficient evidence
to decide the extent to which they should relyon the available social information. For example, there is considerable uncertainty about boththe actor’s knowledgeability and her own. Our
framework can explicitly define the learners’
understanding of the actors’ and their own attributes, as well as the learning situation given
the uncertainty of each. The prediction ofBayesian inference over this formalization of
the learner’s model of the situation provides a
new standard with which to evaluate their social
evidence integration.
Using our framework to form a normativereference point for advice utilization paradigm
we can see if, whereas learners do not incorporate advice as much as they should by the ecological standard, they conform to an ideal observer reasoning over an uncertain learning
situation. We form such models for two experiments in the JAS tradition, which suggest that30

Or, as we will see in Yaniv’s study in the following
text, these inferences can happen simultaneously.

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what was interpreted as an egocentric bias (fromthe ecological standard) can be understood as
Bayesian reasoning over uncertain learning situations.
The two experiments follow the same experimental structure but differ in the information
they provide the learner. Harvey and Fischer(1997) use training on a cue-based judgmenttask as a proxy for knowledgeability. They manipulated the amount of training given to learners (30, 100, or 240 training trial, betweensubjects) and the amount of training the adviser
had (within-subjects) such that each learner saw
an adviser at each level of training. Harvey andFischer’s experiment provides a test of learner’sability to use an actor’s attributes to determine
how each cue should be used.
Yaniv and colleagues asked questions tapping historical knowledge (Yaniv 2004). Learners were only told that the adviser was anotherstudent participant of the study, which left them
uncertain about the adviser’s absolute and relative knowledgeability. As Yaniv pointed out,
this leads to an epistemic asymmetry as learners
have a better understanding of their own knowledgeability than the advisers. Yaniv’s experiment therefore tests the framework’s predictionthat when information about an adviser’s attributes is not present, learners flexibly use theirunderstanding of the situation to reason about
the adviser and how informative his actions are.
Sensitivity to the actor’s relative knowledgeability: Judicious use of evidence.
Modeling Harvey and Fischer (1997). The
learner in Harvey and Fischer’s experiment istasked with making final estimate qf of continuous quantity q on a screen that displays her
initial estimate ql, her number of training trials(proportional to her knowledgeability kl), theadviser’s estimate qa, and his number of training trials (⬀ ka). How should she integrate these
two estimates with their associated knowledgeability cues? A learner using the ToM formalized in Equation 5, could reason about eachestimate as the intentional choice of a personwith belief b and reliability ␣, generalizing
Equation 3:p(qa ⱍ ba, ka, ␣a) ⬀ e␣·⺕p(q ⱍ ba,ka)U(qa;q) ,(17)

where utility of an estimate U(qa; w) is given bya squared error loss function and p共q ⱍ b, k兲 ⬃ᏺ共b, xk兲, where x represents a free parameter49

for each level of k describing the increase inprecision decision makers associate that level oftraining.31 The three values of k that minimize
the mean standard error of the model’s predictions and the data produces the qualitativemodel predictions shown in the lower panel of
Figure 16.
The learner could then integrate the adviser’s
estimate with her own initial estimate using ageneralization of Equation 6 for each social
source s in our case the learner l and adviser a:p(q f ⱍ ql, kl, qa, ka)⬀ p(q f )

兺 兺 p(qs ⱍ bs, ks, ␣s)p(bs ⱍ qf, kl)p(␣s),

s僆兵l,a其 ␣s,bs(18)

where p共b ⱍ q, k兲 ⬃ ᏺ共q, xk兲. This model differs
importantly from the integrative learner model.
It describes learning about a continuous quantity from two social cues (instead of learning
about a discrete variable from one direct andone social cue). It also models a situation inwhich the sources are not understood as updating their beliefs from unknown evidence. The
sources beliefs are understood directly in termsof the underlying quantity with precision proportional to the source’s training p(b|q, k).
With the limited information about the domain and actor attributes the complex social
inferences described in Equation 18 can be wellapproximated by a simple optimal cue integration model (widely used in perception, e.g.,
Knill & Richards, 1996; multimodal sensory
integration, e.g., Bejjanki et al., 2011; social
cognition, e.g., Zaki, 2013):
p(q f ⱍ ql, kl, qa, ka) ⬀ p(q f )p(ql ⱍ q f , kl)p(qa ⱍ q f , ka),(19)
where each source’s estimate is interpreted as acue of the underlying q with Gaussian noise
proportional to the source’s training.
31
Alternatively, the learner may not be reasoning with
her initial estimate shown on the screen, rather she might bedirectly relying on memory traces of cue. Unfortunately, thedata available in this experiment does not allow us todistinguish between this (or a number of other) alternative
formalization of the learner’s understanding of the situation.
That is to say that there are multiple ideal observer models
consistent with the experimental situation.

