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Reasoning About Social Sources to Learn From Actionsand Outcomes
Daniel Hawthorne-Madell and Noah D. GoodmanStanford University
Learning what others know, especially experts, is a crucial shortcut to understandingthe 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 inferwhat others believe and integrate these beliefs with their own. In this paper, we presenta computational account of the inference and integration process that underpinslearning from a combination of social and direct evidence. This account formalizes thelearner’s intuitive understanding of psychology—or theory of mind (ToM)—includingattributes such as confidence, reliability, and knowledgeability. It then shows how ToMis the lens used to interpret another person’s choices, weighing them against thelearner’s own direct evidence. To test this account, we develop an experimentalparadigm 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 betweenevidence 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 racehorse? When confronted with decisions likethese, people are information omnivores, flexi-bly learning from multiple information sources.Stock investors, for example, look for directinformation about a company’s strength, scour-ing 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 presumablyknowledgeable company insider? The choicesthese actors make represent a rich source ofsocial information, even when they are not in-tentionally trying to advise or influence others.1
At its core, social learning involves interpretingintentional actions in the context of the learner’sunderstanding of the situation, her prior beliefs,
and direct evidence. A theory of social learningmust both explain how a learner interprets anactor’s actions and how the results are inte-grated with nonsocial evidence and prior be-liefs. This suggests two core questions of sociallearning. First, what is the relative impact ofsocial sources? How do people integrate socialinformation with other evidence like direct ob-servations? Do they heavily discount advice ashas 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 theyrepresented, and how do they interact to deter-mine an action’s informativity?
To understand the actions of an actor, peoplemust have a framework for relating people’s
1 Although the computational framework we propose isgeneral to multiple kinds of social cues, we focus ourdiscussion on reasoning about intentional actions and referto the social source with the generic actor. For clarity, wesubsequently refer to the actor with a male pronoun and thelearner with a female pronoun.
This article was published Online First September 14,2017.
Daniel Hawthorne-Madell and Noah D. Goodman, De-partment of Psychology, Stanford University.
Correspondence concerning this article should be addressedto Daniel Hawthorne-Madell. E-mail: [email protected]
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Decision © 2017 American Psychological Association2019, Vol. 6, No. 1, 17–60 2325-9965/19/$12.00 http://dx.doi.org/10.1037/dec0000088
17
knowledge, traits, thoughts and behavior—atheory of mind (ToM; Wellman, 1990; Perner,1991; Premack & Woodruff, 1978). A learnercan use her ToM to reason about the informa-tiveness of an actor’s intentional actions, infer-ring the actor’s beliefs and the traits that willaffect learning. Previous research shows thatlearners prefer to learn from people who theythink acted intentionally (e.g., Carpenter et al.,1998; Olineck & Poulin-Dubois, 2005), confi-dently (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 latentproperties as actor attributes. The ToM re-quired to account for social learning thus relatesbeliefs, actions, and actor attributes.
In this article, we address the two core ques-tions of social learning with a computationalmodel that formalizes how learners interpretsocial information and integrate it with directevidence, which we tested with a series of em-pirical studies. First, we review the factors thatlearners use to identify the informativeness ofsocial information. We then discuss the existingempirical work in which learners have beenshown to integrate social information with othersources. Building off these experimental para-digms, we introduce one capable of both quan-titatively manipulating an actor’s attributes andquantifying the information content of a learn-er’s direct and social evidence. We then intro-duce our model, which provides a unified ac-count of how people interpret and learn fromsocial information and integrate it with othersources of knowledge; crucially, this model re-lies on inferences about the actor—his reliabil-ity, knowledge, and other attributes. We per-form four experiments and show their close fitto the predictions of our model. In the firstexperiment, we testeded how learners’ integra-tion of an actor’s choices with their own evi-dence is affected by the actor’s knowledgeabil-ity and perceived reliability. The secondexperiment tested the impact of an actor whoalso reports his degree of confidence in hisaction, providing metacognitive informationabout his choice. The third experiment looks forsignatures of spontaneously attributing actors’confidence when it is not explicitly provided. Inthe final experiment, we manipulated the appar-ent reliability of the actor directly. Following
the experiments, we discuss alternative modelsand then model two previous experiments toshow how our model captures how people learnfrom and about others. We conclude with adiscussion of the model’s role in understandingsocial cognition, learning, and the future empir-ical 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 clearestdomain to study how people infer an actor’sattributes from social cues and how these attri-butes are used to determine from whom to learnfand how to weigh their advice.2 Exploration ofthese complementary questions has been under-taken in two separate intellectual traditions. Theepistemic trust literature has focused on theearly developmental emergence of sensitivity toactor attributes when selecting an actor to learnfrom (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 inte-grated with a person’s own opinions (for a re-view see Bonaccio & Dalal, 2006).3 This sec-tion provides an overview of the key socialfactors these independent lines of inquiry haveconverged 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 extentthat it is independent from, or more extensivethan, what the learner knows. His value as aninformation source could be the result of supe-rior conceptual understanding of the domain, asis the case with Warren Buffet, or he couldsimply have more information (or unique infor-
2 Communicative actions, for example, require recursivereasoning to interpret Shafto et al., 2012; Goodman &Stuhlmüller, 2013.
3 The classic adviser in the JAS literature simply stateswhat they would do akin to the actors in the developmentalliterature.
18 HAWTHORNE-MADELL AND GOODMAN
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mation).4 Learners therefore need to know theactor’s knowledgeability to appropriately learnfrom him. Previous research has shown thatlearners are sensitive to a range of cues of anactor’s knowledgeability, such as what evidencehe has, his past actions, or his claimed knowl-edge.
An actor’s perceptual access is a very clearcue explored extensively in the developmentalliterature. A particularly clear case of an inde-pendently knowledgeable actor is when he hasseen something that the learner has not; that is,when the actor has privileged perceptual access.Sensitivity to this kind of knowledge requiresunderstanding that other people have their ownperceptual access, which is evident in infants asearly as 14 months (Brooks & Meltzoff, 2002,2005). Understanding that this independent per-ceptual access can allow other people to formtheir own beliefs emerges later with the devel-opment of a full-fledged ToM (Flavell, 1999;Wellman et al., 2001). Children with a ToMunderstand that “looking leads to knowing” andcorrectly identify actors who have had percep-tual access to the item in question (Pillow,1989; Pillow & Weed, 1997; Pratt & Bryant,1990). Children are also sensitive to the rele-vance 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; Whit-combe & Robinson, 2000).5
Claimed knowledge or expertise is a moreindirect source of evidence of an actor’s knowl-edgeability. Children prefer to heed actors whoclaim to be knowledgeable (Birch et al., 2010;Jaswal & Malone, 2007; Koenig & Harris,2005; Moore et al., 1989; Moore & Davidge,2009; Sabbagh & Baldwin, 2001), and adultshave been shown to be more influenced byactors who claim to be knowledgeable experts(Jungermann & Fischer, 2005). Another indi-rect source of evidence of an actor’s knowl-edgeability is their past actions. Young childrenare naturally trusting of older actors, but oncean actor makes a mistake they will preferen-tially learn from others (Birch et al., 2008; Har-ris & Corriveau, 2011; Jaswal & Neely, 2006;Koenig et al., 2004; Koenig & Harris, 2005).Similarly, adult learners use an actor’s past per-formances to infer their knowledgeability and
are subsequently less influenced by poorly per-forming actors (Yaniv, 2004).
In the absence of information about percep-tual access, past actions, or claimed knowledge,learners fall back on more indirect cues. Infantsprefer actors that exhibit nonverbal confidencecues (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 con-fident actors (Phillips, 1999; Sniezek & Buck-ley, 1995; Sniezek & Van Swol, 2001; Swol etal., 2005; Yaniv, 1997). Children use additionalheuristics like familiarity (Corriveau & Harris,2009), cultural similarity (Hu et al., 2013; Kin-zler 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 wis-dom (Feng & MacGeorge, 2006).
Reliability
Although reliability is often used inter-changeably with knowledgeability in the epis-temic trust literature, we argue for a distinctmeaning—a reliable actor chooses reasonableactions to achieve their goals, informed by theirbeliefs. Reliable actors’ actions are diagnosticof their knowledge, unlike actors that are noisydecision makers or purposefully deceitful. Asdiscussed previously, preschoolers track actors’past mistakes and prefer accurate, “reliable”actors. Although these actors’ previous mis-takes could be interpreted as stemming fromtheir lack of knowledge, the true cause is un-derspecified in many experiments (e.g., Clé-ment et al., 2004; Jaswal & Neely, 2006; Koe-nig et al., 2004).
There is a fundamental distinction betweenthe possession of knowledge and the reliableuse of it, either of which could be the cause ofan informant’s past mistakes. Although diffi-cult, some experimental paradigms have begun
4 If there is independent noise in the belief formationprocess, 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 thatadults weigh the evidence of actor’s with independent in-formation sources.
19REASONING ABOUT SOCIAL SOURCES
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to dissociate these two potential causes by cre-ating situations where the learner knows theactor is knowledgeable and yet does not actreliably on their understanding (Chow et al.,2008; Poulin-Dubois et al., 2011). They findthat learners are sensitive to an actor’s reliabil-ity, preferentially learning from actors whohave reliably used their knowledge in the past.
Intentionality
A final necessary condition for learning fromthe actions of a knowledgeable and reliableactor is that the action be intentional. Unin-tended actions are effectively independent ofthe actor’s knowledge. Eighteen-month-olds al-ready display sensitivity to intentionality, selec-tively learning from purposeful actions overaccidental actions (Carpenter et al., 1998; Oli-neck & Poulin-Dubois, 2005). Infants preferen-tially reproduced the actions of adults playingwith 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 ac-tion). 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 evidencefrom an intentional action but not from identicalphysical evidence or accidental action (Good-man et al., 2009).
Toward a Cohesive Account of ActorAttributes
We have seen that learners spontaneouslyreason about an actor’s attributes and preferen-tially learn from the actions of knowledgeableand reliable actors who act intentionally andwith confidence. However, we have neither acoherent account of how learners combine theseattributes to interpret social evidence nor of howthis social evidence is integrated with their otherevidence, like direct observations. In the nextsection, we propose such an account. Our ac-count formally defines knowledgeability, reli-ability, and confidence in terms of a learner’sToM. It makes specific predictions about therelative importance of these factors and theirinteractions. We embed this ToM into an inte-grative framework to describe how learnerscombine social and direct evidence. The result-
ing model predicts how a Bayesian learnerwould (a) use available evidence to characterizean actor, that is, infer how knowledgeable, con-fident, and reliable he is and (b) integrate socialevidence from the actor given the learner’scharacterization of him. That is to say that themodel provides a standard to evaluate whetherlearners learn from social evidence as much asthey “should,” given their model of the actorand their characterization of his attributes.
To test our model’s predictions requires anexperimental paradigm in which the influenceof social evidence can be quantitatively mea-sured relative to a well understood referencepoint (be it other social sources, direct evidence,or prior knowledge). Additionally, the learner’scharacterization of the actor’s attributes must bemeasured or manipulated. However, previousexperimental paradigms have not been clear onone or both points, limiting their ability to rig-orously test our models predictions. We brieflyreview some key experiments in the advice in-tegration 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 litera-ture explored advice utilization by comparing alearner’s initial estimate to her estimate afterreceiving advice. The learner’s initial estimateis effectively a second source of social evidence(a decision made by a person) and is used tocalculate how influential the adviser was. Arobust finding using this paradigm is that learn-ers 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 insuffi-cient evidence to decide the extent to whichthey should rely on the advice. For example,there is considerable uncertainty about both theactor’s knowledgeability and their own. Ourframework can formally describe the learner’suncertainty about the adviser and herselfand represents how a Bayesian learner wouldintegrate the information provided in these ex-periments (this is further explored in the appliedsection Learning from and about advisers inJAS experiments). It suggests that the seemingunderutilization could be due to the substantial
20 HAWTHORNE-MADELL AND GOODMAN
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uncertainty the learner has about the adviser,which is compounded by the uncertainty she hasabout herself. Testing this prediction requiresmeasuring the learner’s characterization of boththe adviser’s and her own attributes. Addition-ally, using a less complicated reference pointwould allow for a clear test of whether peopletruly exhibit an egocentric bias.
Direct evidence, like observations or baserate information, can provide a clearer refer-ence point for social evidence. For example,Birnbaum and Mellers (1983) used base rateinformation to measure the impact of socialevidence while manipulating the adviser’s ac-curacy and bias. They found that learnerswere sensitive to both. Learners listened toaccurate advisers more and discounted advicefrom biased sources. Birnbaum and Meller’sparadigm enabled a clean test of how partic-ipants integrated social and direct evidence byparametrically manipulating the adviser’s ac-curacy and the learner’s direct evidence. Dothey weigh advice as much as an ideal Bayes-ian integrator would? Although the magnitudeof advice integration was well-captured bythe Bayesian model, a clear violation ofBayesian reasoning led Birnbaum and Mellersto reject his formulation of an ideal Bayesianintegrator. When the adviser endorsed an op-tion that was highly likely given the base-rate,his advice paradoxically reduced the learner’sestimate of its likelihood.
An alternative explanation to this surprisingresult lies in learners’ characterization of theadviser in Birnbaum and Mellers’ experiment.Although they carefully controlled the adviser’sattributes in his experiment, they did not ma-nipulate or measure the adviser’s confidence. Inthe subsequent experiments, we find evidencethat learners spontaneously attribute confidenceto advisers, and we describe how spontaneouslyattributing low confidence would result in par-adoxical integration of advice. Again, our“learnercentric” perspective that focuses on thelearner’s characterization highlights a new in-terpretation of advice integration findings andpoints to the importance of pinning down thelearner’s characterization of advisers.
Taking stock of the strengths and limita-tions of these previous investigations, we candescribe the desiderata for an empirical par-adigm to test our framework. Social evidenceshould have a clear reference point to gauge
its relative influence— experimentally con-trolled direct observations. Having both thelearner and the adviser’s knowledge clearlyquantified in terms of direct observations en-sures that they are both commensurable andamenable to parametric manipulation. Simi-larly, other actor attributes should be definedin a graded way and either explicitly manip-ulated or the learner’s unmanipulated esti-mate should be measured. We now introducea betting scenario built on these insights, ca-pable of rigorously testing our framework andformalization of ToM.
Social Learning at the Racetrack
Testing how learners combine social and di-rect information requires an experimental de-sign that quantifies both knowledge sources inthe same way. To do this, we use a horse racingscenario in which both the learner and the actorindependently observe some number of races.Hence, the knowledgeability of each is simplythe number of races seen, which can be individ-ually manipulated. Instead of receiving a directreport of the race results from the actor, thelearner only observes the bet placed by theactor. This requires the learner to infer the ac-tor’s beliefs from his actions, and combinethose beliefs appropriately with the learner’sown evidence.
We consider a restricted class of races inwhich the actor, who we refer to as Zach, hasfollowed two racehorses, x and y. Zach ob-served some number, nz, of their previous head-to-head races. The learner has seen some num-ber, 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. Thelearner just sees Zach say which horse he beton, bz, and how many races he saw, nz.
Suppose that the learner sees horse x win twoout of five races and is told that Zach saw 10races and bet on horse x in the next race—giventhe 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’
21REASONING ABOUT SOCIAL SOURCES
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ToM is used to interpret social evidence andintegrate it with other sources of evidence. Themodel makes quantitative predictions for ma-nipulations of bz, kl, and nz that suggest system-atic tests of the model.
