#  Social Learning with Intrinsic Preferences111Support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2117–422037984 is gratefully acknowledged. We would
like to thank Brendan Barrett, Yves Breitmoser, Sebastian Fehrler, Moritz
Janas, Katrin Schmelz, the research group of the Cluster for the Advanced
Study of Collective Behaviour, the Thurgau Institute of Economics (TWI), and
the Department of Economics at the University of Konstanz, as well as
participants of various conferences, workshops and seminars for helpful
comments and suggestions. We are indebted to Charles Efferson for many
insightful comments on an earlier draft of this paper. All remaining errors
are our own.

Fabian Dvorak
University of Konstanz
<EMAIL_ADDRESS>Urs Fischbacher
University of Konstanz
<EMAIL_ADDRESS>
Despite strong evidence for peer effects, little is known about how
individuals balance intrinsic preferences and social learning in different
choice environments. Using a combination of experiments and discrete choice
modeling, we show that intrinsic preferences and social learning jointly
influence participants’ decisions, but their relative importance varies across
choice tasks and environments. Intrinsic preferences guide participants’
decisions in a subjective choice task, while social learning determines
participants’ decisions in a task with an objectively correct solution. A
choice environment in which people expect to be rewarded for their choices
reinforces the influence of intrinsic preferences, whereas an environment in
which people expect to be punished for their choices reinforces conformist
social learning. We use simulations to discuss the implications of these
findings for the polarization of behavior.

Keywords: social learning, experiment, discrete-choice modeling
JEL Classification: C92, D83, D91

##  1 Introduction

Many decisions, such as how to dress, which sources of information to trust,
and what to buy, are influenced not only by intrinsic preferences about choice
alternatives, but also by social learning. This dual nature of decision making
can lead to a fundamental conflict in situations where individual preferences
differ from the behavior suggested by peers. Despite overwhelming evidence for
the influence of both individual preferences and social learning on human
decisions, little is known about how individuals trade off individual
preferences and social learning in different choice tasks and choice
environments.

Separating the influence of intrinsic preferences and social learning on
people’s decisions is notoriously difficult (Manski, 1993). People may behave
similarly because they have similar intrinsic preferences for choice
alternatives or because they are conformist social learners. At the same time,
someone with unusual preferences may be misidentified as anticonformist when
making independent decisions. To further complicate the identification of
preferences and social learning, people tend to surround themselves with
people who have similar preferences, making the information we receive about
the behavior of others endogenous. Thus, to cleanly separate the effects of
intrinsic preferences and social influence on individuals’ decisions, we would
need to measure preferences in the absence of social information and compare
them to decisions made under exogenous social influence.

In this paper, we use a combination of experimental data and discrete choice
modeling to cleanly separate the effects of intrinsic preferences and social
learning on participants’ decisions in two different choice tasks and three
different choice environments. Our main contribution is the finding that
preferences and social learning jointly influence participants’ decisions in
all experimental conditions, suggesting that models of human behavior should
account for both factors. However, the relative importance of the two factors
differs substantially and depends on both the choice task and the choice
environment. Despite this complexity, a tractable pattern emerges. We find
that intrinsic preferences are generally more important in a subjective choice
task in which participants choose between works of art, whereas social
learning determines participants’ decisions in a task in which participants
are instructed to choose the correct answer to a question. A choice
environment in which people expect to be rewarded reinforces the influence of
intrinsic preferences, whereas an environment in which people expect to be
punished reinforces conformist social learning.

Combining experimental data with discrete choice modeling also allows us to
identify participants’ individual social learning strategies. We find that
individual social learning strategies vary substantially across individuals,
experimental tasks, and treatments. Conformist social learning is the norm in
both experimental tasks, but is more common in the experimental task where
participants choose answers to questions with an objectively correct answer.
In the experimental task where participants choose one of two pieces of art,
social learning is much more diverse and anticonformity occurs. When some
participants are rewarded for their choices, individual social learning
strategies cover the full spectrum of social learning in the painting
selection task. Consistent with previous findings (Dvorak et al., 2020),
conformist social learning dominates in the environment where participants
expect punishment and becomes less frequent when participants expect reward.
In the painting selection task, the average social learning strategy shifts
from conformity in the control treatment to nonconformity in the reward
environment, with implications for simulated population dynamics.

In addition to examining how people trade off intrinsic preferences and social
learning, our study makes several methodological contributions. We extend the
experimental design of Dvorak et al. (2020), which allows us to predict
participants’ intrinsic preferences by using past choices of the same
participant, to study social learning in groups larger than three. Predicting
participants’ choices mitigates the problem that participants may want to vote
consistently, which would inflate estimates of the role of preferences in
decision making. In the key phase of the experiment, we inform participants of
the choices of five other (uninfluenced) participants in the same decision. We
then use statistical modeling to disentangle the effects of intrinsic
preferences and social influence on participants’ decisions. The basis of our
modeling approach is a random utility model in which discrete choices depend
on (1) intrinsic preferences and (2) the expected frequency of others’
choices. The model predicts several stylized facts about social learning, but
also generates new insights into when and how social learning works. Perhaps
most importantly, the interplay of preferences and social learning suggests
that social norms are not arbitrary, but have some intrinsic value beyond the
benefits of coordination.

In addition to integrating intrinsic preferences and social learning, our
modeling framework extends existing models of social learning in two important
ways. First, it goes beyond the simple threshold model of collective behavior
(Granovetter, 1978) by allowing for a variety of individual social learning
strategies, including conformity, nonconformity, independence, and
anticonformity. Second, the model incorporates prior beliefs about others’
choices into the decision-making process, which are updated after observing a
random sample of behavior. In the model, individual utility is a function of
the individual’s posterior belief about the frequency of choices in the
reference group and not a function of the frequency of choices in the sample.
This is not only conceptually plausible, as it prevents small samples from
triggering unusually strong social learning, but also avoids numerical
problems in estimating social learning strategies at the individual level.
Furthermore, explicit modeling of prior beliefs can be useful to incorporate
biases in the decision process (Bénabou and Tirole, 2002; Köszegi, 2006;
Zimmermann, 2020).

##  2 Methods

###  2.1 Experiment

#### Choice tasks

In the online experiment, participants make binary choices in two different
choice tasks. The first choice task involves selecting one of two artistic
paintings by well-known artists. The second choice task is to select the
correct answer from two possible answers to a knowledge question. A list of
all paintings, questions and answers can be found in the Tables 3, 4 and 5 in
the appendix.

Each subject makes 60 binary decisions in each choice task. The first 50
decisions in each task are made without information about other participants’
choices. In the last 10 decisions, the participant is informed about the
choices of five other participants in the same decision. We use each
participant’s first 50 decisions to predict her uninfluenced preferences in
the last 10 decisions with social information.

To incentivize participants’ decisions in the painting selection task,
participants received the motif selected in one of the binary choices printed
on a postcard. The postcard was mailed to the participant in an envelope a few
days after the experiment. To increase intrinsic motivation to provide correct
answers in the question task, participants know that they will be informed of
the correct answers at the end of the experiment.

#### Measurement of intrinsic preferences

The left panel of Figure 1 illustrates how we predict participants’
preferences in the last 10 decisions of each choice task. We create sets that
consist of four choice alternatives $A$, $B$, $C$, $D$. Depending on the task,
these are four different paintings or four potential answers to the same
questions. Each participant makes one binary choice for each possible pairwise
combination of the four alternatives.

After each choices, the participant indicates the strength of her preference
using a slider ranging from minus one (strong preference for the alternative
displayed on the left), over zero (indifference) to one (strong preference for
the alternative displayed on the right). The slider appears at the bottom of
the decision screen after each decision. An example of the decision screen
with the slider can be found in the Appendix.

To predict the individual preference $\Delta$ of a participant in the binary
choice between alternatives $A$ and $B$, we use the comparisons of both
alternatives to the remaining alternatives $C$ and $D$. To predict the binary
choice $A$ vs. $B$, we use the four related binary choices $A$ vs. $C$, $B$
vs. $C$, $A$ vs. $D$ and $B$ vs. $D$, which are depicted in the left panel of
Figure 1.

The self-reported preference strength in the two related choices that compare
the alternatives $A$ and $B$ to a common third alternative yield an estimate
for the participant’s intrinsic preferences in the decision $A$ vs. $B$. Since
there are two comparisons of $A$ and $B$ to a common third alternative, we use
the mean of the two estimates as a measure for the participant’s intrinsic
preferences in the decision $A$ vs. $B$.

In the example in the left panel of Figure 1, the participant chooses
alternative $C$ over $A$ and reports a preference strength of $0.5$. In the
binary choice $B$ vs. $C$, the same participant prefers $B$ and reports a
preference strength of $-0.5$. The difference between the two reported values
divided by two produces an estimate for the intrinsic preference of the
participant in the choice $A$ vs. $B$. The division makes sure that the
estimate is between minus one and one. The two other related binary
comparisons $A$ vs. $D$ and $B$ vs. $D$ yield an estimate of $0.3$. In the
example, we would use the expected value $\frac{0.5+0.3}{2}$ of the two
estimates as a measure for the intrinsic preference of the participant in the
decision $A$ vs. $B$.

