Dissertation

Queues, Multiple values,
and Information Cascades
A literature review and experimental design for
investigating the impact of waiting costs and
multiple values on Information cascades.
Student: Jethro Elsden
Student number: 4248665
Supervisor: Professor Sonderegger
This Dissertation is presented in part fulfillment of the requirement for the completion of an
MSc in the School of Economics, University of Nottingham. The work is the sole
responsibility of the candidate.

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Table of Contents
1. Introduction....................................................................2
2. Literature Review .........................................................2
3. Experiment .................................................................. 18
3.1. Outline...................................................................................... 18
3.2. Basic Treatment .................................................................... 19
3.3. Waiting Treatment................................................................ 20
3.4. Multi-value Treatment ........................................................ 23
3.5. Experiment Summary.......................................................... 23
3.6. Technical Analysis............................................................... 24
3.6.1. Analysis of Basic Treatment ........................................ 25
3.6.2. Analysis of Waiting Treatment ................................... 27
3.6.3. Analysis of Multi-value Treatment ............................ 29
3.7. Running the Experiment..................................................... 31
3.8. Extensions and Improvements.......................................... 34
4. Conclusion ...................................................................35
5. Appendix.......................................................................36
6. Bibliography.................................................................43

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1. Introduction
Information cascades are widespread phenomenon that can have large economic
consequences and are important in explaining the large amounts of conformity present in
human societies (Bikhchandani, et al., 1992). There is a large literature existing on the topic
of information cascades, and the subject is examined both theoretically and experimentally.
However there are several interesting aspects of the topic that are relatively unexplored and
the experimental literature have focussed on only a few of the issues concerning the topic.
This study first reviews the key theoretical and experimental literature on the topic of
information cascades. An innovative experiment is then developed which focuses on
exploring the impact on cascade formation and information aggregation of queueing (and the
associated waiting costs) and a larger number of options than have previously been offered to
subjects in experiments concerning information cascades.
2. Literature Review
TheoreticalLiterature
The seminal theoretical paper on information cascades is Bikhchandani, et al (1992); most
theoretical and experimental papers on the topic have been informed and influenced by this
paper. Of particular importance has been the underlying mechanism contained within the
paper. The authors seek to explain conformity and imitation using information cascades,
defined as an agent ignoring their own information and instead basing their decision on the
actions of predecessors. This contrasts with the traditional explanations for conformity:
deviation sanctions, positive payoff externalities, conformity preferences, and
communication. These four factors imply robust conformity which will strengthen the longer
it lasts. But this implication is empirically suspect, conformity is often fragile not robust;
small shocks can lead to large shifts in behaviour.

3
The underlying mechanism:
Imagine a scenario where individuals choose to adopt or reject an action having first observed
the decisions of all prior agents and use Bayesian learning. If the cumulative weight of prior
decisions outweighs the individual’s private information, then it is optimal for them to imitate
these previous decisions and ignore their private information. This individual’s decision is
now no more informative than the prior decisions upon which it was based; because the
decision doesn’t transmit any of the individual’s private information it is useless to
subsequent agents. These subsequent agents will have the same information as the current
individual except for their private signals, and so they may make the same decision and also
imitate. Hence these subsequent decisions will also transmit no additional information, this is
an information cascade.
The above illustrates the primary issue with cascades: It is rational for individuals to imitate,
but imitation doesn’t transmit any of their private information. This can lead to social
inefficiency, because every agent that imitates reduces the volume of private information that
is publicly aggregated. This may result in poorer decision making by subsequent agents and
this will produce a social loss.
This issue is compounded by the fact that cascades can begin on the basis of only a small
amount of information: All that is required for a cascade to begin is for the accumulated
decisions of prior agents to outweigh the present agent’s private information; this might arise
after only a few agents. For instance if the first two agents both adopt, even if the third agent
has private information indicating that they should reject the action, the fact that the first two
agents adopted may be enough to cause the third agent to ignore their private information and
also adopt. If all agents receive signals of similar strength and precision, then all agents
following the third agent are likely to make the same decision and imitate. Therefore an
information cascade has been started based on only the private information of the first two
agents. This is likely to further worsen decisions and increase the social loss; it also means
that cascades are likely to be fragile, with cascades occurring close to the borderline between
adoption and rejection. A small amount of new contrary information (such as a public
information announcement) could break or reverse the cascade.

4
The history of medicine provides many examples of unproven medical fads, and encapsulates
the main issues surrounding information cascades. For many years tonsillectomy was, despite
no conclusive evidence, routinely performed unnecessarily on healthy children, often to their
detriment (Robin, 1984). The prime motivation for surgeons performing these operations
seems to have been the fact that other surgeons were performing them, resulting in naïve
imitation and cascade behaviour. This point is supported by data on the prevalence of
tonsillectomy across England, which shows wide variation between regions, suggesting the
presence of local cascades (Taylor, 1979). However it should be emphasised that imitation
and conformity are not universally disadvantageous; they can be either harmful or beneficial
depending on the context. For instance, inexperienced agents may find it beneficial to copy
the actions of more experienced agents, but these experienced individuals may still make
errors1 and if they do this could result in a large social loss.
It is important to note as Bikhchandani, et al (1992) do that it is entirely rational for
individuals to take account of other agents’ actions when making their own decision, even if
they place no value on conformity for its own sake. Imitation is widespread throughout nature
and society, imitation among animals is common when selecting territory or foraging or
mating. The main contribution of this paper is modelling imitation as information cascades:
The actions of other agents, provided they aren’t imitating, are based on the private
information of those agents, the current agent can infer this private information from the
actions. Therefore, considering the actions of other agents allows the current agent to
accumulate more information than just their own private information. Having more
information should produce improved decisions. Furthermore individuals who act in this
manner don’t need to have a preference for conformity, not caring whether they conform or
deviate from the majority decision. Such individuals decide to imitate not because they want
to conform, but having observed the actions of other agents they decide to copy these actions
and ignore their own information because they judge that this will maximise their payoff.
1 Here an individual makes an error if they fail to choosethe option that would maximisetheir utility.In this
model factors such as conformity preferences etc. are excluded, so utility is purely the return an agent receives
from their action.Therefore an individual would be committing an error if they failed to choose the action that
would maximisetheir return given the information they held at that point. In reality individualsmay derive
utility from conformingor even from deviating,if this is the casethen an individual may maximisetheir utility
if they choose an action that doesn’t maximisetheir return, this would not be an error sincethey are
maximisingtheir utility even though they aren’t maximisingtheir return.

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Of key importance is the form of information transmission: here agents only observe the
actions of other agents. They don’t observe the other agents private information, they have to
infer this from those agents’ actions. A different setup where agents do observe the private
information of other agents will yield significantly different results: even if a cascade starts
private information will continue to be publicly aggregated, so better decisions are likely to
be made with lower social loss.
In real life it is an unrealistic assumption that agents won’t observe other agents private
information. Agents are likely (at least at the localised level such as family and friends), to
tell each other some of their private information. Therefore agents will probably have more
information than just their private information. But they will nevertheless have only a small
fraction of all agents’ private information, so the assumption is still reasonable. Furthermore
it is assumed that “actions speak louder than words”, i.e. actions are more credible than talk.
So agents infer more information from actions than from words.
In an attempt to ensure greater comprehension Bikhchandani et al (1992) first present a
restricted specific model, which they later extend into a general model. Their model consists
in essence of a sequence of agents deciding whether to adopt or reject a certain action. This
action has a cost and value, denoted C and V respectively, which are uniform across agents.
In the specific model, V is restricted to 0 or 1, whereas in the general model, V is drawn from
a finite set of possible values. In both models, C is set constant at ½. An agent should only
adopt if the value of the action exceeds the cost of the action, i.e. V > C. However agents
don’t know what V is, instead they each receive a private signal indicating the level of V (i.e.
what the state of the world is). In the specific model, the signal is either H, implying V = 1, or
it is L, implying V = 0. In the general model the signal is drawn from a finite set
corresponding to the set of V. However these signals are uncertain, they indicate what the
state of the world is with some probability, denoted p. Each agent observes the decisions of
any prior agents and their own private signal and then decides whether to adopt or reject the
action. In cases of indifference there is a tie breaking convention, which in the specific model
is for agents to randomly mix between adoption and rejection, in the general model any
indifferent agents adopt.