A Computational Account of LearningFrom 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 inte-grative learner who infers a horse’s chance ofwinning future races by integrating observa-tions of its previous performance with theway Zach bets given his observations. Wedescribe the learner’s mental model of bothsources in terms of probabilistic generativemodels (Griffiths et al., 2008; Tenenbaum et
al., 2011). These models describe the learn-er’s theory about the causal processes in theworld giving rise to her observations. Fordirect observations, the generative model rep-resents the learner’s theory of the domain, forexample, what latent properties dictate ahorse’s performance (Figure 1a.). The learn-er’s model of an actor’s actions represents hertheory of mind, that is, the learner’s predic-tion of how the actor uses his observations toform beliefs and decide which actions willfurther his goals (Figure 1d.). Combiningthese two mental models results in the unifiedrepresentation used by the integrative learnerto learn from social and direct evidence (Fig-ure 1c). Using Bayesian inference to learnabout the common latent properties of thesetwo information sources, the integrativelearner model represents a coherent referencepoint 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 relationbetween variables. Solid nodes are observed variables; empty nodes represent latent variableswhose values must be inferred. Panel a: The “direct learner” who learns about the environ-ment from observations. Panel b: The “pure-social learner” who learns about the environmentfrom 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 formbeliefs and make decisions given observations, beliefs, and degree of rationality.
22 HAWTHORNE-MADELL AND GOODMAN
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learner described subsequently is an instanceof a general framework for formalizing howsocial information is interpreted and inte-grated with other evidence. The frameworkcan be used to describe learning in compli-cated situations requiring elaborate theoriesof domain and mind, for example, why yourcar whines only when turning slightly left andthe implication of your mechanic’s knowingsmile. The simple racetrack scenario elidesvery little of the general structure and allowsfor a concise treatment of the integrativelearner with clear formalizations of the ac-tor’s knowledgeability and reliability. Thefull integrative learner model can be builtincrementally. We first describe the pure-direct learner, who updates her beliefs aboutthe world from direct observations. We nextintroduce the learner’s ToM, which combinesa belief update model with a decision rule tomodel how Zach forms beliefs from evidenceand then acts on those beliefs. We then showhow 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 observ-ing a number of races, the learner can gaininformation about the probability horse x winsthe next race. The learning process is guided bythe learner’s theory of the domain—what causesa horse to win. Learners likely entertain a num-ber of complicated theories about the factorscontributing to a horse’s performance (jockey,pedigree, betting odds, etc.). However, the re-stricted information the learner and Zach re-ceive admits of a simple domain theory inwhich the probability that a horse wins a race wnis 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 re-stricted to the results of repeated races betweenthe same two horses with no ordering informa-tion provided, sl is sufficient to describe thelearner’s beliefs about the domain (with 1 � sl
representing horse y’s relative skill).6 Given thisframing, races between the two horses are in-dependent and their outcomes can be predictedby flipping a coin weighted by their skill foreach of the nl races:
p(kl�sl, nl) � binomial(sl, nl). (1)
Assuming learners use this simple domain the-ory and Bayesian inference to update their be-liefs, their posterior beliefs after seeing somenumber nl of observations where horse x wins kltimes is
p(sl�kl, nl) � p(kl�sl, nl)p(sl). (2)
where p(sl) is the learner’s prior belief in ahorse’s chance of winning. Equation 2 repre-sents the direct learner (Figure 1a) who reliessolely on direct observations to update her be-liefs.
ToM
To interpret Zach’s actions, a learner musthave a theory of the relationship between Zach’sobservations, beliefs, goals, and choices—a ToM(see Figure 1d). The first component of the learn-er’s ToM is how Zach learns from evidence. Weassume 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 ToMcaptures how Zach makes decisions given hisbeliefs. We understand Zach as an (approxi-mately) rational decision maker, who (softly)maximizes his utility (in the tradition of Good-man et al. (2009) and Baker et al. (2009)). In thehorserace scenario there are no “odds” or dif-ferential payoffs for the two horses, makingZach’s utility function simply “1” if he bets onthe winning horse and “0” otherwise. Whichhorse Zach thinks will win is specified by hisbeliefs p(wn�1 |sz). Given this belief distribu-
6 Indeed, under the assumption that the order of races isunimportant (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 distributionsFinetti (1974) implies that the complex theory of thehorses’ skill can be simplified to a single probability,without losing any information.
23REASONING ABOUT SOCIAL SOURCES
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tion, an optimal decision theoretic Zach wouldchoose the action, bz, that maximizes his ex-pected utility �p�wn�1 � sz�
U�bz; wn�1�, always bet-ting on the horse he thinks is most likely to win.If the learner thinks that Zach may not be en-tirely optimal, she may assume that Zach some-times chooses other actions but does so in a waythat higher expected-utility actions will be morelikely; we model this as soft maximization, sothat the probability of a bet is
p(bz�sz, �) � e�·�p(wn�1 � sz)U(bz;wn�1). (3)
Or, given the simple structure of the racetrackscenario, (p(wn�1 � x |sz) � sz), it is simply
p(bz � x�sz, �) � e�·sz, (4)
where alpha is a “rationality” parameter that cor-responds to the amount of noise in Zach’s decisionmaking. As alpha goes to infinity, the decisionrule approaches a strict max-rule where the actionwith the highest expected utility is always taken.High values correspond to “reasonable” agentswhose 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 pos-iting that learners either think Zach is “reason-able” with a high alpha or “unreasonable” with alow alpha.
Combining these two components— beliefformation (Equation 2) and action selection(Equation 3)—we have the learner’s model ofhow Zach bets given his observations:8
p(bz�kz, nz, �) � �szp(bz�sz, �)p(sz�nz, kz) dsz. (5)
Pure-Social Learner
The pure-social learner does not have anydirect observations and must therefore relysolely on social information. In our racetrackscenario, the pure-social learner is trying toinfer the horse skill most likely to make Zachbet as he did. She uses her ToM (Figure 1d) toreason about Zach’s bet. By Bayes’ rule,
p(sl�bz, nz, �) � p(bz�sl, nz, �)p(sl). (6)
We make the further simplification that thelearner reasons using a fixed assumption aboutZach’s alpha (rather than jointly updating herbeliefs about alpha and other unknown quanti-ties), unless there is direct and salient informa-tion about Zach’s rationality. In our experimen-tal setting this means that alpha is only updatedwhen Zach’s observations and actions are bothobserved. Different learners may have differentbeliefs about Zach’s a priori reasonability, re-sulting in a population (i.e., mixture) of learnerswho think Zach is reasonable with p(�) andunreasonable with 1 � p(�). Each learner hasincomplete knowledge of other latent factorsgoverning Zach’s bet. She knows how manyraces Zach saw (nz), but she doesn’t know thecontent of his observations (kz). The learnermust therefore consider both what Zach is likelyto have seen and his rationality in order toevaluate how likely he is to have made this betbz. For a learner attributing Zach rationality �,the probability Zach bets bz is
p(bz�sl, nz, �)
� �kz
p(bz�sl, nz, kz, �)p(kz�sl, nz, �) (7)
��kz
p(bz�nz, kz, �)p(kz�sl, nz), (8)
where the second line follows by simplifyingusing the independencies evident in Figure 1b(namely, bz does not depend on sl once kz isknown). We see by combining Equations 6 and8 that the pure-social learner can be defined interms of the learner’s theory of mind and do-main (Equations 5 and 2) and prior expectationsabout Zach’s rationality and the horse’s skill.
The pure-social learner model formalizeshow a learner can use Zach’s bet to infer infor-mation about the world. It includes a nestedmodel of how Zach updates his beliefs andmakes decisions (Equation 5), which is inverted
7 We use the terms reliable, reasonable, and rationalinterchangeably in our discussion of the model and ourexperiments. However, in the General Discussion we intro-duce distinctions between reliable and reasonable or ratio-nal.
8 We assume joint update of beliefs and action plans.
24 HAWTHORNE-MADELL AND GOODMAN
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to infer what Zach saw from how he bet. Theouter level of the model must again use Bayesrule to infer horse x’s skill from what Zach waslikely to have seen. Equation 6 therefore re-quires two normalizations to compute—one forthe application of Bayes rule in the inner modelof Zach and one for the application of Bayesrule 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’sbet also has her own direct observations, howshould she combine these two informationsources? It depends on her understanding of thecausal relation between the sources. As seen inFigure 1c, the integrative learner models bothZach’s and her own observations as being gen-erated from horse x’s skill (sl). This common(unobserved) cause allows information to flowbetween the learner’s model of Zach and herown observations. The learner’s belief abouthorse x’s skill constrains what she infers Zachsaw. Zach’s bet, in turn, provides evidenceabout sl. This is to say that an integrative sociallearner forms her belief (sl) by jointly consid-ering her observations (kl) and what she infersabout Zach’s observations (kz). This requireselaborating 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 directand social evidence, given sl,
p(sl�bz, nz, kl, nl, �) � p(bz, kl�sl, nl, nz�)p(sl)
(9)
� p(bz�sl, nz, �)p(kl�sl, nl)p(sl). (10)
Equation 10 combines elements from both thedirect and pure-social learners (Equations 2 and8). This integrative learner model formalizeshow a learner should combine her direct evi-dence with Zach’s bet to estimate horse x’s trueskill. Like the pure-social learner, the integra-tive learner model includes a nested model ofhow Zach makes decisions (Equation 5). How-ever, the top level of the integrative model is
more complex, using Bayes rule to infer sl fromboth bz and kl.
Predictions of the integrative learner.The integrative learner model makes quantita-tive predictions about the relationship betweenZach’s attributes and the learner’s estimate ofhorse x’s skill. These predictions are made clearby the concise formal definitions of Zach’s at-tributes within the horserace scenario. The num-ber of races observed represents both Zach’sand the learner’s knowledgeability (i.e., horsex’s record in the races) and the content of theseobservations. The learner’s estimate of Zach’srationality is simply the amount of decisionnoise (alpha) she thinks he has. Finally, thelearner understands Zach’s confidence (dis-cussed subsequently) simply as his estimate ofthe likelihood that his chosen action will resultin the desired outcome. To explore the predictedimpact of each of these actor attributes, weconducted a series of simulations leading to fourcore testable predictions. All simulations wereconducted in the probabilistic programminglanguage, Church (Goodman et al., 2008).9
The first three predictions concern the effectof Zach’s knowledgeability and reliability onthe learner’s estimate of horse x’s skill. Wesimulated the learner’s estimate of horse x’sskill when Zach’s knowledgeability and the di-rectly observed horse records were manipulated(see Figure 2).10 We simulated this for learnerswho thought that Zach was reliable and forthose who thought he was unreliable. Themodel predicts that all three variables—Zach’sknowledgeability, the learner’s estimate ofZach’s reliability, and the content of the learn-er’s observations—should have an impact onher estimate of horse x’s skill. First, the simu-lation makes the straightforward prediction that,all else being equal, as horse x becomes moredominant 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 knowledge-
9 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 reliabil-ity and horse skill parameters. When Zach was reliable, hemaximized his utility (� � �); when he was unreliable, hebet randomly (� � 0). The learner had a uniform prior overhorse strengths and she imagined Zach had the same uni-form prior.
25REASONING ABOUT SOCIAL SOURCES
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ability and perceived reliability. When Zachhasn’t seen any races, and therefore isn’t knowl-edgeable about horse x’s skill, the learnershould ignore Zach and rely on her own evi-dence. The following is the first major predic-tion of the integrative learner model:
Prediction 1: When Zach hasn’t seen anyraces, 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 ef-fect; thus we predict the following:
Prediction 2: Learners who think Zachdoes not bet reasonably, that is, has a lowrationality parameter, should be less influ-enced by his bet than those who think hebets reasonably. In the extreme case whenZach is completely unreliable, his betshould be ignored.
As Figure 2 shows, there is an interactionbetween Zach’s reliability and knowledgeabil-ity on his influence. Zach must be both reliableand knowledgeable.
Prediction 3: There should be an interac-tion between knowledgeability and reli-ability such that knowledgeability is onlyrelevant to the extent that Zach’s bets arediagnostic of his knowledge, that is, he hasa high rationality parameter. Conversely,Zach’s reliability is only relevant whenZach has some relevant knowledge.
These model predictions are consistent withthe finding that children and adult advice-takerslearn more from reliable and knowledgeableactors (Harris, 2007; Bonaccio & Dalal, 2006).Thus, we testeded the model’s quantitative pre-dictions for these effects. Beyond these factorsidentified in the epistemic trust and JAS litera-tures, the model also predicts that a less ex-plored variable should also modulate howknowledgeable learners incorporate informationfrom other’s actions: the learner’s estimate ofthe actor’s observations.
Learners infer Zach beliefs about horse x’sskill from his bet, and understand this belief asa reflection of his observations through Equa-tion 2. Zach’s observations, in turn, provide thelearner with information about horse x’s skill.Indeed, Zach’s observations screen off his other
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’sknowledgeability. The learner saw horse x win zero, one, or two of five races, and Zach sawzero, 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.
26 HAWTHORNE-MADELL AND GOODMAN
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properties from the learner’s beliefs. Zach’s ob-servations therefore form the link between hisactions and horse x’s skill. This leads to thefollowing prediction:
Prediction 4: The learner’s estimate ofZach’s observations should fully mediatethe effect of his knowledgeability.
We tested all four predictions within thehorse betting experimental paradigm in Exper-iment 1. By manipulating the number of obser-vations Zach sees we can test Prediction 1. Byasking learners whether they think Zach wasbetting reliably we can test Prediction 2 and theinteraction, Prediction 3. By asking learners toestimate what Zach saw while manipulating hisbet and the proportion of horse x wins in theirsample, we can test Prediction 4. The modelspredict that Zach’s bet will have the largestimpact in inconsistent trials, where the evidencelearners receive is inconsistent with Zach’s bet.In particular, these situations maximize the dif-ference between the direct, pure-social, and in-tegrative models (assuming for the moment thatZach is reliable). Thus, in the following exper-iments 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 ToMto allow her to interpret an actor’s confidence, acue 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 in-tegrative learner can use her ToM to inferZach’s reliability directly from Zach’s bet inobserved races. More realistically, the learner’smental model allows her to simultaneouslylearn from and about an actor (as is evidentearly in social reasoner’s development; seeKushnir, 2013), across partially observed situa-tions.
Learning from confidence cues. The de-gree of confidence an actor has in his action isa highly influential cue for social learners. Wemodel the reliable actor’s confidence, cz, as hismetacognitive judgment about the likelihoodthat his action will result in the desired outcome(where 0 � cz � 1). In the horserace scenario,Zach’s confidence in his bet is his estimate of
the chances the horse he bets on will win, whichis equivalent to the probability he bets on thehorse:
p(cz � ��bz, sl, nz, �) � ��p(bz � sl,nz,�). (11)
Confidence is often expressed with naturallanguage as a gradable adjective, for example,“I am [not/somewhat/very] confident horse xwill win.” Similarly, we bin the continuous con-fidence variable cz into a discrete confidencevariable with low, medium, and high levels. Areliable person sends confidence signal � whenhis confidence, cz, falls within the intervalcmin(�) � cz � cmax(�). We understand Zach’sconfidence signal as a function of his belief czand his reliability alpha vis-a-vis the decisionrule described by Equation 4. The probabilitythat Zach with reliability � would send confi-dence signal � in their bet bz is then
p(�bz, sl, nz, �) � e�·�cmin()cmax()p(,cz �bz,sl,nz) dcz (12)
� e�·�cmin()cmax()p( � cz)p(cz �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 sociallearner with this understanding of how Zachdetermines his confidence level allows us tomodel learning from variably confident actors:
p(sl�u, bz, nz, kl, nl, �)
� p(, bz, kl�sl, nl, nz, �)p(sl) (14)
� p(�bz, sl, nz, �)p(bz�sl, nz, �)p(kl�sl, nl)p(sl).
(15)
Given this formulation, learners know that areliable and highly confident Zach is morelikely to have seen a sample strongly dominatedby his chosen horse. Learners are thereforemore influenced by Zach’s bet when he ishighly confident in it.
In Figure 3 we examined the model’s predic-tions for how Zach’s confidence in his bet af-fects his influence on learners. We consideredthe situation where Zach bet on horse x and has
27REASONING ABOUT SOCIAL SOURCES
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seen 10 races, varying the amount of directevidence the learner has and the confidenceZach expresses in his bet. Consistent with pre-vious empirical results (e.g., Sniezek & VanSwol, 2001) the model predicts the following:
Prediction 5: When Zach is highly confi-dent his bet should be more influential thanwhen he has low confidence or doesn’tprovide his confidence.
