# The Bandwagon Effect: Not Just Another Bias

Norman Knyazev  Radboud UniversityNijmegenThe Netherlands<EMAIL_ADDRESS>and  Harrie Oosterhuis  Radboud UniversityNijmegenThe Netherlands
<EMAIL_ADDRESS>
(2022)

###### Abstract.

Optimizing recommender systems based on user interaction data is mainly seen
as a problem of dealing with selection bias, where most existing work assumes
that interactions from different users are independent. However, it has been
shown that in reality user feedback is often influenced by earlier
interactions of other users, e.g. via average ratings, number of views or
sales per item, etc. This phenomenon is known as the _bandwagon effect_.

In contrast with previous literature, we argue that the bandwagon effect
should not be seen as a problem of statistical bias. In fact, we prove that
this effect leaves both individual interactions and their sample mean
unbiased. Nevertheless, we show that it can make estimators inconsistent,
introducing a distinct set of problems for convergence in relevance
estimation. Our theoretical analysis investigates the conditions under which
the bandwagon effect poses a consistency problem and explores several
approaches for mitigating these issues. This work aims to show that the
bandwagon effect poses an underinvestigated open problem that is fundamentally
distinct from the well-studied selection bias in recommendation.

Bandwagon Effect, Consistency, Statistical Bias, Recommendation

††copyright: rightsretained††journalyear: 2022††copyright:
rightsretained††conference: Proceedings of the 2022 ACM SIGIR International
Conference on the Theory of Information Retrieval; July 11–12, 2022; Madrid,
Spain††booktitle: Proceedings of the 2022 ACM SIGIR International Conference
on the Theory of Information Retrieval (ICTIR ’22), July 11–12, 2022, Madrid,
Spain††doi: 10.1145/3539813.3545142††isbn:
978-1-4503-9412-3/22/07††submissionid: 11††ccs: Information systems Social
recommendation††ccs: Mathematics of computing Time series analysis††ccs:
Information systems Learning to rank Figure 1.  Impact of selection bias and
bandwagon effect on average rating estimation. Under popularity bias, users
are less likely to interact with unpopular items, leaving most of their
ratings unobserved. Under the bandwagon effect, new ratings are skewed in the
direction of previous users’ opinion, with larger group size eliciting a
stronger effect.

##  1\. Introduction

One of the main challenges in optimizing Recommender Systems (RecSys) is
correctly capturing user preferences from historical interaction data (Ricci
et al., 2015; Hu et al., 2008; Su and Khoshgoftaar, 2009; Ilievski and Roy,
2013; Jannach et al., 2018). A widely adopted approach is to train a model to
predict known ratings and then apply it to previously unobserved user-item
pairs (Pradel et al., 2012; Steck, 2013; Tang et al., 2015). However, in the
last decade it has become apparent that observed interactions are often a very
skewed representation of the actual user preferences (Steck, 2011; Pradel et
al., 2012; Cañamares and Castells, 2018). In particular, ratings are known to
be affected by various forms of _selection_ bias, such as _popularity_ bias
(Steck, 2011), _positivity_ bias (Pradel et al., 2012) or _position_ bias
(Oosterhuis and de Rijke, 2020). Popularity bias, for instance, occurs because
as most interactions involve only a small number of very popular items, the
patterns in observed ratings may not necessarily extrapolate to the data as a
whole (Cañamares and Castells, 2018; Steck, 2011; Pradel et al., 2012;
Schnabel et al., 2016). It is now well understood that ignoring statistical
biases during optimization is likely to lead to suboptimal recommendation
performance (Huang et al., 2020; Schnabel et al., 2016; Cañamares and
Castells, 2018) and numerous solutions have been proposed to address various
forms of selection bias in recommendation data (Schnabel et al., 2016; Huang
et al., 2020; Saito et al., 2020; Wang et al., 2019; Saito, 2020; Saito et
al., 2020; Yuan et al., 2020; Sato et al., 2021; Jeunen and Goethals, 2021).

In this work, we will investigate the related but fundamentally distinct
_bandwagon effect_ , also referred to as _herding_ or _conformity bias_.
Despite being observed by multiple studies, this often-overlooked effect
occurs because users tend to follow the behavior and opinions of others
(Simon, 1954; Flanagin and Metzger, 2013). That is, if a user is told that
many other users have liked an item, they themselves become more likely to
give a positive rating. Conversely, when a user is told that many other users
dislike an item, their own rating is likely to become lower. Several studies
have observed this behavior when exposing the average rating from previous
users (Flanagin and Metzger, 2013; Sundar et al., 2008; Cosley et al., 2003;
Adomavicius et al., 2016) and analogous trends have also been reported for
implicit feedback (Qin et al., 2021; Hanson and Putler, 1996). Consequently,
the bandwagon effect can pose a serious problem when trying to infer user
preferences from interaction data.

Despite its well-known existence, there have only been limited attempts at
reducing the influence of the bandwagon effect (Wang et al., 2014; Zhang et
al., 2017; Liu et al., 2016) and few authors have theoretically analyzed it in
statistical terms (Xie and Zhong, 2020; Xie et al., 2021). Our work will build
on a mathematical model proposed by Xie et al. (Xie and Zhong, 2020; Xie et
al., 2021), who in line with earlier literature (Wang et al., 2014; Zhang et
al., 2017) interpret the bandwagon effect as a problem of statistical bias.

In this work, we provide a rigorous theoretical analysis of the bandwagon
effect and its statistical impact on rating and Click-Through Rate (CTR)
estimation. In stark contrast with previous work, we prove that a simple form
of bandwagon effect of Xie et al. does _not_ introduce statistical bias, as
both individual ratings and their sample mean yield unbiased estimates under
the effect. Nonetheless, our analysis also reveals that the bandwagon effect
can create serious problems for convergence by making the sample mean
inconsistent. Therefore, despite not being a bias issue, the bandwagon effect
can still introduce significant errors during optimization. To understand when
this may pose a problem, we theoretically prove a condition under which
inconsistency is guaranteed to arise for the sample mean. Furthermore, our
theoretical and simulated results show that even when an estimator is
consistent, the convergence can still be so slow that finding the true user
preferences becomes practically infeasible. We therefore investigate whether
existing debiasing methods can be adapted to handle these convergence issues.
However, as none of our methods are able to combine theoretical guarantees
with robust empirical performance, we conclude that the bandwagon effect
currently remains an open problem that needs more attention in future work.