The sign of $\Delta$ indicates the preferred alternative of the participant.
In the example, the sign of the expected value is positive which indicates
that alternative $B$ is intrinsically preferred. The absolute difference in
intrinsic utility between the two alternatives is measured by $\Delta$, the
absolute value of the expected value. The variable $\Delta$ ranges between
zero and one. In the example, the estimated absolute difference in intrinsic
utility between alternatives $A$ and $B$ is 0.4 - which is less than half of
the maximum preference strength.

The method to infer intrinsic preferences illustrated in Figure 1 rests on the
assumption of transitivity across the four choice alternatives. The method has
the advantage that the participants of the experiment do not have to make the
same binary choice twice - once without and once with social learning. This
reduces the concerns that participants want to make consistent choices, which
would lead to an underestimation of the effect of social learning. The
technique to measure intrinsic preferences presented in the left panel of
Figure 1 produces estimates that are proportional to the true intrinsic
preferences plus some independent, identically distributed random error as
long there is no systematic violation of transitivity.

Figure 1: Experimental design

measurement of $\Delta$related choicesACBCADBD0.5-0.50.80.2$A$ vs.
$B$$\frac{0.5-(-0.5)}{2}=0.5$$\frac{0.8-0.2}{2}=0.3$$0.4$ sequence of
decisionsAcomparisonsABCDC1C2C3C4C5C6sbj123456phase
1C1,C2,C3,C4,C5C1,C2,C3,C4,C6C1,C2,C3,C5,C6C1,C2,C4,C5,C6C1,C3,C4,C5,C6C2,C3,C4,C5,C6phase
2C6C5C4C3C2C1

Notes: Left panel illustrates the method to measure the intrinsic preference a
participant in the binary decision $A$ vs. $B$. The estimate $\Delta$ of the
intrinsic preference is the expected value of two normalized estimates that
are derived by comparing the two alternatives $A$ and $B$ to a common third
alternative - either $C$ or $D$. The division by two normalizes the two
estimates such that $\Delta$ ranges from -1 to 1. Right panel illustrates how
the six pairwise comparisons of four alternatives are distributed over two
phases of the experiment. In phase 2, each participant is informed about the
uninfluenced choices of the five other participants in phase 1.

#### Social information

Before a participant makes the the decision with social learning between $A$
and $B$, the participant receives information about the decisions of five
other participants in the same choice. Depending the experimental task,
participants see the selected paintings or the selected answers of the five
other participants on the decision screen. An example of the decision screen
can be found in the Appendix.

At the time of their decisions, the five other participants themselves have no
information about the decisions of other participants. The right panel of
Figure 1 illustrates how the experimental design efficiently makes sure that
this condition is fulfilled for a group of six subjects. The pairwise
comparisons of four alternatives $A$, $B$, $C$ and $D$ generate six binary
decisions. In a first phase of the experiment, each participant of a group of
six completes five of the six binary decisions without social information. In
phase two of the experiment, the remaining sixth decision is completed by each
participant. Since this sixth decisions differs for each participant of the
group, it is possible to inform the participant about the choices of the other
five participants from phase one.

In the example in the right panel of Figure 1, the decisions highlighted in
red mark the decision C5 which is the comparison of alternatives $B$ and $D$.
All subjects except subject 2 complete this decision in phase 1. Subject 2 can
therefore be informed about the decisions of the five other subjects before
she makes her decision in phase 2. The right panel of Figure 1 illustrates
that this works for all six subjects.

#### Environments with reward or punishment

As in Dvorak et al. (2020), we conduct two experimental treatments, one with
reward and one with punishment, and a control treatment. In the treatments
with reward or punishment, participants are informed that their decisions with
social information, together with the choices of the five other participants,
will be shown to a participant who is not part of the group of six. This
participant takes the role of an evaluator. The six choices are displayed on
the screen of the evaluator in a randomized order. The task of the evaluator,
who does not know which of the displayed choices is the social choice, is to
select one of the six participants by clicking on one of the six choices. At
the end of the experiment, one decision of the evaluator is randomly chosen
for each experimental task. In the reward treatment, the payoff of the
participant who was selected by the evaluator decision is increased by 10
Euro. In the punishment treatment, the payoff of the selected participant is
reduced by 10 Euro. An example of the evaluation screen can be found in the
Appendix Study materials and instructions.

Dvorak et al. (2020) analyze the incentives in environments with reward or
punishment based on a simple model of social evaluation in which the evaluator
selects a participant based on her own intrinsic preference. This means the
evaluator selects a participant who has chosen her preferred alternative for
reward and a participant who has not chosen her preferred alternative for
punishment (if possible). They show that, with information about others
behavior, the incentive to choose alternative $j$ depends the number $n_{ij}$
of other participants that have chosen alternative $j$ in phase 1 of the
experiment. In the reward treatment, the incentive to choose alternative $j$
generally decreases in $n_{ij}$. This incentivizes anti-conformity. In the
punishment treatment, the incentive to choose alternative $j$ increases in
$n_{ij}$. This incentivizes conformity. The intuition behind this result is
that, since the social evaluation is only based on partici pants’ choices, the
probability of being rewarded or punished is evenly distributed among those
who display the same behavior. Based on the theoretical analysis of Dvorak et
al. (2020), we expect that the choice environments with reward or punishment
influence the model parameter $f$ that captures the reaction of a participant
to $n_{ij}$ \- the number of other individuals that have chosen alternative
$j$. Compared to the control treatment, the social learning parameter $f$
should be larger under punishment and smaller under reward.

#### Implementation

We conducted 10 experimental sessions online between December 11-16, 2020 with
174 students of various fields of the University of Konstanz in Germany. The
mean age of the participants was 21.6 years. We recruited the participants of
the experiment with the recruitment software hroot (Bock et al., 2014). The
experiments were conducted with z-Tree (Fischbacher, 2007) and z-Tree
unleashed (Duch et al., 2020). The duration of the experimental sessions was
between 1.5 and 2 hours. The data set contains 3475 binary choices under
social learning. Participants received a fixed amount of experimental currency
for each decision in the experiment. On average, a participant earned 23 Euro
for participating in the experiment.

###  2.2 Modeling framework

Let $Y_{i}$ denote the discrete choice of individual $i$ among several
alternatives with index $j\in\\{1,\dots,J\\}$. Let $U_{ij}$ represent the
utility of alternative $j$ to individual $i$. We assume that the utilities
$U_{ij}$ is the sum of two systematic components and one random component.

$\displaystyle U_{ij}=\lambda_{i}v_{ij}+f_{i}\cdot log(s_{ij})+\epsilon_{ij}$
(1)

The two systematic components are the elements $\lambda_{i}v_{ij}$, and
$f_{i}\cdot log(s_{ij})$ of the utility function. The component
$\lambda_{i}v_{ij}$ reflects the intrinsic utility of alternative $j$ for the
individual. As in the conditional logit model (McFadden, 1974), it is assumed
that intrinsic utility arises from several observable characteristics of the
alternative multiplied by the individual marginal utilities of these
characteristics. To simplify the notation, we consider a single characteristic
$v_{ij}$ that yields marginal utility $\lambda_{i}$.

The second systematic component $f_{i}\cdot log(s_{ij})$ reflects the social
utility of the alternative. Social utility is a function of the natural
logarithm of the expected share $s_{ij}$ of alternative $j$ in the reference
group. The individual-specific parameter $f_{i}$ reflects individual $i$’s
social-learning strategy, which is our main parameter of interest. The social-
learning strategy of individual $i$ is conformist for $f_{i}>1$, linear for
$f_{i}=1$, and non-conformist $0<f_{i}<1$, independent for $f_{i}=0$, and
anti-conformist for $f_{i}<0$.222As in frequency-dependent social learning, we
define conformity as disproportional copying of behavior in form of an
s-shaped response of the individual to the observed frequency of choices in
the population (Efferson et al., 2008). If all individuals are conformists,
the s-shaped response implies polarization of behavior over time social
learning goes beyond - which can explain a wide variety of behavioral
phenomena ranging from the emergence and persistence of social norms (Akerlof,
1980; Jones, 1984; Bernheim, 1994; Efferson et al., 2020) over inter-dependent
consumer demand (Gaertner, 1974; Pollak, 1976; Alessie and Kapteyn, 1991) and
neighborhood effects (Schelling, 1971), to the effectiveness of nudges and
peer effects (Allcott, 2011; Allcott and Rogers, 2014; Alpizar et al., 2008;
Bobek et al., 2007; Coleman, 2007; Nolan et al., 2008; Schultz et al., 2007;
Smith et al., 2015).