Looking at the left half of Figure 3 whereZach’s bet is inconsistent with the learner’sevidence, we see that when Zach has high con-fidence in his bet, the learner is highly influ-enced by him. Even when the leaner saw horsex win only one of five races, she thinks there isa moderate chance horse x will win the next oneif Zach bet with high confidence. When Zachbet but did not convey confidence, the learnerthinks that horse x is likely to lose the next race.This seemingly muted influence of Zach stemsfrom the nature of the information he provides,a binary bet. A binary bet only licenses a weakinference about Zach’s knowledge; even if Zachwere 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 whereZach’s bet is consistent with the learner’s evi-dence, we see that Zach’s low confidence betsbecome more influential with the strength of thelearner’s evidence. When Zach says he has lowconfidence 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 xparadoxically decreases the learner’s estimateof x’s strength, thus we predict the following:
Prediction 6: If Zach states or is interpretedas having low confidence, the learner hasstrong evidence for an outcome, and Zachbets on that outcome his bet can decreaseher belief in that outcome. That is to saythat when these conditions hold and he betsconsistently with her evidence, his bet hasa paradoxical effect.
If learners fill in Zach’s confidence when hedoesn’t state it and some non-negligible propor-tion attributing Zach’s low confidence, thenPrediction 6 predicts that “paradoxical integra-tion” should be evident in Experiment 1 becauseconfidence cues were not provided. Birnbaumand Mellers (1983) similarly observed this par-
Figure 3. Simulation of the integrated learner model’s estimate of horse x’s skill whenZach’s confidence is varied (Equation 14). The simulation was conducted when Zach saw 10races and said he had high confidence, low confidence, or did not provide his confidence inhis 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 articlefor the color version of this figure.
28 HAWTHORNE-MADELL AND GOODMAN
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adoxical integration in their 1983 experiment,and used this effect to reject Bayesian models oftheir data. However, if some of his participantssimilarly attributed to their advisers low confi-dence, their findings would be predicted by ourBayesian integrative learner model.
We testeded Prediction 5 in Experiment 2 byhaving Zach provide either high or low confidencein his bets. In addition to testing Prediction 5,having a condition in which Zach should be highlyinfluential helps distinguish between possible ex-planations for the predicted modest effect ofZach’s bet in Experiment 1. For example, al-though both our computational account andaccounts that find “under-utilization” of ad-vice predict the muted impact of Zach inExperiment 1, our account more naturallypredicts the large relative impact of Zach’sconfidence in Experiment 2.
We testeded Prediction 6 in Experiment 3where we provide the learner with strong directevidence that horse x is stronger and then see howZach betting on horse x influences her estimate.We testeded Zach’s influence when he has high orlow confidence and compare these results to thecorresponding trials in Experiment 1 where Zachdoesn’t provide confidence information and base-line trials where there is only direct evidence.
Inferring reliability. The generativemodel underlying the integrative sociallearner allows reasoning about arbitrary latentproperties of the actor. For example, in pre-vious experiments, learners inferred the reli-ability of an actor from the way they behavedtoward known items (e.g., Poulin-Dubois etal., 2011) or named known words (e.g., Clé-ment et al., 2004; Jaswal & Neely, 2006;Pasquini et al., 2007). Did they call a cow aduck or seem pleased with the contents of anempty box? The equivalent situation in thehorserace scenario is seeing how Zach betswhen the races he saw are known. Did Zachbet on horse x when she only saw him win oneout of 10 races? To infer Zach’s reliability,learners consider which alpha value is mostconsistent with his behavior. The probabilitythat Zach has reliability level alpha is
p(��bz, kz, nz) � p(bz�kz, nz, �)p(�). (16)
Although learners could consider Zach havingany � � ��, we simplify by considering only
whether Zach is reasonable or unreasonable,each corresponding to a particular alpha value.
The model indicates that learners should inferZach’s reliability from the consistency of hisaction and the information available to him,thus we predict the following:
Prediction 7: When Zach’s observationsare more consistent with his bet, learnersare more likely to infer he is reliable.
The strength of this reliability inference de-pends on the learner’s model of how Zach betswhen he is reliable (a reliable Zach’s alpha). IfZach maximizes his expected utility (� � �),his consistency is highly influential on the learn-er’s estimate of his reliability. However if Zachprobability matches (� � 1), his consistency isless diagnostic. There is still some chance hecould be reliable and bet on horse x when x’srecord is poor.
We tested Prediction 7 in Experiment 3 byextending the horserace scenario with a com-plete information trial in which the learner seesZach’s observations and his bet. By manipulat-ing the consistency of Zach’s bet with his ob-servations we can test the model’s quantitativeprediction about his reliability inference. Wecan further test whether learners’ updated esti-mate of Zach’s reliability affects Zach’s influ-ence in subsequent trials.
Applying the Integrative Learner Model
The use of the integrative model outlined inthe preceding text to make quantitative predic-tions 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 maxi-mizes the fit of the model to the behavioral datait is trying to account for, or on the basis ofindependent empirical data. Of the three op-tions, using independent data as an empiricalprior distribution of the unspecified variableprovides the most stringent test of a theoreticalframework. We therefore strove to constrainunspecified variables with empirical data wher-ever possible. We used data from prior experi-mentation when available and otherwise lever-aged the ease of online experimentation toperform additional “prior elicitation experi-ments” to determine empirical priors. Followingthis method, no model predictions were the re-
29REASONING ABOUT SOCIAL SOURCES
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sult of “fitting” to the data they are trying toaccount for.
In each of the four experimental sections is amodel specification subsection that describesthe methods used to define the unspecified vari-ables required to make quantitative predictions.The resulting completely specified models wereimplemented in the probabilistic programminglanguage, Church.11,12 We used exact inferencemethods to calculate the posterior estimates.The variability in the resulting model predic-tions is therefore a reflection of the variability ofthe empirical priors used, for example, p(sl).
13
Experiment 1: Effect of Knowledgeabilityand Reliability
We set out to test the first four predictions ofthe integrative learner model by manipulatingthe direct and social evidence available to par-ticipants in our simple racetrack scenario.
Method
For an overview of the flow of the experi-ment, see Figure 4, and for the details abouthow the evidence was provided, see Figure 5.The primary stimuli consisted of a table sum-marizing the results of a series of races betweentwo horses. For each trial, the horses’ nameswere randomly selected from a list of fictionalhorse 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 tothe betting game. Participants watched an ani-mation of a series of head-to-head racesbetween two horses. As the winning horsepassed the finish line, its results were added to aresults table summarizing the series. Partici-pants were then told that they were going to seethe results of a series of different head-to-headmatch-ups summarized like the results tablethey had just seen. They were then given sixbaseline 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 possi-ble number of times—zero through five. Theorder 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 directlearner model. It was used to calculate theirprior beliefs about horse skills.
Following the baseline trials, participantswere introduced to the (fictional) last participantto complete the experiment. The last participantwas given a random male name (i.e., Zach).Participants were told that Zach “saw differentraces from the same match-ups,” giving himindependent information about them. AfterZach was introduced, the test trials were pre-sented. 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 vari-ables—the number of races the participant saw(nl), the proportion of times horse x won inthose races (kl), the number of races Zach saw(nz), and which horse Zach bet on (bz). Theindependent variables kl and bz were manipu-lated within participants, whereas nz was ma-nipulated between participants, and nl was heldconstant at 5.
Participants always saw the results of fiveraces (each with five matches). Zach’s knowl-edgeability (nz) was manipulated by randomlyplacing 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 possiblenumber of times—zero through five. In half ofthese combinations Zach’s bet was consistentwith the participants’ observations, for exam-ple, “I saw [horse x] win four out of fiveraces, and Zach bet on him,” whereas in theother half, his bet was inconsistent, for exam-ple, “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). Partici-pants saw each of the three unique horserecords twice for a total of six trials: one trialin which Zach bets consistently with the re-cord and one in which he bets inconsistently.
11 See https://probmods.org/v1/lfz.html.12 The model can be edited interactively to see its sensi-
tivity to the unspecified parameters.13 However, the bootstrapped confidence intervals calcu-
lated were sufficiently small to be invisible on our plots.This holds true for all subsequent model results.
30 HAWTHORNE-MADELL AND GOODMAN
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The target horse (whose skill was queried)was counterbalanced such that it was not al-ways the weaker horse, and it was not alwaysthe 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 maintainingthis constraint were used.
Following the presentation of all the evidence(see Figure 5), participants were asked to fill inZach’s empty results table with what theythought Zach saw. They clicked on the questionmarks and freely typed their response (Figure5b). Then the participants were asked “What doyou think is the chance that [horse x] wins thenext race?” They indicated their response byclicking on a continuum, ranging from certainlywill to certainly won’t, which was coded on ascale from 0 to 100, respectively. The evi-dence—participant s’ observed races table andZach’s bet—were presented when making thisassessment to ensure that participants’ judg-ments were not limited by memory.
After completing all the test trials, the par-ticipants answered a debriefing question(“Which best described Zach?”) on a four-point scale, ranging from very unreliable tovery reliable.14 For an overview of the param-eters used in each trial of all the experiments,see Table 1.
Participants
Two hundred participants (mean age � 29years, 57% female) were recruited through Ama-zon’s Mechanical Turk crowd-sourcing service.They completed the experiment in 6 min to 9 minfor a small payment. Forty participants were ran-domly assigned to each of five conditions in whichZach saw zero, five, 10, 20, or 30 races.
Data Processing and CounterbalanceTesting
The data were put into a standard form foranalysis. Counterbalanced trials were trans-formed such that Zach always bet on horse x(the horse the participant was asked about). Thismade all Zach-inconsistent trials occur whenhorse x was dominated in the participants’ ob-served races. Consider a Zach-inconsistent trialin 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 trialinto the standard form, all quantities were in-verted such that the participant now observed xwin 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.
Figure 4. Outline of Experiment 1. The number of races Zach saw was manipulated betweenparticipants, from zero to 30. One trial sequence is pictured. Each participant saw six test trialsvarying horse x’s record and Zach’s bet. See the online article for the color version of thisfigure.
31REASONING ABOUT SOCIAL SOURCES
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The transformed trials were not distinguishablefrom the untransformed counterbalanced trails(�df�1
2 � 7.4 10�2, p .05).15 Therefore theresults neither significantly depended on whichhorse was the subject of the query nor onwhether the horse that participants were askedabout was the same as the one that Zach had beton. Additionally, the different trial orderingsdid not significantly influence participants’ re-sponses (�df�3
2 � 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 learnerwill think that Zach is reasonable, p(�). Wediscuss the empirical bases for specifying eachof these pieces in turn.
We used the nonsocial baseline trials toestimate p(sl). We bootstrapped an estimate ofthe participant’s mean baseline scores with1,000 samples. We discretized p(sl) into amultinomial distribution on 10 evenly spacedpoints in [0, 1], and for each sampled meanwe assigned the points probability such thatthe mean absolute error between the directlearner model and the sampled mean wasminimized (see Figure 6 for fit prior).16 Tolower the dimensionality of this optimization,we added the constraint that the resultingdistribution be symmetrical around p(sl) � .5(a simplification supported by the symmetryof responses seen in Figure 7). We fit our
model to the resulting distribution of fittedp(sl), enabling our model predictions to reflectthe variability in this empirically derived pri-or.
We set the probability of a learner thinkingZach is reasonable, p(�), to the proportionmeasured in the debriefing question (.74). Asdescribed in the model section, we are assum-ing that participants’ estimate of Zach’s reli-ability is constant throughout their trials, notupdated within or between trials. This is con-sistent with the ordering of the trials havingno effect (discussed in the previous section—Data Processing and Counterbalance Test-ing). This is not to say that participants areunable to update their beliefs about Zach’sreliability, 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 Exper-iment 4, we focus on participants’ inferences
15 Chi-square statistic calculated from the difference indeviance of the full model and the (nested) model withoutthe predictor (Pinheiro & Bates, 2009; Barr et al., 2013).
16 It would have been natural to assume a beta distributionprior on sl, resulting in a beta-binomial model for the baseline(nonsocial) trials. However, participants’ baseline scores devi-ated from a beta-binomial model, motivating our use of thebaseline data to directly estimate a less parametric prior. Par-ticipants’ baseline scores also deviated from canonical proba-bility weighting functions (Tversky & Kahneman, 1992). Theempirically estimated prior exhibits the classic overweightingof extreme probabilities but has high fidelity for moderateprobabilities, contradicting both the central tendency predictedby the beta-binomial model and the shallower slope of stan-dard probability weighting functions.
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.
32 HAWTHORNE-MADELL AND GOODMAN
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about Zach’s alpha by providing stronger andmore salient evidence.
Finally, we fixed the alpha values to de-scribe a “reasonable” and “unreasonable”Zach to empirical values measured in a sep-arate experiment. We elicited 40 participants’intuitive theories regarding how Zach wouldbet, given his observations and reliability.Participants were randomly split into twoconditions. In one, they were told Zach was“reasonable,” and in the other, they were toldhe was “unreasonable.” Zach always saw 10races, and participants were asked how Zachwould bet when he saw horse x win five, six,seven, eight, nine, or 10 of them. When Zachwas reliable, all 20 participants said he wouldbet on horse x as soon as he won a majority ofthe races (six to ten). This corresponds to
imagining that Zach used a strict maximizingrule (a very high �). When Zach was unreli-able, participants said Zach would bet onhorse x exactly half of the time. A logisticregression found no effect of horse x’s recordon how participants’ thought Zach would bet(�df�1
2 � .82, p .05). This corresponds toimagining that an unreliable person just flipsa coin to decide who to bet on (� � 0),effectively ignoring their observations.
All unspecified parameters of the integra-tive model are therefore determined by exper-imental measurements that were independentof the target judgments in integrative trials—thus all parameters were measured rather thanfit in the usual sense. We use the same pa-rameters where they are required to modellater experiments.
Table 1Overview of the Manipulated Parameters in the Four Experiments
Parameters
Participant Zach
Experiment
Numberparticipant
observations
Observedproportion ofhorse x wins
Number ofZach’s
observations
Observedproportion ofhorse x wins
Chosenhorse Confidence
Totalnumberof trials
Baseline 5 a — – — — 61 5 a (5, 10, 15, 20, 30) — a — 62 5 a 10 — a (low, high) 63 5 5/5 10 — x (low, high) 64 5 4/5 ¡ 1/5 10 8/10 ¡ — (x, y) ¡ — — 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 onehalf 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.
Figure 6. Empirically derived prior distribution over ahorse’s skill.
Figure 7. Mean of participants’ estimates of horse’schances on the basis of a summary table (i.e., the nonsocial“baseline trials”).
33REASONING ABOUT SOCIAL SOURCES
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Results and Discussion
The four predictions from the integrativelearner model guide our analysis. All predictionswere tested with linear mixed-effects regressionmodels to account for subject-level effects andavoid issues of multiple independent comparisons(Bates et al., 2014; Pinheiro & Bates, 2009). Theeffect of Zach’s bet was predicted to get larger ashis bet became more inconsistent with the partic-ipants’ observations, so we focus our analysis onthese Zach-inconsistent trials.17
We first verified that the influence of Zach’sbet was dependent on the races he observed. Weused a planned comparison of when Zach sawno races, when Zach saw 10 races, and thebaseline scores. Looking at the inconsistent tri-als, we would expect participants who werelearning from Zach’s bet to move their estimateof horse x’s chances toward Zach’s bet, relativeto the baseline scores, which are solely based ontheir own observed races. If Zach’s influence isbased on the fact that he saw some races theparticipant didn’t, then we expect him to beinfluential when he saw 10 races but not whenhe saw none. We used a mixed-effects model totest the planned comparison. The model alsoincluded the participants’ observed race resultsand a (random) term for each participant. In theresulting model, participants’ estimates of horse
x’s chances were more consistent with Zach’sbet 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 in-distinguishably from those in the baseline trialswhere Zach was not yet introduced (� � 6.7 10�4, t � 2.6 10�2, p .05). Consistent withthe first prediction, participants listened to Zachwhen he had independent information and ig-nored him when he didn’t.19
Figure 8 shows how the integrative learnermodel and participants respond to the full quan-titative manipulation of Zach’s knowledgeabil-ity. The integrative learner model captures howlearners combine their own observations withZach’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 integrationeffect 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 Chris-tensen, 2014).