##  2\. Related Work

###  2.1. Debiasing Interactions for Recommendation

Learning from user interactions has a long tradition in the field of
recommender systems (Koren et al., 2009; Pradel et al., 2012; Schnabel et al.,
2016; Hernández-Lobato et al., 2014; Marlin and Zemel, 2009). A common
approach is to cast the recommendation problem as a rating prediction task
where the goal is to predict the ratings a user would give to items (Koren et
al., 2009; Marlin and Zemel, 2009; Hernández-Lobato et al., 2014). However,
one should not straightforwardly base the optimization on the ratings observed
in interaction logs as they are often not representative of actual user
preferences due to selection bias (Hernández-Lobato et al., 2014; Marlin and
Zemel, 2009). The most well-known and well-studied form of bias in rating data
is popularity bias: a small selection of popular items receives the large
majority of user ratings and is thus grossly overrepresented (Cañamares and
Castells, 2018; Pradel et al., 2012; Steck, 2011). Positivity bias is also
prevalent: users may be more likely to leave a rating on items they like;
consequently, low ratings become underrepresented in the resulting interaction
data (Pradel et al., 2012). As seen in Figure 1 (top), if popular items that
attract the majority of ratings also happen to be more liked by users (i.e.
receive higher ratings), the average rating estimate from observed ratings
becomes skewed towards a higher value. Schnabel et al. (2016) showed that not
accounting for selection bias makes the optimization procedure focus on
statistically overrepresented user-item combinations. As a solution, they
propose to use Inverse Propensity Scoring (IPS) to unbiasedly estimate and
optimize the performance of a system (Little and Rubin, 2019). Debiasing for
recommendation has become a very popular topic in recent years: for example,
Wang et al. (2019) propose a doubly robust estimator that combines IPS with
regression estimates; Saito et al. (2020) adapted the IPS matrix factorization
approach to debias click data instead of ratings; and Jeunen and Goethals
(2021) incorporate a prior distribution to avoid overestimation during
prediction.

Selection bias is also widespread in clicks. Its most well-studied form is
position bias, which occurs when the position at which an item is displayed
heavily impacts the attention the item will receive, and thus how many users
will click on it (Craswell et al., 2008). Similar to ratings, IPS is a popular
method to correct for selection bias in clicks (Joachims et al., 2017;
Oosterhuis, 2020).

The problem of selection bias in explicit feedback, i.e. ratings, and implicit
feedback, e.g. clicks, is thus well recognized and has been addressed
extensively by the existing literature (Joachims et al., 2017; Oosterhuis,
2020; Jeunen and Goethals, 2021; Saito et al., 2020; Wang et al., 2019;
Schnabel et al., 2016; Koren et al., 2009; Pradel et al., 2012; Hernández-
Lobato et al., 2014; Marlin and Zemel, 2009). This can lead to the impression
that learning from interactions is mainly a selection bias problem.
Nevertheless, our current work on the bandwagon effect will show that this is
definitely not the case.

###  2.2. The Bandwagon Effect

The _bandwagon effect_ , or _herding_ , describes a phenomenon whereby
individuals tend to exhibit a greater affinity towards an item when they
believe other individuals also like it (Simon, 1954; Raafat et al., 2009). The
decisions of earlier individuals can propagate through _information cascades_
to individuals also taking the same decision later, leading to snowballing
effects, as well documented in social psychology, economics and finance, and,
more recently, web information systems (Raafat et al., 2009; Caparrelli et
al., 2004; Borghesi and Bouchaud, 2010; Cosley et al., 2003).

Online platforms frequently include various social cues next to items to
assist users with their choice, e.g. the number and the average value of
previous user ratings. Numerous studies have shown that users actively
leverage such information when making purchasing or consumption decisions
(Hanson and Putler, 1996; Dholakia and Soltysinski, 2001; Sundar et al., 2008;
Flanagin and Metzger, 2013; Qin et al., 2021; Salganik et al., 2006; Salganik
and Watts, 2008; Fu and Sim, 2011; Kim and Gambino, 2016; Huang and Chen,
2006; Adomavicius et al., 2016). Hanson and Putler (1996) found that
artificially inflating download counts for some of the products on a software
download service led to those products receiving more subsequent downloads
than their unaltered counterparts. Similarly, Qin et al. (2021) report that
Chrome Web Store users rarely click on items displayed with a low download
count. Flanagin and Metzger (2013) noted that users exposed to the average
rating also rated movies closer to that average. Analogously, Sundar et al.
(2008) found that participants reported higher purchasing intent for high-
rated items over low-rated items and that this effect arose from an improved
perception of quality and value. Both studies also reported that the strength
of the effect was positively correlated with the item’s number of ratings,
consistent with earlier works associating larger group size with stronger
herding (Latané and Wolf, 1981; Asch, 1951; Huang and Chen, 2006). Altogether,
these observations suggest that users incorporate the experience of others
into their own judgements and the strength of this effect is correlated with
the size of the group.

This phenomenon can have a number of negative consequences. On an individual
level, the bandwagon effect can be so strong that users may act against their
own interests: Dholakia and Soltysinski (2001) highlight that eBay users
frequently bid on items that already have many bids, whilst ignoring
comparable or superior listings with fewer bids. However, another consequence
of the bandwagon effect – and the focus of this paper – is the difficulty it
brings to estimating item relevance from user interactions. In the pioneering
experiment of Salganik et al. (2006), a large number of sequentially arriving
participants browsed a randomly shuffled grid of the same $48$ songs and could
interact with songs of their choice. The participants were randomly assigned
to one of nine groups, with users in groups $2-9$ additionally presented with
the download count from previous users _from their group_ alongside each song.
The authors observed that songs had a higher probability of being downloaded
in a group where they already had a high number of downloads. In contrast, a
song shown next to a low download count was less likely to get downloaded
compared to the same song in groups where it had a high or no presented
download count (group $1$). Moreover, with the exception of the very best and
worst songs, as determined by the control group $1$, the final ranking
according to download counts varied wildly across different groups. In their
follow-up work, Salganik and Watts (2008) further showed that the bandwagon
effect may be so strong that artificially altering the presented download
count can make unpopular songs popular and vice-versa, with only the best
songs eventually recovering their true relevance ranking. Taken together, the
above results suggest that under the bandwagon effect even a large amount of
user feedback is not guaranteed to yield an accurate estimate of true user
preference. This is also demonstrated in Figure 1 (bottom): users’ true
preferences are intertwined with the ratings of earlier users and this
influence grows as the number of ratings increases. As a consequence, even
when assuming no selection bias, true average rating cannot be determined via
the sample mean.