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The decision rule agents use will vary depending on their position in the sequence, because
the information agents have varies according to their position. The first agent has no
predecessors to observe, so will decide purely on the basis of their private signal. The second
agent observes this choice and infers the first agents signal. If the first and second agents
signals match then they make the same decision, if the signals differ then the second agent
computes the expected value of adoption, which equals ½, so the second agent randomly
mixes between adoption and rejection.
The third agent observes the actions of the first two agents, if both prior agents acted
identically then the third agent imitates, ignoring their own private signal and thus a cascade
begins.2 However if the prior agents choose different actions, then the third agent faces the
same problem as the first agent, so their private signal determines their choice. In this
scenario the fourth agent faces the same problem as the second agent and the fifth agent faces
the same problem as the third agent and so on.
The decision rules outlined above are used by Bikhchandani et al to compute the probability
of an ‘up’ cascade, ‘down’ cascade, and no cascade occurring after an even number of agents,
which can be seen in the set of equations (1). They also compute the probability of a correct
cascade (here an ‘up’ cascade), an incorrect cascade, and no cascade occurring after an even
number of agents, which can be seen in the set of equations (3).
The authors use these equations to show that as the number of agents (n) and the precision (p)
of the information agents receive increases, cascades become more likely and are more likely
2 Cascades in Bikhchandani et al (1992) are split into ‘up’ where agents adopt,and ‘down’ where agents reject.
1 − (𝑝 + 𝑝2
) 𝑛/2
2
,
1 − (𝑝 + 𝑝2
) 𝑛/2
2
, (𝑝 − 𝑝2
) 𝑛/2
, (1)
𝑝(𝑝 + 1)[1 − (𝑝 − 𝑝2
) 𝑛/2
]
2(1 − 𝑝 + 𝑝2)
,
(𝑝 − 2)(𝑝 − 1)[1 − (𝑝 − 𝑝2
) 𝑛/2
]
2(1 − 𝑝 + 𝑝2)
, (𝑝 − 𝑝2
) 𝑛/2
, (3)

7
to occur earlier3. Furthermore as the volume of information agents receive increases the
likelihood of the correct cascade occurring increases.
The most significant result from the general model is that after applying only a few mild
assumptions it can be shown that a cascade will eventually start: Provided an agent is far
enough along the sequence, then by the strong law of large numbers, they will be able to infer
the true value of V from prior actions with near certainty, thus the agent will ignore their
private signal and start a cascade. But as the authors emphasise cascades will tend to begin
before the agent can infer V with near certainty, meaning they will often be incorrect4 (see
also Banerjee (1992) and Welch (Welch, 1992)).
Bikhchandani et al (1992) emphasise the fragile nature of cascades, and this is the key
distinction of conformity arising from information cascades compared to the traditional
explanations of conformity (deviation sanctions etc). This fragility is down to the lack of
information aggregation within a cascade: Agents in a cascade imitate their predecessors, so
none of their own information is transmitted through their action; consequently subsequent
agents make identical inferences from the decision history and will also imitate. Therefore
decisions in a cascade are based on less information than if no prior agent had imitated.
Additionally, cascades can begin on the basis of a small amount of information, and the
authors suggest this will be true for many cascades, in their model a cascade starts if the first
two agents receive the same signal. But if cascades begin on the basis of only a small amount
of information, the opposite is also true, a small amount of new contradictory information can
cause the cascade to breakdown or even reverse. However this fragility helps partially to
alleviate the lack of information aggregation, since it means that multiple cascades are more
likely, which will increase the amount of information aggregation. Additionally fragile
cascades will mean that incorrect cascades will be more likely to be reversed.
Bikhchandani et al (1992) suggest that the fragility of cascades and the resulting cascade
reversals can explain fads, sudden often inexplicable changes in social behaviour. If agents
initially converge on an incorrect cascade, this is obviously non-optimal, but because of the
fragility of cascades any new information suggesting the current cascade is incorrect can lead
to the rapid breakdown and reversal of the cascade. This process can occur multiple times
3 As n increases, the probability of no cascade declines exponentially.
4 Even for very informative signals, high p.

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with agents potentially alternating between adoption and rejection cascades. Additionally, if
there is some possibility (even if minute), that the underlying value will change, this can also
cause fads, since agents will try to anticipate when the underlying value will change.
Therefore behaviour may change more often than the underlying value.
What form could this new information take? It is unlikely to be the private information agents
receive, since this can only affect one agent at a time, a more likely candidate would be
public information releases such as government advice or the widely publicised findings of a
scientific study. A key conclusion of Bikhchandani et al (1992) is that public information
releases may not always be socially beneficial. If before a cascade, ‘noisy’ public information
is released, i.e. an inconclusive study whose findings may subsequently be overturned, then
this can result in harmful transient fads. The authors illustrate this point using real life
examples, for instance oat bran: an initial study suggested that consuming oat bran could
lower cholesterol levels; this led to a fad, however after a later study contradicted this finding
the fad died. A further study later found that the original findings may have been somewhat
correct, suggesting the original cascade may not have been incorrect. Although agents were
not directly harmed, this example does illustrate how ‘noisy’ public information releases can
result in non-beneficial behaviour change, which may be costly in itself.
Alternatively public information releases will definitely be socially beneficial if they are
conclusive, since they then won’t trigger a harmful fad. Furthermore, once in a cascade all
public information releases are unambiguously beneficial, because they are a source of new
information. The release of even a small amount of public information (e.g. less than a single
agent’s signal) can be sufficient to shatter a cascade. Here the authors underline the fragility
of cascades, by pointing out that even a long lasting cascade with a large number of agents,
can be shattered, provided that the public information release can offset the information
transmitted by the last agent prior to the start of the cascade.
Significantly the authors also find that even if public information releases are infrequent,
provided there is a positive probability that one occurs, then the correct cascade will
eventually occur: By the law of large numbers, as public information releases accumulate the
correct choice becomes clearer and the correct cascade becomes more likely. The authors
support this point using a numerical example, which shows that even if public information
releases occur at only 10 in every 1000 agents, the correct cascade is substantially more likely
to occur.

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The authors examine the predictions of their model in a number of contexts. For instance they
examine the effect of agents with heterogeneous precision levels. They find that if high
precision agents decide first, then a cascade will arise sooner and even less information will
be aggregated. This is because low precision agents are liable to imitate their high precision
peers, believing this will result in a better choice (further supported by findings from
experimental psychology (Deutsch & Gerard, 1955)). The authors interpret this finding as
implying that sudden, inexplicable changes in social behaviour may be due to cascades
arising from the actions of so called “early adopters” or community leaders, rather than grand
causal forces. If the wider community perceives such individuals as better decision makers,
they will likely imitate their actions, resulting in a cascade.
Furthermore the authors posit that socially, it may be optimal for less precise agents to be
near the start of the sequence and more precise agents to be near the end. High precision
agents trust their signal more, so are more likely to deviate and break a cascade. This is
important since if the initial cascade is wrong and all higher precision agents are at the start
of the sequence, then the incorrect cascade won’t reverse. In other words, having high
precision agents towards the end of the sequence is socially beneficial since it increases the
chances of multiple cascades, meaning greater information aggregation and thus better
choices. However the authors concede that such a sequence is unlikely to arise naturally
given the propensity for imitation by low precision agents.
The authors challenge a common explanation for conformity: peer pressure, which can be
interpreted as a coercive non-monetary sanction against deviants. Instead they argue that
rather than peer pressure, inexperienced agents imitate their more experienced peers in order
to obtain better information. This is how Deutsch & Gerard (1955) interpret the results of
Asch’s (1952) line-length group evaluation experiment.
The authors also examine the issue of stigma, negative typecasting of agents who deviate
from social norms. Drawing on Ainlay, et al (1986) stigma is a localized phenomenon, which

10
agents learn from other group members, especially parents. The authors argue that stigma
could result from a ‘down’ cascade: Initial bad information leads to rejection of a job
applicant, this rejection transmits a signal leading to further rejection and so on resulting in a
cascade and the stigmatisation of the job applicant. The authors point out that the reverse may
be true: Good early performance could earn an agent a reputation as a star performer, other
firms would then be more likely to hire the agent simply based on their reputation, thus an
‘up’ cascade could occur. However as the earlier analysis of the fragility of cascades
demonstrated, such a reputation is likely to be brittle, a few poor performances could soon
break the ‘up’ cascade.
A criticism that the authors admit is that their model doesn’t effectively address fashions, i.e.
where there is no true underlying value, instead the value of adoption depends on how other
agents act. However, as the authors point out: in such situations, where agents are attempting
to forecast the actions of other agents, information cascades will still play a prominent role. It
is a weakness of the paper that the authors do not propose how their model could be modified
to account for such scenarios.
Another criticism of the paper would be that the authors do not address the issue of learning.
Potentially this could have a big impact; since cascades are widespread they will affect a
diverse range of contexts, so there would be plenty of scope for learning. The paper also does
not examine situations in which agents face more than just two options. Such situations are
widespread, for instance there is a large range of models to choose from when buying a car.
The paper also doesn’t address the importance of agent error or communication, both of
which could have a substantial impact.
The authors conclude their paper by summarizing the main findings and implications; they
also suggest how the paper could be extended. For practical applications they advise
including other factors alongside information cascades, such as deviation sanctions or
conformity preference, so as to provide a richer analysis. An extension the authors suggest is
heterogeneous but correlated values of adoption, i.e. V is no longer uniform across agents. An
alternative avenue of investigation would be ‘liaison’ agents, who straddle multiple groups
and can therefore spread or break local cascades across groups. They also advocate the use of
experimental tests to investigate how cascades form and change in order to deepen our
understanding of social change.