19 Subsequent analyses focused on exploring the factorsmodulating Zach’s influence will therefore omit the conditionin which Zach saw no races, as it was indistinguishable fromthe baseline trials.
Figure 8. Model’s and participants’ mean estimate of horse x winning the next race whenZach 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.
34 HAWTHORNE-MADELL AND GOODMAN
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from the number of races Zach observed, thenumber of times horse x won in his direct ob-servations, its estimate of how many times Zachsaw horse x win, and its assessment of Zach’sreliability. The best mixed-effects model of thedata had precisely these predictors with the ad-dition of a (random) participant term.
Analyzing the mixed-effects model we firstsee that, like the integrative model, participantsused both Zach’s bet and their observed races toinfer horse x’s chances. When x was more dom-inant in the participants’ sample, participantsrated him more likely to win (� � .42, �df�1
2 �7.5, p � .001).20 The number of races Zach sawwas also predictive; as Zach saw more races,participants estimated horse x’s chances more inline with his bet (� � 4.9 10�3, �df�1
2 � 19,p � .001). In line with the first prediction,Zach’s influence increases with his knowledge-ability (see Prediction 1 in Table 2). WhetherZach was thought to be reliable was also pre-dictive (see Prediction 2 in Table 2). The 74%of participants who thought Zach was reliablegave responses more consistent with his betthan did the 26% who did not think he wasreliable (� � .12, �df�1
2 � 25, p � .001).21
Finally, participants’ estimate of the percentageof the races horse x won in Zach’s observationswas also predictive of their estimate of horse x’schances (� � 9.8 10�2, �df�1
2 � 7.5, p �.001). The better they thought horse x’s chanceswere, the more they thought x dominated Zach’sobservations.
As the model predicted, participants’ esti-mate of Zach’s observations mediated the effectof his knowledgeability (see Prediction 4 inTable 2). Including both the number of racesZach saw and the participants’ estimate ofZach’s observations resulted in a significant ef-fect for his observations (� � 7 10�3, t477 �3.5, p � .05), and the previous effect of Zach’sknowledgeability is absent (� � 1 10�3,t477 � .20, p .05). The Sobel test showed asignificant reduction in the effect of Zach’sknowledgeability on participants’ estimate ofhorse x’s chances (z � 3.5, p � .05).
Finally, Figure 9 shows that the effect ofZach’s knowledgeability is dependent on himbetting reasonably in both the data and the in-tegrative learner model (see Prediction 3 in Ta-ble 2). The model captures this interaction(R2 � .88, MSE � 2.5 10�3). Adding theinteraction of Zach’s reliability with his knowl-
edgeability to our previous mixed-effects modelyields a significant predictor and a better modelfit (� � 5.0 10�3, �df�1
2 � 4.6, p � .05). Asis evident in Figure 9, participants who saidZach bet unreasonably showed no effect ofZach seeing more races (� � 1.5 10�3,�df�1
2 � .84, p .05). For participants who saidZach bet reasonably, the more he saw, the morethey listened to him (� � 6.4 10�3, �df�1
2 �21, p � .001).
In the integrative learner model, Zach’s ob-servations are the common currency betweenhis bet and what the learner already knowsabout the world. They form the bridge betweenthe learner’s observations and her final estimateof horse x’s chances. To test the model’s pre-dictions about the crucial role played by Zach’sunseen observations, participants were asked topredict what Zach saw. Explicitly posing thisquestion could have induced participants toconsider Zach’s observations, which they maynot 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 impliesthat learners spontaneously appreciate the im-portance of what Zach saw and form an estimateof this quantity in the course of reasoning abouthorse 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 withoutthe predictor (Pinheiro & Bates, 2009; Barr et al., 2013).
21 The proportion of participants that thought Zach wasreliable was independent of the number of races he saw(�df�3
2 � 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 wasidentical to the Zach sees 10 races condition, except thatparticipants did not report what they thought Zach saw. The30 participants were not distinguishable from those in thecorresponding Zach sees 10 races condition of Experiment1. An exact Wilcoxon’s Mann–Whitney rank sum test wasperformed that found that the estimate of horse x’s chanceswas 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 vari-able was applied to this control and the correspondingExperiment 1 data. In the resulting data model, the dummycoded condition was neither predictive of the results, norwas its interaction with participants’ observed proportion orreliability attribution (respectively, �df�1
2 � 1.8, p .05;�df�1
2 � 2.3, p .05; �df�12 � 2.0 10�3, p .05).
35REASONING ABOUT SOCIAL SOURCES
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Participants were influenced by Zach’s bet,which reflected the knowledge he derived fromhis observations. However, the influence of hisbet, even when he saw six times as many racesas 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 beton him, how many observations would Zachneed to have seen before you would believe himand bet on horse x (i.e., estimate horse x’schances greater than .5)? Our initial intuitionwas that Zach would have to have seen twice asmany. This was corroborated by the free re-sponses of 34 Amazon Mechanical Turk re-sponders. The mean number of observationsZach would have to have seen to persuade themwas 12.3 when they saw zero of five, with theamount decreasing as their observations becamemore consistent with his bet. Hence, thoughparticipants’ behavior conformed closely to thatof the normative integrative learner model inExperiment 1, explicit intuitions were highlyinaccurate. To put this another way, partici-pants’ online behavior was consistent with anagent reasoning rationally over the causal struc-ture of the integrative learner model (see Figure8, Panels c and d), but the same reasoning wasnot used to explicitly predict the outcome of theexperiment. This décollage is an interestingsubject for future exploration.
The full integrative model predicts the modestobserved effect of Zach’s bet because a single betonly licenses a limited inference about Zach’s
observations. Although Zach betting multipletimes 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 confi-dence ratings, the model predicts that participantsshould be able to infer what Zach saw with greaterprecision and, subsequently, weigh his high con-fidence bets more heavily. Experiment 2 tests thisprediction.
In Experiment 3 we examine a surprisingresult observed in the Zach-consistent tri-als—we find that Zach’s bet on horse x causesparticipants to think that horse x winning is lesslikely than if they had not seen his bet andsimply had the same direct evidence. This par-adoxical effect of Zach’s bet seems to clashwith the measured way in which participantsintegrated his bet on the inconsistent trials,which was well-captured by our Bayesian inte-grative model. However, this effect is consistentwith our model if some learners spontaneouslyattributed Zach low confidence, similar to howsome learners seemed to spontaneously attri-bute him low reliability. We explore this possi-bility in Experiment 3.
The 24% of participants who spontaneouslythought that Zach was betting unreasonably alsocontributed to the relatively muted effect. WhenZach 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 expla-nation: Participants with low estimates of horse
Table 2Results of the Major Predictions of the Integrative Learner Model Simulations
Prediction Description Result
1 The more knowledgeable Zach is, the more influential he should be. �df�12 � 16, p � .001
2 The more reliable Zach is, the more influential he should be. �df�12 � 22, p � .001
3 Zach’s influence is dependent upon him being both reliable andknowledgeable (interaction).
�df�12 � 4.6, p � .05
4 The learner’s estimate of Zach’s observations should mediate the effectof his knowledgeability.
z � 3.5, p � .05
5 When Zach is highly confident his bet should be more influential thanwhen he has little confidence or doesn’t provide his confidence.
�df�12 � 16, p � .001
6 Zach’s bet can have a paradoxical effect when he has low confidenceand his bet is consist with strong direct evidence.
t238 � 2.74, p � .05
7 When Zach’s bet is inconsistent with his observations, learner’s shouldinfer he is unreliable.
�df�12 � 16, p � .001
Note. All statistics are derived from relevant mixed-effects models.
36 HAWTHORNE-MADELL AND GOODMAN
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x’s chances could simply be more likely to saythat Zach is unreliable, as his bet is dissonantwith their estimate. Conversely, those whothink horse x’s chances are good would be morelikely to say Zach is reliable. This alternativeexplanation would not explain the observed in-teraction between reliability and knowledge-ability, but the nature of the reliability inferencemerits further study. In Experiment 4, we ex-plicitly manipulate the evidence of Zach’s reli-ability to explore both the source and flexibilityof participants’ reliability estimates.
Experiment 2: Effect of Confidence
Method
All procedures were identical to those in Ex-periment 1, with one addition to the stimuli. InZach’s speech bubble, where he stated whichhorse 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 betsconsistent with the participants’ information andhad moderate confidence on the first trial, whichhad ambiguous direct evidence (horse x winingtwo of four races). When Zach’s bet was incon-sistent 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’sMechanical Turk crowd-sourcing service. Theycompleted the experiment in 6 to 9 min for asmall payment. Forty participants were ran-domly assigned to each of the high and lowconfidence conditions.
Model Specification
For this experiment, the integrative learnermodel was extended with a straightforwarddefinition of confidence—what Zach thinkshis chances of success are, as described byEquation 14. Given this definition we mustassess how participants interpreted the utter-ances, “I bet on horse x with [high/low] con-fidence,” that is, what learners think Zachthinks about the chance that his bet is correctgiven his stated confidence level. This corre-sponds to the utterance thresholds used inEquation 14. We conducted a separate exper-iment with 40 participants to measure thismapping. Participants were shown a (fic-
Figure 9. Model’s and participants’ mean estimate of horse x winning the next race whenZach 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.
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tional) previous “Zach,” who made a bet on ahorse with either high, moderate, or low con-fidence. Participants saw one trial at eachlevel of confidence. No information beyondZach’s confidence was provided. Participantsresponded 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 betbeing correct. We calculated the median pointat which participants thought Zach would sayhe 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 proba-bility that Zach was reliable was fixed to theproportion measured in this experiment (.83).
Results and Discussion
Did Zach’s high confidence in his bet in-crease its influence on participants’ subsequentestimate of horse x’s chances (vis-a-vis Predic-tion 5)? We tested for this effect by expandingthe additive mixed-effects model described inExperiment 1 with a dummy coded confidencevariable (and removing the knowledgeabilityvariable because that was held constant in thisexperiment). As seen in Figure 10, Zach’s con-fidence did have a significant effect (� � .13,�df�1
2 � 11, p � .05), which the integrativelearner model captured (R2 � .99, MSE �1.6 10�3). When Zach had high confidence inhis bet, it was 2.6 times as influential as when hehad low confidence in it (comparing the impactof Zach’s bet in each condition to the baselinescores). Compared with the corresponding trialsin 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 influen-tial.
All the significant predictor variables fromExperiment 1 were again predictive of partici-pants’ estimate of horse x’s chances. Partici-pants were receptive to the proportion of timeshorse x won in their five observed races (� �.37, �df�1
2 � 34, p � .001). Their assessment ofZach’s reliability influenced how much theyincorporated his bet (� � .15, �df�1
2 � 8.7, p �.05), and participants’ estimate of Zach’s ob-served proportion of x wins was also a signifi-
cant predictor of his bet (� � .17, �df�12 � 4.7,
p � .05).Participants’ assessment of Zach’s reliability
was not driving the effect of confidence: Theproportion of participants who thought Zachwas reliable did not differ between confidenceconditions (�df�1
2 � 0, p .05). However,Zach’s perceived reliability did interact with theeffect of his confidence, just as his perceivedreliability interacted with the effect of hisknowledgeability in Experiment 1. Adding theinteraction of Zach’s reliability and his confi-dence to the additive mixed model yields asignificant predictor (� � .15, �df�1
2 � 4.0, p �.05). The integrative learner model also capturesthis interaction (R2 � .99, MSE � 1.6 10�3
see Figure 11). Looking at the participants whosaid Zach bet unreasonably, there was no effectof Zach’s confidence (� � 8.7 10�3, �df�1
2 �1.5 10�2, p .05). For participants who saidZach bet reasonably, his bet had a larger impactwhen he was more confident (� � .15, �df�1
2 �12, p � .001).
As the integrative model predicted, whenZach had high confidence he was much moreinfluential, even more iso than if he had seenthree times as many races. This finding indi-cates that the modest effect of Zach’s bet inExperiment 1 was not due to a general discount-ing of social information. The integrative modelpredicted the small effect of knowledgeabilityin Experiment 1 from the limited inferencesabout Zach’s beliefs that his binary bet permit-ted. In the model, the addition of confidenceenabled a stronger inference about Zach’s be-liefs, resulting in his bet being more informa-tive, consistent with participants’ responses.Again, participants’ estimate of Zach’s reliabil-ity mediated his influence.
We next examine the impact of Zach’s con-fidence when his bet is consistent with strongevidence. Do his low confidence bets have aparadoxical effect on participants as the modelpredicts?
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 confi-dence were averaged to yield the moderate confidencebound. The median of these within subject estimates wasused.
38 HAWTHORNE-MADELL AND GOODMAN
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Experiment 3: Paradoxical Social EvidenceIntegration
As is shown in Figure 12, participants fromExperiment 1 thought that horse x had worsechances when they heard Zach bet on horse x(with unspecified confidence) than in the corre-sponding baseline trial (where they had no social
evidence). Looking at the figure from left to right,we observe that the “paradoxical” effect of socialevidence increased as the direct evidence becamestronger with the largest effect in the maximallyconsistent trials in which participants saw horse xwin all five races.
This paradoxical updating of beliefs is incon-sistent with the integrative learner model’s predic-
Figure 10. Outline of Experiment 2 and 3. Experiment 2 was identical to Experiment 1except that Zach always saw 10 races and stated his confidence in his bet. Zach’s confidencewas manipulated between participants such that he had either “low” or “high” confidence inthe inconsistent trials. As in Experiment 1, each participant saw six test trials, varying horsex’s record and Zach’s bet. See the online article for the color version of this figure.
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.
39REASONING ABOUT SOCIAL SOURCES
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tion that seeing Zach’s bet would produce a (verymodest) increase in the learner’s estimate of horsex’s chances (see the middle panel of Figure 12).The only way our framework would capture thispattern of reasoning would be if people were au-tomatically thinking about Zach’s confidence inhis bet and some people thought he bet with lowconfidence.24 We therefore tested whether para-doxical integration can be replicated by explicitlyproviding confidence cues (vis-a-vis Prediction 6)in the maximally consistent trials in which theeffect expected to be the strongest. Crucially, themodel predicts that the effect should occur whenZach bets with low confidence but not when hebets with high confidence, and only if Zach is alsothought to be reliable.
Method
The procedure was similar to that of Ex-periment 2 in which Zach provides his confi-dence in his bets, but it focused on the max-imally consistent trials in which Zach bet onhorse x and participants had strong directevidence supporting horse x winning (horse xwon five of the five the trials they saw). Thecrucial comparison was between the maxi-mally consistent trial in which Zach bets withhigh confidence and the maximally consistenttrial 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 participantsknew the alternative confidence levels Zachcould have stated prior to these two trials.Therefore, after the baseline trials, partici-pants first saw three trials in which Zach betwith low, moderate, and high confidence. Af-ter these three trials, the two trials of interestwere interspersed with other consistent trialsto lessen demand characteristics. Thus, par-ticipants saw eight pseudorandom trials afterthe baseline trials. First, they saw the three“alternatives” trials in which Zach bet onhorse x with each of the confidence levels. Forthese trials, the direct evidence was randomlyselected among horse x winning two, three, orfour out of five races. Then participants hadone of the two trials of interest (randomlyselected), which were followed by three ran-domly selected consistent trials in which Zachbet on horse x with randomly selected confi-dence and direct evidence randomly selectedfrom horse x wins three or four out of five.The final trial was the remaining trial of in-terest.
24 They automatically reasoned about Zach’s reliability inExperiment 1.
Figure 12. Model’s and participants’ mean estimate of horse x winning the next race whenZach bets on him with either high or low confidence and split by participants’ assessment ofZach’s reasonableness. By marginalizing across the participants’ observations, we can clearlysee 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.