The component $\epsilon_{ij}$ captures randomness in the preferences of
individual $i$ for the $j$th alternative. We will assume that the errors
$\epsilon_{ij}$ are iid and follow a standard Gumbel distribution, which
results in choice probabilities in the logit form (Luce, 1959; McFadden,
1974). The probability that individual $i$ chooses alternative $j$ has the
following closed-form solution:

$\displaystyle
Pr(Y_{i}=j)=\frac{e^{\lambda_{i}v_{ij}}(s_{ij})^{f_{i}}}{\sum_{k}e^{\lambda_{i}v_{ik}}(s_{ik})^{f_{i}}}$
(2)

Equation (2) shows that the systematic components of the utility function
influence the choice probabilities based on different functional forms. For
the characteristics $v_{ij}$, the functional form is exponential. The
exponential form has the advantage that the characteristics $v_{ij}$ can be
specified in absolute or relative terms.

Using the logarithm of the expected shares in the utility function (1) yields
the power form for the expected shares. A practical advantage of the power
form is that is invariant to multiplying all shares by a constant. As a
result, the model makes the same predictions if one uses expected counts
instead of expected shares, which is a convenient feature.

#### Beliefs about shares

We assume that the share of individuals in the reference group that chooses
alternative $j$ is ex-ante unknown. Individual $i$ holds a prior belief about
the share of alternative $j$ that follows a Dirichlet distribution with
parameters $\alpha=\alpha_{i1},\dots,\alpha_{iJ}$. The $\alpha$-parameters can
be interpreted as obseved counts and define the shape and the expected value
of the prior belief $s_{ij}=\alpha_{ij}/\sum\alpha_{ik}$ . If
$\alpha_{ij}=1~{}\forall j\in\\{1,\dots,J\\}$, the prior belief is uniform,
i.e. all shares in the unit interval are believed to be equally plausible. In
this case, the expected share is $1/J$. The larger $\alpha_{ij}$ is relative
to the other $\alpha$-parameters, the more likely are large shares of
alternative $j$.

After observing a random sample of $N$ individuals, of which
$n_{ij}\in\\{0,\dots,N\\}$ prefer alternative $j$, individual $i$ updates the
prior belief about the share of alternative $j$ in the reference group. Since
the Dirichlet distribution is the conjugate prior of the multinomial
distribution, the resulting posterior also follows a Dirichlet distribution
with updated parameters $\alpha_{ij}+n_{ij}$. After the belief update, the
expected shares of the alternatives are

$\displaystyle
s_{ij}=\frac{\alpha_{ij}+n_{ij}}{\sum^{J}_{k}\left(\alpha_{ik}+n_{ik}\right)}.$
(3)

An important remark is that the utility defined in Equation (1) differs from
the expected utility of the alternative given the posterior belief of the
individual. Hence, Equation (1) suggests that the individual does not form an
expectation about the social utility of the alternatives. Instead, it models
the following sequential cognitive process: the individual first updates the
her belief when exposed to social information, calculates the expected share
of each alternative in the reference group, and then decides based on the
utilities of the alternatives that result from the expected shares.333The
advantage compared to expected utility maximization is that $s_{ij}$ can be
interpreted as the share of alternative $j$ in Equation (1). Expected utility
maximization requires that the individual maximizes $E(log(X))$ where $X$ is a
Dirichlet distributed random variable with parameters $\alpha_{ij}+n_{ij}$
and:  $\displaystyle
E(log(X))=\psi(\alpha_{ij}+n_{ij})-\psi\left(\sum_{k}\alpha_{ik}+n_{ik}\right)$
where $\psi(\cdot)$ is the digamma function which has the logistic
approximation $\psi(x)\approx log\left(x-\frac{1}{2}\right)$ (Johnson et al.,
1994). Using this approximation:  $\displaystyle E(log(X))\approx
log\left(\frac{\alpha_{ij}+n_{ij}-\frac{1}{2}}{\sum_{k}(\alpha_{ik}+n_{ik})-\frac{1}{2}}\right)\approx
s_{ij}$ if the quantity $\alpha_{ij}+n_{ij}$ is large for all alternatives.

#### Biased beliefs

An advantage of explicitly modeling individuals’ beliefs is that the model can
be extended to account for biases in the belief-formation process. From
Equation (1) it is clear that an individual might be in a situation in which
her intrinsic preferences and social learning suggest different alternatives.
For example, consider a conformist individual who strongly prefers one
alternative but, at the same time, believes that the other alternative is more
frequent in the reference group. Depending on the strength of her intrinsic
preferences and the strength of social learning, the individual will either
stick to the intrinsically preferred alternative at the cost of less social
utility or switch to the more popular alternative at the cost of less
intrinsic utility. One way to resolve such conflicts is by manipulating
beliefs in a way that the social utility penalty for the intrinsically
preferred alternative is less severe. Such acts of self-deception are known as
false-consensus effect (Mullen et al., 1985) or motivated beliefs (Bénabou and
Tirole, 2006).

One possibility to incorporate motivated beliefs is to define the parameters
of the prior belief distribution as:

$\displaystyle\alpha_{ij}=\phi_{i}\cdot
J\cdot\frac{e^{\delta_{i}v_{ij}}}{\sum_{k}e^{\delta_{i}v_{ik}}}$ (4)

where $\phi_{i}>0$ is the strength of prior belief and $\delta_{i}$ is a
parameter for the preference bias. If the parameters $\delta_{i}$ and
$\lambda_{i}$ have the same sign, the prior is biased such that large shares
of the intrinsically preferred alternative are more likely. If the signs of
the two parameters $\delta_{i}$ and $\lambda_{i}$ differ, the prior is biased
such that small shares of the intrinsically preferred alternative are more
likely. If $\delta_{i}$ is zero, the individual’s prior belief is unbiased and
the expected shares are $1/J$ for every alternative.

Other variants of preference-biased prior beliefs are also plausible.444An
alternative way to introduce preference-biased beliefs would be to assume that
the updating process of the individual is biased by her intrinsic preferences
(Zimmermann, 2020). This can be achieved by introducing updating weights
proportional to the intrinsic utility of each alternative. We use the form
outlined in Equation (4) because of the convenient property that the prior
expected share is:

$\displaystyle
s_{ij}=\frac{\alpha_{ij}}{\sum_{k}\alpha_{ik}}=\frac{e^{\delta_{i}v_{ij}}}{\sum_{k}e^{\delta_{i}v_{ik}}}$

If $\delta_{i}\approx\lambda_{i}$, this means that the expected prior belief
corresponds to the own behavior in the absence of social learning.

#### Comparative statics

The modeling framework conveys several stylized facts about social learning.
The four graphs of Figure 2 illustrate the effects of the parameters $\lambda$
and $f$ on the probability to choose the preferred alternative. We use the
index $j^{*}$ to indicate the preferred alternative. The probability to choose
alternative $j^{*}$ is a function of the expected share of alternative
$s_{j^{*}}$ in the reference group depicted on the x-axis. In Figure 2, this
function is plotted for a binary choice. Note that we omit the individual
index $i$ for better readability.

The four different panels illustrate the shape of the probability function for
different values of the social learning parameter $f$. Conditional on the
value of the social learning parameter, the individual is classified as
anticonformist, independent, non-conformist or conformist. While
anticonformity (left panel) implies that the probability to choose the
preferred alternative decreases in the expected frequency of the preferred
alternative in the reference group, non-conformity and conformity (central-
right and right panels) both imply a positive relation between the probability
to choose alternative $j^{*}$ and the expected share of $j^{*}$ in the
reference group. Independence (central-left panel) implies that the
probability to choose alternative $j^{*}$ is not affected by the expected
frequency of the alternative in the reference group.

Figure 2: Predicted response to social learning

Notes: The four graphs show the probability of choosing the intrinsically
preferred alternative $j^{*}$ in a binary choice as a function of the expected
frequency of the preferred alternative $j^{*}$ in the reference group
($s_{j^{*}}$). From left to right, the graphs show this probability for an
anticonformist, independent, non-conformist and conformist individual,
characterized by the social learning parameter ($f$). The solid lines assume
that $v_{j}$ is a dummy for the preferred alternative and vary the $\lambda$
parameter.
---

The solid red lines in Figure 2 show how the probability to choose alternative
$j^{*}$ is influenced by the model parameter $\lambda$. The more red the color
of a line is, the stronger is the utility difference between the alternatives.
Figure 2 illustrates that intrinsic preferences bend the probability curves
upwards which makes it more likely that the individual chooses the
intrinsically preferred alternative.

Figure 2 illustrates three stylized facts about social learning:

  1. 1.

Intrinsic preferences immunize against social information. Figure 2 shows how
intrinsic preferences mediate the response to social information. The key
aspect to observe is that intrinsic preferences for alternative $j^{*}$ (solid
red lines) create a larger region of possible beliefs for which the predicted
response curve is relatively high and flat, i.e. insensitive to changes in the
belief. This implies that (1) the individual almost certainly chooses the
intrinsically preferred alternative $j^{*}$ and (2) needs a relatively large
and homogeneous sample to overcome this tendency.

  2. 2.