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There are several other theoretical papers on information cascades that make contributions
that are worth considering. Bikhchandani, et al (1998) return to the topic of information
cascades in a later paper in which they cover much of the same ground and use an identical
model as in their seminal paper (Bikhchandani, et al., 1992). They reach broadly similar
conclusions, however they do arrive at some new findings. They discuss the case of more
than just two alternatives, concluding that as the number of alternatives increases, cascades
will still arise but will take longer to start and more information will be aggregated.
Furthermore they conclude that a degree of finiteness is key to cascade formation, if the set of
alternatives is continuous (e.g. all points in the interval [1,10]) then a cascade cannot arise
because all agents, even those late in the sequence, will adjust their choice based on their
private signal even if only slightly. This means that subsequent agents can infer the private
information of prior agents and this prevents a cascade from arising.
Chamley (2004) devote an entire chapter of “Rational Herds: Economic Models of Social
Learning” to the topic of information cascades. In particular the author discusses the
difference between a cascade and a herd, which can be summarised as: A cascade occurs
when all agents herd on the basis of a sufficiently strong public belief and private information
is disregarded, there is no social learning and the cascade lasts forever. Whereas in a herd all
agents take the same decision but not all agents may be herding, some may make their
decision based on private information. Because some agents may not be herding and there is
some probability that the herd may be broken, some social learning is possible although the
amount is likely to be low. In other words, agents who deviate from the herd imply that they
have received a strong private signal, subsequent agents infer this and act accordingly. The
author also raises an interesting point, which is that queues may increase social welfare. This
is because queues carry a cost this may induce agents to deviate from the cascade and
increase experimentation with alternative options. This experimentation may lead to useful
discoveries and increased information aggregation which will increase social welfare.

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Banerjee (1992) also developed a theoretical model of information cascades and herding.
This model differs somewhat from that of Bikhchandani, et al (1992). For example in the
general model of Bikhchandani, et al (1992) indifferent agents follow their own information,
whereas in Banerjee (1992) an indifferent agent ignore their own information and copies their
predecessor. Banerjee justifies this tie-breaking convention by arguing that if a second agent
receives a signal different to that which the first agent received then precisely because the
first agent has already made their decision the second agent will disregard their own
information and imitate.
Banerjee (1992) reaches similar conclusions to Bikhchandani (1992), they find that in
equilibrium in their model herding will be extensive. Further as the probability of an agent
receiving a signal and the precision of the signal increases, then the likelihood of the correct
cascade occurring increases. Herding makes agents less responsive to their own information,
this makes the agents’ decisions less informative to subsequent agents, which can result in
inefficient outcomes. So as to prevent this it may be in the social interest that some agents
are constrained so that they have to use their own information.
However Banerjee (1992) also raises some interesting points. For instance the author
suggests that often there is a payoff to being an early adopter of the correct option, this
creates an incentive to deviate and go against the majority, which acts against herding.
Banerjee (1992) also suggests that waiting costs may impact the result, since they may alter
the order of choice: If waiting costs are low enough, the order could rearrange so that agents
that are of higher precision or are better informed choose first, and lower precision or worse
informed agents choose later, leading to an efficient outcome. However, even if waiting costs
are low some low precision agents may choose before their high precision peers, i.e. the order
of choice doesn’t fully rearrange, this would result in an inefficient outcome.
However Banerjee (1992) has been criticised because the idiosyncrasies of the structure of
the model means the robustness of its properties can’t be analysed (Chamley, 2004). Further
Banerjee (1992) doesn’t emphasise the fragility of cascades to the extent that Bikhchandani,
et al (1992) does thus it may be less useful in explaining the sudden changes that are
characteristic of fads.

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Experimental Literature
Since the publication of Bikhchandani, et al (1992) there have been a number of papers
published involving experimental tests of information cascades. The most influential
experiment is that developed by Anderson and Holt (1996), the following is a summary. In
the experiment there are two urns, denoted A and B. In urn A there are 2 red marble and 1
blue marble, in urn B there are 2 blue marbles and 1 red marble. Prior to the start of the
experiment a coin flip is used to select one of the urns, the contents of which are then
transferred into a third urn. Subjects, having been randomly sorted into a sequence, observe a
private draw from the third urn and then guess which urn has been selected; these guesses are
recorded and made public. This experimental design has formed the basis for most
subsequent experimental investigations of information cascades. Several other papers have
been authored by Anderson & Holt (1997; 2000; 2008) in which they build upon the design,
adding modifications such as incentivising subjects (Anderson & Holt, 1997).
Çelen & Kariv (2004) adapt the experiment by using a continuous rather than a discrete
signal, i.e. signal is drawn from the range [-10, 10]. They also elicit subject beliefs by asking
subjects to choose a cut-off point for their signal which will determine their action. These
alterations are made so as to be able to distinguish between information cascades and
herding. This is difficult because a subject in a herd or cascade act identically and it is
therefore necessary to know the beliefs of subjects. In two further papers (Çelen & Kariv,
2003; 2005) the same authors make further modifications: Investigating the different
outcomes arising under perfect versus imperfect information, the authors simulate imperfect
information by limiting subjects to observing only their immediate predecessor.
Hung & Plott (2001) modify the payoff institutions of the standard design: they include the
usual individual payoff institution where the payoff of subjects is determined solely by that
subject’s actions. Additionally a majority payoff institution is included, under which payoffs
of subject depend on whether the majority decision was correct. Further they include a
conformity payoff institution, subjects payoffs depend on whether their decision is the same
as the majority.
Other studies that also adapt the Anderson and Holt experimental setup include Ziegelmeyer,
et al (2010), Nöth & Weber (2003), and Kübler & Weizsäcker (2004). Despite its widespread
use in experiments investigating information cascades and herding, there are alternatives to

14
the Anderson and Holt urn experiment. For instance Camerer & Weigelt (1991) conduct an
experiment into very transient cascades in the financial markets. Their experimental setup
involved subjects taking part in a laboratory double-oral auction having been given an
endowment of assets. A certain number of subjects were designated as ‘insiders’ and given
information about the returns on assets that other subjects were ignorant of.
Camerer & Weigelt also provide an important justification for using experiments to
investigate information cascades, despite being widespread it would be difficult to study
cascades using real world data since many cascades are very short term. Furthermore
laboratory experiments allow the experimenters to control the flow of information, which is
key for distinguishing between a herd and a cascade.
Huck & Oechssler (2000) develop an interesting experimental design to test the link between
rational Bayesian learning and information cascades. They design a series of decision tasks
which they include as part of an undergraduate economics exam. To test whether those
subjects whose decisions conform to Bayesian learning are intentionally doing so, subjects
are asked to explain the reasoning process behind their decision.
In terms of the results of experiments concerning information cascades, there is strong
support for many of the theoretical conclusions of Bikhchandani, et al (1992) provided by
Anderson & Holt (1996; 1997; 2000; 2008). Their results are that cascades are commonplace,
in 80% of the occasions where it is possible for cascades to form they do. They also find that
cascades are fragile and easily broken by errors. These results have been replicated in further
studies for instance Hung & Plott (2001).
However they also find that errors, when incorporated into the model as a possibility (which
is not the case in Bikhchandani, et al (1992)) help to explain the experimental results. Further
when subjects are incentivised (Anderson & Holt, 1997; 2008) this substantially reduces the
number of errors subjects make; it is key that there is an incentive, but the size of the
incentive doesn’t seem to matter.
Anderson & Holt also raise questions about the use of Bayesian learning in explaining subject
behaviour, previous controlled experiments have found subjects deviate from Bayesian

15
learning and employ other methods such as simple counting heuristics. Furthermore much of
the evidence used to support the standard theoretical explanation is anecdotal, and hence open
to scrutiny (Anderson & Holt, 1997). However they conclude from their results that Bayesian
learning is widely employed by subjects, although it is not universal; additionally, deviations
from Bayesian learning are mainly due to error.
Çelen & Kariv (2004) find that cascades occur frequently and that incorrect cascades are very
rare (Hey & Allsopp (1999) make the same finding). They also find that subjects tend to
overvalue their own information over public information but over time this weakens and
subjects move closer to Bayesian updating. Building upon Anderson & Holt (1997; 2008)
they incorporate errors into a Bayesian learning model, the resulting model successfully
predicts subject behaviour in the experiment. This suggests that the theoretical assumption
that subjects use Bayesian learning is generally correct and that subject error can largely
explain deviations.
In later papers (Çelen & Kariv, 2003; 2005) the same authors examine a different aspect of
information cascades. They compare perfect information, under which subjects observe all
prior decisions, with imperfect information, under which subjects observe only a fraction of
all prior decisions. The theoretical discussion of information cascades (including
Bikhchandani, et al (1992)) assumes perfect information, but in reality imperfect information
is more likely. Their results show that under imperfect information, imitation is substantially
reduced; hence the frequency of cascades is reduced. The authors conclude that Bayesian
learning performs well in explaining subject behaviour under perfect information; however
when there is imperfect information its performance is reduced. The authors also find that
imperfect information can help to improve modelling of cascades; in particular imperfect
information explains the episodic instability characterising fads and fashions. Under
imperfect information agents will face uncertainty about whether the cascade is correct; this
will result in sharp reversals and because subjects will continue to operate under imperfect
information these reversals may occur many times.
The study of Camerer & Weigelt (1991) looks at very short term cascades in the financial
markets called ‘mirages’, i.e. trades based on the belief that another trade was motivated by