40 HAWTHORNE-MADELL AND GOODMAN
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Participants
Forty participants (mean age � 26 years,59% female) were recruited through Amazon’sMechanical Turk crowd-sourcing service. Theycompleted the experiment in 6 min to 9 min fora small payment.
Model Specification
The same model described in Experiment 2,and Equation 14 also applies to this experimentwithout modification. We used the same empir-ically derived priors described earlier: the horseskill prior and alpha values used in Experiment1, and the meaning of the confidence levelsfrom Experiment 2. The probability that Zachwas reliable was fixed to the proportion mea-sured in this experiment (.85).
To apply our model to Experiment 1, wesimilarly needed to determine the probabilitythat Zach was confident p(�). We fixed theprobability that Zach was confident to the pro-portion measured in a separate experiment. Thisexperiment was a modified version of Experi-ment 1 in which we elicited 40 participants’confidence attributions. The experiment wasidentical to Experiment 1 except that after thethird test trial participants were asked whetherZach had high or low confidence in his bet (atwo alternative forced choice). On this randomtrial, 21 of the 40 participants (53%) said theythough Zach had high confidence in his bet.
The resulting model deals with a fully spec-ified Zach with known knowledgeability andwith confidence and reliability spontaneouslyattributed to him. This fully specified integrativelearner model, like the integrative learnermodel, has all parameters measured indepen-dently.
Results and Discussion
Participants who saw Zach bet on horse xwith low confidence exhibited paradoxical inte-gration, believing that horse x had worsechances than when they had not seen Zach’s bet(t39 � 2.5, p � .05).25 Conversely, when Zachbet on horse x with high confidence, there wasno significant difference between his bet and thebaseline (t39 � �.83, p .05). We thereforesee paradoxical updating from low confidencebets but not from high confidence bets (as ex-pected from Prediction 6). We can see this
effect in Figure 13 as the low-confidence pointbeing deflected significantly below the blackline representing the baseline score and thehigh-confidence point.26
We further expect that the paradoxical inte-gration should only be evident when partici-pants attribute low confidence to Zach and thinkthat he is reasonable (as seen in the interactionin Experiment 2, Figure 11). Indeed, Zach’slow-confidence bet on horse x only led to par-adoxical integration in the reasonable condition(reasonable, t33 � 3.5, p � .001; unreasonable,t5 � �.42, p .05, as seen in the differentialeffect of confidence between the columns ofFigure 13). The paradoxical integration fromExperiment 1 was consistent with this pattern.Only participants who thought Zach was rea-sonable showed the discounting effect (for allconditions where Zach saw some number ofraces: reasonable, t118 � 3.8, p � .001; unrea-sonable, t40 � 1.0, p .05).27
The observed evidence is therefore consistentwith paradoxical integration being driven bylow confidence attribution. If participants attrib-uted either high or low confidence to Zach in theabsence of explicit confidence cues (Experi-ment 1), then averaging this mixture shouldyield an intermediate estimate between condi-tions in which all participants knew Zach hadhigh 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 Ex-periment 1 data) fell between their estimateswhen Zach bet with high and low confidence.
In conclusion, the paradoxical discountingobserved in Experiment 1 and its interaction
25 All t tests conducted in this section were testing one-sided predictions derived from our model.
26 Additionally, we replicate the main effect from Exper-iment 2 finding that when Zach bet on horse x with highconfidence, participants thought horse x was more likely towin than when Zach had low confidence (t39 � 3.9, p �.001).
27 Interestingly, Zach’s reasonableness had the oppositeeffect than it had for the inconsistent trials in Experiment 1.When Zach is unreasonable and bet on horse x, participantshad 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 mayreflect this, however we simplify our discussion by assum-ing that participants only thought Zach had high or lowconfidence.
41REASONING ABOUT SOCIAL SOURCES
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with perceived reasonableness is well describedby a model that posits that participants sponta-neously fill in both Zach’s reliability and hisconfidence (R2 � .90, MSE � 9.2 10�4).Importantly, if we use this model to fit all thedata from Experiment 1, then we see that itspredictions only significantly diverge from theintegrative model (see Equation 10 and Figure8) when Zach’s bet is consonant with verystrong direct evidence, which is consistent withour data and Birnbaum and Mellers’ (1983)finding. The model that includes learners attrib-uting confidence to Zach therefore fits the Ex-periment 1 data in Figure 8 when Zach’s betwas inconsistent (R2 � .94) and the paradoxicalupdating observed when Zach’s bet was consis-tent with direct observations.
In our last experiment, we tested whether par-ticipants update their initial attributions when pro-vided with strong evidence, inferring Zach’s latent
states from his actions and the their own knowl-edge. Will evidence of Zach acting inconsistentlywith his observations cause people to revise theirestimate of his reliability, consistent with our mo-del’s predictions?
Experiment 4: Inferring Reliability FromAction
Method
Stimuli in Experiment 4 were similar to thosein Experiment 1, with one major alteration (seeFigure 15). In the first test trial, the completeinformation trial, the participants were able tosee Zach’s unobscured results table in additionto his bet. Zach always saw horse x win eightout of 10 times in this trial, which was consis-tent with the four out of five wins the partici-pants saw. Participants were randomly placed
Figure 13. Participant’s and model’s mean estimate of horse x winning the next race whenhe 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 onhorse 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.
42 HAWTHORNE-MADELL AND GOODMAN
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into either the consistent condition in whichZach’s bet was supported by his observations orthe inconsistent condition in which he betsagainst his observations.
The trial following the complete informationtrial, the transfer trial, was the same for bothconditions and identical to the standard testtrials in Experiment 1. The participants sawhorse x win one of five trials and Zach bet onhorse x. The transfer trial was followed by thesame debriefing question from Experiment 1assessing the participants’ judgment of whetherZach was reliable.
Participants
Eighty participants (mean age � 31 years,51% female) were recruited through Amazon’sMechanical Turk crowd-sourcing service. Theycompleted the experiment for a small paymentin 4 min to 7 min. Forty participants wererandomly assigned to each of the Zach betsconsistently 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 Experi-ment 1 and 2, the posterior probability of Zachbeing reliable was fixed to the measured propor-tion. For this experiment, we let the model inferZach’s reliability given his behavior in the com-plete information trial, like the participants (asdescribed in Equation 16). The prior of Zach be-ing reliable was set to the average measured pro-portion from Experiments 1 and 2 (.77).
Results and Discussion
The manipulation of the consistency ofZach’s bet led to the predicted assessment ofZach’s reliability (see Prediction 7). In the con-sistent condition, participants said Zach wasreliable 88% of the time, compared with just33% of the time in the inconsistent condition(see Table 3; �df�1
2 � 23, p � .001). The inte-grative learner model makes a similar inference,concluding that Zach is more likely to be reli-able 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
with consistency condition as a predictor ofhorse x’s chances in the transfer trial, we canconclude that the consistency manipulation hadthe predicted effect on the transfer trial (t78 �2.1, p � .05). When Zach previously bet con-sistently, he was subsequently more influentialthan when he bet inconsistently as the integra-tive learner model predicted (MSE � 3.8 10�2; see Table 4).
In the integrative learner model, the completeinformation trial influences the transfer trialthrough the learner’s estimate of Zach’s reliabil-ity. Because Zach’s reliability is assumed to beconstant across his bets, forming a strong im-pression that he is reliable or unreliable shouldinfluence the amount of information his betsconvey in subsequent trials. Indeed, the effect ofZach’s (in)consistency in the complete informa-tion trial on judgments in the transfer trialshould therefore be completely mediatedthrough the manipulation-induced reliability at-tribution. A mediation analysis concluded thatthe effect of condition on the transfer trial wasmediated by the manipulation-induced reliabil-ity attribution (Baron & Kenny, 1986). Includ-ing both condition and reliability attribution inthe linear model yielded a highly significanteffect for reliability (� � .14, t77 � 3.0, p �.05), and the previous effect of condition isabsent (� � .009, t77 � .20, p .05). The Sobeltest showed significant reduction in the condi-tion’s effect on participants’ estimate of horsex’s chances (z � 2.6, p � .05).
Participants were more likely to infer that Zachwas reliable when his bet was consistent with hisinformation, and they subsequently use this in-ferred quality to determine how much they shouldlearn 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 supportfor different key components of the full integrativelearner model (Figure 1c; Equations 10, 15, and17) to simpler models with these components “le-sioned” (see Table 5). The integrative learnermodel posits that learners reason about Zach’s betwith a generative model of behavior that describeshow Zach’s observations influence his beliefs andhow his beliefs, desires, and reliability guide his
43REASONING ABOUT SOCIAL SOURCES
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actions (see Figure 1d). It interprets Zach’s betwith this intuitive ToM and integrates it withdirect observations to learn about the commoncause—the horse’s skill (see Figure 1c). Are all ofthe components of the integrative learner modelnecessary to explain the pattern of results acrossthe three experiments? Specifically, do the datasupport 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 attri-butes like reliability and confidence such that theZach is “fully specified”, (d) updates these spon-taneous 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 5with a model number (M1, M2, etc.) where itsability to fit the data across our four experimentswas assessed. The integration of direct and so-cial evidence of the integrative learner–M7 wascontrasted with the nonintegrative direct–M1and pure-social–M2 models (Figure 1, Panels aand b, respectively). Each of these models ex-clusively relies on one source of evidence. Thedirect model simply infers the horse’s skill fromthe direct observations, ignoring social cues,
Figure 14. Participant’s and model’s mean estimate of horse x winning the next race whenthey 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, whereasthose who thought he was unreasonable is displayed on the right. The horizontal black linerepresents 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 unspecifiedconfidence 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 attributinglow confidence and should only occur when participants also attribute reasonability, which isobserved in the paradoxical integration of the low confidence estimate in the reasonablecolumn but not the unreasonable column. See the online article for the color version of thisfigure.
44 HAWTHORNE-MADELL AND GOODMAN
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whereas the pure-social model ignores directevidence and instead uses Zach’s bet to infer thehorse’s skill. The direct model uses the horse-skill prior empirically fit to independent data, asdiscussed in Experiment 1’s Model Specifica-tion section. The best pure-social model sharesall seven parameters of the integrative learnermodel, fit to independent data as described ear-lier.
The fully specified integrative learner–M7 pos-its that learners fill in Zach’s unobserved attributeslike his reliability and confidence. We contrast thiswith simple-reliability–M3 that assumes thatlearners do not attribute reliability (instead mod-eling all actors with a fixed degree of decisionnoise, fit as a free parameter). M7 claims that, foreach attribute, there is heterogeneity in the popu-lation of learners, for example, some believe Zachis a priori reasonable with p(�), whereas othersbelieve that he is unreasonable with 1 � p(�).Similarly, some think a priori Zach bets with high
confidence with p(�). The probability that learnersmake these attributions—p(�) and p(�)—is fixedto the observed proportion in the data. Similarly,the interpretation of high and low confidence anda reliable and unreliable actor are fixed to inde-pendently collected data. The fully specified–M7therefore has seven measured parameters, thesesix and the horse-skill prior, and no free parame-ters.
When Zach’s confidence and reliability arenot observed, the model’s characterization ofthese attributes reflects its priors. The modeltherefore has to specify when this belief is up-dated, particularly reliability, which is definedas a stable attribute of Zach’s that impacts sub-sequent decisions. An ideal Bayesian reasonerwould always update his or her estimate ofZach’s reliability with new evidence as the up-date-reliability–M6 does. Departing from thisideal reasoner, the flexibly update-reliability–M5/M7 posits that, perhaps because of resource
Table 3Mean Probability Zach is Reliable With StandardError of the Mean
Condition Data Simulation
Unreliable .33 .074 .17Reliable .88 .052 .80
Table 4Mean Estimate of Horse X’s Chances WithStandard Error of the Mean
Condition Data Simulation
Unreliable .30 .023 .26Reliable .38 .034 .37
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.
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limitations, learners only update their beliefsabout Zach’s reliability when provided withstrong evidence.29 We contrast the ideal update-reliability–M6, the flexibly update-reliability–M5/M7, and the fixed-reliability–M4 that neverupdates its belief about Zach’s reliability. Thesemodels share the same representation of Zach’sreliability but differ in when this representationis 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 experi-mental sections. Across these summaries of thefour experiments, the fully specified integrativelearner–M7 provided the best average fit withno free parameters. Looking at the pattern offailures of the simpler models reveals the com-ponents of M7 that capture different aspects ofthe learner’s social reasoning about Zach andthe world.
Participants’ responses across the experi-ments reflected both the races they directly ob-
served and Zach’s bet. This integration of socialand nonsocial cues is evident in the poor fit ofthe direct–M1 and the pure-social–M2. Partici-pants were influenced by Zach’s bet and showedsensitivity to Zach’s knowledgeability (see Fig-ure 8), confidence (see Figure 10), and reliabil-ity (see Figures 9 and 11 and Tables 3 and 4)that direct–M1 couldn’t account for. Con-versely, 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 ofZach’s reliability affected how much they werepersuaded by his bet. This effect of reliability
29 This situation-dependent updating of the learner’s es-timate of Zach’s reliability was motivated by the data inExperiment 1’s Data Processing and Counterbalance Test-ing section. The model itself does not specify what circum-stances will lead learners to update their estimate of adviserattributes; we make the auxiliary hypothesis that this hap-pens in our setting only when explicit and strong evidenceis available. This is clearly an important direction for futurework, which is further discussed in the Reliability subsec-tion of the Limitations and Future Work section.
Table 5Comparison Between Learner Models That Make Different Assumptions About the Structure and Contentof the Representations Employed by Participants Across the Three Experiments
Simple model
Model KindFigure
8Figure
9Figure
11Figure
12Average
fit
1 Direct .56 0 .38 0 .222 Pure-social .36 .74 .57 .17 .467 Integrative .96 .88 .99 .99 .96
Integrative model
Reliability ConfidenceFigure
8Figure
9Figure
11Figure
12Figure
13Figure
14Tables 2
and 3Average
fitAttribute Update Attribute
3 No No .96 .45 .99 .54 .85 .35 0 .594 Yes Never No .96 .88 .99 .99 .85 .86 0 .795 Yes Flexibly No .96 .88 .99 .99 .85 .86 .98 .936 Yes Always No .73 .17 .74 .17 .85 .86 .98 .647 Yes Flexibly Yes .96 .88 .99 .99 .97 .90 .98 .95
Note. Each column represents the R2 of the model’s predictions for the data represented in the named figure. The data inFigure 8 show the effect of the manipulation of Zach’s knowledgeability. The data in Figure 9 Illustrate the interactionbetween 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.
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could not be captured by no-reliability–M3. Themodel’s one free parameter allowed it to capturethe main effect of Zach’s knowledgeability (seeFigure 8) and confidence (see Figure 10), but itcouldn’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 reliabilityin Experiment 4. All models that had two levelsof reliability that were spontaneously attributedto Zach (M3 through M7) were able to capturethe influence of participants who have differentbeliefs about Zach’s reliability. Similarly, themodels that did not spontaneously attribute con-fidence to Zach (M3 through M6) were unableto explain the paradoxical discounting evidentin Figure 12 and 13. Only fully specified inte-grative learner–M7 that spontaneously attrib-uted both reliability and confidence could fullyaccount for the data across the four experi-ments.