Recently, there has been some interest in modelling herding and disentangling
the effects of the crowd on the individual. Wang et al. (2014) propose an
estimator maximizing a surrogate function of the likelihood of the data and
demonstrate the effectiveness of their approach for prediction of subsequent
ratings. Zhang et al. (2017) show that subsequent rating estimation can be
further improved by considering that some users may actually choose to
distance their opinion from the crowd. Liu et al. (2016); Zhang et al. (2021);
Wei et al. (2021) all propose various extensions to classical recommender
models by using the rating mean and/or count as additional inputs. Finally,
Xie, Zhong and colleagues formulate the bandwagon process as either a rating
reweighting or hidden state prediction problem and show theoretically that the
bandwagon effect slows down the convergence of the sample mean (Xie and Zhong,
2020; Xie et al., 2021; Zhong et al., 2021).

Notably, most of the above works treat the bandwagon effect as a problem of
statistical bias. Furthermore, to the best of our knowledge, only the works of
Xie et al. (Xie and Zhong, 2020; Xie et al., 2021) and Zhong et al. (2021)
attempt to establish a link between the bandwagon effect and statistical
theory. Our theoretical analysis will contrast with the existing literature by
showing that the bandwagon effect can affect relevance estimation without
introducing statistical bias.

##  3\. Background

Estimated Value
---
Number of Samples
Figure 2. Three estimators with varying bias and consistency; bold lines show
the mean estimate over 1,000 independent runs, shaded areas indicate 90%
confidence intervals. Top - unbiased and consistent: its expected value and
point of convergence are both correct. Middle - biased and inconsistent: its
expected value and point of convergence are both incorrect. Bottom - unbiased
and inconsistent: its expected value is correct but its confidence interval
will never converge.

###  3.1. Preliminaries

Before we address the bandwagon effect specifically, this section will
introduce our problem setting and the statistical properties that our analysis
will consider. For the sake of generality, we focus on _unpersonalized_ rating
estimation, as we expect the issues observed for this simple task to also
apply to the more involved forms of recommendation. Furthermore, our findings
are not specific to recommendation and can be readily applied to other
problems involving relevance estimation such as CTR estimation.

In our generic unpersonalized recommendation setting, the goal is to estimate
the average preference of all users for each item. We assume that this value
$p\in(0,1)$ exists for each item and represents the percentage of users that
truly like the item. Furthermore, users interact with the items by assigning
them binary ratings s.t. $r_{i}\in\\{0,1\\}$ is the $i$’th rating for an item.
While we refer to $r_{i}$ as a rating, it can equally well represent many
other binary interactions such as _clicks_ , _purchases_ , _shares_ , etc. As
discussed in Section 2.2, in real systems users are often able to see how many
other users have rated an item and what their average rating was, such that
past ratings can affect the behaviour of the current user. We define $p$ to be
the _true preference_ or the percentage of users who like an item when they
have no knowledge of the preferences of any other users. Our goal is then to
estimate $p$ for a single item from the given set of observed ratings
$\\{r_{1},r_{2},r_{3}\dots\\}$.

Let $\hat{p}_{n}$ indicate an estimate of $p$ based on $n$ observed ratings.
Our analysis will investigate three important beneficial properties that we
generally desire from such an estimate. To start, we normally want the
estimate to be unbiased. In other words, in expectation it should be equal to
the true value of $p$:

(1)
$\text{Unbiased}\big{(}\hat{p}_{n}\big{)}\longleftrightarrow\mathbb{E}\big{[}\hat{p}_{n}\big{]}=p.$

Understandably, bias and unbiasedness have received a lot of attention in
previous selection bias literature, as it is desirable to avoid systematic
errors in rating estimation (Schnabel et al., 2016; Joachims et al., 2017;
Oosterhuis and de Rijke, 2020; Sato et al., 2020; Wang et al., 2020). However,
unbiasedness on its own does not guarantee that an estimator is actually
usable in practice.

The second important and often overlooked property that we may want from an
estimator is consistency. An estimator is considered consistent if the
estimate converges in probability to the true value of $p$ as the size of
available data approaches infinity:

(2)
$\text{Consistent}\big{(}\hat{p}_{n}\big{)}\longleftrightarrow\lim_{n\rightarrow\infty}\hat{p}_{n}=p.$

Whilst at a first glance unbiasedness and consistency may appear to refer to
the same property, this is certainly _not_ the case. For example, estimators
can be biased yet consistent (Swaminathan and Joachims, 2015b) or unbiased and
inconsistent. As illustrated in Figure 2, the problem with an unbiased
inconsistent estimator is that its confidence intervals do not converge to a
zero error even as the size of the data increases indefinitely. This means its
estimation errors will not disappear even after gathering more data, thus
preventing accurate convergence.

Nevertheless, whilst consistency guarantees that eventually the estimate will
be correct, the amount of data required for convergence may still be
unattainable in practice. Therefore, the final property we will consider is
(data) efficiency, measured by the expected squared error (equivalent to
variance for unbiased estimators):

(3)
$\text{Efficiency}\big{(}\hat{p}_{n}\big{)}=\mathbb{E}\big{[}\big{(}\hat{p}_{n}-p\big{)}^{2}\big{]}.$

Overall, efficiency represents a more practical side of an estimator: the
error we can expect it to have w.r.t. the true value $p$ after $n$ samples.
The effect the number of samples has on this measure can be seen as the speed
of convergence. While most existing work on learning from user interactions
focuses on correcting for bias, they often also apply propensity clipping or
self-normalization which introduce a little bias to greatly reduce variance
(Joachims et al., 2017; Sato et al., 2020; Yang et al., 2018; Saito et al.,
2020; Swaminathan and Joachims, 2015a).

Our theoretical investigation will consider how the bandwagon effect impacts
all three properties: bias, consistency and efficiency.

###  3.2. A Formal Model of the Bandwagon Effect

The mathematical model of the bandwagon effect we will use in this paper is a
simplified version of that of Xie et al. (Xie and Zhong, 2020; Xie et al.,
2021). This model represents a sequential unpersonalized recommendation
scenario where users rate the item in succession and one at a time. It assumes
there are two factors that influence the probability of a positive value for
the $n$’th rating ($r_{n}=1$):  (i) $p$, the true proportion of users who
would like the item given that they do not know anything of the other users’
preferences (no bandwagon effect); and  (ii) $\bar{p}_{n-1}$, the mean of the
ratings that have taken place previously.  The rating probability is modelled
as a linear combination of these two factors:

(4)
$P(r_{n}=1)=\lambda_{n}p+(1-\lambda_{n})\bar{p}_{n-1},\qquad\bar{p}_{n-1}=\dfrac{1}{n-1}\sum_{i=1}^{n-1}r_{i}.$

$\lambda_{n}\in[0,1]$ captures the degree to which users adjust their behavior
to match that of the crowd. For example, $\forall\lambda_{n}=1$ indicates a
process with no bandwagon effect and $\forall n>1:\lambda_{n}=0$ a process
where true preferences play no role except for the very first user. As
previously mentioned, earlier works have shown a positive correlation with the
strength of herding and the size of the group (Latané and Wolf, 1981; Asch,
1951; Huang and Chen, 2006). We capture this increase in effect by assuming
that the influence of the sample mean cannot decrease as the number of ratings
increases:

(5) $\lambda_{1}=1\land\left(\forall
n>1:0\leq\lambda_{n}\leq\lambda_{n-1}\right).$

In addition, this assumption states that the first rating is completely
unaffected by the bandwagon effect (i.e. $\lambda_{1}=1$), as the first user
has no information about the preferences of others. We have chosen the above
formulation over others (e.g. (Wang et al., 2014; Zhang et al., 2017)) due to
its amenability to statistical analysis whilst still capturing the key aspects
of the bandwagon effect.