16
private information even if it wasn’t. Their results suggest that such cascades occur
seemingly at random, without any tell-tale signs. The authors conclude that ‘mirages’ may be
caused by behavioural factors such as the representativeness bias: Subjects overgeneralise
from their past experiences of mirages. Alternatively ‘mirages’ may arise if Bayesian
learning is compromised, which the presence of noise traders would accomplish. This
supports the argument of Anderson & Holt (1997; 2008) and Çelen & Kariv (2004) that
deviations from Bayesian learning are mainly due to error.
Kübler & Weizsäcker (2004) conduct an experiment in which subjects may purchase signals.
They find that early subjects over purchase signals but that later subjects don’t. The
explanation for this is that subjects believe that they are less likely to make errors than are
other subjects, so they place more faith in their own information over public information (this
overconfidence of subjects is supported by the findings of Nöth & Weber (2003)). Early
subjects therefore lack confidence in the decisions of predecessors and so purchase a signal,
but eventually enough subjects will have made their decision that subjects will have enough
confidence and so won’t buy a signal. This suggests that contrary to the theoretical
assumptions of Bikhchandani, et al (1992) the fragility of a cascade will vary, as it lengthens
subjects grow more confident that it is correct and it therefore becomes less fragile, this
argument is further supported by the findings of Ziegelmeyer, et al (2010).
The experimental evidence of subject overconfidence ( (Çelen & Kariv, 2004), (Kübler &
Weizsäcker, 2004), (Nöth & Weber, 2003)) suggests the tie-breaking convention of
Bikhchandani, et al (1992) that indifferent agents follow their own information5, is more
realistic than the tie breaking convention of Banerjee (1992) where indifferent agents ignore
their own information and imitate their predecessor. The evidence suggests that subjects put
greater weight on their own private information than public information, which suggests that
if indifferent they will rely on their own information.
Huck & Oechssler (2000) conclude from their study that a simple heuristic such as following
own signal, is better able to explain experimental data than Bayesian learning (This is also a
finding of Hey & Allsopp (1999)). In the study about 50% of subjects made decisions
consistent with Bayesian rationality, however when asked to explain their decision very few
could successfully explain Bayes rule. The authors re-examine Anderson & Holt (1996) and
find that a simple heuristic would work just as well as Bayes rule much of the time. This
5 This is the tie breaking convention for the general model.

17
suggests that although it may appear that Bayesian rationality is being employed by subjects,
in fact it is not and simpler heuristics are being used. However a counter point to this would
be that although subjects aren’t using Bayesian learning it is as if they are (Friedman, 1966),
evidence for this is provided by the fact that the decisions of subjects are the same as if they
had applied Bayesian learning.
In Summary the key theoretical paper concerned with information cascades is Bikhchandani,
et al (1992). The authors show that while it is individually rational to imitate, this limits the
transmission of private information, which means that less information is publicly
aggregated. This reduces the quality of later decisions, leading to a reduction in social
welfare. Furthermore they emphasise that cascades are innately fragile precisely because only
a little information is aggregated. The experimental evidence supports many of the
conclusions of Bikhchandani, et al (1992), in particular Anderson & Holt (1996; 1997; 2000;
2008) who develop the standard experiment that has been used and modified in most later
experimental studies of the topic. However other experimental results raise questions about
the rationality assumptions, Bayesian learning doesn’t appear to be universally applied,
simpler heuristics are often used by subjects (Huck & Oechssler, 2000) and evidence of
overconfidence is also found (Kübler & Weizsäcker, 2004).

18
3. Experiment
3.1. Outline
We now present an experiment designed to test the following hypotheses:
The first hypothesis is developed to address a point made by Chamley (2004, p. 84), which is
that since queuing carries an opportunity cost it will increase experimentation by agents,
which should improve information discovery. Thus a queue could improve social efficiency.
The hypothesis is stated as:
i) H0: The imposition of a queue in which subjects face a cost of waiting, will result
in increased deviation from the cascade and more information aggregating.
Ha: The imposition of a queue in which subjects face a cost of waiting, will not
result in increased deviation from the cascade and more information aggregating.
The second hypothesis arises from a point raised in Bikhachandani, et al (1998), that if there
are more than just two options, cascades will take longer to form and more information will
be aggregated. The hypothesis is stated as:
ii) H0: If the number of options increases from two to three, then this will result in
cascades arising later and more information aggregating.
Ha: If the number of options increases from two to three, then this will not result
in cascades arising later and more information aggregating.
We first give an outline of the three treatments we run. We then present a section detailing
the technical analysis outlining our predictions for subject behaviour in each of our
treatments. A further section details how the experiment can be run and any practical issues
which may arise. Finally we explore how the experiment could be further extended.

19
In all treatments in this experiment we define a cascade as: a subject ignoring their own
information and instead imitating the choice of prior subjects. In this setup cascades are
infinitely long, since precision of subjects and signals doesn’t vary, thus the information and
inferences of the subject who started the cascade will be the same for all subsequent subjects,
who should therefore make an identical choice thus the cascade is propagated. The only way
a cascade can be broken in our setup is through subject error.
3.2. Basic Treatment
This treatment is based on the urn experiments of Anderson and Holt (1996; 1997; 2008),
however those experiments are here modified so that other treatments may be run. Anderson
and Holt had subjects guess which urn they were observing a draw from, whereas in this
experiment subjects choose the urn which they wish to receive a draw from and following
their draw they guess the ‘true value’ of that urn. Subjects know the probability that a draw
from each urn is the true value, in this experiment the probability is constant at 80% for both
urns. We run this treatment to act as a control for the other treatments in our experiment.
This treatment can be summarised by the following: There are 2 urns containing 10 balls of
two types (red and blue). In each of these urns one of the ball types is the ‘true value’ and
yields an increased payoff. Each urn contains a certain number of each ball type; the
proportion that are the true value ball type represents the probability of the urn, i.e. the
likelihood that a draw from the urn will be the true value. In all the treatments of this
experiment 4/5 or 80% of the balls in an urn are the true value; hence the probability of the
urn is 80%. Subjects are informed of the probability of the urns prior to the start of the
experiment.
Subjects sequentially choose one of the urns from which they then observe a private draw,
based on this draw and the past decision of other subjects they then select the ball type that
they believe is the true value of the urn. The subject’s urn and true value choices are made
public and subsequent subjects can hence aggregate this information to make more accurate
judgements. Once all subjects have performed this process the round ends and subjects are

20
privately informed of their earnings from that round. Prior to the start of the next round new
true values are randomly chosen for the urns, the process then starts again, and this continues
until all rounds are complete.
3.3. Queueing Treatment
This treatment was developed to test the first hypothesis that if subjects face a queue with a
waiting cost, then deviation from the cascade will increase and more information will be
aggregated.
There are several motivations for researching this aspect of information cascades. Queueing
is a widespread real life phenomenon; the opportunity costs of queueing can be high. Many
people are willing to queue for hours or even days to gain access to a music festival or to be
able to purchase the latest technological innovation. Furthermore, many of these queues are
motivated at least in part by fads and fashions, thus analysing queueing from an information
cascade perspective may yield useful and interesting findings. However there has been little
of no investigation into queueing in either the theoretical or experimental literature.
A further motivation is that in the literature the main inefficiency of information cascades is
that while it is rational for individual agents to conform, such conformity at the aggregate
level results in social loss. However if as Chamley (2004, p. 84) suggests queueing acts to
reduce the social loss of cascades, then it represents a ready-made institution for alleviating
the costs of cascades. Given the high occurrence of queueing, it also suggests that the current
literature may have exaggerated the inefficiency arising from cascades. Thus investigating
queueing allows us to provide clarity about the social costs of cascades, and to what extent
these are alleviated by queueing.
The following analogy, similar to that used by Easley and Kleinberg (2010, pp. 425 - 430),
will help to further explore the queueing aspect of information cascades: Imagine a street
with two new restaurants, which we denote A and B respectively. Customers have no
information about the relative quality of the restaurants, but they infer that the busier
restaurant is better and so choose it. Assume the first customer chooses restaurant A, the next
customer observes this and infers that restaurant A is superior and so imitates, as does the