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50

HAWTHORNE-MADELL AND GOODMAN

Figure 16. Participant’s and model’s percentage shift in judgment after advice as a functionof participant’s and advisers training (data from Harvey & Fischer, 1997, Figure 4a). Thereis a strong interaction between the participant’s level of training and the adviser’s level oftraining on how the participant used the adviser’s estimate, which is captured by the model.
More highly trained advisers were more influential, and this effect increased as the participants were less well trained. The model captures these qualitative patterns (R2 ⫽ .92). See theonline article for the color version of this figure.

Results. As seen in Figure 16, participants
were sensitive both to their own level of knowledgeability and to their adviser’s. Both Harvey
and Fischer’s (1997) data and our model’s qualitative predictions are shown in terms of weightof advice—the difference between the original
and final judgment expressed as a percentage ofthe difference between the original judgment
and the advice. As predicted by the model,
advice utilization reflects an interaction of theknowledgeability of the two sources (R2 ⫽ .92).
The more knowledgeable an adviser is, themore influential they are, but as learners themselves become more knowledgeable, the adviser’s impact diminishes. When the learner thinksthat she has low accuracy interpreting the cue,as in the top line of Figure 16, advisers arehighly influential. However, when the learner
thinks she can interpret the cue with high fidelity, an adviser (who sees the exact same cue)has muted influence.

Discussion. The model qualitatively captures the interaction of the learner’s and adviser’s training on advice integration, but themodel consistently underestimates the extent towhich the learner is influenced by the adviser(seen in the 20% increased utilization seen in all
the data points when compared to the model in
Figure 16). This is instructive of the differences
between the ecological and the Bayesian learnerreference points. Participants underutilize theadvice in that they would have been more accurate if they had shifted more toward adviser’s
estimates. However, compared with the Bayesian learner model, learners erred toward overutilization, not underutilization. The Bayesiananalysis indicates how learners trying to form
accurate beliefs would integrate advice, and thefact that learners exceeded this amount is consonant with Harvey and Fischer’s (1997) discussion that there are additional motivational

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REASONING ABOUT SOCIAL SOURCES

factors for advice utilization like sharing responsibility and accepting offered help.32
Inferring actor attributes from their actions and knowledge of the domain.
Modeling (Yaniv, 2004). Yaniv’s experiment focuses on how two factors impact learner’s advice utilization: the learner’s knowledgeability and the distance of an adviser’s estimate
from the learner’s initial estimate (the advice⌬). Yaniv divided his participants into high andlow knowledge groups according to the accuracy of their preadvice judgments (mediansplit). The distance of advice was binned into
near, intermediate, and far. As we shall see, ourframework predicts that these two factors
should interact, which results from the learnerlearning both from and about the adviser. Thisprediction emerges from applying the samemodel used for Harvey and Fischer’s (1997)
experiment, except that in Yaniv’s experiment
the learner does not have any information about
the adviser’s knowledge and therefore has torely on internal cues of their own knowledgeability:p(q f ⱍ ql, kl, qa)⬀ p(q f )p(ql ⱍ q f , kl)p(qa ⱍ q f , ka)p(ka), (20)兺kawhere k, although potentially continuous, is discretized into the “high” and “low” knowledge
categories Yaniv used in his analyses, and, as inEquation 20, p共q ⱍ q f , k兲 ⬃ ᏺ共q f , xk兲, where x
represents a free parameter for each level of k
describing the precision learners associate withsources with high and low knowledge.
The resulting model shows how learners canuse the advice ⌬ to infer an adviser’s knowledgeability, and how the strength of this inference
depends on how accurate they think their estimate
is. When the learner is highly knowledgeable andthinks that her estimate is close to the right answer, she can draw a strong inference that anadviser with an implausible estimate is notknowledgeable. She can then discount his estimate. A learner who guessed, and therefore
does not think that her estimate was accurate,would not be able to make such a strong inference about an adviser based on the his advice ⌬and would utilize advisers similarly, regardlessof the distance of their estimate from hers.