Before we begin our theoretical analysis, we will briefly discuss two claims
from earlier work by Xie and Zhong (2020) so that they can later be compared
with our findings in Section 4. First, Xie and Zhong refer to the bandwagon
effect as _persuasion bias_ , by which they mean that the bandwagon effect
biases the rating distribution such that it no longer represents the true
underlying preference. Second, they show that as long as the lower bound of
$\lambda_{n}$ is above $0$, the bandwagon effect will eventually be eliminated
since the sample mean $\bar{p}_{n}$ is guaranteed to converge to the true
parameter value:

(6) $\inf_{n\in
N^{+}}{\lambda_{n}}>0\longrightarrow\lim_{n\to\infty}\bar{p}_{n}=p.$

Our theoretical analysis will take another critical look at the bias and
consistency of $\bar{p}_{n}$. Surprisingly, our findings regarding bias
heavily contrast with previous work on the bandwagon effect.

##  4\. Theoretical Analysis:
The Bandwagon Effect

Now that Section 3.2 has introduced our mathematical model of the bandwagon
effect, we will begin our theoretical analysis of its effects on bias,
consistency and efficiency. First, we want to again consider the claim of
previous work that the bandwagon effect introduces bias to the rating
distribution (Xie and Zhong, 2020; Xie et al., 2021). From our bandwagon model
(Equation 4) it appears that the probability of a positive rating is indeed a
distortion of the true preference $p$. However, Theorem A.1 proves that the
expected values of all sample means $\bar{p}_{n}$ and all ratings $r_{n}$ are
actually equal to the true preference:

(7) $\forall
n:\mathbb{E}\left[\bar{p}_{n}\right]=\mathbb{E}\left[r_{n}\right]=p.$

In other words, under our model, both the sample means and actual ratings
provide _unbiased_ estimates of the true preference, corresponding to either
top or bottom, but not center of Figure 2. This finding contrasts heavily with
how previous work has approached the bandwagon effect as a bias problem (Zhang
et al., 2017, 2020; Xie and Zhong, 2020; Xie et al., 2021; Zhong et al., 2021;
Wang et al., 2014; Liu et al., 2016; Wei et al., 2021) and similarly, how
learning from user interactions is generally also framed as a bias problem
(Joachims et al., 2017; Oosterhuis, 2020; Jeunen and Goethals, 2021; Saito et
al., 2020; Wang et al., 2019; Schnabel et al., 2016; Koren et al., 2009;
Pradel et al., 2012; Hernández-Lobato et al., 2014; Marlin and Zemel, 2009).
The bandwagon effect _may_ be seen as an increased impact of earlier
interactions on the estimate of the item’s relevance. However, in contrast
with selection bias, there is no predetermined rating value that is
disproportionally represented for the purposes of average relevance estimation
or loss minimization.

Nevertheless, the lack of bias does not mean that the bandwagon effect does
not introduce other problems. As discussed in Section 3.1, there are other
statistical properties besides unbiasedness that one should consider. Theorem
A.4 proves the following efficiency of the sample mean, i.e. its expected
(squared) error:

$\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=p\left(1-p\right)\bigg{(}\dfrac{1}{n^{2}}+\sum_{i=1}^{n-1}\dfrac{1}{i^{2}}\prod_{j=i+1}^{n}\dfrac{(j-1)(j+1-2\lambda_{j})}{j^{2}}\bigg{)}.$

Consistent with our prior discussion, the above formulation reveals that when
$\forall\lambda_{n}=1$ , the variance of $\bar{p}_{n}$ is equivalent to the
variance of sampling from the true preference $p$. Conversely, when $\forall
n>1:\lambda_{n}=0$, the expected error does not decrease after the first
interaction i.e. $\forall
n:\mathbb{E}[(\bar{p}_{n}-p)^{2}]=\mathbb{V}[\bar{p}_{1}]=p(1-p)$. Obviously,
an error that never decreases provides a serious convergence problem.

To investigate whether the sample mean is consistent, we look at the value of
the expected error in the limit of infinite samples. Theorem A.5 proves the
following expected error in the limit:

$\lim_{n\to\infty}\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=\lim_{n\to\infty}p\left(1-p\right)\bigg{(}\sum_{i=1}^{n-1}\dfrac{1}{i\left(i+1\right)}\prod_{j=i+1}^{n-1}\left(1-\dfrac{2\lambda_{j}}{j+1}\right)\bigg{)}.$

Importantly, if the expected error is not zero in the limit as in Figure 2
(bottom), the estimator is not consistent and vice-versa. Based on this fact,
we prove a novel necessary condition for the consistency of $\bar{p}_{n}$ in
Theorem A.6:

(8)
$\forall\epsilon:\lim_{n\to\infty}\text{Pr}\left(\left|\bar{p}_{n}-p\right|>\epsilon\right)=0\longrightarrow\lim_{n\to\infty}\sum_{i=1}^{n}\lambda_{i}^{2}=\infty.$

In other words, if the sample mean is consistent (corresponds to Figure 2,
top), the sum of $\lambda_{i}^{2}$ terms is unbounded. Equivalently,
convergence of $\lim_{n\to\infty}\sum_{i=1}^{n}\lambda_{i}^{2}$ leads to
inconsistency. We note that whilst the sufficient condition of Xie and Zhong
(2020) (Equation 6) highlights the situations where we can guarantee
consistency, it cannot be used to guarantee inconsistency. For instance, one
cannot use it to show whether $\lambda_{n}=n^{-1}$ leads to issues with
convergence as $\inf_{n\in N^{+}}\lambda_{n}=0$. On the other hand, as
$\lim_{n\to\infty}\sum_{i=1}^{n}i^{-2}$ is finite, Theorem A.6 shows that
$\bar{p}_{n}$ is inconsistent. Our condition can thus be seen as complementary
to the sufficient condition of earlier work, qualifying the set of $\lambda$
sequences where inconsistency is certain.