21
next and a cascade occurs with all customers choosing restaurant A over restaurant B. The
restaurant soon fills up and a queue forms outside the restaurant. However queueing carries
costs, first there is the opportunity cost of waiting and secondly if the queue is long enough
then customers at the back of it will not get served, thus for each customer in the queue there
is some probability that they will not get served. These costs of queueing can be interpreted
as a negative externality of the information cascade. They arise because it is rational for each
individual agent to conform and imitate the choices of prior agents; however this results in
too many agents making the same choice, leading to the queue. To summarise, the aggregate
conformity of agents forces individual agents to pay a cost in the form of a queue.
It is important to note that the externality doesn’t affect all agents, only those agents in the
queue, early agents get to the restaurant before a queue forms and thus avoid the externality.
Further, the effect of the externality on queueing agents varies depending on the agent’s
position in the queue: An agent toward the start of the queue will queue for only a short
period of time and the probability that they don’t get served will be very low, so the
externality will be small. Whereas an agent near the end of the queue will have to wait a long
time and face a high probability (if the queue is long) that they won’t get served, hence the
externality will be large.
If the costs of queueing are sufficiently high customers will view it as in their interest to leave
the queue (deviation from the cascade) and enter the empty restaurant B. Of course since
restaurant B is empty they have no indication of its relative quality. However it may be that
the first customer chose restaurant A at random or by error in which case the cascade is
incorrect. If this is the case and both restaurants are of equal quality then deviating customers
will be better off since they will avoid the cost of queueing and they will consume a meal of
equal quality, so they reduce their costs without lowering their benefits. In other words
deviating customers increase their individual efficiency, but how will this impact social
efficiency? The answer to this depends on how the information transmitted by deviating
customers is interpreted by other customers still in the queue, i.e. do they view it as true or
due to error. If the information is interpreted as arising from erroneous decisions then it will
be ignored and any social efficiency increase will be minor. But if it is interpreted as true then
it will be aggregated and further agents will deviate, thus social efficiency will increase
significantly.

22
Whether information arising from deviations is interpreted as true and thus aggregated may
depend on how many customers are willing to deviate and experiment. If only one customer
is willing to deviate then this is more likely to be interpreted as an error, but if several
customers are willing to experiment when faced by a queue, then other customers are more
likely to interpret this as an accurate reflection of quality and also deviate.
This raises a question: Why would some customers be more open to experimenting than
others? One answer to this would be that behavioural biases will have a considerable impact
when subjects decide whether to deviate, in particular the degree that an agent is risk averse
will be important. A highly risk averse agent won’t want to switch since this would mean
losing all the aggregated information about restaurant A and switching to restaurant B where
no information has been accumulated. Whereas a less risk averse or risk loving agent will be
much more willing to switch, they will be less concerned about the loss of aggregated
information and more attracted by avoiding the costs arising from queueing.
Risk aversion will reduce experimentation to a lesser extent as the queue lengthens and the
probability of missing a turn increases. The risk of missing a turn has increased, while the
risk of switching to the other restaurant has remained constant. Thus risk aversion will act in
two ways: first to discourage experimentation due to the risk of lost information from
switching to the other restaurant, secondly to encourage experimentation due to the risk of
losing a turn because of a long queue. If the queue is long enough then the later effect will
become more important and risk aversion will act to encourage experimentation.
Therefore differences in the degree of risk aversion between agents will cause some agents to
experiment by switching, while others will prefer to queue. Of course other biases may also
have an impact, for instance agents may have conformity preferences which will discourage
switching even if the queue is extremely long.
Here note that the information transmitted by deviations from the cascade isn’t simply
constrained to the act of the deviator leaving the queue and entering the other restaurant. In a
practical sense the information could also include how satisfied these deviators seem in the
restaurant and on leaving the restaurant, if they seem satisfied then other queueing customers
can infer that the other restaurant is of adequate quality and thus it is in their own interest to
also deviate.

23
3.4. Multi-value Treatment
This treatment was designed to test the second hypothesis that if the number of options
subjects can choose from increases from two to three, then cascades will arise later and more
information will be aggregated. It is the same as the basic treatment except for one
modification, which is that there are now three ball types (red, blue, and yellow) instead of
two (red and blue). It arises from a point raised in Bikhachandani, et al (1998), that if there
are more than just two options, cascades will take longer to form and more information will
be aggregated. This treatment is designed to test this assertion by adding a third ball type
(Yellow). Furthermore, the treatment also acts as a robustness test of prior experimental
results, i.e. will the additional option curtail imitation and prevent cascades from arising?
Many real life situations involve more than two options, for instance when buying a car there
is usually a large range to choose from. If experimental results can’t be replicated when the
range of options rises from two to three, this implies that they can’t be generalised and their
practical applications are limited.
3.5. Experiment Summary
- 2 Urns: Urn A, Urn B.
- Each urn contains 10 balls of either 2 or 3 types depending on treatment.
- Series of subjects (10 – 20).
- 3 treatments: Basic, Waiting, Multi-value.
- Series of rounds for each treatment.
- Series of rooms: one for subjects to wait in, another containing the urns, and a third for
subjects to wait until the experiment is over. The two waiting rooms should be separate so
that communication between subjects who have observed a draw and subjects who are
waiting to observe a draw is not possible.

24
Treatments:
- Basic, this is the control treatment, 2 ball types: Red and Blue, each type has a ½ probability
of being randomly chosen as the true value. Out of the 10 balls in the urn 8 will be the true
value and 2 will be the non-true value, meaning there is a 4/5 probability of the true value
being drawn.
- Waiting, same setup as in Basic, except that once a cascade develops later agents are forced
to queue.
- Multi-value, 3 ball types: Red, Blue, and Yellow, each type has a 1/3 probability of being
randomly chosen as the true value. Out of the 10 balls 8 will be the true value and 1 ball each
will be the two non-true values, so there is a 4/5 probability of the true value being drawn.
Thus if Red is the true value then P(R) = 4/5, P(B) = 1/10, P(Y) = 1/10.
Summary:
Subjects sequentially observe prior history, they then choose an urn from which they observe
a private draw, which they combine with prior history to choose a value which they think is
the urns true value. At the end of the round subjects learn their earnings from that round.
3.6. TechnicalAnalysis
In the following analysis we follow the example of Bikhchandani, et al (1992) by using
Bayesian learning to model how subjects make their urn and value decisions. Subjects know
the prior probabilities of each value being the true value and the true value being drawn from
each urn, Bayes theorem is used to calculate the posterior probability of a value being the true
value given the observations the subject has observed. For a fuller exposition on the use of
Bayesian Learning in this study see appendix A at the end of this paper, the appendix also
includes the calculations of the posterior probabilities used in this study. Subjects also are
assumed to be rational utility maximisers. Furthermore in our technical analysis subjects are
assumed to derive no utility from conformity etc, they derive utility only from the return they
receive from their decisions in the experiment. So as in Bikhchandani, et al (1992) a rational
utility maximiser aims to solely maximise the return they receive.

25
3.6.1. Analysis of Basic Treatment:
Prior to the start of the round, red or blue is randomly and independently selected as the true
value of urns A and B, for convenience let us assume that the true values are red and blue
respectively.
There is no prior history of choices for the first agent to observe, so they randomly choose
between the urns6, and this choice will determine which urn subsequent agents choose, since
it will give the chosen urn an information advantage. For convenience let us assume that they
select urn A. The first agent determines their value choice by the signal they observe, i.e. if
they observe a red ball they will select red as the true value.
The second subject observes the first subject’s choices and infers their signal. We predict
they will also choose urn A, because the first subject choosing urn A gives it an information
advantage over urn B: If the second subject switches to urn B the probability that their signal
is the true value is 80%, however if the second subject stays with urn A and receives the same
signal, the probability is 94.12%, a significant difference. Of course if the second agent
receives a different signal from urn A then they would be better off switching, however the
choice of urn must be made before the subject receives their private signal. We judge that the
high degree of certainty the subject enjoys if they receive the same signal will outweigh the
lower degree of certainty if the signal differs, therefore we feel our assumption is justified7.
The second agent will determine their value choice according to the signal they receive. This
assumption is backed up by the experimental findings of Kübler & Weizäcker (2004) who
found that subjects generally believe other subjects are more error prone than they are, so
they place more weight on their own information than public information. Thus this implies
that if the second subject received a different signal from the first subject, they would follow
their own signal.
6 The only exception to this,would be if the probabilities of the urns differed, in which casethe subjectwould
select the urn with the higher probability,however here both urns have probabilities of 80%.
7 It may be that risk aversion will affectwhich urn the agent chooses:subjects exhibitinghigher risk aversion
will bemore likely to switch to urn B and viceversa.However other behavioural biases may also havean
impact, for instanceconformity preferences. Tryingto incorporatethese aspects into our analysis would
unnecessarily complicatethings,and given that the different biases may actagainsteach other, we have left
them out.