51

Results. As seen in Figure 17, there was amain effect of dividing the learners into low andhigh knowledgeability groupings. Learners withhigh confidence were less influenced by advisers, regardless of the advice ⌬. This is the same
main effect observed in the model of Harvey
and Fischer’s (1997) experiment where participants had external cues of their knowledgeability, and is consistent with the model’s assumption that learners’ internal cues allowed forsimilar (coarse) awareness of their own knowledgeability. Looking beyond the main effect we
see the striking interaction predicted by themodel. High knowledge learners progressively
discount advisers as their advice ⌬ increases.
The advice utilization of low knowledge learners, in contrast, is unaffected by the adviser’s
advice ⌬. The model qualitatively captured thisinteraction well (R2 ⫽ .98).
Discussion. This pattern of results would
be expected from learners who reason about
an adviser’s attributes and use their conclusions to titrate the relative influence of hisestimates in their final judgment. The model
prediction is therefore dependent upon thelearner’s estimate of the adviser’s knowledge,
which should mediate the interactions observed in Figure 17. Yaniv’s experiments(Yaniv, 1997, 2004), like others in the JAS
tradition, do not measure the learner’s estimates of their own attributes or their adviser’s. Because learner with high and low
knowledgeability were simply grouped by average score it is unknown whether the learners had accurate trial by trial estimates of
their own knowledgeability. Alternatively,
they could have just had a coarse estimateakin to “I am [good/bad] at these kind ofquestions.”
Learners with a coarse estimate could be systematically underutilizing advice by making
poor inferences about advisers when they incorrectly think themselves knowledgeable. Although this learner may be integrating information rationally with respect to their beliefs about
32
Motivational effects are outside the scope of our present analysis of learning about the world from social anddirect evidence, but such an effect could be incorporated
into our paradigm by including an explicit utility function
for the learner that includes social goals (that are themselvesdefined using a ToM; see Ullman et. al [2010] for an
example).

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52

HAWTHORNE-MADELL AND GOODMAN

Figure 17. Participant’s and model’s percentage shift in judgment after advice as a functionof participant’s accuracy (median split; data from Yaniv, 2004). Knowledgeable participants
use adviser’s estimates less when the estimates are far from their own initial estimate. Less
knowledgeable advisers do not show this pattern, instead using all estimates to the same
extent. This interaction is consistent with the qualitative predictions of a model of participantsin which they infer the adviser’s knowledgeability using their own knowledge (R2 ⫽ .98).
When they feel knowledgeable, they make strong inferences about the adviser (shown in thelower curve of the graphs). See the online article for the color version of this figure.

the learning situation, their false beliefs leadthem to overweight their own evidence. This
points to an important limitation of the narrowly
defined Bayesian learner we present. Our model
simply indicates whether learner’s combine information rationally given their beliefs about
the situation.
Yaniv and colleagues (Yaniv, 1997; Yaniv& Kleinberger, 2000; Harvey & Fischer,1997) explain the “discounting” observed in
the experiment and others as arising from “aninformational asymmetry inherent in any decision-making process that involves the use ofadvice” (Yaniv, 2004, p. 2). Indeed, in his
experiment the learner has no direct evidence
about the adviser’s knowledgeability, butdoes seem to have a sense of their own knowledge. Our cognitive model formalizes thisasymmetry and predicts the interaction observed in Figure 17.33

Conclusion From Application of
Framework to JAS Studies
The two JAS experiments highlight two of
the hallmarks of social learning. Learners inHarvey and Fischer’s (1997) experiment weresensitive to parametrically varied (cues of) the
sources knowledgeability. They flexibly used
the available evidence about the relative diagnosticity of the two sources—judiciously integrating the two sources. Learners in Yaniv’s
experiment used their own knowledge to draw
inferences about the adviser, filling in his unknown attributes. Our framework shows how
understanding social evidence in terms of a33
This is an example of how formalizing theories withcomputational models illuminates the theory’s subtle consequences, as Yaniv proposed in the asymmetry theory, butdoes not predict the observed interaction.

REASONING ABOUT SOCIAL SOURCES

generative model of behavior can predict thesepatterns of inference.