Comparing the predictions of M4 throughM6—the integrative learner, constant-reliabil-ity, and update-reliability models—can revealwhen learners updated their estimate of Zach’sreliability. Fixed-reliability:M4 captured learn-ers’ social information integration across thefirst two experiments (see Figures 8, 9, 10, and11). In these experiments the participants’ be-havior is consistent with the model’s assump-tion that learners do not update their beliefabout Zach’s reliability but do use this fixedbelief to interpret Zach’s bet. However, themodel fails to account for the impact of thetransparent trial in Experiment 4 in which par-ticipants update their belief about Zach’s reli-ability 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 can-not account for the participant’s estimate in thefirst two experiments. On the trials in whichZach’s bet is strongly inconsistent with the par-ticipants’ direct observations, update-reliabili-ty–M6 infers that Zach is likely unreliable anddiscounts his bet. Participants’ integration doesnot exhibit this effect, and instead is substan-tially influenced by Zach on these incongruenttrials, 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
circumstances, like flexibly updating M5. How-ever, the data presented here are insufficient tobuild a formal understanding of the circum-stance in which learners should update theirbeliefs about latent actor attributes like reliabil-ity. One major difference between Experiment 1and 2, in which participants did not update theirbelief about Zach’s reliability, and Experiment4, in which they did, was that the evidence inExperiment 4 was more diagnostic of Zach’sreliability (according to update-reliability–M6).Given this difference, one hypothesis is thatlearners are flexible “resource-rational reason-ers” who conserve computational resources—only reasoning about reliability when the addedinferential complexity results in substantiallyimproved accuracy (Lieder et al., 2012). Alter-natively, learners could have a fixed cognitivelimitation such that they are unable to perform ajoint inference over the horse’s skill, Zach’sobservations, and his reliability. Under this al-ternative hypothesis, learners update their beliefin Zach’s reliability in Experiment 4 not be-cause of the strong inference it licensed, butbecause Zach’s observations were known,which simplified the complexity of the infer-ence to Equation 16.
In conclusion, the fully specified integrativelearner–M7 provides the best fit to participantssocial reasoning and information integrationacross the four experiments (overall fit R2 �.96). Simpler models that did not integrate so-cial and direct evidence or fully specify Zach’sattributes could not account for all the majoreffects in the data. Neither could models thatdid not selectively update Zach’s reliability.The success of M7 is not likely due to overfit-ting the data. The full model’s predictionsacross the four experiments had seven parame-ters fit to independent data (as described in eachof their Model Specification sections), and thesefit values were reused across the four experi-ments with no remaining free parameters(though the choice of which of the four exper-imental situations led to reliability updatescould be seen as a single free parameter).
Insights From a Computational Account ofLearning From Others
Learners’ flexible social reasoning is evidentin their ability to use what they know to inferwhat they do not. We saw this clearly across our
47REASONING ABOUT SOCIAL SOURCES
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four experiments; learners used their model ofZach to interpret his actions to learn from him.Similarly, we saw in Experiment 4 that learnerscould learn about Zach’s latent attributes whenthey 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 treat-ment of “learning to trust and trusting to learn”,p. 1).30
The fit of the fully specified integrativelearner model across our experiments demon-strate that reasoning over a generative modelof action and outcomes naturally capturesthese flexible patterns of inference. Further,by precisely defining the learner’s ToM andwhat she knows about the situation— her di-rect evidence, prior knowledge, and theory ofdomain— our framework describes howlearners learn from and about people and thefactors that determine how much they learnfrom and about them. It allows us to distilllearning from and about people into two in-terlocking patterns of social learning. First,learners use evidence judiciously, that is, theyintegrate different sources according to theirdiagnosticity. Because an actor’s attributesdetermine their diagnosticity, learners useavailable information about them to weigh therelative value of his actions. Second, learnersuse available evidence to infer the unknownattributes of an actor and in the absence ofevidence fill these attributes in. Our frame-work describes how these patterns inform oneanother—learners use available evidence(e.g., the discrepancy between the learner’sinitial 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 totwo experiments that exhibit these patterns ofreasoning. The first experiment by Harvey andFischer (1997) demonstrated that learners useall available information about an adviser toguide how they integrate that adviser’s actionswith their own beliefs. The second experimentby Yaniv (2004) shows that when informationabout an adviser’s attributes is not present,learners flexibly use their understanding of thesituation to reason about the adviser and howinformative his actions are.
Learning From and About Advisers in JASExperiments
Both Yaniv and Harvey and Fischer’s exper-iments follow the same basic structure com-monly used in the JAS literature. First theyelicit an initial estimate from the learner. Theythen show the actor’s estimate, and while dis-playing both the initial and actor’s estimate,they elicit a final estimate.
The difference between the learner’s esti-mates before and after receiving advice is theiradvice utilization. Researchers have consis-tently found that learners under-utilize adviceacross a range of situations and methods forcalculating learners’ advice utilization (as de-scribed previously in the section Toward a Co-hesive Account of Actor Attributes). That is tosay that learners weigh their initial opinionsmore 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 movedher estimate toward the actor’s estimate to ar-rive 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 evidenceto decide the extent to which they should relyon the available social information. For exam-ple, there is considerable uncertainty about boththe actor’s knowledgeability and her own. Ourframework can explicitly define the learners’understanding of the actors’ and their own at-tributes, as well as the learning situation giventhe uncertainty of each. The prediction ofBayesian inference over this formalization ofthe learner’s model of the situation provides anew standard with which to evaluate their socialevidence integration.
Using our framework to form a normativereference point for advice utilization paradigmwe can see if, whereas learners do not incorpo-rate advice as much as they should by the eco-logical standard, they conform to an ideal ob-server reasoning over an uncertain learningsituation. We form such models for two exper-iments in the JAS tradition, which suggest that
30 Or, as we will see in Yaniv’s study in the followingtext, these inferences can happen simultaneously.
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what was interpreted as an egocentric bias (fromthe ecological standard) can be understood asBayesian reasoning over uncertain learning sit-uations.
The two experiments follow the same exper-imental structure but differ in the informationthey provide the learner. Harvey and Fischer(1997) use training on a cue-based judgmenttask as a proxy for knowledgeability. They ma-nipulated the amount of training given to learn-ers (30, 100, or 240 training trial, between-subjects) and the amount of training the adviserhad (within-subjects) such that each learner sawan adviser at each level of training. Harvey andFischer’s experiment provides a test of learner’sability to use an actor’s attributes to determinehow each cue should be used.
Yaniv and colleagues asked questions tap-ping historical knowledge (Yaniv 2004). Learn-ers were only told that the adviser was anotherstudent participant of the study, which left themuncertain about the adviser’s absolute and rela-tive knowledgeability. As Yaniv pointed out,this leads to an epistemic asymmetry as learnershave a better understanding of their own knowl-edgeability than the advisers. Yaniv’s experi-ment therefore tests the framework’s predictionthat when information about an adviser’s attri-butes is not present, learners flexibly use theirunderstanding of the situation to reason aboutthe adviser and how informative his actions are.
Sensitivity to the actor’s relative knowl-edgeability: Judicious use of evidence.
Modeling Harvey and Fischer (1997). Thelearner in Harvey and Fischer’s experiment istasked with making final estimate qf of contin-uous quantity q on a screen that displays herinitial estimate ql, her number of training trials(proportional to her knowledgeability kl), theadviser’s estimate qa, and his number of train-ing trials (� ka). How should she integrate thesetwo estimates with their associated knowledge-ability cues? A learner using the ToM formal-ized in Equation 5, could reason about eachestimate as the intentional choice of a personwith belief b and reliability �, generalizingEquation 3:
p(qa�ba, ka, �a) � e�·�p(q � ba,ka)U(qa;q), (17)
where utility of an estimate U(qa; w) is given bya squared error loss function and p�q�b, k� ���b, xk�, where x represents a free parameter
for each level of k describing the increase inprecision decision makers associate that level oftraining.31 The three values of k that minimizethe mean standard error of the model’s predic-tions and the data produces the qualitativemodel predictions shown in the lower panel ofFigure 16.
The learner could then integrate the adviser’sestimate with her own initial estimate using ageneralization of Equation 6 for each socialsource s in our case the learner l and adviser a:
p(qf �ql, kl, qa, ka)
� p(qf) �s��l,a�
��s,bs
p(qs�bs, ks, �s)p(bs�qf, kl)p(�s),
(18)
where p�b�q, k� � ��q, xk�. This model differsimportantly from the integrative learner model.It describes learning about a continuous quan-tity from two social cues (instead of learningabout a discrete variable from one direct andone social cue). It also models a situation inwhich the sources are not understood as updat-ing their beliefs from unknown evidence. Thesources beliefs are understood directly in termsof the underlying quantity with precision pro-portional to the source’s training p(b|q, k).
With the limited information about the do-main and actor attributes the complex socialinferences described in Equation 18 can be wellapproximated by a simple optimal cue integra-tion model (widely used in perception, e.g.,Knill & Richards, 1996; multimodal sensoryintegration, e.g., Bejjanki et al., 2011; socialcognition, e.g., Zaki, 2013):
p(qf �ql, kl, qa, ka) � p(qf)p(ql�qf, kl)p(qa�qf, ka),
(19)
where each source’s estimate is interpreted as acue of the underlying q with Gaussian noiseproportional to the source’s training.
31 Alternatively, the learner may not be reasoning withher 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) alternativeformalization of the learner’s understanding of the situation.That is to say that there are multiple ideal observer modelsconsistent with the experimental situation.
49REASONING ABOUT SOCIAL SOURCES
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Results. As seen in Figure 16, participantswere sensitive both to their own level of knowl-edgeability and to their adviser’s. Both Harveyand Fischer’s (1997) data and our model’s qual-itative predictions are shown in terms of weightof advice—the difference between the originaland final judgment expressed as a percentage ofthe difference between the original judgmentand 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 them-selves become more knowledgeable, the advis-er’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 learnerthinks she can interpret the cue with high fidel-ity, an adviser (who sees the exact same cue)has muted influence.
Discussion. The model qualitatively cap-tures the interaction of the learner’s and advis-er’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 allthe data points when compared to the model inFigure 16). This is instructive of the differencesbetween the ecological and the Bayesian learnerreference points. Participants underutilize theadvice in that they would have been more ac-curate if they had shifted more toward adviser’sestimates. However, compared with the Bayes-ian learner model, learners erred toward over-utilization, not underutilization. The Bayesiananalysis indicates how learners trying to formaccurate beliefs would integrate advice, and thefact that learners exceeded this amount is con-sonant with Harvey and Fischer’s (1997) dis-cussion that there are additional motivational
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 partici-pants were less well trained. The model captures these qualitative patterns (R2 � .92). See theonline article for the color version of this figure.
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factors for advice utilization like sharing re-sponsibility and accepting offered help.32
Inferring actor attributes from their ac-tions and knowledge of the domain.
Modeling (Yaniv, 2004). Yaniv’s experi-ment focuses on how two factors impact learn-er’s advice utilization: the learner’s knowledge-ability and the distance of an adviser’s estimatefrom the learner’s initial estimate (the advice�). Yaniv divided his participants into high andlow knowledge groups according to the accu-racy of their preadvice judgments (median-split). The distance of advice was binned intonear, intermediate, and far. As we shall see, ourframework predicts that these two factorsshould 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 experimentthe learner does not have any information aboutthe adviser’s knowledge and therefore has torely on internal cues of their own knowledge-ability:
p(qf �ql, kl, qa)
� p(qf)p(ql�qf, kl)�ka
p(qa�qf, ka)p(ka), (20)
where k, although potentially continuous, is dis-cretized into the “high” and “low” knowledgecategories Yaniv used in his analyses, and, as inEquation 20, p�q�qf, k� � ��qf, xk�, where xrepresents a free parameter for each level of kdescribing the precision learners associate withsources with high and low knowledge.
The resulting model shows how learners canuse the advice � to infer an adviser’s knowledge-ability, and how the strength of this inferencedepends on how accurate they think their estimateis. When the learner is highly knowledgeable andthinks that her estimate is close to the right an-swer, she can draw a strong inference that anadviser with an implausible estimate is notknowledgeable. She can then discount his esti-mate. A learner who guessed, and thereforedoes not think that her estimate was accurate,would not be able to make such a strong infer-ence about an adviser based on the his advice �and would utilize advisers similarly, regardlessof the distance of their estimate from hers.
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 advis-ers, regardless of the advice �. This is the samemain effect observed in the model of Harveyand Fischer’s (1997) experiment where partici-pants had external cues of their knowledgeabil-ity, and is consistent with the model’s assump-tion that learners’ internal cues allowed forsimilar (coarse) awareness of their own knowl-edgeability. Looking beyond the main effect wesee the striking interaction predicted by themodel. High knowledge learners progressivelydiscount advisers as their advice � increases.The advice utilization of low knowledge learn-ers, in contrast, is unaffected by the adviser’sadvice �. The model qualitatively captured thisinteraction well (R2 � .98).
Discussion. This pattern of results wouldbe expected from learners who reason aboutan adviser’s attributes and use their conclu-sions to titrate the relative influence of hisestimates in their final judgment. The modelprediction is therefore dependent upon thelearner’s estimate of the adviser’s knowledge,which should mediate the interactions ob-served in Figure 17. Yaniv’s experiments(Yaniv, 1997, 2004), like others in the JAStradition, do not measure the learner’s esti-mates of their own attributes or their advis-er’s. Because learner with high and lowknowledgeability were simply grouped by av-erage score it is unknown whether the learn-ers had accurate trial by trial estimates oftheir 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 sys-tematically underutilizing advice by makingpoor inferences about advisers when they incor-rectly think themselves knowledgeable. Al-though this learner may be integrating informa-tion rationally with respect to their beliefs about
32 Motivational effects are outside the scope of our pres-ent analysis of learning about the world from social anddirect evidence, but such an effect could be incorporatedinto our paradigm by including an explicit utility functionfor the learner that includes social goals (that are themselvesdefined using a ToM; see Ullman et. al [2010] for anexample).
51REASONING ABOUT SOCIAL SOURCES
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the learning situation, their false beliefs leadthem to overweight their own evidence. Thispoints to an important limitation of the narrowlydefined Bayesian learner we present. Our modelsimply indicates whether learner’s combine in-formation rationally given their beliefs aboutthe situation.
Yaniv and colleagues (Yaniv, 1997; Yaniv& Kleinberger, 2000; Harvey & Fischer,1997) explain the “discounting” observed inthe experiment and others as arising from “aninformational asymmetry inherent in any de-cision-making process that involves the use ofadvice” (Yaniv, 2004, p. 2). Indeed, in hisexperiment the learner has no direct evidenceabout the adviser’s knowledgeability, butdoes seem to have a sense of their own knowl-edge. Our cognitive model formalizes thisasymmetry and predicts the interaction ob-served in Figure 17.33
Conclusion From Application ofFramework to JAS Studies
The two JAS experiments highlight two ofthe hallmarks of social learning. Learners inHarvey and Fischer’s (1997) experiment weresensitive to parametrically varied (cues of) thesources knowledgeability. They flexibly usedthe available evidence about the relative diag-nosticity of the two sources—judiciously inte-grating the two sources. Learners in Yaniv’sexperiment used their own knowledge to drawinferences about the adviser, filling in his un-known attributes. Our framework shows howunderstanding social evidence in terms of a
33 This is an example of how formalizing theories withcomputational models illuminates the theory’s subtle con-sequences, as Yaniv proposed in the asymmetry theory, butdoes not predict the observed interaction.
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 participantsuse adviser’s estimates less when the estimates are far from their own initial estimate. Lessknowledgeable advisers do not show this pattern, instead using all estimates to the sameextent. 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.
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generative model of behavior can predict thesepatterns of inference.
General Discussion
How do people form a coherent view of theworld from both social and direct informationsources? Using a novel quantitative experimen-tal paradigm, we found that people integratedsocial information and direct observations inaccordance to the predictions of a sophisticatedBayesian computational model. People dis-played a sensitivity to the actor’s knowledge-ability, reliability, and confidence consistentwith our model’s qualitative and quantitativepredictions.
The core of the computational model is aformalization of an intuitive ToM—people’smental 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 so-cial evidence. As an explicit theory of the struc-ture of people’s ToM, the model makes a num-ber of quantitative predictions about the roleactor attributes play in our social reasoning.These predictions build upon previous findingsof people’s sensitivity to these actor attributesby providing a unified account of their relativeimpact and interrelationship.
In our computational model, the actor’s in-fluence is dependent on the precise extent of hisknowledgeability. Experiment 1 confirmed thisprediction. It found that people were influencedby actors in proportion to their knowledge andignored actors with no knowledge. The evi-dence in our experiments was graded and quan-titative, which allowed us to parametrically varythe actor’s knowledge. This manipulation re-vealed that people integrated the (variablyknowledgeable) actor’s bet with their directlyobserved evidence as predicted by our Bayesianintegrative model.