Strong prior beliefs immunize against social information. Figure 2 also
illustrates the role of prior beliefs for individuals choices. Before
observing a sample of others’ behavior, the choice probability is determined
the expected share of the preferred alternative in the reference group. When
observing the sample, the individual updates the prior belief. If the observed
frequencies differ from the individual’s prior belief, the expected share will
also differ and the choice probability will change accordingly. The strength
of the individual’s prior belief defines the distance between the prior and
the posterior expected share on the x-axis of Figure 2. If choices are not
independent ($f\neq 0$), a strong prior implies a small change in the choice
probability before and after observing the sample. Conversely, this means
that, given some strength of the prior belief ($\phi>0$), the effect of social
information increases in the size of the sample.

  3. 3.

Social norms are not arbitrary. Each response curve depicted in Figure 2
implies different choice dynamics in a group of repeatedly interacting
individuals. The intersections of the solid lines with the dashed lines fix
points of the choice dynamics, assuming that all individuals are the same and
individuals’ beliefs will eventually converge towards the true choice
frequencies. For $f<1$ the red lines suggest converges to a state in which the
majority of individuals chooses the intrinsically preferred alternative. In
this state of convergence, the frequency of the majority behavior is
indicative of the strength of the intrinsic preference for the majority
choice. For conformity ($f>1$) the strength of the intrinsic preference
increases the basin of attraction of the majority choice which means that the
group converges to the intrinsically preferred choice with higher probability.
Independent of the type of social learning, the behavior of the group contains
information about intrinsic preferences of the group after convergence. We run
simulations to assess whether the behavior of the majority contains also
information about the average preferences in a population with heterogeneous
preferences. Figures 7 and 8 in the Appendix show that this is the case for a
broad range of model parameters, and also, if a minority with strong
preferences against the majority behavior exists.

The model nests other models of collective behavior. Frequency-dependent
social learning model (Boyd and Richerson, 1985) is obtained for
$\lambda_{i}:=0$ and $\phi_{i}:=0$. The influential threshold model of
collective behavior (Granovetter, 1978) is obtained by assuming
$f_{i}\rightarrow\inf$ and $\phi=0$.

We fit Bayesian multilevel models to the data of the online experiment using R
(R Core Team, 2021), Stan (Stan Development Team, 2021) and the packages RStan
(Stan Development Team, 2020) and rethinking (McElreath, 2020). Details about
the Bayesian parameter estimation can be found in the Appendix.

##  3 Results

###  3.1 Population parameters

Table 1 shows estimated population means of the model parameters along with
the 5th and 95th percentiles of their posterior distributions. For the social
learning parameter $f$, the table shows the estimates for the control
treatment, without reward or punishment. The posterior means of all population
parameters are larger than zero which suggests that participants’ decisions
are generally influenced by intrinsic preferences, social learning and
preference-biased prior beliefs. Table 1 also shows that the parameter
estimates vary substantially between the two experimental tasks.

Table 1: Population means of model parameters

| $\lambda$ | $f$ | $\phi$ | $\delta$
---|---|---|---|---
task | quest | paint | quest | paint | quest | paint | quest | paint
mean | 2.89 | 5.09 | 2.61 | 1.22 | 3.29 | 1.98 | 2.89 | 2.36
5th percentile | 0.99 | 3.25 | 1.81 | 0.57 | 1.66 | 0.48 | 1.47 | 0.32
95th percentile | 4.72 | 6.75 | 3.55 | 2.13 | 5.72 | 5.93 | 4.36 | 4.18

Notes: Table shows means and percentiles of the posterior distribution of the
model parameters. The values for the social learning parameter $f$ are
reported for the control treatment.

The posterior means of the parameter $\lambda$ indicate that the measured
intrinsic preferences influence the participants’ decisions. All estimates are
clearly positive, indicating a tendency to choose the alternative with greater
intrinsic utility. The posterior means of the social learning parameter $f$
indicate conformity in both experimental tasks for the control treatment. The
estimates indicate more conformity in the questioning task. The estimates of
the belief parameters $\phi$ and $\delta$ suggest that participants hold
preference-biased prior beliefs. The estimates of $\delta$ are positive. On
average, participants believe that it is more likely that their intrinsically
preferred alternative will be chosen by more individuals in the reference
group.

###  3.2 Variation of social learning and beliefs

Figure 3 depicts individual heterogeneity in social learning (left graphs) and
prior beliefs (right graphs) for the two experimental tasks. Each line
represents an individual estimate. The red curves indicate the mean of the
individual curves.

Figure 3: Individual social-learning strategies and beliefs

Notes: Figure depicts individual estimates for social learning (left graphs)
and prior beliefs (right graphs). Red lines indicate the average of the
individual estimates.

The upper left graphs of Figure 3 suggests that participants’ conformity
prevails in the questions task with little individual heterogeneity. The
inter-individual differences are more pronounced in the painting selection
task. In this task, the majority of individuals is conformist and displays an
S-shaped reaction to the expected frequency of an alternative. A minority is
non-conformist with an inverse S-shaped reaction.

The right graphs depict individuals’ prior beliefs about the frequency of
their intrinsically preferred alternative in the reference group. The graphs
suggest that majority of individuals discount the information presented to
them about the frequency of choices. Some individuals hold a U-shaped prior
which suggests no discounting of social information. The depicted prior
beliefs also show that most individuals hold the preference-biased prior
belief that larger frequencies are more plausible for their intrinsically
preferred alternative.

###  3.3 Social learning under reward and punishment

As in Dvorak et al. (2020), we find that the evaluators allocate punishment
and reward based on their own intrinsic preferences. Based on their theory and
experimental findings, we expect that punishment induces conformity while
reward induces anticonformity. This means that, compared to the control
treatment, the social learning parameter $f$ should be larger under punishment
leading to more conformist social learning and smaller under reward leading to
less conformist social learning.

Figure 4: Individual social-learning strategies

Notes: Individual social-learning strategies for the two experimental tasks
(rows) in different choice environments (columns). Red lines indicate the
average social-learning strategy.

Figure 4 illustrates the effects of the reward and punishment treatment on
average (red lines) and individual (grey lines) social learning strategies.
For the painting task, represented by the three panels in the second row, the
effects are in line with our prior expectations. Social learning is indeed
less conformist (negative effect on $f$) in the reward environment and more
conformist (positive effect on $f$) in the environment with punishment. In the
reward treatment (left panel), the average behavior indicated by the red line
is non-conformist, and a small fraction of individual estimates even indicates
anticonformity $f<0$. The average social learning strategy almost linear in
the control treatment (center panel), with a slight conformist twist. In the
environment with punishment, average social learning is clearly conformist.
This suggests that environments with reward can induce different choice
dynamics compared to environments with punishment, in which standing out makes
little sense.

For the questions task, the treatments effects are not consistent with prior
expectations. The different treatments have no effect on social-learning,
which is strongly conformist in all three treatments.

###  3.4 Model fit and model comparisons

A natural question is how well the estimated model fits the data, and how the
model fit compares to models with fewer parameters. Figure 5 illustrates the
model fit for average and individual behavior. The three plots in the first
row of Figure 5 plot model predictions (circles) against observed average
behavior (dots) for the two experimental tasks in each of the three
experimental treatments. The filled dots depict the average probability that a
participant chooses her intrinsically preferred alternative $j^{*}$
conditional on $n_{ij^{*}}$ \- the number of other participants that have
chosen this alternative. The three plots illustrate why it is virtually
impossible to infer the average social-learning strategy from average
behavior. The model predictions are fairly close to the observed average
behavior in the two experimental tasks of each treatment. The fit is generally
better for larger values of $n_{ij^{*}}$. There seems to be no systematic
error in the model predictions.

To illustrate the fit of the model to the individual behavior, we consider all
choices from the same individual in the same task and calculate the mean
posterior probability of these choices for different sets of model parameters
sampled from the posterior distributions. The three graphs in the lower part
of Figure 5 depict the distribution of the mean posterior probability of
participants’ choices conditional on the experimental task and treatment. In
all three graphs, most of the density mass of is above one-half - which is the
expected benchmark for random predictions indicated by the dashed line. Only
few individuals fall below this benchmark. The mode of the mean posterior
probability is around 70%.

Figure 5: Fit for average and individual behavior

Notes: Upper part shows model fit to average behavior. Circles represent model
predictions. Dots represent average behavior observed conditional on the
information about others behavior. Lower part shows the distribution of the
mean posterior probability of all the choices from the same participant.

We compare the model to three nested models. The first model does not include
a parameter for the preference-bias (PSI model). Instead, it uses
$s_{ij}=(\phi_{it}+n_{ij})/(2\phi_{it}+5)$. The second nested model is a pure
preference model that only takes the intrinsic preferences over the choice
alternatives into account (P model). The third model is a pure social learning
model in which participants’ choices depend on the expected frequencies of the
choice alternatives (SI model). The three nested models are obtained from the
preference-biased social learning model (PBSI model) outlined in Section
Bayesian multilevel modeling for the restrictions $\delta:=0$, $f:=0$ and
$\delta:=0\wedge\lambda:=0$ respectively.