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General Discussion
How do people form a coherent view of theworld from both social and direct information
sources? Using a novel quantitative experimental paradigm, we found that people integratedsocial information and direct observations inaccordance to the predictions of a sophisticated
Bayesian computational model. People displayed a sensitivity to the actor’s knowledgeability, reliability, and confidence consistent
with our model’s qualitative and quantitative
predictions.
The core of the computational model is aformalization of an intuitive ToM—people’s
mental model of how intentional actors makedecisions given their beliefs, desires, traits, andother attributes. This generative model of anactor describes what it means for him to beknowledgeable, reliable, and confident, andhow these attributes guide reasoning about social evidence. As an explicit theory of the structure of people’s ToM, the model makes a number of quantitative predictions about the role
actor attributes play in our social reasoning.
These predictions build upon previous findingsof people’s sensitivity to these actor attributes
by providing a unified account of their relative
impact and interrelationship.
In our computational model, the actor’s influence is dependent on the precise extent of his
knowledgeability. Experiment 1 confirmed thisprediction. It found that people were influenced
by actors in proportion to their knowledge andignored actors with no knowledge. The evidence in our experiments was graded and quantitative, which allowed us to parametrically vary
the actor’s knowledge. This manipulation revealed that people integrated the (variably
knowledgeable) actor’s bet with their directly
observed evidence as predicted by our Bayesian
integrative model.
Experiment 2 found that people were more
influenced by confident actors than those who
expressed low confidence in their actions. Although this sensitivity to an actor’s confidence
is well documented, our experiment allowed us
to measure the relative importance of an actor’s
confidence and knowledgeability. Both actor attributes are formalized in our model, providing

53

quantitative predictions for their relative impact
and relationship. For instance, the model predicted that an actor who saw 10 races would be
more influential if he expressed confidence in
his action than if he was three times as knowledgeable but didn’t express confidence (fromthe Predictions of the Integrative Learner andLearning from Confidence Cues subsections of
our Computational Account of Learning From
Others section). Comparing Experiment 1 and
2, we see precisely this relationship: Confidence
dramatically increases an actor’s influence compared with increases in his knowledgeability.
Both the effect of Zach’s knowledgeability
and confidence are predicted to be modulated byhis reliability. In the extreme, if the actor iscompletely unreliable, he should be ignored regardless of his knowledgeability and confidence. Comparing people who thought the actor
was reliable to those who though he was unreliable in Experiment 1 and 2 reveals these predicted interactions. In Experiment 1 people who
thought the actor was unreliable were not influenced by his knowledgeability. Those who
thought the actor was reliable showed gradedsensitivity to his knowledgeability. Similarly, in
Experiment 2, only people who thought Zach
was reliable showed sensitivity to Zach’s confidence.
In both Experiment 1 and Experiment 2, theactor’s reliability was not mentioned until theend of the experiment, when participants wereasked to assess it. The fact that reliability judgments still mediated the effects of knowledgeability and confidence indicates that people
spontaneously used their estimate of the actor’s
reliability to determine how informative his actions were. Similarly, we found evidence thatparticipants spontaneously estimate an actor’s
confidence and use these estimates when learning from him. In Experiment 1, when Zach’s
bets were consonant with the participants’strong evidence for horse x, his bet had theparadoxical effect of decreasing their estimateof horse x’s likelihood. Zach did not state his
confidence in Experiment 1, but this paradoxical effect could be the result of some participants who spontaneously attribute Zach low
confidence in his bet. Experiment 3 found thatthe paradoxical effect observed in Experiment 1
is consistent with participants who fully specify
Zach’s unobserved reliability and confidence.