Experiment 2 found that people were moreinfluenced by confident actors than those whoexpressed low confidence in their actions. Al-though this sensitivity to an actor’s confidenceis well documented, our experiment allowed usto measure the relative importance of an actor’sconfidence and knowledgeability. Both actor at-tributes are formalized in our model, providing
quantitative predictions for their relative impactand relationship. For instance, the model pre-dicted that an actor who saw 10 races would bemore influential if he expressed confidence inhis action than if he was three times as knowl-edgeable but didn’t express confidence (fromthe Predictions of the Integrative Learner andLearning from Confidence Cues subsections ofour Computational Account of Learning FromOthers section). Comparing Experiment 1 and2, we see precisely this relationship: Confidencedramatically increases an actor’s influence com-pared with increases in his knowledgeability.
Both the effect of Zach’s knowledgeabilityand confidence are predicted to be modulated byhis reliability. In the extreme, if the actor iscompletely unreliable, he should be ignored re-gardless of his knowledgeability and confi-dence. Comparing people who thought the actorwas reliable to those who though he was unre-liable in Experiment 1 and 2 reveals these pre-dicted interactions. In Experiment 1 people whothought the actor was unreliable were not influ-enced by his knowledgeability. Those whothought the actor was reliable showed gradedsensitivity to his knowledgeability. Similarly, inExperiment 2, only people who thought Zachwas reliable showed sensitivity to Zach’s con-fidence.
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 judg-ments still mediated the effects of knowledge-ability and confidence indicates that peoplespontaneously used their estimate of the actor’sreliability to determine how informative his ac-tions were. Similarly, we found evidence thatparticipants spontaneously estimate an actor’sconfidence and use these estimates when learn-ing from him. In Experiment 1, when Zach’sbets 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 hisconfidence in Experiment 1, but this paradoxi-cal effect could be the result of some partici-pants who spontaneously attribute Zach lowconfidence in his bet. Experiment 3 found thatthe paradoxical effect observed in Experiment 1is consistent with participants who fully specifyZach’s unobserved reliability and confidence.
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Participants who fully specify Zach’s unob-served attributes should not take their attributedvalues as given, but rather update their priorestimates with available information. Departingfrom a fully Bayesian approach, the behavioralevidence suggests that these estimates areformed and used but not always updated: Thebest-fitting model used people’s prior expecta-tions 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 abet that was (in)consistent with his observationsproviding clear evidence that he was (un)reli-able. People used this evidence to infer that theactor was (un)reliable in the proportion pre-dicted by our model, and people used theirupdated estimate of the actor’s reliability whensubsequently learning from him. The experi-ments taken together therefore suggest that peo-ple spontaneously attribute actors with attri-butes like reliability to reason about them, butonly update their attributions in certain circum-stances. We discuss the implications more in thefollowing passages.
Our model accurately predicted people’s re-sponses in all four experiments. The experi-ments manipulated the actor’s attributes andprobed people’s beliefs about the actor and theworld. The same core model was used to ac-count for these manipulations and the differentdependent variables, illustrating the productiv-ity of our generative model of behavior (ToM).The model’s flexibility emerges from this gen-erative 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 Alterna-tive Model Comparison section, only the fullyspecified 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 exper-iments.
Connections to Other Work
The results of our computational analysisprovide a new perspective into two documentedirrationalities of advice utilization—that advicecan have a paradoxical effect (Birnbaum &
Mellers, 1983) and that it is systematically un-derutilized (e.g., Harvey & Fischer, 1997;Yaniv, 1997; Yaniv & Kleinberger, 2000). Ourframework is built around a formal model of thelearner’s understanding of the adviser. Takingthis 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 optimallyintegrating the social evidence given their char-acterization of the adviser.
Given the information provided to participantsin Birnbaum and Mellers’ (1983) experiment andour own Experiment 1, the actor selecting optionx should have increased the participant’s belief inx, but it has the opposite, paradoxical effect. Cru-cially, in neither case was the participant’s esti-mate of the adviser’s confidence measured. Ourframework describes how this pattern would beexpected from a learner who, in the absence ofreceiving confidence information, spontaneouslyattributes the adviser confidence. Experiment 3supported this account of the paradoxical effectand could similarly explain the effect in Birnbaumand Mellers’ experiment (which could be testedwith manipulation of the adviser’s confidence ormeasurement of participants unmanipulated attri-butions).
Similarly, our model indicates that what hasbeen characterized as underutilization can beunderstood as Bayesian reasoning under uncer-tainty about the adviser’s knowledgeability, re-liability, and confidence. Indeed, in many of theclassic experiments finding underutilization, thesource or extent of the adviser’s knowledge wasuncertain for the learner, which our model pre-dicts would significantly reduce the potentialinfluence of the adviser (Yaniv, 1997; Yaniv &Kleinberger, 2000). However, we have shownthat even knowing the adviser’s exact knowl-edgeability, as people did in Experiment 1, doesnot ensure that he is highly influential.
This points to a strength of our computationalapproach: it can predict the precise cause of thelow impact (seeming underutilization) of advis-ers—what the “bottleneck” in a given learningsituation is. For Experiment 1, our model pre-dicted that it was the nature of the actor’s ac-tion—a discrete, binary choice—and the uncer-tainty about his reliability that limited hisinfluence. To increase the adviser’s influence,we widened the communication channel in Ex-
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periment 2 and found the predicted jump in theadviser’s influence. Similarly, our model canmake predictions of what would further weakenan actor’s influence. By providing strong evi-dence that the actor was unreliable in Experi-ment 4 we found that he was effectively ig-nored. Our modeling framework thereforeprovides a tool for both generating realisticlearner-based normative reference points for ad-vice utilization and for making predictionsabout which changes to a situation or adviserwill have the largest impact on advice utiliza-tion. Both applications can buttress existing in-vestigations in the advice-utilization and deci-sion support systems literatures (Gönül et al.,2006; Harvey & Fischer, 1997).
Our framework uses the tools of computationalcognitive psychology to synthesize two distinctintellectual traditions—epistemic trust and judg-ment adviser systems. The framework formalizedthe actor attributes identified in the epistemic trustliterature and combined it with the insights andempirical methodologies for testing advice utili-zation of the JAS literature. The framework wasable to captures some major findings in the respec-tive fields, but a number of questions central to theepistemic trust and JAS literature remain to beexplored. The developmental trajectory of the in-tegration of social information with other evi-dence remains open. Similarly, we haven’t ad-dressed the quantitative dynamics of integratinginformation from multiple actors. Despite theirdifferent focuses, the epistemic trust and JASfields pursue complimentary questions—how dopeople decide whom to trust, and how do theydecide how far to trust them relative to otherinformation sources. Our framework highlightsthe common cognitive substrate involved in thesequestions. As such we hope that our frameworkfor understanding social information integrationfosters connections between these complimentaryliteratures and fosters a broader dialogue about thesocial and cognitive dynamics underlying sociallearning.
Our computational framework uses a generalprobabilistic learning framework (Tenenbaumet al., 2011), sharing and elaborating on featuresof a number of previous accounts of socialreasoning and learning (Baker et al., 2009,2011; Goodman et al., 2009; Shafto et al.,2012). Our model shares the basic model ofToM as a formalization of belief-desire psy-chology with Baker et al. (2009, 2011) and
Goodman et al. (2009; see Butterfield et al.,2008 for a synergistic approach using Markovrandom fields). However, this basic conceptionof ToM is elaborated in a number of ways. Ouraccount used a graded notion of knowledgeabil-ity 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 completelyknowledgeable). Our account also relaxed afoundational assumption of previous ac-counts—that agent’s act rationally—allowinglearners to reason about the extent to which anactor was reliable. Finally, our model formal-ized and explored the effect of an actor’s con-fidence. We used this elaborated ToM to ex-plain both how people learn about the worldfrom actors, like Goodman et al. (2009) andShafto et al. (2012), and how they infer latentstates of the actor, like Baker et al. (2009, 2011)and Pantelis et al. (2014). The resulting modelprovides a unified account of how actor attri-butes guide social learning, bridging the epis-temic trust and judgment adviser systems liter-atures (as Eaves and Shafto’s [2012] modelprovided a unified account of pedagogical rea-soning and epistemic trust).
Limitations and Future Work
Our experimental scenario provided a simpledomain for measuring social information integra-tion where the situation makes the nature of theactor’s attributes transparent, for example, knowl-edge is simply the number of races observed. Thescenario enabled a number of simplifying assump-tions in our integrative learner model and allowedus to cleanly test the effect of actor attributes onsocial learning. Exploring more complicated situ-ations where these assumptions no longer holdmay expose additional complexity of people’s nat-uralistic social reasoning. Each of these simplifi-cations therefore represents an interesting avenuefor further research; we next describe several ofthese possible extensions.
Amount of direct evidence. Across thefour experiments the participants’ knowledgeabil-ity from direct evidence, that is, the number ofraces they observed, was held constant. The modelpredicts that the actor’s influence should diminishas the learner has more direct observations, con-sistent with Harvey and Fischer’s (1997) findings.The model also predicts an interaction between
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the effect of the actor and learner’s knowledge-ability: An actor’s knowledgeability should havegreater impact for less knowledgeable learners.Extending beyond Harvey and Fischer, the modelpredicts that learners with more knowledge wouldbe able to draw stronger inferences about latentactor attributes, like reliability. For example, inExperiment 1, when the learner only had a fewobservations and the actor bet on the horse thatwas dominated in them, it did not license a stronginference. From the learner’s limited observations,it was not unlikely that the actor saw a conflictingsample and was therefore acting reliably. How-ever, a learner that saw many observations wouldhave a better estimate of what the actor was likelyto 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 reli-ability is likely richer than was exposed in ourexperimental set-up. For instance, gradation inreliability was limited by the four-point scale weused to assess it. We then further binned responsesinto two categories—people who thought the ac-tor was reliable or unreliable—to simplify theanalysis. However, people’s use of the scale andintegration behavior indicated that their represen-tation of reliability was more graded than the twosimple categories. Additional explorations will beneeded to determine how fine-grained reliabilityis. More importantly, our model treated reliabilityas simply the noise of a person’s decision rule. Itis likely that people also consider other ways inwhich actors could be (un)reliable, such as theway they form beliefs. We were unable to distin-guishing between these two sources of (un)reli-ability from the limited evidence that people re-ceived in our experiments, but other paradigmscould 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 there-fore could be reliable for recently acquired knowl-edge but uninformative at greater temporal re-move. These more nuanced ways of beingunreliable could be formalized in more elaboratemodels of mind; incorporating these more elabo-rate models of mind into an integrative sociallearner would highlight the unique behavioral sig-natures of each trait.
Another question suggested by our results iswhen learners update their attribution of traits to
an actor. Across our four experiments, learnersonly updated their estimate of the actor’s reliabil-ity in Experiment 4. One explanation for this isthat the evidence provided about the actor’s reli-ability was relatively strong (from our computa-tional model’s predictions) and salient (from theauthors’ intuition) relative to the evidence pro-vided in Experiment 1 and 2. An alternate expla-nation identifies the more complex inference re-quired to infer the actor’s reliability from theevidence provided in Experiment 1 and 2 (see theAlternative Model Comparison section for furtherdiscussion of these theories). Both of these mayamount to resource-efficient processing strategiesfor approximating a complex fully-Bayesian be-lief update. The current evidence cannot deter-mine which explanation best accounts for the dif-ferential reasoning about reliability across ourexperiments. Future investigations are thereforenecessary to determine the cause of this pattern ofreasoning and interrelated questions like whetherthe effect of the strength and salience of evidenceis graded or whether there is a threshold at whichparticipants will begin to reason about (rather thanjust using) reliability and other traits.
Prior knowledge. The learner’s understand-ing 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 as-sumed learners thought the actor shared their priordistribution over horse skill.34 We also assumedcommon knowledge of the causal structure of thedomain, which meant that the actor’s observationswere the sole source of his knowledge. Althoughthis correspondence is desirable for parametricallymanipulating social evidence, people often learnfrom actors who have a different understanding ofthe 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 stockfundamentals as everyone else, his bets are still
34 Although expedient, it is unclear when such assump-tions 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 foractors (Meltzoff, 2007)? This relationship between a learn-er’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 providesa new tool with which to approach this classic question.
56 HAWTHORNE-MADELL AND GOODMAN
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highly influential. This situation corresponds tothose often explored in the JAS literature, whichhas found that advice utilization is variable anddependent on a number of heuristic actor attri-butes like age and experience (Feng & Mac-George, 2006). Developing a formal theory ofhow learners understand these ambiguouslyknowledgeable actors, and attribute knowledge onthe basis of other traits, represents an importantelaboration of the simple ToM we presented.
Communication. Finally, as we saw in Ex-periment 2, the nature of the actor’s communi-cation has a large impact on the influence oftheir message. In our experiments, we restrictedthe “communication” to an actor’s goal-directedaction, which is equivalent to watching the actorinteract directly with the world (perhaps hesi-tantly or with confidence). Although our actorinteracted directly with the world, actors areoften learner-oriented, interacting with thelearner with communicative, pedagogical, ornefarious intent. Learning from a teacher re-quires a model of how they make pedagogicaldecisions because the teacher’s goal is not aboutthe state of the world. It is about the beliefs ofthe learner. This requires the learner to thinkabout the teacher who is, in turn, thinking aboutthe 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 “pure-social” case in which learners do not have al-ternative sources of information. Incorporatingtheir more elaborate ToM into the integrativelearner model would provide a good test of theextensibility of our framework. The resultingmodel could provide insights into how studentslearn from a mix of instruction and exploration.Does a student’s self-confidence get in the wayof her learning? What can teachers do to man-age the student’s perception of their attributes,like reliability and knowledgeability?
Conclusion
Other people represent an important source ofinformation, 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
sources and showed how it formalizes the integra-tion of social and direct evidence by describing thethree major attributes of social sources—theirknowledgeability, confidence, and reliability.We found that this model explained the quan-titative data from our horserace paradigm ex-tremely well. Such rigorous models of sociallearning provide a framework for both under-standing previous empirical investigationsand generating future experiments. Develop-ing our understanding of the theories andcompetencies underlying learning from othersprovides insights into the big questions ofpersuasion, pedagogy, and culture. How can Iconvince someone to heed my advice? Howdoes a student’s understanding of the teacherand situation impact their learning? How isknowledge transmitted from one generation tothe next?
References
Baker, C. L., Saxe, R., & Tenenbaum, J. B. (2009).Action understanding as inverse planning. Cogni-tion, 113, 329–349.
Baker, C. L., Saxe, R. R., & Tenenbaum, J. B.(2011). Bayesian theory of mind: Modeling jointbelief-desire attribution. In C. Laura, H. Christoph,& F. S. Thomas (Eds.), Proceedings of the 33rdAnnual Conference of the Cognitive Science Soci-ety (pp. 2469–2474). Boston, Massachusetts: Cog-nitive Science Society.
Baron, R. M., & Kenny, D. A. (1986). The modera-tor–mediator variable distinction in social psycho-logical research: Conceptual, strategic, and statis-tical considerations. Journal of personality andsocial psychology, 51, 1173–1182.
Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J.(2013). Random effects structure for confirmatoryhypothesis testing: Keep it maximal. Journal ofMemory and Language, 68, 255–278.
Bates, D., Maechler, M., Bolker, B. M., & Walker, S.(2014). lme4: Linear mixed-effects models usingEigen and S4. R package version, 1, 1–23.
Bejjanki, V. R., Clayards, M., Knill, D. C., & Aslin,R. N. (2011). Cue integration in categorical tasks:Insights from audio-visual speech perception.PLoS ONE, 6(5), e19812. http://dx.doi.org/10.1371/journal.pone.0019812
Birch, S. A., Akmal, N., & Frampton, K. L. (2010).Two-year-olds are vigilant of others’ non-verbalcues to credibility. Developmental Science, 13,363–369.
Birch, S. A., Vauthier, S. A., & Bloom, P. (2008).Three-and four-year-olds spontaneously use oth-
57REASONING ABOUT SOCIAL SOURCES
Thi
sdo
cum
ent
isco
pyri
ghte
dby
the
Am
eric
anPs
ycho
logi
cal
Ass
ocia
tion
oron
eof
itsal
lied
publ
ishe
rs.