Table 2 summarizes the posterior means of the model parameters for all four
models. The posterior means of the preference parameter $\lambda$ are larger
in the PSI and P models and do not differ between the experimental tasks. The
posterior means of the social learning parameters $f$ estimated for the
control treatment are smaller in the PSI and SI models and suggest a non-
conformist reaction to social information in the painting selection task. The
estimates of the parameter $\phi$ \- which indicates prior experience - are
larger in these models.

Table 2: Posterior means

| $\lambda$ | $f$ | $\phi$ | $\delta$ | WAIC
---|---|---|---|---|---
task | quest | paint | quest | paint | quest | paint | quest | paint |
PBSI | 2.89 | 5.09 | 2.61 | 1.22 | 3.29 | 1.98 | 2.89 | 2.36 | 3157
PSI | 6.28 | 6.81 | 2.08 | 0.58 | 1.55 | 0.29 | - | - | 3180
P | 6.36 | 6.68 | - | - | - | - | - | - | 3652
SI | - | - | 1.94 | 0.69 | 1.70 | 0.39 | - | - | 4192

Notes: Table compares posterior means of the model parameters of four
different models. The last column depicts the value of the widely applicable
information criterion (Watanabe, 2010) for each model.

The last column of Table 2 indicates the values of the widely applicable
information criterion (WAIC, Watanabe, 2010) for the four models.555 The
formula of the WAIC is:  $\displaystyle
WAIC(Y|\Psi)=-2\left(\sum_{m}log\frac{1}{S}\sum_{s}Pr(Y_{m}|\Psi_{s})-\sum_{m}var\left(log\left(Pr(Y_{m}|\Psi_{s})\right)\right)\right)$
where $Y$ are the observed choices and $\Psi$ are the samples used to
approximate the posterior (McElreath, 2020).  The WAIC approximates the out-
of-sample prediction accuracy of a Bayesian model. It is asymptotically
equivalent to leave-one-out cross-validation (Vehtari et al., 2017). Hence,
models with low WAIC values are preferable. The WAIC values indicate that the
model with intrinsic preferences, social learning and preference-biased prior
beliefs generates more accurate out-of-sample predictions than the nested
models.

We also fit models with fixed effects of the treatment condition for all model
parameters. We find that these models do not yield better out of sample
predictions according to the WAIC. This means that the treatment effect is
best explained by the social learning parameter. Removing the task fixed
effect of any model parameter also results in larger values of the WAIC.

###  3.5 Simulation of choice dynamics

The fitted model can be used to simulate choice dynamics of the binary choices
used in our experiment. We can go beyond the limitations of our experimental
setup and explore choice dynamics in larger groups and over a longer time
horizon. For the simulation of choice dynamics, we will have to assume that
individual model parameters and intrinsic preferences do not change over time
or when interacting in larger groups. It is important to stress that these
assumptions cannot be checked with data of our experiment. For the results
presented in the following, the reader should keep this limitation in mind.

Figure 6 shows the simulated choice dynamics for the 120 unique binary choices
of the online experiment over 100 periods. In each period, we simulate a
choice for each participant based on the measured intrinsic preferences and
the expected frequencies of others’ decisions. After each period, each
participant is informed about the decisions of five participants randomly
samples from the same choice environment in the same choice. To simulate
participants’ choices and update their beliefs, we use the posterior means of
individual parameter estimates.

Figure 6: Simulation of choice dynamics

Notes: Simulation of choice dynamics for the 120 unique binary choices using
fitted individual choice parameters. After each period, each participant is
informed about the decisions of five other participants in the same binary
choice, randomly sampled from the same choice environment. Each line
represents one binary choice and shows the evolution of the frequency of one
of the two alternatives, the target alternative, over 100 periods. Red lines
indicate an average preference for the target alternative.

Each line in the six panels of Figure 6 shows the choice dynamics of one
binary choice. On the vertical axis, we plot the share of one of the two
alternatives, the target alternative. For the questions task, the target
alternative is the correct answer to the question. For the painting selection
task, we randomly select one of the two paintings that then represents the
target alternative for every participant. The color of each line reflects the
average preferences of the interacting participants in each panel. A red line
indicates an average preferences for the target alternative. A black line
indicates the opposite, i.e. average preferences for the other alternative.

The panels for the questions task in the first row of Figure 6 show many
instances of convergence of opinions after 100 periods. The panels also show
considerable sorting according to the average preferences of the participants.
Red lines tend to converge to a target share of one, and black lines tend to
converge to target share of zero. This is particularly the case for choices
that converge early. In the painting selection task depicted in the lower
three panels, the fitted model parameters suggest few instances convergence
after 100 periods and the effect of the choice environment on the diversity of
opinions is clearly visible. Sorting takes place nevertheless as red lines
tend to lie above one-half, and black lines below one-half, which indicates
that the majority choice corresponds to the average preferences in the
population.

In summary, the results of the simulation suggest that the descriptive norm
after several periods of social interaction can be used to infer the average
preferences of the participants. This is true for both choice tasks and all
three choice environments.

##  4 Discussion

Understanding individual social learning from observational data is
notoriously difficult. Manski (1993) highlights the fundamental problem of
separating social learning from correlated effects, i.e., the fact that people
behave similarly because they have similar intrinsic preferences over choice
alternatives. Thus, to isolate intrinsic preferences from social learning, one
must control for individuals’ intrinsic preferences. This raises the problem
that in observational settings, individuals’ intrinsic preferences are often
unknown and can only be inferred from their choices - leading to a fundamental
identification problem.

In this paper, we use a combination of experimental data and discrete choice
modeling to understand the complexity and determinants of the interplay
between preferences and social learning. We show that preferences and social
learning jointly influence participants’ decisions across experimental
conditions, suggesting that models of human behavior must account for both
factors. Participants follow a diverse set of social-learning strategies that
are shaped by features of the choice task as well as features of the choice
environment. The individual variation we find in social learning strategies is
important because it goes beyond different facets of conformity. At the same
time, contextual variation in social learning strategies follows a clear
pattern. Social learning is predominantly conformist in tasks with an
objectively correct solution. This changes little when incentives for
anticonformity are introduced. In contrast, individuals display a variety of
social learning strategies in tasks that are subjective and do not have an
objectively correct solution. In such tasks, the incentives provided by the
environment play a crucial role in the nature of social learning strategies,
shifting toward conformism in punishment environments and toward nonconformism
in reward environments.

Methodologically, we derive these results based on a modeling framework that
is inspired by different disciplines: we combine random utility theory
developed in the economics literature with the idea of frequency-dependent
social learning which is used in psychology, biology and behavior ecology to
study collective behavior in humans and other animals (Boyd and Richerson,
1985; McElreath et al., 2005, 2008; Efferson et al., 2008; Aplin et al., 2017;
Barrett et al., 2017; Toyokawa et al., 2019; Deffner et al., 2020; Toyokawa
and Gaissmaier, 2021). The advantage of the combined model is that it can
disentangle the role of individual preferences and beliefs and social
learning.

Our modeling approach differs in several important ways from the influential
threshold model of collective behavior Granovetter (1978), which can be
considered the standard approach to integrating individual variation in
preferences and conformity. Interpreted as a model of intrinsic preferences
plus conformity, the threshold model assumes that each individual can be
characterized by an individual choice frequency threshold for following the
majority. Individuals with strong minority preferences are characterized by
large thresholds because they face higher intrinsic costs of conformity. The
equilibria of collective behavior are defined by the cumulative distribution
function of thresholds in the population. Our modeling framework allows for a
more detailed analysis of individual social learning, featuring linear and
nonlinear responses to social influence, and also accounts for variation in
individual beliefs about the behavior of others.

Beliefs are important when people have prior experience with others’ behavior
or extrapolate from similar experiences, but also if the individual cares
about the entire reference group and only observes random sample. A similar
argument applies if others’ decisions are subject to errors and observations
can thus be surprising because they are not in line with the individual’s
prior belief. The seminal studies of Asch (1952; 1956) on conformity provide a
good example. One explanation why most participants did not conform to the
wrong majority opinion is that it was in strong contrast to their prior
expectation about the majority opinion. Asch’s finding that the number of the
majority substantially increases conformity to the majority can also be
interpreted along those lines. Prior experience plays an important role in
field experiments, and particularly, in natural field experiments on peer
influence.

The modeling framework used in this study can explain several stylized facts
about social learning. The first concerns the limits of social learning. The
model suggests that those who do not learn socially are either idealists with
strong intrinsic preferences or strong believers with substantial prior
knowledge about peer behavior. Idealists do not respond to social information
because their intrinsic preference is too strong for social learning. Strong
believers, on the other hand, are unresponsive to social information because
their prior belief is too strong to be significantly affected by information
about others’ behavior. Both these mechanisms can also differences in
individual behavior across situations.