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54

HAWTHORNE-MADELL AND GOODMAN

Participants who fully specify Zach’s unobserved attributes should not take their attributed
values as given, but rather update their priorestimates with available information. Departingfrom a fully Bayesian approach, the behavioral
evidence suggests that these estimates areformed and used but not always updated: The
best-fitting model used people’s prior expectations of the actor’s reliability, held constantthroughout the experiment, rather than updatingin response to (weak, indirect) evidence fromthe actors actions. However, people did updatethese estimates in Experiment 4 where weshowed stronger evidence—the actor placed a
bet that was (in)consistent with his observations
providing clear evidence that he was (un)reliable. People used this evidence to infer that theactor was (un)reliable in the proportion predicted by our model, and people used their
updated estimate of the actor’s reliability whensubsequently learning from him. The experiments taken together therefore suggest that people spontaneously attribute actors with attributes like reliability to reason about them, but
only update their attributions in certain circumstances. We discuss the implications more in thefollowing passages.
Our model accurately predicted people’s responses in all four experiments. The experiments manipulated the actor’s attributes andprobed people’s beliefs about the actor and theworld. The same core model was used to account for these manipulations and the different
dependent variables, illustrating the productivity of our generative model of behavior (ToM).
The model’s flexibility emerges from this generative model, not free parameters fit to thespecifics of the experiments; our model used thesame fitted parameters for all its predictions,leaving none free. As discussed in the Alternative Model Comparison section, only the fully
specified integrative model that reasoned aboutboth direct and social evidence and filled inunobserved attributes of the actor could accountfor the pattern of data across these four experiments.
Connections to Other WorkThe results of our computational analysis
provide a new perspective into two documented
irrationalities of advice utilization—that advice
can have a paradoxical effect (Birnbaum &Mellers, 1983) and that it is systematically underutilized (e.g., Harvey & Fischer, 1997;
Yaniv, 1997; Yaniv & Kleinberger, 2000). Our
framework is built around a formal model of thelearner’s understanding of the adviser. Taking
this perspective and formally considering thelearner’s characterization of the adviser— hisknowledgeability, reliability, and confidence—
reveals that the inexplicable behavior in bothcases could emerge from learner’s optimally
integrating the social evidence given their characterization of the adviser.
Given the information provided to participantsin Birnbaum and Mellers’ (1983) experiment andour own Experiment 1, the actor selecting option
x should have increased the participant’s belief inx, but it has the opposite, paradoxical effect. Crucially, in neither case was the participant’s estimate of the adviser’s confidence measured. Our
framework describes how this pattern would be
expected from a learner who, in the absence ofreceiving confidence information, spontaneously
attributes the adviser confidence. Experiment 3
supported this account of the paradoxical effectand could similarly explain the effect in Birnbaumand Mellers’ experiment (which could be tested
with manipulation of the adviser’s confidence ormeasurement of participants unmanipulated attributions).
Similarly, our model indicates that what has
been characterized as underutilization can beunderstood as Bayesian reasoning under uncertainty about the adviser’s knowledgeability, reliability, and confidence. Indeed, in many of theclassic experiments finding underutilization, thesource or extent of the adviser’s knowledge was
uncertain for the learner, which our model predicts would significantly reduce the potential
influence of the adviser (Yaniv, 1997; Yaniv &Kleinberger, 2000). However, we have shownthat even knowing the adviser’s exact knowledgeability, as people did in Experiment 1, does
not ensure that he is highly influential.
This points to a strength of our computational
approach: it can predict the precise cause of the
low impact (seeming underutilization) of advisers—what the “bottleneck” in a given learning
situation is. For Experiment 1, our model predicted that it was the nature of the actor’s action—a discrete, binary choice—and the uncertainty about his reliability that limited his
influence. To increase the adviser’s influence,
we widened the communication channel in Ex-

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REASONING ABOUT SOCIAL SOURCES

periment 2 and found the predicted jump in theadviser’s influence. Similarly, our model canmake predictions of what would further weaken
an actor’s influence. By providing strong evidence that the actor was unreliable in Experiment 4 we found that he was effectively ignored. Our modeling framework therefore
provides a tool for both generating realistic
learner-based normative reference points for advice utilization and for making predictions
about which changes to a situation or adviser
will have the largest impact on advice utilization. Both applications can buttress existing investigations in the advice-utilization and decision support systems literatures (Gönül et al.,2006; Harvey & Fischer, 1997).
Our framework uses the tools of computational
cognitive psychology to synthesize two distinct
intellectual traditions— epistemic trust and judgment adviser systems. The framework formalized
the actor attributes identified in the epistemic trust
literature and combined it with the insights andempirical methodologies for testing advice utilization of the JAS literature. The framework wasable to captures some major findings in the respective fields, but a number of questions central to theepistemic trust and JAS literature remain to be
explored. The developmental trajectory of the integration of social information with other evidence remains open. Similarly, we haven’t addressed the quantitative dynamics of integrating
information from multiple actors. Despite theirdifferent focuses, the epistemic trust and JAS
fields pursue complimentary questions— how dopeople decide whom to trust, and how do theydecide how far to trust them relative to otherinformation sources. Our framework highlights
the common cognitive substrate involved in thesequestions. As such we hope that our framework
for understanding social information integration
fosters connections between these complimentary
literatures and fosters a broader dialogue about thesocial and cognitive dynamics underlying social
learning.
Our computational framework uses a generalprobabilistic learning framework (Tenenbaum
et al., 2011), sharing and elaborating on featuresof a number of previous accounts of social
reasoning and learning (Baker et al., 2009,2011; Goodman et al., 2009; Shafto et al.,2012). Our model shares the basic model of
ToM as a formalization of belief-desire psychology with Baker et al. (2009, 2011) and55