We can also leverage Theorem A.5 to quantify the asymptotic expected absolute
error for any specific $\lambda$ values. For instance, Proposition A.8 proves
that for $\lambda_{i}=c^{i-1}$ with $c\in(0,1)$:

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\left|\bar{p}_{n}-p\right|\right]\geq
2p\left(1-p\right)\exp\big{(}-c\phi\left(c,1,2\right)-c^{2}\phi\big{(}c^{2},2,2\big{)}\big{)}$
(9) $\displaystyle\hskip 113.81102pt\text{ s.t.
}\phi\left(z,s,a\right)\equiv\sum^{\infty}_{k=0}\dfrac{z^{k}}{\left(a+k\right)^{s}}.$

Similar bounds may also be derived for other $\lambda$ formulations.

Our analysis proved that neither the sample mean $\bar{p}_{n}$ nor the
individual ratings $r_{n}$ are biased, a finding that appears to heavily
contrast with earlier work that casts the bandwagon effect as a problem of
bias (Zhang et al., 2017, 2020; Xie and Zhong, 2020; Xie et al., 2021; Zhong
et al., 2021; Wang et al., 2014; Liu et al., 2016; Wei et al., 2021). We
speculate that the bias framing may at least in part stem from the dependency
between ratings. More specifically, conditioning the expectation of
$\bar{p}_{n}$ and $r_{n}$ on an earlier $\bar{p}_{m}$ by Theorem A.1 and
Proposition A.7 leads to:

(10) $\displaystyle\forall
n>m>1:\mathbb{E}[\bar{p}_{n-1}\mid\bar{p}_{m}]-p=\left(\bar{p}_{m}-p\right)\prod_{i=m+1}^{n-1}\Big{(}1-\frac{\lambda_{i}}{i}\Big{)},$
(11) $\displaystyle\forall
n>m>1:\mathbb{E}[r_{n}\mid\bar{p}_{m}]-p=\left(1-\lambda_{n}\right)\mathbb{E}[\bar{p}_{n-1}-p\mid\bar{p}_{m}].$

This can be seen as a _conditional_ bias, i.e. conditioned on any
$\bar{p}_{m}$ being incorrect, all subsequent ratings and sample means are
biased:

(12) $\forall n>m\geq 1:\mathbb{E}[\bar{p}_{n}\mid\bar{p}_{m}\neq p]\neq p.$

We speculate that previous work has interpreted this conditional bias as bias
in the rating distribution and sample mean. Nonetheless, as discussed above,
our bandwagon effect model does not lead to systematic errors in relevance
estimation. In Section 7, we will argue that therefore the bandwagon effect is
best interpreted as a problem of consistency and not statistical bias.

To summarize the main findings of our theoretical analysis: we have proven
that the bandwagon effect does not introduce bias to the rating distribution
but can make the sample mean inconsistent. We have also derived a necessary
condition for consistency, expanding on prior work. Lastly, our formulation of
the expected error can quantify the asymptotic error for any specific
$\lambda$ values.

##  5\. Bandwagon Effect Mitigation

###  5.1. Correcting for the Bandwagon Effect

We have established that while the bandwagon effect does not make ratings
biased, it can lead to convergence problems. As possible solutions, inspired
by analogous approaches in click debiasing (Vardasbi et al., 2020; Chuklin et
al., 2015), we will propose an affine and a maximum likelihood estimator.

Firstly, similar to earlier unbiased Learning to Rank work (Vardasbi et al.,
2020), we notice that the model in Equation 4 is invertible. We therefore
propose an intermediate affine estimate based on a single rating:

(13)
$\hat{r}_{i}=\dfrac{r_{i}-\left(1-\lambda_{i}\right)\bar{p}_{i-1}}{\lambda_{i}}.$

Theorems A.9 and A.10 prove that our intermediate estimator is both unbiased
and conditionally unbiased w.r.t. previous ratings:

(14) $\forall i>j\geq
1:\mathbb{E}[\hat{r}_{i}\mid\hat{r}_{j}]=\mathbb{E}[\hat{r}_{i}\mid\bar{p}_{j}]=\mathbb{E}[\hat{r}_{i}]=p.$

Therefore, using $\hat{r}_{i}$ instead of $r_{i}$ in an averaging estimator
may allow us to speed up convergence by incorporating only the information
about the user’s true preference without reinforcing the previous $i-1$
ratings. With this in mind, we propose our affine estimator that uses all of
the observed ratings via a weighted mean:

(15)
$\hat{p}_{n}=\dfrac{1}{\sum_{i=1}^{n}\omega_{i}}\sum_{i=1}^{n}\omega_{i}\hat{r}_{i},\quad\text{
s.t.\ }\forall\,1\leq i\leq n:\omega_{i}>0.$

Clearly $\hat{p}_{n}$ is also unbiased as it is simply a weighted average over
the unbiased estimates $\hat{r}_{i}$. Additionally, Theorem A.14 proves it is
consistent if $\sum_{i=1}^{n}(\omega_{i}/\lambda_{i})^{2}$ grows slower in $n$
than $(\sum_{i=1}^{n}\omega_{i})^{2}$ , with Corollary A.15 providing specific
conditions $\forall\omega_{i}=1$ and $\forall\omega_{i}=\lambda_{i}$.
Furthermore, Theorem A.13 proves $\hat{p}_{n}$ has the following variance:

(16)
$\mathbb{V}\left[\hat{p}_{n}\right]=\dfrac{1}{\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}}\sum_{i=1}^{n}\dfrac{\omega_{i}^{2}}{\lambda_{i}^{2}}\left(p\left(1-p\right)-\left(1-\lambda_{i}\right)^{2}\mathbb{V}\left[\bar{p}_{i-1}\right]\right).$

Unfortunately, interpreting $\mathbb{V}\left[\hat{p}_{n}\right]$ is not
straightforward as in addition to the choice of $\omega_{i}$ it depends on
both the rating values as well as their order. Analogous to its use in
selection bias (Joachims et al., 2017; Strehl et al., 2010), one can also
apply clipping to the bandwagon effect, e.g.
$\hat{\lambda}_{i}=\max{(\lambda_{i},\tau)}$ s.t. $\tau>0$. However, in
contrast with IPS, $\hat{r}_{n}$ and $\hat{p}_{n}$ would remain unbiased,
though not conditionally unbiased (Corollary A.11): clipping or misestimation
s.t. $\hat{\lambda}_{i}\neq\lambda_{i}$ may still affect the rate of
convergence.