The model can also explain the stylized fact that conformity increases with
the size of the majority - a finding that pervades the experimental literature
on conformity (Bond and Smith, 1996; Sasaki, 2019). The reason is that, given
the same prior belief, large samples have a greater effect on the individual’s
posterior belief.666Prior experience can also explain social inertia in
behavior which is sometimes observed when social learning is supposed to
induce social change. If the status-quo exist for a long time, individuals
develop strong beliefs that, in combination with conformity, stabilize the
status-quo (Smerdon et al., 2020). Another advantage of explicitly modeling
individuals’ beliefs is that models of social learning can be extended to
incorporate biases in the formation of beliefs (Bénabou and Tirole, 2002;
Köszegi, 2006; Zimmermann, 2020). Such biases seem plausible since individuals
frequently face a trade-off between intrinsic preferences and social learning
and one way to resolve the conflict is to engage in self-deception. For
example, a conformists could overestimate the frequency of her preferred
alternative to resolve the conflict that arises when she intrinsically prefers
the minority choice.  This also has an econometric advantage. When fitting
frequency-dependent social learning models to the decisions of individuals
influenced by small samples, the models predict some decisions with certainty
because observing small homogeneous samples generates exceptionally strong
social learning. A single decision in the data that does not match the model’s
prediction means that the model has a likelihood of zero, making it impossible
to estimate parameters. Prior beliefs effectively reduce the impact of small
samples and circumvent this problem.

Finally, and perhaps most importantly, the modeling framework developed in
this study can explain why it can be reasonable to infer intrinsic preferences
from average behavior even when individuals are social learners who have
interacted in the past. The reason is that the basin of attraction of the
behavior preferred by the majority is larger. If all individuals are
conformists and have the same intrinsic preferences, the group’s choices are
more likely to converge on the intrinsically preferred behavior. If the
individuals are nonconformists, the choices will not converge, but the
intrinsically preferred behavior will still be performed more often. Thus, the
model supports the view that social norms are not arbitrary, but have some
intrinsic value beyond the mere benefits of coordination.

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## Appendix

#### Bayesian multilevel modeling

We fit Bayesian multilevel models to the data of the online experiment using R
(R Core Team, 2021), Stan (Stan Development Team, 2021) and the packages RStan
(Stan Development Team, 2020) and rethinking (McElreath, 2020). The advantage
of the multilevel specification is that the amount of information pooling
between individual parameter estimates is defined by the data. This
effectively prevents over-fitting of individual data, which would lead to
overestimation of the variation in social-learning strategies and other
individual differences.

The multilevel models include individual random effects (index $i$) and fixed
effects for the experimental task (index $t$) for all model parameters.
Importantly, the random effects are drawn from a common distribution
characterized by a population mean and some finite population variance. We
introduce additional fixed effects for the effect of the treatment conditions
(index $c$) on the social learning parameter $f$ to estimate the effect of the
experimental treatment on this model parameter. Hence, we assume that the
effects of the choice task and the choice environment have a similar effect on
all individuals and do not affect the individual variation of social learning.

We use standard priors that result in expected effects of zero which
corresponds to random choice and Hamiltonian Monte-Carlo (Neal, 2011;
Betancourt, 2013) to approximate the posterior distributions of the
parameters.777 We use the standard normal distribution for the prior of the
fixed effects of all model parameters except $\phi$ which cannot be negative.
For the parameter $\phi$, we use $exp(N(0,1))$. As prior for the individual
varying effects, we use the multivariate normal distribution $MVN(0,\Sigma)$
with:  $\displaystyle\Sigma=\begin{pmatrix}\sigma_{\lambda}&\dots&0\\\
\vdots&\ddots&\vdots\\\
0&\dots&\sigma_{\delta}\end{pmatrix}\Omega\begin{pmatrix}\sigma_{\lambda}&\dots&0\\\
\vdots&\ddots&\vdots\\\ 0&\dots&\sigma_{\delta}\end{pmatrix}$ where
$(\sigma_{\lambda},\sigma_{f},\sigma_{\phi},\sigma_{\delta})\sim
Exponential(1)$ and $\Omega\sim LKJ(2)$ (Lewandowski et al., 2009). Our main
results are not sensitive to the specification of priors we use.

We estimate the model with preference biased prior beliefs. In the statistical
model, the probability that participant $i$ chooses alternative $j$ is:

$\displaystyle
Pr(Y_{i}=j)=\frac{e^{\lambda_{it}\nu_{ij}}(s_{ij})^{f_{itc}}}{\sum_{k}e^{\lambda_{it}\nu_{ik}}(s_{ik})^{f_{itc}}}$

The model parameter $\lambda_{i}$ reflects the effect of the estimated
intrinsic utility $\nu_{ij}$ on the individual’s choice. The exponential form
implies that the choice probability is a function of the difference in
intrinsic utility of both alternatives that we measure by $\Delta$. For
convenience we use $\nu_{ij}:=\frac{\Delta}{2}$ if $j$ is predicted to be the
preferred alternative, and $\nu_{ij}:=-\frac{\Delta}{2}$ otherwise. The
exponential form of the intrinsic preference component assures that different
distributions of $\Delta$ over the two choice alternatives will give the same
results.

The model parameter $f_{itc}$ reflects the effect of social learning based on
$s_{ij}$, the frequency of alternative $j$ expected by the participant. The
expected frequency of alternative $j$ is:

$\displaystyle
s_{ij}=\frac{2\phi_{it}\frac{e^{\delta_{it}\nu_{ij}}}{\sum_{k}e^{\delta_{it}\nu_{ik}}}+n_{ij}}{2\phi_{it}+5}$

The parameter $\phi_{it}$ is the strength of the prior and can be interpreted
as the number of past observations of each alternative. The parameter $\delta$
reflects the preference bias of the prior belief. As before, $n_{ij}$
indicates the number of observed uninfluenced participants that chooses
alternative $j$.

### Simulations on social norms and preferences

We run simulations to assess whether the behavior of the majority contains
information about the average preferences in a population with heterogeneous
preferences. We simulate the choices of $N=50$ individuals over 50 periods
between a commonly preferred alternative and another alternative. For a
specific combinations of model parameters, we repeat the simulation 50 times,
and report the average share of a target alternative across the 50
simulations.

We assume a standard normal distribution of preferences. Across the
simulations we vary the mean preferences in steps of 0.1 ranging from 0 (no
common preference) to 4 (strong common preferences).

For the other model parameters, we assume no heterogeneity. We vary the social
learning parameter $f$ from -2 (anticonformity) to 2 (conformity) in steps of
0.1. As possible strength of the prior beliefs we use values of 1, 5, and 20.
The preference-bias of prior beliefs is either 0, 0.5 or 1. To also assess the
influence of an extreme minority, we use a second set of simulation, for which
we assume that every fifth individual has preferences of -4, with no
heterogeneity.

Figure 7: Simulation Figure 8: Simulation with minority

Figures 7 and 8 show that the average share of the commonly preferred
alternative in the last period of the 50 independent simulation runs. The
color coding of the graphs shows averages above one-half in green and yellow
colors. These colors are usually more frequent, which suggests that the
descriptive social norm usually contains information about the average
preferences of the population. An exception to this rule applies with an
extreme minority and weak majority preferences. In these cases, conformist
social learning induces convergence on the alternative preferred by the
minority.