Goodman et al. (2009; see Butterfield et al.,2008 for a synergistic approach using Markov
random fields). However, this basic conceptionof ToM is elaborated in a number of ways. Our
account used a graded notion of knowledgeability which allowed it to reason about actors withdifferent epistemic relationships to the world(whereas Goodman et al. (2009) and Baker et al.
(2009, 2011) assumed agents were completely
knowledgeable). Our account also relaxed afoundational assumption of previous accounts—that agent’s act rationally—allowing
learners to reason about the extent to which anactor was reliable. Finally, our model formalized and explored the effect of an actor’s confidence. We used this elaborated ToM to explain both how people learn about the worldfrom actors, like Goodman et al. (2009) and
Shafto et al. (2012), and how they infer latentstates of the actor, like Baker et al. (2009, 2011)
and Pantelis et al. (2014). The resulting model
provides a unified account of how actor attributes guide social learning, bridging the epistemic trust and judgment adviser systems literatures (as Eaves and Shafto’s [2012] model
provided a unified account of pedagogical reasoning and epistemic trust).
Limitations and Future WorkOur experimental scenario provided a simple
domain for measuring social information integration where the situation makes the nature of theactor’s attributes transparent, for example, knowledge is simply the number of races observed. The
scenario enabled a number of simplifying assumptions in our integrative learner model and allowed
us to cleanly test the effect of actor attributes onsocial learning. Exploring more complicated situations where these assumptions no longer hold
may expose additional complexity of people’s naturalistic social reasoning. Each of these simplifications therefore represents an interesting avenue
for further research; we next describe several ofthese possible extensions.
Amount of direct evidence. Across thefour experiments the participants’ knowledgeability from direct evidence, that is, the number ofraces they observed, was held constant. The model
predicts that the actor’s influence should diminish
as the learner has more direct observations, consistent with Harvey and Fischer’s (1997) findings.
The model also predicts an interaction between

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56

HAWTHORNE-MADELL AND GOODMAN

the effect of the actor and learner’s knowledgeability: An actor’s knowledgeability should havegreater impact for less knowledgeable learners.
Extending beyond Harvey and Fischer, the model
predicts that learners with more knowledge would
be able to draw stronger inferences about latentactor attributes, like reliability. For example, inExperiment 1, when the learner only had a few
observations and the actor bet on the horse thatwas dominated in them, it did not license a strong
inference. From the learner’s limited observations,
it was not unlikely that the actor saw a conflicting
sample and was therefore acting reliably. However, a learner that saw many observations would
have a better estimate of what the actor was likely
to see, and his inconsistent bet would license amuch stronger inference that he is unreliable.
These predictions provide a further empirical testof the integrative learner model.
Reliability. The general conception of reliability is likely richer than was exposed in our
experimental set-up. For instance, gradation inreliability was limited by the four-point scale weused to assess it. We then further binned responses
into two categories—people who thought the actor was reliable or unreliable—to simplify the
analysis. However, people’s use of the scale andintegration behavior indicated that their representation of reliability was more graded than the two
simple categories. Additional explorations will beneeded to determine how fine-grained reliability
is. More importantly, our model treated reliability
as simply the noise of a person’s decision rule. It
is likely that people also consider other ways inwhich actors could be (un)reliable, such as theway they form beliefs. We were unable to distinguishing between these two sources of (un)reliability from the limited evidence that people received in our experiments, but other paradigms
could distinguish between these, and other, formsof reliability. There is likely a rich intuitive theoryof reliability itself to be explored. For example,
the actor could have a terrible memory and therefore could be reliable for recently acquired knowledge but uninformative at greater temporal remove. These more nuanced ways of being
unreliable could be formalized in more elaboratemodels of mind; incorporating these more elaborate models of mind into an integrative social
learner would highlight the unique behavioral signatures of each trait.
Another question suggested by our results iswhen learners update their attribution of traits toan actor. Across our four experiments, learners
only updated their estimate of the actor’s reliability in Experiment 4. One explanation for this isthat the evidence provided about the actor’s reliability was relatively strong (from our computational model’s predictions) and salient (from theauthors’ intuition) relative to the evidence provided in Experiment 1 and 2. An alternate explanation identifies the more complex inference required to infer the actor’s reliability from theevidence provided in Experiment 1 and 2 (see theAlternative Model Comparison section for further
discussion of these theories). Both of these may
amount to resource-efficient processing strategies
for approximating a complex fully-Bayesian belief update. The current evidence cannot determine which explanation best accounts for the differential reasoning about reliability across our
experiments. Future investigations are therefore
necessary to determine the cause of this pattern ofreasoning and interrelated questions like whether
the effect of the strength and salience of evidenceis graded or whether there is a threshold at which
participants will begin to reason about (rather thanjust using) reliability and other traits.
Prior knowledge. The learner’s understanding of the actor’s knowledge and prior beliefs wasconstrained in our experiments. This allowed us toadopt a simple model of knowledge that will notnecessarily hold in general. For example, we assumed learners thought the actor shared their priordistribution over horse skill.34 We also assumed
common knowledge of the causal structure of thedomain, which meant that the actor’s observations
were the sole source of his knowledge. Although
this correspondence is desirable for parametricallymanipulating social evidence, people often learn
from actors who have a different understanding of
the domain. In fact, superior understanding of thedomain is often precisely why others’ opinions aresought, not their privileged evidence— eventhough Warren Buffet looks at the same stock
fundamentals as everyone else, his bets are still
34