Secondly, inspired by the approach of click models (Chuklin et al., 2015), we
also propose the approximate maximum likelihood estimator $p^{*}_{n}$,
obtained via iterative application of Newton’s method (Galántai, 2000) to
maximize the log-likelihood of the data $L$:

(17)
$\begin{split}L=\sum_{i=1}^{n}{}&{}r_{i}\ln\left(\lambda_{i}p^{*}_{n}+(1-\lambda_{i})\bar{p}_{i-1}\right)\\\\[-6.45831pt]
&+\left(1-r_{i}\right)\ln\left(1-\left(\lambda_{i}p^{*}_{n}+(1-\lambda_{i})\bar{p}_{i-1}\right)\right).\end{split}$

Because both the likelihood and the solution found by Newton’s method appear
intractable, we are unable to prove this estimator is unbiased. Nonetheless,
maximum likelihood estimators have been very effective when applied to user
interactions in the past (Zhuang et al., 2021; Chuklin et al., 2015) and can
be consistent under certain feasible conditions (Crowder, 1976).

###  5.2. Bandwagon Effect Estimation

Both our proposed estimators require accurate $\lambda$ values to capture the
bandwagon effect well. Inspired by the experiments of Salganik et al. (2006),
we propose the following parameter estimation scheme that can be applied when
dealing with one or more items affected by the same bandwagon effect.

We will assume that $\lambda$ values are shared across a subset of $K\geq 1$
items and that all users are randomly assigned to one of $M$ bins s.t. users
give $n$ ratings per item and are only presented with rating information
regarding users in the same bin. Let $p^{k}$ indicate the true preference of
all users w.r.t. item $k$ (analogous to $p$) and $\hat{\lambda}_{i}$ an
estimate of $\lambda_{i}$. The log-likelihood of this data is then:

(18)
$\begin{split}L=\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{i=1}^{n}{}&{}r_{kmi}\ln\big{(}\hat{\lambda}_{i}p^{k}+\big{(}1-\hat{\lambda}_{i}\big{)}\bar{p}_{i-1}^{km}\big{)}\\\\[-7.5347pt]
&+\left(1-r_{kmi}\right)\ln\big{(}1-\hat{\lambda}_{i}p^{k}-\big{(}1-\hat{\lambda}_{i}\big{)}\bar{p}_{i-1}^{km}\big{)}.\end{split}$

The maximum likelihood estimate can be found by maximizing $L$, for instance
through the iterative application of Newton’s method. In order to get
$\hat{\lambda}_{i}$ values for $i>n$, we propose fitting them to a curve, e.g.
$\hat{\lambda}_{i}=a+(1-a)\cdot b^{i-1}$ where $a\in[0,1]$ and $b\in(0,1)$.
Note that as $p^{k}$ may be unknown, one may instead estimate $\lambda_{i}$
values by using its unbiased estimate
$\bar{p}^{k}=\frac{1}{Mn}\sum_{m=1}^{M}\sum_{i=1}^{n}r_{kmi}$.

##  6\. Experiments and Results

###  6.1. Experimental Setup

To investigate how the bandwagon can affect convergence and whether our
proposed estimators can mitigate negative effects, we performed several
experiments with simulated user interactions.

For an item with true relevance $p=0.4$, we generated user interactions under
three conditions: a _no bandwagon_ setting where $\forall\lambda_{i}=1$, a
_weak bandwagon_ setting where $\lambda_{i}=0.6+0.4\cdot 0.9^{i-1}$ and a
_strong bandwagon_ setting where $\lambda_{i}=0.1+0.9\cdot 0.95^{i-1}$. Our
comparison includes the following four estimators:  (i) the sample mean
$\bar{p}_{n}$ (Equation 4);  (ii) affine mean $\hat{p}_{n}$, which is the
affine estimator (Equation 15) with uniform weights $\forall i:\omega_{i}=1$;
(iii) affine weighted $\tilde{p}_{n}$, which is the affine estimator but with
the weights $\forall i:\omega_{i}=\lambda_{i}$; and  (iv) the maximum
likelihood estimate $p_{n}^{*}$ (Equation 17).  For statistical significance,
the entire sampling process was repeated $1{,}000$ times independently, all
our graphs include 90% confidence bounds over these independent runs.

Furthermore, to investigate how sensitive our estimators are to misestimated
$\hat{\lambda}$ values, we computed the estimator values with both the actual
$\lambda$ values and estimated $\hat{\lambda}$ values. We followed the
estimation scheme of Section 5.2, generating additional ratings with $K=1$,
$p_{1}=0.4$, $M=20$ and $n=100$ to represent a plausible production-level
scenario (Salganik et al., 2006). The result obtained from fitting a curve:
$\hat{\lambda}_{i}=a+(1-a)\cdot b^{i-1}$ where $a\in[0,1]$ and $b\in(0,1)$ to
the data was: $\hat{\lambda}_{i}=0.33+0.67\cdot 0.92^{i-1}$. We only report
the results for misestimated $\hat{\lambda}$ values in the strong bandwagon
setting, due to high similarity with the weak setting.

###  6.2. Estimator Bias and Convergence

Estimated Value |
---|---
Number of Samples
Figure 3. Convergence of $\bar{p}_{n}$ under various levels of bandwagon. Solid and dashed lines indicate points where a $90\%$ confidence interval over $1{,}000$ independent runs is within $0.05$ and $0.01$ of $p$ respectively, $p=0.4$. _Misestimated_ _Weak Bandwagon_ Estimated Value |  |
---|---|---
_Misestimated_ _Strong Bandwagon_ Estimated Value |  |
_Misestimated_ _Strong Bandwagon_ Estimated Value |  |
x x x |  |
x x x Number of Samples | Number of Samples | Number of Samples
Figure 4.  Convergence of $\hat{p}_{n}$ (left), $\tilde{p}_{n}$ (center) and
$p_{n}^{*}$ (right) compared to $\bar{p}_{n}$ with $p=0.4$ and
$\lambda_{i}=0.6+0.4\cdot 0.9^{i-1}$ (top), $\lambda_{i}=0.1+0.9\cdot
0.95^{i-1}$ (center and bottom) and misestimated
$\hat{\lambda}_{i}=0.33+0.67\cdot 0.92^{i-1}$ (bottom); bold lines show the
mean estimate over $1{,}000$ independent runs, shaded areas indicate 90%
confidence intervals.

Our results are visualized in Figure 3, which displays the convergence rate of
the sample mean under various levels of bandwagon effect, and Figure 4, which
shows the behavior of all estimators with known $\lambda$ in the weak and
strong bandwagon settings.