### Study materials and instructions

Figure 9: Decision screen with slider Figure 10: Evaluation screen

Table 3: List of paintings

Set | Theme | Painting 1 | Painting 2 |
---|---|---|---|---
1 | Marc Chagall | Bride and Groom | The couple |
2 | Lyonel Feininger | The Grain Tower at Treptow on the Rega | Village Pond of Gelmeroda |
3 | Claude Monet | The Artist’s Garden in Giverny | The artist’s Garden in Giverney |
4 | Wassily Kandinsky | Improvisation 26 | Improvisation 28 (2nd version) |
5 | August Macke | Garden Restaurant | Large Bright Walk |
6 | Franz Marc | Yellow Cow | The Dog in Front of the World |
7 | Caspar D. Friedrich | The Churchyard Gate | Hutten’s Grave (Ruin of a Church Choir ) |
8 | Houses | Egon Schiele, House with drying Laundry | Albrecht Duerer, The Castle of Trient |
9 | Work | Paul Cézanne, The Mowers | Vincent van Gogh, Field with Farmer and Mill |
10 | Women | August Macke, Portrait with Apples | Pablo Picasso, The Absinthe-Drinker |
11 | Trees | Paul Cézanne, The Chestnut Trees at Jas de Bouffan | Rudolph von Alt, Landscape in the Prater in Vienna |
12 | Hands | Albrecht Duerer, The Hands of Jesus Christ | Pablo Picasso, Crossed Hands |
13 | Ships | Egon Schiele, Fishing Boats in Trieste | Berthe Morisot, The Harbour of Nice |
14 | Flowers | Paul Cézanne, Flowers in a Vase and Fruit | Vincent van Gogh, Bouquet of Irises |
15 | Bridges | Claude Monet, Bridge over the Seine near Argenteuil | Vincent van Gogh, The Bridges of Asnières |
Set | Theme | Painting 3 | Painting 4 |
1 | Marc Chagall | The Newly-Married of the Eiffel-Tower | Lovers |
2 | Lyonel Feininger | Gelmeroda IX | The Church of Halle |
3 | Claude Monet | The artist’s Garden in Vétheuil | Resting under the Lilac |
4 | Wassily Kandinsky | Improvisation 34 (Orient II) | Improvisation Gorge |
5 | August Macke | Girls under Trees | Sunny Path |
6 | Franz Marc | The Tiger | Fox (Blue Black Fox) |
7 | Caspar D. Friedrich | Ruins of the Monastery Eldena | The Graveyard Door (The Churchyard) |
8 | Houses | Rudolph von Alt, The ”Goldene Dachl” in Innsbruck | Carl Spitzweg, A Hypochondriac |
9 | Work | Carl Spitzweg, The Walk of the Boarding School | Pond at the Forest |
10 | Women | Egon Schiele, Peasants-Girl | Gustav Klimt, Johanna Staude |
11 | Trees | Vincent van Gogh, Road with Cypress and Star | Alfred Sisley, The Path to the Old Ferry at By |
12 | Hands | Egon Schiele, Clasped Hands | August Rodin, The Cathedral - Hands |
13 | Ships | Redon Odilon, The Mystical Boat | Raoul Dufy, The old Harbour of Marseille |
14 | Flowers | Claude Monet, Vase of Flowers | Pierre-Auguste Renoir, Bouquet of Chrysanthemums |
15 | Bridges | Alfred Sisley, Bridge near Hampton Court | William Turner, Old Welsh Bridge, Shrewsbury |

Before each session, we selected 10 sets and for each set three of the
paintings listed based on availability. Postcards of the paintings were
ordered from Kunstverlag Reisser, Braunschweigstrasse 12, 1130 Vienna,
Austria. Names are translations from German and taken from http://www.reisser-
kunstpostkarten.de.

Table 4: Question sets 1-9 Set | Question | Option 1 | Option 2 | Option 3 | Answer 1 | Answer 2 | Answer 3
---|---|---|---|---|---|---|---
1 | Which country is larger (2015, m2̂)? | Canada | USA | China | 9984k | 9826k | 9596k
1 | Which country is larger (2015, m2̂)? | Portugal | Czech Republic | Austria | 92090 | 78867 | 83871
1 | Which country is larger (2015, m2̂)? | Estonia | Denmark | Netherlands | 45228 | 43094 | 41543
1 | Which country is larger (2015, m2̂)? | Lithuania | Croatia | Latvia | 65300 | 56594 | 64589
1 | Which country is larger (2015, m2̂)? | Sudan | Indonesia | Mexico | 1,861k | 1,904k | 1,964k
2 | Which country has more inhabitants (2014)? | France | Italy | UK | 65,835k | 60,782k | 64,351k
2 | Which country has more inhabitants (2014)? | Spain | Ukraine | Poland | 46,512k | 45,245k | 38,017k
2 | Which country has more inhabitants (2014)? | Greece | Belgium | Czech Republic | 10,926k | 11,203k | 10,512k
2 | Which country has more inhabitants (2014)? | Austria | Switzerland | Bulgaria | 8,506k | 8,139k | 7,245k
2 | Which country has more inhabitants (2014)? | Malta | Luxemburg | Iceland | 425k | 549k | 325k
3 | Which company had more employees (2014)? | Bosch | Daimler | Metro | 290,183 | 279,972 | 24,9150
3 | Which company had more employees (2014)? | Bayer | ThyssenKrupp | Continental | 118,900 | 160,745 | 189,168
3 | Which company had more employees (2014)? | Lufthansa | BASF | BMW | 118,781 | 113,292 | 116,324
3 | Which company had more employees (2014)? | RWE | E.ON | MAN | 59,784 | 58,503 | 55,903
3 | Which company had more employees (2014)? | Bertelsmann | SAP | TUI | 112,037 | 74,406 | 77,309
4 | Who was born earlier? | Konrad Adenauer | F.D. Roosevelt | Theodor Heuss | 1876 | 1882 | 1884
4 | Who was born earlier? | Willy Brandt | John F. Kennedy | Walter Scheel | 1913 | 1917 | 1919
4 | Who was born earlier? | Helmut Schmidt | Richard Nixon | R. Weizsaecker | 1918 | 1913 | 1920
4 | Who was born earlier? | Horst Koehler | Gerhard Schroeder | Bill Clinton | 1943 | 1944 | 1946
5 | Which harbor is bigger (2014, TEU)? | Shanghai | Hong Kong | Singapore | 35.3 | 22.30 | 33.9
5 | Which harbor is bigger (2014, TEU)? | Hamburg | Antwerp | Los Angeles | 9.7 | 9 | 8.3
5 | Which harbor is bigger (2014, TEU)? | Guangzhou | Dubai | Rotterdam | 16.2 | 15.2 | 12.3
6 | Which airline had more passengers? | United Airlines | American Airlines | Ryanair | 90,440k | 87,830k | 86,370k
6 | Which airline had more passengers? | Lufthansa | Easyjet | Air China | 59,850k | 62,310k | 54,580k
6 | Which airline had more passengers? | Air Berlin | Brithish Airlines | Air France | 29,910k | 41,160k | 45,410k
6 | Which airline had more passengers? | KLM | Aeroflot | SAS | 27,740k | 23,600k | 27,390k
7 | Which country discharges more CO2 (2010, pp)? | Germany | Netherlands | Austria | 12.3 | 10.1 | 12.1
7 | Which country discharges more CO2 (2010, pp)? | Poland | Slovakia | Hungary | 7.7 | 7.8 | 7.3
7 | Which country discharges more CO2 (2010, pp)? | Lithuania | Latvia | Estonia | 5.9 | 6.5 | 13.5
7 | Which country discharges more CO2 (2010, pp)? | France | Portugal | Spain | 9 | 6.9 | 8.5
7 | Which country discharges more CO2 (2010, pp)? | Finland | Norway | Sweden | 18.7 | 10.1 | 9.3
8 | Which country has more inequality (2012, GINI)? | France | Belgium | Austria | 33.1 | 27.6 | 30.5
8 | Which country has more inequality (2012, GINI)? | Norway | Finland | Sweden | 25.9 | 27.1 | 27.3
8 | Which country has more inequality (2012, GINI)? | Bolivia | Ecuador | Peru | 46.7 | 46.6 | 45.1
8 | Which country has more inequality (2012, GINI)? | Costa Rica | Brazil | Argentina | 48.6 | 52.7 | 42.5
8 | Which country has more inequality (2012, GINI)? | Thailand | Laos | Vietnam | 39.3 | 37.9 | 38.7
9 | Which soccer club is worth more (2016)? | Manchester City | FC Chelsea | M. United | 501.75 | 490 | 411.25
9 | Which soccer club is worth more (2016)? | AS Rom | FC Valencia | SSC Neapel | 250.7 | 282 | 284
9 | Which soccer club is worth more (2016)? | Bayer 04 Leverkusen | VfL Wolfsburg | FC Schalke 04 | 211.1 | 183.1 | 199.8
9 | Which soccer club is worth more (2016)? | Zenit St.Petersburg | AC Mailand | FC Sevilla | 198.6 | 188.1 | 186.2