Although expedient, it is unclear when such assumptions of correspondence between a learner and their modelof an actor is justified. Are actors assumed to be “like me”
until proven otherwise, or is there a separate default used for
actors (Meltzoff, 2007)? This relationship between a learner’s conception of herself and her understanding of others isa central question in social psychology (Malle, 2004; Ross,
1977). By explicitly representing the learner’s beliefs andher beliefs about others, our modeling framework provides
a new tool with which to approach this classic question.

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REASONING ABOUT SOCIAL SOURCES

highly influential. This situation corresponds tothose often explored in the JAS literature, which
has found that advice utilization is variable anddependent on a number of heuristic actor attributes like age and experience (Feng & MacGeorge, 2006). Developing a formal theory of
how learners understand these ambiguously
knowledgeable actors, and attribute knowledge on
the basis of other traits, represents an importantelaboration of the simple ToM we presented.
Communication. Finally, as we saw in Experiment 2, the nature of the actor’s communication has a large impact on the influence of
their message. In our experiments, we restricted
the “communication” to an actor’s goal-directedaction, which is equivalent to watching the actor
interact directly with the world (perhaps hesitantly or with confidence). Although our actor
interacted directly with the world, actors areoften learner-oriented, interacting with thelearner with communicative, pedagogical, or
nefarious intent. Learning from a teacher requires a model of how they make pedagogicaldecisions because the teacher’s goal is not about
the state of the world. It is about the beliefs of
the learner. This requires the learner to think
about the teacher who is, in turn, thinking about
the learner. This more elaborate ToM has beenformalized and tested by Goodman and Shafto(Eaves & Shafto, 2012; Shafto & Goodman,
2008; Shafto et al., 2012, 2014) in the “puresocial” case in which learners do not have alternative sources of information. Incorporating
their more elaborate ToM into the integrative
learner model would provide a good test of theextensibility of our framework. The resulting
model could provide insights into how students
learn from a mix of instruction and exploration.
Does a student’s self-confidence get in the wayof her learning? What can teachers do to manage the student’s perception of their attributes,
like reliability and knowledgeability?
Conclusion
Other people represent an important source of
information, but learning from them requires anunderstanding of how they tick—what they know,
how they choose actions, and what they want.
Formal models of ToM are therefore crucial tounderstanding social learning. We presented ahighly extensible computational framework formodeling a learner’s understanding of social

57sources and showed how it formalizes the integration of social and direct evidence by describing thethree major attributes of social sources—theirknowledgeability, confidence, and reliability.
We found that this model explained the quantitative data from our horserace paradigm extremely well. Such rigorous models of social
learning provide a framework for both understanding previous empirical investigations
and generating future experiments. Developing our understanding of the theories andcompetencies underlying learning from others
provides insights into the big questions of
persuasion, pedagogy, and culture. How can I
convince someone to heed my advice? How
does a student’s understanding of the teacher
and situation impact their learning? How isknowledge transmitted from one generation to
the next?

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Received October 20, 2016
Revision received June 2, 2017
Accepted June 8, 2017 䡲