26
The third subject observes the prior choices and infers the prior signals. Again we predict that
they will also choose urn A because of its information advantage which is more pronounced
for the third subject: Unlike the second agent there are no circumstances in which switching
to urn B confers an advantage over urn A: for instance the sequence R,B,R, yields an 80%
probability of the true value being red which is the same as the probability if the subject
switched. While if the sequence were R,R,R, the probability would be 98.46%. The third
subject can begin a cascade if the first two choices are identical, in which case the subject
should imitate these prior choices and ignore their own signal. For instance Bayesian
inference implies the sequence R, R, B, has an 80% probability that red is the true value, thus
the subject should ignore their own blue signal and choose red, initiating a cascade. If
however the prior choices differ then the third subject will determine their choice based on
their private signal.
If the third agent initiates a cascade then all subsequent subjects should imitate and join the
cascade. Since only the first two signals are aggregated then all subsequent subjects are in the
same position as the third subject and will make the same inferences, resulting in the same
choice to imitate. Of course in an alternative setup this may not be the case, if the precision of
subjects or the signals they receive varies, then subjects may deviate from the cascade,
however in our setup the precision of subjects and signals is constant. A further point is worth
noting, that only odd numbered subjects can begin a cascade, even numbered subjects can’t.
The fourth subject observes prior history and infers prior signals. Again we predict, because
of the information advantage, that they will choose urn A. If a cascade is in progress then the
fourth agent should imitate and join it. In the absence of a cascade the fourth agent should
choose according to their private signal.
The fifth subject observes prior history and again chooses the same urn. If a cascade is in
progress then they imitate. If no cascade exists, they can start one if the two prior subjects
received identical signals, as in the sequence R,B,R,R, whatever signal the fifth subject
receives, Bayesian inference implies that they should ignore it and instead imitate by
choosing red. However if the two prior subjects choices differ then the fifth subject should
follow their own signal.

27
The above analysis can be applied to all subsequent subjects: If there is a cascade, then
imitate. If there is no cascade, then depending on prior choices and the subjects position
either start one by being the first agent to imitate, or choose according to your private signal.
Although it is possible that a cascade never arises (for example the sequence:
RBRBRBRBRB…..), it is highly likely as Bikhchandani et al (1992) conclude that one will
eventually occur. This is intuitive, since all a cascade requires is that the two subjects prior to
an odd numbered subject receive an identical signal, given that the true value will be drawn
with an 80% probability, this is very likely to occur at some point, and hence cascade is
highly likely to occur.
3.6.2. Analysis of Waiting Treatment
This is identical to the basic treatment until the point at which a cascade develops, after
which all subsequent subjects have to queue behind the ‘cascade urn’. Queueing carries a
penalty of a 50% probability that subjects will miss their turn8; however they can avoid this
by switching to the other urn. Thus up to the point that a cascade develops and subjects have
to queue the technical analysis is identical to that in the basic treatment9.
We will illustrate this using an example, for convenience let us assume that by the fifth
subject a cascade has developed in urn A. Thus from the sixth subject a queue develops
behind urn A. The sixth agent knows that if they queue then there is a 50% probability that
they will miss their turn and forgo any potential earnings from the round. Alternatively they
8
It might be more realisticif theprobability of missinga turn varied with the subject’s position in thequeue,
i.e. subjects at the startof the queue facea low/negligibleprobability,as the queue lengthens the probability
rises.Eventually the queue will beso longthat those subjects atthe end will be certain to miss their turn.
However for the purposes of simplicity and convenience a constantprobability of 50%is used in this
experiment.
9 Note here that risk aversion will havea different effect on urn selection in this treatment compared to the
basic treatment. This is becausethere is now a risk for queueing agents of missingtheir turn, which was not
present in the basic treatment. Hence risk aversion to have two contrastingeffects: Firstly,there is the risk of
switchingdue to the loss of aggregated info, this will discouragethe agent from switching.Secondly,there is
the risk of queueing due to the 50% probability of missingtheir turn; this will encouragethe agent to switch.
Furthermore, the above is only true if a cascadehas already formed,if a cascadehas yetto form then risk
aversion,as noted in footnote 2 in the basic treatment, can encourage switching.For instancethe second
agent can be 80% surethat their signal will bethe true valueif they switch,but if they don’t switch a nd receive
a different signal to the firstagent then they can only be 50% certain that their signal is thetrue value.In this
situation a risk aversesecond agent may choose to switch.

28
can avoid this by switching to urn B, guaranteeing them a turn, but at the cost of losing all the
information that has aggregated in urn A. We predict that if the sixth agent switched, they
would be the first agent to receive a draw from that urn, thus urn A has a substantial
information advantage over urn B: If they switch they face a 80% probability that their signal
is the true value, however queueing and joining the cascade in urn A yields either a 98.46%
probability of red being the true value if they receive a red signal, or a 80% probability if they
receive a blue signal. Thus the sixth subject must calculate if the cost of the 50% probability
of missing a turn is larger or smaller than the information cost of switching urns. If the former
cost is larger than the later, then the subject should switch and vice versa.
If we assume the sixth subject switches and observes a blue signal and chooses blue. The
seventh agent infers from this that if they also switch and receive a blue signal then there is a
94.12% probability that the true value of urn B is blue, in this scenario it is better to be the
second switcher than the first. However if the seventh subject received a red signal
contradicting the sixth subjects signal, then the probability is equal at 50% that blue or red is
the true value, in this scenario it is better to be the first switcher than the second.
If both the sixth and seventh subjects switched and received the same signal, then the eighth
subject should switch and start a cascade, from this point on all queueing subjects should
switch and join the cascade, since urn A no longer has an information advantage over urn B.
Alternatively if the sixth and seventh subjects switched and received contradictory signals,
then the eight subject is in the same position as the sixth agent: whatever signal they receive
there is an 80% probability that is the true value.
The above analysis can be applied to all subsequent subjects. A major conclusion we draw is
that the more subjects that switch the more likely a cascade is to occur and thus the more
attractive it becomes to switch. Furthermore it is better to be a latter switcher than an earlier
switcher, since more information will have been aggregated and it is more likely that a
cascade will have already begun.
A scenario worth exploring is the second subject choosing a different urn to the first subject;
this could be due to error or subject preference. How would this alter the above analysis? We
conclude that while it would have some effect on the waiting treatment it would not
fundamentally affect the analysis of any of the three treatments included in this experiment.

29
Our explanation is thus: imagine the first subject chooses urn A and receives a red signal. The
second subject for whatever reason chooses urn B and receives a blue signal. The third
subject should be indifferent between the two urns, so randomly chooses urn A and also
receives a red signal. The fourth subject faced with the choice between urn A and urn B,
should choose urn A, imitate and begin a cascade. Alternatively the third agent could choose
urn A and receive a contrary signal to the first subject, if the fourth subject chooses urn A
then whatever their signal is the probability will be 80% whereas choosing urn B and
receiving the same blue signal as the second subject will result in a probability of 94.12%.
Let us assume the fourth subject chooses urn B, if they receive a blue signal then the fifth
agent should also choose urn B and start a cascade by imitating. If they receive a red signal
the fifth subject will be indifferent between the two urns, whichever they choose will give
that urn an information advantage so the sixth agent should then choose that urn. Eventually
two consecutive subjects will receive the same signal from one of the urns, this will start a
cascade and all subsequent subjects will herd into this urn. Our analysis would only be
significantly affected if subjects kept making errors and failing to herd into one urn thus a
cascade could develop in both urns; however we judge the likelihood of such an occurrence
as negligible.
Thus even if the first two subjects choose different urns, eventually a cascade will begin in
one of the urns, resulting in all other subjects choosing that urn. Although the information
advantage will be smaller than in our standard analysis it will still exist. In other words if the
second subject chooses urn B this will reduce the cost of switching for later subjects, but our
main conclusions are unaffected.
3.6.3. Analysis of Multi-value treatment
This is identical to the basic and waiting treatments except for one modification: 3 rather than
2 ball types. This means that the probability of each type being the true value is 33. 3̇ % rather
than 50%. The probability of the true value being drawn remains 80% but because there are
now two non-true values each has a 10% probability of being drawn (i.e. if red is the true
value then out of 10 balls, 8 will be red, 1 will be blue and 1 will be yellow). This compares
to the 20% probability of the single non-true value being drawn in the basic or waiting
treatments.