First, we consider the influence of the bandwagon effect on the sample mean
($\bar{p}_{n}$): from Figure 3 it is immediately clear that the effect’s
strength has a significant impact on the rate of convergence. Whilst under no
bandwagon effect the confidence intervals are within $0.05$ of $p$ after fewer
than $300$ samples and within $0.01$ after roughly $8{,}000$ samples, under
the weak bandwagon effect these numbers exceed $1{,}000$ and $33{,}000$
respectively. Under the strong bandwagon effect the confidence intervals
increase even further, exceeding $0.05$ after $10^{6}$ samples. However, in
all cases the confidence bounds still appear to decrease steadily, suggesting
that $\bar{p}_{n}$ will converge eventually, as also predicted by the
sufficient condition for consistency (Equation 6). Nevertheless, it is also
clear that the bandwagon effect introduces a significant loss in data
efficiency. In practice, this could mean that an infeasibly large amount of
data is required to get a decent estimate under a strong bandwagon effect.

Next, we examine the impact on the bias of the sample mean and other
estimators. As shown in Figure 4 (top, middle), even prior to convergence, all
estimators including $\bar{p}_{n}$ make negligible errors in expectation,
strongly suggesting that they are all unbiased. Whilst the unbiasedness was
already theoretically proven for $\bar{p}_{n}$ and the affine estimators
$\hat{p}_{n}$ and $\tilde{p}_{n}$, it is surprising to see that even at low
numbers of samples the expected error of the maximum likelihood estimate
$p_{n}^{*}$ appears negligible. Our results thus seem to indicate that all the
estimators are unbiased or have an extremely small bias, further confirming
that the bandwagon effect is not a bias problem.

We then focus on the weak bandwagon condition (Figure 4 top). Here, all
estimators converge to $p$, at a rate similar to that of $\bar{p}_{n}$. That
the sample mean and affine estimators converge is not surprising, as the
setting meets the sufficient condition for consistency. However, in this
scenario none of the estimators actually provide a clear improvement over the
sample mean, indicating that they are unable to apply significant beneficial
corrections to the bandwagon effect. The lack of improvement over
$\bar{p}_{n}$ is particularly concerning as this indicates that our estimators
are similarly affected by the loss of data efficiency induced by the bandwagon
effect.

Nonetheless, one may expect the convergence to slow down even further in the
strong bandwagon setting shown in the middle row of Figure 4. Here, both the
affine and maximum likelihood estimators are actually able to converge on the
true value much sooner than the sample mean. However, confidence bounds of the
former also become wider after roughly $10$ sampled ratings and stay wider
until roughly $1{,}000$ samples, with the weighting of $\tilde{p}_{n}$
seemingly reducing this effect when compared with $\hat{p}_{n}$. Thus during
this period, the sample mean actually appears to be a more reliable choice in
terms of expected error. This observation illustrates that choosing between
estimators is not always straightforward, as in this case the optimal choice
between $\bar{p}_{n}$ and $\tilde{p}_{n}$ depends on the number of samples
that are available. Importantly, the maximum likelihood estimator does not
have this drawback: its confidence bounds never appear wider than the sample
mean while still converging quicker.

Across both settings, we have thus observed no noticeable bias problems for
any estimator. Importantly, the sample mean has serious convergence issues
under the strong bandwagon, where the other estimators are able to converge
much sooner. Nonetheless, despite its high efficiency, the affine estimator
was also found to initially have the highest variance. It appears that when
$\lambda$ values are known, the maximum likelihood estimator is the most
reliable in spite of the lack of proven theoretical guarantees for this
estimator. However, even the maximum likelihood estimator is unable to fully
undo the slowdown introduced by either level of the bandwagon effect,
highlighting the substantial impact this effect still has on effective sample
size and relevance estimation.

###  6.3. Bandwagon Misestimation

To investigate the effect of misestimated $\hat{\lambda}$ values, we look at
the bottom row in Figure 4. In particular, it shows what happens when we
underestimate a strong bandwagon effect. Clearly, we see that no noticeable
bias has been introduced to any estimator. Interestingly, the affine
estimators actually suffer less from initial variance: however, we speculate
that overestimating the bandwagon effect could instead enlarge the variance.
Nevertheless, all our estimators converge substantially slower than when
$\lambda$ was known (cf. middle row), with their confidence bounds now only
moderately smaller than those of $\bar{p}_{n}$. We thus conclude that the
ability of our estimators to accelerate convergence is very sensitive to
correct estimation of $\lambda$, where we do not observe an advantage for any
of the estimators.

The results with misestimated $\hat{\lambda}$ values demonstrate that
$\lambda$ estimation is a crucial part of effective bandwagon correction.
Researching bandwagon estimation methods is therefore an important task for
future work: since without robust estimation, correcting for the bandwagon
effect appears to remain an open problem.

##  7\. Discussion

So far, we have investigated the bandwagon effect in both theoretical and
empirical terms. In contrast with the well-studied selection bias (Schnabel et
al., 2016; Yuan et al., 2020; Sato et al., 2021; Jeunen and Goethals, 2021),
our experimental results indicate that the bandwagon effect mainly poses a
problem for convergence without introducing noticeable statistical bias. Yet
the effect is generally referred to as a bias by previous work (Xie and Zhong,
2020; Xie et al., 2021; Zhong et al., 2021; Wang et al., 2014). This raises a
higher-level conceptual question: _should the bandwagon effect be interpreted
as a bias?_ The remainder of this section will contrast three possible
perspectives on this question that we deem valuable for the understanding of
the field.

On one hand, both our theoretical and empirical results clearly indicate that
statistical bias is not the issue. The expected value of both the sample mean
and individual ratings is equal to the true relevance, with no noticeable
error in the mean value of each estimator in our experiments. Therefore, in
statistical terms, it seems that we are purely dealing with convergence and
consistency issues and not with bias. This distinction is important as
solutions to inconsistency may be very different than those for bias: i.e. our
experimental results suggest that methods based on debiasing techniques are
not particularly effective at solving convergence issues.

On the other hand, our theoretical analysis also revealed that the bandwagon
effect introduces conditional bias: given that there is an error in the sample
mean after $m$ ratings, the expected value of any subsequent rating or sample
mean will be incorrect. In other words, under the bandwagon effect any errors
in the sample mean, which are unavoidable in practice, will subsequently
introduce (conditional) bias. A proponent of the first perspective could argue
that this view is mistaking convergence issues for bias. Nevertheless, we
argue that as long as both parties understand the difference between _bias_
and _conditional bias_ , their disagreement is mainly about semantics and not
on the underlying nature of the issue.

Alternatively, one could consider the bandwagon effect as actually changing
the _true_ preferences of users. This view poses that the observed ratings are
not a distortion of the true preferences but that the actual preferences of
users have changed. One could argue that this is in line with the users’
experience as they do not make a conscious choice to alter their ratings in
response to others’ feedback but aim to assign ratings according to their
satisfaction. Instead, their enjoyment of a product may still be
subconsciously influenced by the enjoyment of others. Thus the advantage of
this view is the apparent alignment with the users’ own experience. However,
their experience is precisely unaware of any subconscious factors such as the
bandwagon effect. The danger is that by not viewing the bandwagon effect as a
misrepresentation of preferences, one may be less inclined to correct for it.
However, when training a recommender system, it surely seems critical to know
whether high ratings were given because a user was affected by previous users,
or whether they made this decision solely based on the item itself.