Thi
sar
ticle
isin
tend
edso
lely
for
the
pers
onal
use
ofth
ein
divi
dual
user
and
isno
tto
bedi
ssem
inat
edbr
oadl
y.
ers’ past performance to guide their learning. Cog-nition, 107, 1018–1034.
Birnbaum, M. H., & Mellers, B. A. (1983). Bayesianinference: Combining base rates with opinions ofsources who vary in credibility. Journal of Person-ality and Social Psychology, 45, 792–804.
Bonaccio, S., & Dalal, R. S. (2006). Advice takingand decision-making: An integrative literature re-view, and implications for the organizational sci-ences. Organizational Behavior and Human Deci-sion Processes, 101, 127–151.
Brooks, R., & Meltzoff, A. N. (2002). The impor-tance of eyes: How infants interpret adult lookingbehavior. Developmental Psychology, 38, 958–966.
Brooks, R., & Meltzoff, A. N. (2005). The develop-ment of gaze following and its relation to lan-guage. Developmental Science, 8, 535–543.
Butterfield, J., Jenkins, O. C., Sobel, D. M., & Schw-ertfeger, J. (2008). Modeling aspects of theory ofmind with Markov random fields. InternationalJournal of Social Robotics, 1, 41–51.
Carpenter, M., Akhtar, N., & Tomasello, M. (1998).Fourteen-through 18-month-old infants differen-tially imitate intentional and accidental actions.Infant Behavior and Development, 21, 315–330.
Chow, V., Poulin-Dubois, D., & Lewis, J. (2008). Tosee or not to see: Infants prefer to follow the gazeof a reliable looker. Developmental Science, 11,761–770.
Clément, F., Koenig, M., & Harris, P. (2004). Theontogenesis of trust. Mind & Language, 19, 360–379.
Corriveau, K., & Harris, P. L. (2009). Choosing yourinformant: Weighing familiarity and recent accu-racy. Developmental Science, 12, 426–437.
Eaves, Jr., B. S., & Shafto, P. (2012). Unifying ped-agogical reasoning and epistemic trust. Advancesin Child Development and Behavior, 43, 295–319.http://dx.doi.org/10.1016/b978-0-12-397919-3.00011-3
Feng, B., & MacGeorge, E. L. (2006). Predictingreceptiveness to advice: Characteristics of theproblem, the advice-giver, and the recipient.Southern Communication Journal, 71, 67–85.
Finetti, B. (1974). Theory of probability; a criticalintroductory treatment. New York, NY: Wiley.
Flavell, J. H. (1999). Cognitive development: Chil-dren’s knowledge about the mind. Annual ReviewPsychology, 50, 21–45.
Gardner, P. H., & Berry, D. C. (1995). The effect ofdifferent forms of advice on the control of a sim-ulated complex system. Applied Cognitive Psy-chology, 9, S55–S79.
Gönül, M. S., Önkal, D., & Lawrence, M. (2006).The effects of structural characteristics of expla-nations on use of a DSS. Decision Support Sys-tems, 42, 1481–1493.
Goodman, N. D., Baker, C., & Tenenbaum, J. B.(2009). Cause and intent: Social reasoning incausal learning. In T. Niels & V. R. Hedderik(Eds.), In Proceedings of the 31st annual confer-ence of the cognitive science society (pp. 2759–2764). Amsterdam, the Netherlands: CognitiveScience Society.
Goodman, N. D., Mansinghka, V. K., Roy, D. M.,Bonawitz, K., & Tenenbaum, J. B. (2008).Church: A language for generative models. Re-trieved from https://arxiv.org/abs/1206.3255
Goodman, N. D., & Stuhlmüller, A. (2013). Knowl-edge and implicature: Modeling language under-standing as social cognition. Topics in CognitiveScience, 5, 173–184.
Griffiths, T. L., Kemp, C., & Tenenbaum, J. B.(2008). Bayesian models of cognition. In Ron Sun(Ed.), Cambridge handbook of computational cog-nitive modeling (pp. 59–100). Cambridge, UK:Cambridge University Press.
Harris, P. L. (2007). Trust. Developmental Science,10, 135–138.
Harris, P. L., & Corriveau, K. H. (2011). Youngchildren’s selective trust in informants. Philosoph-ical Transactions of the Royal Society B: Biolog-ical Sciences, 366, 1179–1187.
Harvey, N., & Fischer, I. (1997). Taking advice:Accepting help, improving judgment, and sharingresponsibility. Organizational Behavior and Hu-man Decision Processes, 70, 117–133.
Hu, J. C., Buchsbaum, D., Griffiths, T. L., & Xu, F.(2013). When does the majority rule? Preschool-ers’ trust in majority informants varies by domain.In M. Knauff, N. Sebanz, M. Pauen, & I. Wa-chsmuth (Eds.), In Proceedings of the 35th annualconference of the cognitive science society (pp.2584–2589). Austin, TX: Cognitive Science Soci-ety.
Jaswal, V. K., & Malone, L. S. (2007). Turningbelievers into skeptics: 3-year-olds’ sensitivity tocues to speaker credibility. Journal of Cognitionand Development, 8, 263–283.
Jaswal, V. K., & Neely, L. A. (2006). Adults don’talways know best: Preschoolers use past reliabilityover age when learning new words. PsychologicalScience, 17, 757–758.
Jungermann, H., & Fischer, K. (2005). Using exper-tise and experience for giving and taking advice. InT. Betsch & S. Haberstroh (Eds.), The routines ofdecision making (pp. 157–173). Mahwah, NJ:Lawrence Erlbaum.
Kinzler, K. D., Corriveau, K. H., & Harris, P. L.(2010). Children’s selective trust in native-accented speakers. Developmental Science, 14,106–111.
Knill, D. C., & Richards, W. (1996). Perception asBayesian inference. New York, NY: CambridgeUniversity Press.
58 HAWTHORNE-MADELL AND GOODMAN
Thi
sdo
cum
ent
isco
pyri
ghte
dby
the
Am
eric
anPs
ycho
logi
cal
Ass
ocia
tion
oron
eof
itsal
lied
publ
ishe
rs.
Thi
sar
ticle
isin
tend
edso
lely
for
the
pers
onal
use
ofth
ein
divi
dual
user
and
isno
tto
bedi
ssem
inat
edbr
oadl
y.
Koenig, M. A., Clément, F., & Harris, P. L. (2004).Trust in testimony: Children’s use of true and falsestatements. Psychological Science, 15, 694–698.
Koenig, M. A., & Harris, P. L. (2005). Preschoolersmistrust ignorant and inaccurate speakers. ChildDevelopment, 76, 1261–1277.
Kushnir, T. (2013). How children learn from andabout people: The fundamental link between socialcognition and statistical evidence. In M. Banaji, &S. Gelman (Eds.), The development of social cog-nition (pp. 191–196). New York, NY: Oxford Uni-versity Press.
Kuznetsova, A., Bruun Brockhoff, P., & HauboBojesen Christensen, R. (2014). lmerTest: Testsfor random and fixed effects for linear mixed effectmodels (lmer objects of lme4 package). R packageversion 2.0 –11. Retrieved from https://cran.r-project.org/web/packages/lmerTest/index.html
Landrum, A. R., Eaves, B. S., Jr., & Shafto, P.(2015). Learning to trust and trusting to learn: atheoretical framework. Trends in Cognitive Sci-ences, 19, 109–111.
Lieder, F., Griffiths, T., & Goodman, N. (2012).Burn-in, bias, and the rationality of anchoring. InP. Bartlett, F. C. N. Pereira, Leon Bottou, ChrisJ. C. Burges, & K. Q. Weinberger (Eds.), Advancesin Neural Information Processing Systems (Vol.25, pp. 2699–2707). Cambridge, MA: MIT Press.
Lutz, D. J., & Keil, F. C. (2002). Early understandingof the division of cognitive labor. Child Develop-ment, 73, 1073–1084.
Malle, B. F. (2004). How the mind explains behavior:Folk explanations, meaning, and social interac-tion. Cambridge, MA: MIT Press.
Matsui, T., Yamamoto, T., & McCagg, P. (2006). Onthe role of language in children’s early understand-ing of others as epistemic beings. Cognitive De-velopment, 21, 158–173.
Meltzoff, A. N. (2007). ‘Like me’: A foundation forsocial cognition. Developmental Science, 10, 126–134.
Moore, C., Bryant, D., & Furrow, D. (1989). Mentalterms and the development of certainty. Child De-velopment, 60, 167–171.
Moore, C., & Davidge, J. (2009). The developmentof mental terms: Pragmatics or semantics? Journalof Child Language, 16, 633.
Nurmsoo, E., & Robinson, E. J. (2009). Children’strust in previously inaccurate informants who werewell or poorly informed: When past errors can beexcused. Child Development, 80, 23–27.
Olineck, K. M., & Poulin-Dubois, D. (2005). Infants’ability to distinguish between intentional and ac-cidental actions and its relation to internal statelanguage. Infancy, 8, 91–100.
Pantelis, P. C., Baker, C. L., Cholewiak, S. A., Sanik,K., Weinstein, A., Wu, C.-C., . . . Feldman, J.
(2014). Inferring the intentional states of autono-mous virtual agents. Cognition, 130, 360–379.
Pasquini, E. S., Corriveau, K. H., Koenig, M., &Harris, P. L. (2007). Preschoolers monitor the rel-ative accuracy of informants. Developmental Psy-chology, 43, 1216–1226.
Perner, J. (1991). Understanding the representa-tional mind. Cambridge, MA: MIT Press.
Phillips, J. M. (1999). Antecedents of leader utiliza-tion of staff input in decision-making teams. Or-ganizational Behavior and Human Decision Pro-cesses, 77, 215–242.
Pillow, B. H. (1989). Early understanding of percep-tion as a source of knowledge. Journal of Exper-imental Child Psychology, 47, 116–129.
Pillow, B. H., & Weed, S. T. (1997). Preschoolchildren’s use of information about age and per-ceptual access to infer another person’s knowl-edge. The Journal of Genetic Psychology, 158,365–376.
Pinheiro, J., & Bates, D. (2009). Mixed-effects mod-els in S and S-PLUS. New York, NY: Springer.
Poulin-Dubois, D., Brooker, I., & Polonia, A. (2011).Infants prefer to imitate a reliable person. InfantBehavior and Development, 34, 303–309.
Pratt, C., & Bryant, P. (1990). Young children un-derstand that looking leads to knowing (so long asthey are looking into a single barrel). Child Devel-opment, 61, 973–982.
Premack, D., & Woodruff, G. (1978). Does the chim-panzee have a theory of mind? Behavioral andBrain Sciences, 4, 515–629.
Robinson, E. J., Butterfill, S. A., & Nurmsoo, E.(2011). Gaining knowledge via other minds: Chil-dren’s flexible trust in others as sources of infor-mation. British Journal of Developmental Psychol-ogy, 29, 961–980.
Robinson, E. J., & Whitcombe, E. L. (2003). Chil-dren’s suggestibility in relation to their under-standing about sources of knowledge. Child De-velopment, 74, 48–62.
Ross, L. (1977). The intuitive psychologist and hisshortcomings: Distortions in the attribution pro-cess. Advances in Experimental Social Psychol-ogy, 10, 173–220.
Sabbagh, M. A., & Baldwin, D. A. (2001). Learningwords from knowledgeable versus ignorant speak-ers: Links between preschoolers’ theory of mindand semantic development. Child Development,72, 1054–1070.
Shafto, P., & Goodman, N. (2008). Teaching games:Statistical sampling assumptions for learning inpedagogical situations. In B. C. Love, K. McRae,& V. M. Sloutsky (Eds.), In Proceedings of the30th annual conference of the cognitive sciencesociety (pp. 1632–1637). Washington DC: Cogni-tive Science Society.
59REASONING ABOUT SOCIAL SOURCES
Thi
sdo
cum
ent
isco
pyri
ghte
dby
the
Am
eric
anPs
ycho
logi
cal
Ass
ocia
tion
oron
eof
itsal
lied
publ
ishe
rs.
Thi
sar
ticle
isin
tend
edso
lely
for
the
pers
onal
use
ofth
ein
divi
dual
user
and
isno
tto
bedi
ssem
inat
edbr
oadl
y.
Shafto, P., Goodman, N. D., & Frank, M. C. (2012).Learning from others the consequences of psycho-logical reasoning for human learning. Perspectiveson Psychological Science, 7, 341–351.
Shafto, P., Goodman, N. D., & Griffiths, T. L. (2014).A rational account of pedagogical reasoning:Teaching by, and learning from, examples. Cogni-tive Psychology, 71, 55–89.
Sniezek, J. A., & Buckley, T. (1995). Cueing andcognitive conflict in judge-advisor decision mak-ing. Organizational Behavior and Human Deci-sion Processes, 62, 159–174.
Sniezek, J. A., & Van Swol, L. M. (2001). Trust,confidence, and expertise in a judge-advisor sys-tem. Organizational Behavior and Human Deci-sion Processes, 84, 288–307.
Sobel, D. M., & Corriveau, K. H. (2010). Childrenmonitor individuals’ expertise for word learning.Child Development, 81, 669–679.
Stock, H. R., Graham, S. A., & Chambers, C. G.(2009). Generic language and speaker confidenceguide preschoolers’ inferences about novel ani-mate kinds. Developmental Psychology, 45, 884.
Stuhlmüller, A., & Goodman, N. D. (2013). Reason-ing about reasoning by nested conditioning: Mod-eling theory of mind with probabilistic programs.Journal of Cognitive Systems Research, 28, 80–99.
Swol, V. M. L., & Sniezek, J. A. (2005). Factorsaffecting the acceptance of expert advice. BritishJournal of Social Psychology, 44, 443–461.
Tenenbaum, J. B., Kemp, C., Griffiths, T. L., &Goodman, N. D. (2011). How to grow a mind:Statistics, structure, and abstraction. Science, 331,1279–1285.
Tversky, A., & Kahneman, D. (1992). Advances inprospect theory: Cumulative representation of un-certainty. Journal of Risk and Uncertainty, 5, 297–323.
Ullman, T. D., Baker, C. L., Macindoe, O., Evans,O., Goodman, N. D., & Tenenbaum, J. B. (2010).
Help or hinder: Bayesian models of social goalinference. Advances in Neural Information Pro-cessing Systems, 22, 1874–1882.
Wellman, H. M. (1990). The child’s theory of mind.Cambridge, MA: MIT Press.
Wellman, H. M., Cross, D., & Watson, J. (2001).Meta-analysis of theory of mind development: Thetruth about false belief. Child Development, 72,655–684.
Whalen, A., Buchsbaum, D., & Griffiths, T. L.(2013). How do you know that? Sensitivity tostatistical dependency in social learning. In M.Knauff, N. Sebanz, M. Pauen, & I. Wachsmuth(Eds.), In Proceedings of the 35th annual confer-ence of the cognitive science society (pp. 1593–1598). Austin, TX: Cognitive Science Society.
Whitcombe, E. L., & Robinson, E. J. (2000). Chil-dren’s decisions about what to believe and theirability to report the source of their belief. Cogni-tive Development, 15, 329–346.
Yaniv, I. (1997). Weighting and trimming: Heuristicsfor aggregating judgments under uncertainty. Or-ganizational Behavior and Human Decision Pro-cesses, 69, 237–249.
Yaniv, I. (2004). Receiving other people’s advice:Influence and benefit. Organizational Behaviorand Human Decision Processes, 93, 1–13.
Yaniv, I., & Kleinberger, E. (2000). Advice taking indecision making: Egocentric discounting and rep-utation formation. Organizational Behavior andHuman Decision Processes, 83, 260–281.
Zaki, J. (2013). Cue integration a common frame-work for social cognition and physical perception.Perspectives on Psychological Science, 8, 296–312.
Received October 20, 2016Revision received June 2, 2017
Accepted June 8, 2017 �
60 HAWTHORNE-MADELL AND GOODMAN
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