30
The theoretical analysis of this treatment is very similar to that in the basic treatment: The
first agent has no history to observe, so will choose randomly between urn A and B, this
choice will then determine which urn all subsequent agents choose due to the information
advantage of the chosen urn. The first agent will choose their value according to the signal
they observe. The second subject will choose the same urn and also determine their value
according to their signal. Subsequent agents observe prior history and their own private
signal, through Bayesian inference they determine which ball type is most likely the true
value, which they choose.
A difference between this treatment and the basic treatment is that the true value probabilities
change, i.e. the sequence R,R, B yields an 80% probability of red being the true value in the
basic treatment but a 87.67% probability in the multi-value treatment. This results from there
being an extra ball type which lowers the probability of each type being the true value.
However these differences don’t fundamentally alter our analysis, since the choices of
subjects remains the same as in the basic treatment.
A more important difference between treatments is that in the basic or waiting treatment
cascades can only begin with odd numbered agents, but in the multi-value treatment an even
agent can sometimes begin a cascade. This is best illustrated using an example: Table 1
shows the signal sequence RYRB, Bayesian inference implies that the 4th subject should
choose according to their signal if they receive a R or Y signal (if they get a Y signal they
will be indifferent but we assume that in that case they follow their signal), but if they receive
a B signal as they do here then they should ignore it and imitate by choosing R. Thus they
begin a cascade. This implies that cascades are more likely when there are three possible
values than when there are only two. It is beyond the scope of this paper to establish whether
in general cascades become more likely as the number of options increases.
Table 1
Agent
1 2 3 4
Private Signal Red Yellow Red Blue
Public Choice Red Yellow Red Red

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3.7. Running the Experiment
Each subject is allocated an I.D. and the order of subjects is randomly determined, subjects
are given the instructions for the experiment and experimenters answer any questions subjects
may have. At the end of the experiment subjects privately collect payment of their earnings
from the experimenter.
Basic Treatment:
1) Prior to the start of the experiment, the experimenters randomly choose a true value
independently for urn A and B (This could be done using a coin toss, or by a draw
from a third urn with equal red and blue balls).
2) Subjects are brought into a waiting room, the urns are located in the next room so as
to ensure privacy when subjects observe their private draw.
3) Subjects sequentially leave the waiting room and enter the urn room, they choose one
of the urns.
4) Subjects observe a private draw from their chosen urn.
5) Subjects then choose a value.
6) Subjects leave into a third room, their choices are publicly recorded so that all other
subjects may view them (It may be simplest to simply write the choices on a large
whiteboard at the front of the waiting room). The choices of all subjects remain
publicly recorded for the entirety of the round so that subsequent subjects are aware of
the whole choice history.
7) Once all subjects have been through this process the round is over and subjects are
privately informed of their earnings from the round (this could be achieved by the use
of sealed envelopes).
8) Prior to the start of the following round, new true values are selected and the choice
histories are erased.

32
Waiting treatment:
Same as the basic treatment until the point that a cascade starts, then
1) All subsequent subjects are sequentially asked to choose between queueing behind the
cascade urn, or switching to the non-cascade urn. They are informed that if they queue
there is a 50% chance that they will miss their turn this round.
2) If subjects switch then they proceed with their turn as normal, receiving a draw from
the non-cascade urn and then choosing a value which is publicly recorded.
3) Once all subjects have chosen whether to queue or switch, a coin is flipped to decide
if all queueing subjects miss their turn. If they don’t miss their turn, then they proceed
as normal with their turn.
4) As in basic setup, at the end of the round subjects are informed of their earnings, new
true values are selected and the choice histories are erased.
Note: queue doesn’t have to start as soon as a cascade begins, the experimenter can start the
queue at any point once a cascade has started, i.e. if a cascade began with the third subject the
experimenter could wait until the sixth subject before implementing the queue.
Also note that subjects in this treatment are informed prior to the start of the experiment
about the queue and the resulting possibility that they may miss their turn and lose any
earnings from the round. It is an open question whether results would be different if they
were uninformed about the queue, perhaps there actions would be more instinctual and closer
to how they would actually behave in the real world. Whatever benefits that might arise from
failing to inform subjects about the queue, it represents the defining characteristic of this
treatment. Therefore failing to inform subjects of it or its consequences is, in our opinion
tantamount to deception or close enough as to be unpalatable.
Multi-Value treatment:
This is the identical to the basic treatment but with 3 values instead of 2, thus to select the
true value the experimenters should conduct a random draw from a nine ball urn with 3 balls
of each value.

33
Treatment Rounds:
How many rounds should be run in each treatment? One approach is for subjects to
participate in 10 rounds of whichever of the three treatments they are randomly chosen for.
The advantage of this approach is that only one set of instructions have to be given and the
experiment can proceed without interruption, however there is a potential issue in that
differences in the results of the experimental treatments may be caused by subject effects, i.e.
differences between subjects lead to differences between treatments.
An alternative approach that avoids this problem is for the experiment to be split into two
parts: In the first part, subjects take part in 5 rounds of the basic treatment. In the second part,
subjects then take part in another 5 rounds of either the waiting treatment or the multi-value
treatment. Any subject effects should be substantially weakened under this approach, instead
differences in results between treatments should be due to treatment effects. However this
approach would mean giving instructions for the basic treatment at the start of the experiment
and then half way through interrupting to give a further set of instructions for the other
treatment.
Furthermore learning effects may have some impact, by the time the second treatment starts,
subjects may have learnt enough to alter how they would otherwise have behaved in the
second treatment. This could be controlled for by running some subjects in a version where
the basic treatment comes first and is then followed by the waiting or multi-value treatment,
and then running other subjects so that the order is reversed
Payment:
Subjects are paid at the end of the experiment; they earn 2 ECU (Experimental Currency
Unit) per round for correctly guessing the true value, or 1 ECU for guessing a non-true value.
Each subject takes part in 10 rounds in total. In the basic and multi-value treatments this
means that each subject earns a minimum of 10 ECU’s and a maximum of 20 ECUs.
However in the waiting treatment, because there is a 50% probability that a queueing subject
will miss their turn a subject could potentially earn 0 ECUs, to ensure that all subjects earn
something each subject receives a showup fee of 10 ECU’s. Including this showup fee means
that subjects can earn a maximum of 30 ECUs.

34
3.8. Extensions and Improvements
There are a number of extensions and improvements that could be made to our experiment.
Firstly, the multi-value treatment could be extended to establish whether the contention of
Bikhchandani, et al (1998) that as the number of alternatives increases, cascades start later
and more information is aggregated, is generally true: does the point at which a cascade
begins continue to be pushed back each time an extra alternative is added and does the
amount of information aggregated continue to increase, or is the effect more specific i.e. once
you move beyond three or four alternatives the effect disappears or is so small as to be
negligible.
Secondly, the number of urns that subjects can choose between could be increased. In this
experiment the number of alternative values that subjects could choose from was increased,
but subjects also choose between urns and varying the number of urn might have a different
impact.
Thirdly, the probabilities of the urns could be adjusted. For example urn A would have a 9/10
probability of the true value being drawn, while urn B could have only a 3/5 probability. This
would be especially useful in the waiting treatment, since the experimenter, by adjusting the
probabilities during the experiment, could investigate how willing subjects are to switch from
a high probability urn to a low probability urn, if this means they avoid a waiting cost.
Finally, the waiting cost could be varied. The waiting cost here is a 50% probability that a
queuing subject will lose their turn, but it is unrealistic for this to remain constant throughout
the queue. It would therefore be more descriptively realistic if the probability varied with the
length of the queue, starting low but steadily rising as the queue length increases. Clearly at
the extremes, where the probability is very low or very high, the behaviour of subjects is easy
to predict, but it is much more difficult to predict how subjects will behave towards the
middle of the probability range.

35
4. Conclusion
This study reviews the theoretical and experimental literature on the topic of information
cascades. It then presents the design for an experiment that builds on existing experiments,
but adds important innovations so as to be able to extend the range of issues surrounding
information cascades that can be investigated experimentally. Three treatments are
developed: A control treatment similar to the experimental design of Anderson & Holt (1996;
1997; 2008). A waiting treatment designed to investigate the effects of queueing on
information cascades. A multi-value treatment designed to investigate whether increasing the
options subjects can choose from will have an impact on the timing of when cascades occur.
Technical analysis is presented for each of these treatments detailing our predictions for how
subjects should behave, and the practical steps necessary to run the experiment are discussed.
Finally, we examine how the experiment might be extended and improved.