Table 5: Question sets 10-19 Set | Question | Option 1 | Option 2 | Option 3 | Answer 1 | Answer 2 | Answer 3
---|---|---|---|---|---|---|---
10 | Which country won more medals (2014 Olympics? | Netherlands | France | Germany | 24 | 15 | 19
10 | Which country won more medals (2014 Olympics? | Switzerland | Sweden | Austria | 11 | 15 | 17
10 | Which country won more medals (2014 Olympics? | Canada | Norway | USA | 26 | 28 |
10 | Which country won more medals (2014 Olympics? | Finland | UK | Ukraine | 5 | 4 | 2
10 | Which country won more medals (2014 Olympics? | Belarus | Kazakhstan | Australia | 6 | 1 | 3
11 | Which airport has more passengers (2014)? | Atlanta Int | L Heathrow | Dubai Int | 96,178k | 73,408k | 70,475k
11 | Which airport has more passengers (2014)? | Singapore Changi | Kuala Lumpur | Shanghai Int | 54,093,000 | 48,930k | 51,687k
11 | Which airport has more passengers (2014)? | Charles de Gaulles | Frankfurt | A Schiphol | 63,813k | 59,566k | 54,978k
11 | Which airport has more passengers (2014)? | Madrid Barajas | SP-Guarulhos | Miami Int | 41,822k | 39,765k | 40,941k
12 | Who sold more records in Germany? | The Beatles | Michael Jackson | Madonna | 7,600k | 11,275k | 12,300k
12 | Who sold more records in Germany? | ACDC | ABBA | R. Williams | 10,475k | 10,800k | 9,275k
12 | Who sold more records in Germany? | Helene Fischer | Pur | Die Aerzte | 9,150k | 9,425k | 7,850k
12 | Who sold more records in Germany? | Britney Spears | Bon Jovi | Xavier Naidoo | 5,050k | 5,150k | 5,525k
13 | In which language is the letter ”a” more frequent? | German | English | French | 6.51 | 8.167 | 7.636
13 | In which language is the letter ”a” more frequent? | Spanish | Italian | Swedish | 12.53 | 11.740 | 9.300
13 | In which language is the letter ”a” more frequent? | German | English | French | 17.4 | 12.702 | 14.715
13 | In which language is the letter ”a” more frequent? | Spanish | Italian | Swedish | 13.68 | 11.790 | 9.900
13 | In which language is the letter ”a” more frequent? | Spanish | Italian | Swedish | 6.71 | 6.880 | 8.800
14 | Which initial letter is more common in German? | E | I | W | 7.8 | 7.1 | 6.8
14 | Which initial letter is more common in German? | H | I | O | 7.232 | 6.286 | 6.264
14 | Which initial letter is more common in German? | C | D | F | 3.511 | 2.670 | 3.779
14 | Which initial letter is more common in German? | J | K | V | 0.597 | 0.590 | 0.649
15 | Which country has more prisoners (2016, per 100k)? | USA | Cuba | Seychelles | 698 | 510 | 799
15 | Which country has more prisoners (2016, per 100k)? | Thailand | Russia | Ruanda | 468 | 447 | 434
15 | Which country has more prisoners (2015, absolute)? | Berlin | Saxony | Rhineland | 3806 | 3385 | 3102
15 | Which country has more prisoners (2015, absolute)? | Saxony-Anhalt | Thuringia | Hamburg | 1670 | 1600 | 1559
15 | Which country has more prisoners (2015, absolute)? | Schleswig-Holstein | Mecklenburg | Brandenburg | 1162 | 1057 | 1324
16 | Which food has more calories (per 100g)? | Paprika Yellow | Paprika Red | Paprika Green | 28 | 33 | 20
16 | Which food has more calories (per 100g)? | Rhubarb | Radicchio | Peperoni | 14 | 13 | 20
16 | Which food has more calories (per 100g)? | Zucchini | Spinach | Pak Choi | 18 | 15 | 16
16 | Which food has more calories (per 100g)? | Leek | Broccoli | Red cabbage | 24 | 26 | 22
16 | Which food has more calories (per 100g)? | Wild garlic | Eggplant | Artichoke | 19 | 17 | 22
17 | Which country is older? | Albania | Finland | Hungary | 1912 | 1917 | 1918
17 | Which country is older? | New Zealand | Norway | Panama | 1907 | 1905 | 1903
17 | Which country is older? | Ghana | Niger | Togo | 1957 | 1958 | 1960
17 | Which country is older? | Tanzania | Ruanda | Mali | 1964 | 1962 | 1960
17 | Which country is older? | Brazil | Uruguay | Costa Rica | 1822 | 1825 | 1821
18 | Which country has more internet users (2015)? | Austria | Germany | UK | 83.1 | 88.4 | 91.6
18 | Which country has more internet users (2015)? | Luxemburg | Netherlands | Denmark | 94.7 | 95.5 | 96
18 | Which country has more internet users (2015)? | Portugal | Italy | Greece | 67.9 | 62 | 63.2
18 | Which country has more internet users (2015)? | Myanmar | Laos | Nepal | 12.6 | 14.3 | 18.1
18 | Which country has more internet users (2015)? | Jamaica | Peru | Panama | 53.6 | 52.6 | 52
19 | Which country has more analphabets (relative)? | Guinea | Niger | Burkina Faso | 74.7 | 84.5 | 71.3
19 | Which country has more analphabets (relative)? | Mali | Chad | Ethiopia | 66.4 | 62.7 | 61
19 | Which country has more analphabets (relative)? | Liberia | Haiti | Sierra Leone | 57.1 | 51.3 | 55.5
19 | Which country has more analphabets (relative)? | Pakistan | Bhutan | Senegal | 45.3 | 47.2 | 47.9
19 | Which country has more analphabets (relative)? | Nigeria | Mozambique | Gambia | 48.9 | 49.4 | 48

### Instructions of the online experiment

Instructions

We ask you to keep quiet at your computer workstation and not to communicate
with others during the experiment. Anyone who intentionally violates this rule
is requested to leave the experiment without payment. If you have any
questions, please contact us and wait until an experimenter comes to you.
Income is calculated in points. At the end of the experiment, the total amount
of points you have earned will be converted into euros at the following rate:

1 point = 1 Euro

You will receive your total income in cash at the end of the experiment.

Please read the instructions carefully. When all the instructions have been
read, answer some comprehension questions. Then make your decisions in the
experiment. Your decisions will be treated anonymously.

General procedure

This experiment consists of two parts, each consisting of three stages. In
each step you make several decisions.

You have a certain number of seconds for each stage. The remaining time is
always displayed on the top right of the screen. The time for each stage is
such that it should be sufficient to process all decisions. When the time is
up, the experiment automatically continues. The remaining decisions are then
randomly generated by the computer. If all participants have completed all the
decisions of a stage before the end of the specified time, the experiment
continues immediately.
For each decision you work on within the given time, they will be credited a
certain amount of Experimental Currency Units. Your total income is the sum of
the EWE from all parts in Euro.

In this experiment, we share with other participants some of your decisions.
Since the other participants will not receive any other information about you,
your decisions will remain anonymous. You will also be shown decisions from
other participants. If other participants have not processed a decision in the
given time, you will be informed that the displayed decisions are randomly
generated decisions.

Overview Part 1

Below are the instructions for Part 1. You will receive the instructions for
Part 2 when Part 1 is completed. Your decisions in Part 1 do not affect your
income in Part 2.

Which postcard do you choose?

For the first part, you will be randomly divided into groups of 6 people. In
this part, you will each decide between two art postcards. To do this, you
will be shown the designs of the two cards on the screen and you will choose
which of the two designs you would prefer to have. These choices are made for
multiple pairs of postcards. The other members of your group also make such
decisions.

At the end of the experiment, a decision situation will be randomly selected
for your group. You will then be sent your preferred design as a postcard per
mail.

This part consists of 3 stages that you will work on one after the other. The
stages are described in more detail below.

Stage 1

On the screen you will see two postcard motifs A and B. You decide which of
the two motifs you prefer. To do so, you simply click the corresponding motif
and confirm your choice later.

After each decision, we ask you to indicate how strong your preference is for
the motif you selected. To do this, a slider will appear below the paintings
after your decision. Please try this out by clicking on one of the motifs
below.

You make these choices 50 different combinations of postcards. For each
decision processed within the given time, you are credited 0.5 EWE.

Stage 2

Also in this stage, you decide between two postcard motifs and indicate how
strong your preference is for the chosen motif.

Before each decision, you will be informed about the decisions of the five
other members of your group in the same choice. However, the other
participants have already gone through the decision situation in stage 1. That
is, they had no information about the decisions of other participants at the
time of their decision. The paintings chosen by the others are displayed in
random order on your screen. An example of the display can be found below.

You make this decision one by one for 10 pairs of postcards. You will be
credited 0.5 EWE for each edited decision of the second part within the given
time.

Stage 3

In Stage 3, the 6 selected paintings of each group are transmitted to a member
of another group for evaluation. Based on the selected paintings, the member
of the other group selects one person in the group to receive a Bonus of 10
points. You will also judge the choices of participants of another group.

All evaluators work on several decision situations of the other group. You do
not know which decisions were made in stage 1 and which in stage 2. Also, the
positions where you see the paintings of the six group members are randomized
for each decision situation.

In addition, all evaluators process several more randomly generated decision
situations. They do not know which decisions are real and which are randomly
generated decision situations. At the end of the first part, a real decision
situation is drawn for each group. The person selected in this decision
situation will then receive the bonus. Below is an example of the judgment
situation that results from the example in Stage 2. One member of your group
has chosen painting A and five have chosen painting B. Your decision appears
at the bottom right next to center. You can now take the role of the member of
the other group and click on one of the six paintings. The painting of the
selected person will be highlighted by a green frame.

control questions

Please use the control questions to check that you have understood the
instructions correctly. Your answers will have no effect on the experiment.

Q1: How many blocks does this experiment have? Q2: How many parts does each
block have? Q3: The decisions of how many other participants do you see in
part 2? Q4: In which part did the other participants make their decisions? Q5:
What happens to the unprocessed decisions of a person when the time for a
stage has expired? These will be…? (randomly generated, repeated later) Q6:
What applies always when a decision was randomly generated? (I will be
informed, I will not be informed)

conclusion

Finally, you will receive an overview of your income from the experiment. We
will transfer the money to the account you specified.

In addition, you will receive the design you chose in the randomly selected
decision situation as a postcard. The postcard will be sent to you by mail in
the next few days.