36
5. Appendices
Appendix A: BayesianLearning
Bayesian learning or inference involves adjusting prior beliefs given new data, posterior
probabilities are calculated using Bayes’s theorem. A simple version of Bayes theorem as
stated in Upton & Cook (2002) is:
𝑃( 𝐴| 𝐵) =
𝑃( 𝐴) 𝑃( 𝐵| 𝐴)
𝑃( 𝐴) 𝑃( 𝐵| 𝐴) + 𝑃( 𝐴′) 𝑃( 𝐵| 𝐴′)
P(A|B) is the probability that event A occurs given that event B has been observed, for
instance this could be the probability that red is the true value given that the first agent
observes a red signal. P(A)P(B|A) is the prior probability of event A occurring, multiplied by
the probability of event B occurring if event A has occurred. 𝐴′
is the alternative event that
could occur instead of event A. Thus 𝑃( 𝐴′) 𝑃( 𝐵| 𝐴′) is the prior probability that alternative
event 𝐴′
occurs, multiplied by the probability of event B occurring if alternative event 𝐴′
has
occurred.
The general form of the theorem is:
𝑃(𝐴𝑗|𝐵) =
𝑃(𝐴𝑗)𝑃(𝐵|𝐴𝑗)
∑ 𝑃( 𝐴 𝑘 ) 𝑃( 𝐵| 𝐴 𝑘)𝑛
𝑘=1
Where events A1, A2, . . . , An are mutually exclusive and exhaustive, i.e. each event is
independent and two such events cannot occur simultaneously. In our context A1, A2, . . . , An
represent all the possible true values of the urn. While the simple form can be used in the
basic and waiting treatments, in the multi-value treatment where there are three ball types it is
necessary to use the general form.
To illustrate further imagine that the first three observations in the multi-value treatment are,
Red, Blue, Red. The posterior probability that the true value is red is thus:
𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅) =
𝑃( 𝑡 = 𝑅) 𝑃( 𝑅𝐵𝑅| 𝑡 = 𝑅)
𝑃( 𝑡 = 𝑅) 𝑃( 𝑅𝐵𝑅| 𝑡 = 𝑅) + 𝑃( 𝑡 = 𝐵) 𝑃( 𝑅𝐵𝑅|𝑡 = 𝐵) + 𝑃( 𝑡 = 𝑌) 𝑃( 𝑅𝐵𝑅| 𝑡 = 𝑌)
The probability of the true value being drawn is 4/5, and the probability of each of the non-
true value being drawn is 1/10 for each value, so if red is the true value there is an 80%

37
chance that it will be drawn and a 10% chance that either blue or yellow will be drawn, these
probabilities remain constant as the number of draws increases because each draw is
independent. The prior probability of each value being the true value is 1/3. Thus the
posterior probability that red is the true value given the sequence of observations: Red, Blue,
Red, is calculated as:
𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅) =
1
3
× (
4
5
×
1
10
×
4
5
)
(
1
3
× (
4
5
×
1
10
×
4
5
)) + (
1
3
× (
1
10
×
4
5
×
1
10
)) + (
1
3
× (
1
10
×
1
10
×
1
10
))
=
64
375
(
64
375
) + (
1
3000
) + (
1
3000
)
=
256
257
= 99.61%
𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅) =
8
375
(
8
375
) + (
1
375
) + (
1
3000
)
=
64
73
= 87.67%
It is unnecessary to calculate every single possibility, for instance if the first three signals are
blue, red, blue, then the third agent clearly doesn’t need to calculate the probability that the
true value is red or yellow, since it is obvious that blue is more likely the true value.
Furthermore once you have calculated the probabilities for a sequence such as Red, Blue,
Red, i.e. P(t = R|RBR) = 4/5. You don’t need to recalculate if the next sequence is Blue, Red,
Blue, since the sequences mirror each other, i.e. P(t = B|BRB) = P(t = R|RBR) = 4/5. For this
reason in the calculations below we only show calculations for the true value being red.
Additionally if we calculate the posterior probability of a sequence such as: Red, Blue, Red,
Blue, Yellow, we don’t also calculate the probability for the sequence: Red, Yellow, Red,
Blue, Yellow. It doesn’t matter if there are 2 blue signals and 1 yellow or 2 yellow’s and 1
blue, the probability of the true value being red will be the same.
Basic and waiting treatments:
In these treatments there are two ball types, Red and Blue. The probability that either is the
true value is 1/2, the probability that the true value will be drawn is 4/5, the probability that
the non-true value will be drawn is 1/5.

40
iii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅𝐵𝑅) =
𝑃( 𝑡=𝑅) 𝑃( 𝑅𝐵𝑅𝐵𝑅| 𝑡 = 𝑅)
𝑃( 𝑡=𝑅) 𝑃( 𝑅𝐵𝑅𝐵𝑅| 𝑡 = 𝑅)+𝑃( 𝑡=𝐵) 𝑃( 𝑅𝐵𝑅𝐵𝑅| 𝑡 = 𝐵)
=
4
5
= 80%
Multi-value treatment:
There are three types of ball in this treatment so the calculations change: P(t = R) = P(t
= B) = P(t = Y) = 1/3
The probability that the true value is drawn from the urn remains the same at 4/5, but because
there are now two instead of only one alternative ball type the probability that a non-true
value is drawn from the urn is 1/10 for each of the two non-true value ball types:
P(t = R|R) = 4/5, P(t = R|B) = 1/10, P(t = R|Y) = 1/10
First agent
i) 𝑃( 𝑡 = 𝑅| 𝑅) =
𝑃( 𝑡=𝑅) 𝑃( 𝑅| 𝑡 = 𝑅)
𝑃( 𝑡=𝑅) 𝑃( 𝑅| 𝑡 = 𝑅)+𝑃( 𝑡=𝐵) 𝑃( 𝑅| 𝑡 = 𝐵)+𝑃( 𝑡=𝑌) 𝑃( 𝑅| 𝑡 = 𝑌)
=
1
3
×
4
5
(
1
3 ×
4
5) + (
1
3 ×
1
10) + (
1
3 ×
1
10)
=
4
15
(
4
15) + (
1
30) + (
1
30)
=
4
5
= 80%
Second Agent
i) 𝑃( 𝑡 = 𝑅| 𝑅𝑅) =
𝑃( 𝑡=𝑅) 𝑃( 𝑅𝑅| 𝑡 = 𝑅)
𝑃( 𝑡=𝑅) 𝑃( 𝑅𝑅| 𝑡 = 𝑅)+𝑃( 𝑡=𝐵) 𝑃( 𝑅𝑅| 𝑡 = 𝐵)+𝑃( 𝑡=𝑌) 𝑃( 𝑅𝑅| 𝑡 = 𝑌)

41
=
1
3
×
16
25
(
1
3
×
16
25
) + (
1
3
×
1
100
) + (
1
3
×
1
100
)
=
16
75
(
16
75
) + (
1
300
) + (
1
300
)
=
32
33
= 96.97%
ii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵) =
1
3
×
2
25
(
1
3
×
2
25
)+(
1
3
×
2
25
)+(
1
3
×
1
100
)
=
2
75
(
2
75
)+(
2
75
)+(
1
300
)
=
8
17
= 47.06%
Third agent
i) 𝑃( 𝑡 = 𝑅| 𝑅𝑅𝑅) =
1
3
×
64
125
(
1
3
×
64
125
)+(
1
3
×
1
1000
)+(
1
3
×
1
1000
)
=
64
375
(
64
375
)+(
1
3000
)+(
1
3000
)
=
256
257
= 99.61%
ii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅) =
1
3
×
8
125
(
1
3
×
8
125
)+(
1
3
×
1
125
)+(
1
3
×
1
1000
)
=
8
375
(
8
375
)+(
1
375
)+(
1
3000
)
=
64
73
= 87.67%
iii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑌) =
1
3
×
1
125
(
1
3
×
1
125
)+(
1
3
×
1
125
)+(
1
3
×
1
125
)
=
1
375
(
1
375
)+(
1
375
)+(
1
375
)
=
1
3
= 33. 3̇ %
Fourth agent
i) 𝑃( 𝑡 = 𝑅| 𝑅𝑅𝑅𝑅) =
256
1875
(
256
1875
)+(
1
30000
)+(
1
30000
)
=
2048
2049
= 99.95%
ii) 𝑃( 𝑡 = 𝑅| 𝑅𝑅𝑅𝐵) =
32
1875
(
32
1875
)+(
1
3750
)+(
1
30000
)
=
512
521
= 98.27%
iii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅𝐵) =
4
1875
(
4
1875
)+(
4
30000
)+(
1
30000
)
=
64
129
= 49.61%

42
iv) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑌𝑅) =
4
1875
(
4
1875
)+(
1
3750
)+(
1
3750
)
=
4
5
= 80%
Fifth agent
i) 𝑃( 𝑡 = 𝑅| 𝑅𝑅𝑅𝑅𝑅) =
1024
9375
(
1024
9375
)+(
1
300000
)+(
1
300000
)
= 99.99%
ii) 𝑃( 𝑡 = 𝑅| 𝑅𝑅𝑅𝑅𝑅) =
128
9375
(
128
9375
)+(
1
37500
)+(
1
300000
)
=
4096
4105
= 99.78%
iii) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅𝐵𝑅) =
16
9375
(
16
9375
)+(
2
9375
)+(
1
300000
)
=
512
577
= 88.73%
iv) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅𝑌𝑅) =
16
9375
(
16
9375
)+(
1
37500
)+(
1
37500
)
=
32
33
= 96.97%
v) 𝑃( 𝑡 = 𝑅| 𝑅𝐵𝑅𝑌𝑅) =
2
9375
(
2
9375
)+(
2
9375
)+(
1
37500
)
=
8
17
= 47.06%

43
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