Apart from these conceptual perspectives, we strongly argue that in
statistical terms the bandwagon effect is a problem of convergence and
consistency and not of statistical bias. Nevertheless, outside of pure
statistical terms, we think that each of the three views brings a valuable
perspective to the field. Most importantly, we think the field should be aware
of the difference between the bandwagon effect and statistical bias. As long
as this distinction is understood, we think it is acceptable if one chooses to
conceptually categorize the bandwagon effect as another form of bias, although
we would advise against it. Overall, our work aims to promote a more complete
statistical understanding of the issues surrounding user interactions that
goes beyond bias to include variance, consistency and convergence.

##  8\. Conclusion

In this paper we examined the bandwagon effect: an underexplored phenomenon
where users’ interactions are affected by interactions of earlier users. While
previous work has approached the bandwagon effect as problem of bias (Xie and
Zhong, 2020; Xie et al., 2021), our theoretical analysis revealed that it does
not add bias to the rating distribution nor to the sample mean. Instead, we
have shown that it is better viewed as a problem of convergence and
consistency. We derived a necessary condition for consistency under the
effect, thereby also adding to earlier consistency claims (Xie and Zhong,
2020). We then proposed several novel estimators to mitigate the convergence
issues. While our empirical analysis revealed that our novel estimators can
substantially increase the convergence rate under a strong bandwagon effect,
all estimators also carry their own limitations. Moreover, even when using the
empirically best-performing estimator, our purposely simple model of the
bandwagon effect still significantly increases the number of samples required
for accurate relevance estimation. Future work may wish to examine the impact
of more complex bandwagon models (e.g. (Zhang et al., 2017; Zhong et al.,
2021)) as well as parameter and rating scale choices on bias and convergence,
both in unpersonalized as well as _personalized_ settings. For the latter case
in particular, user preference may be further associated with the time of
arrival (e.g. item’s fans are both more likely to like it and to rate it
early) or $\lambda$ values for one timestep may vary across users.
Nevertheless, even under our simple model, our findings lead us to conclude
that there is no clear and well-understood solution for the bandwagon effect.

### Acknowledgments

This work was supported by the Google Research Scholar Program. All content
represents the opinion of the author, which is not necessarily shared or
endorsed by their employers and/or sponsors.

### Code and data

To facilitate the reproducibility of the reported results, our experimental
implementation is publicly available at
https://github.com/NKNY/bandwagonictir2022.

##  Appendix A Appendix

###### Theorem A.1 (Bias of sample mean and single rating).

$\bar{p}_{n}$ and $r_{n}$ (Equation 4) are unbiased estimators of parameter
$p$ for all $n$:

(19)
$\mathbb{\forall}n:\mathbb{E}\left[\bar{p}_{n}\right]=\mathbb{E}\left[r_{n}\right]=p.$

###### Proof.

$\mathbb{E}[\bar{p}_{1}]=\mathbb{E}[r_{1}]=p$ by Equations 4-5. Then by
induction:

$\displaystyle\mathbb{E}\left[\bar{p}_{n}\right]=\dfrac{\left(n-1\right)\mathbb{E}\left[\bar{p}_{n-1}\right]+\mathbb{E}\left[r_{n}\right]}{n}=\dfrac{\lambda_{n}p+\left(n-\lambda_{n}\right)\mathbb{E}\left[\bar{p}_{n-1}\right]}{n}=p.$

$r_{n}$ is unbiased as it is a weighted sum of $p$ and unbiased
$\mathbb{E}\left[\bar{p}_{n-1}\right]$:

(20)
$\displaystyle\mathbb{E}\left[r_{n}\right]=\mathbb{E}\left[\lambda_{n}p+\left(1-\lambda_{n}\right)\bar{p}_{n-1}\right]=\lambda_{n}p+(1-\lambda_{n})\mathbb{E}[\bar{p}_{n-1}]=p.\hskip
14.22636pt\qed$

###### Lemma A.2 (Variance of $r_{i}$).

Under the bandwagon effect the variance of any rating $r_{i}$ is $p(1-p)$.

###### Proof.

By Theorem A.1:

(21) $\displaystyle\forall
i:\mathbb{V}\left[r_{i}\right]=\mathbb{E}\left[r_{i}^{2}\right]-\mathbb{E}^{2}\left[r_{i}\right]=\mathbb{E}\left[r_{i}\right]-\mathbb{E}^{2}\left[r_{i}\right]=p-p^{2}.\qed$

###### Lemma A.3.

Covariance of $\bar{p}_{n-1}$ and $r_{n}$ is
$\left(1-\lambda_{n}\right)^{2}\mathbb{V}\left[\bar{p}_{n-1}\right]$.

###### Proof.

(22) $\displaystyle\mathbb{C}\text{ov}\left[\bar{p}_{n-1},r_{n}\right]$
$\displaystyle=\mathbb{E}\left[\left(\bar{p}_{n-1}-p\right)\left(\lambda_{n}p+\left(1-\lambda_{n}\right)\bar{p}_{n-1}-p\right)\right]$
(23)
$\displaystyle=\left(1-\lambda_{n}\right)^{2}\mathbb{V}\left[\bar{p}_{n-1}\right].\qed$

###### Theorem A.4 (Efficiency of sample mean).

The efficiency of the sample mean under the bandwagon effect defined in
Equation 4 is

$\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=p\left(1-p\right)\bigg{(}\dfrac{1}{n^{2}}+\sum_{i=1}^{n-1}\dfrac{1}{i^{2}}\prod_{j=i+1}^{n}\dfrac{(j-1)(j+1-2\lambda_{j})}{j^{2}}\bigg{)}.$

###### Proof.

By Theorems A.1 and Lemmas A.2, A.3:

$\displaystyle\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=\dfrac{\left(n-1\right)^{2}}{n^{2}}\mathbb{V}\left[\bar{p}_{n-1}\right]+\dfrac{2\left(n-1\right)}{n^{2}}\mathbb{C}\text{ov}\left[r_{n},\bar{p}_{n-1}\right]$
(24) $\displaystyle\hskip
15.6491pt+\dfrac{\mathbb{V}\left[r_{n}\right]}{n^{2}}=\dfrac{\left(n-1\right)\left(n+1-2\lambda_{n}\right)}{n^{2}}\mathbb{V}\left[\bar{p}_{n-1}\right]+\dfrac{p(1-p)}{n^{2}}.$

The proof is completed by unrolling the process to
$\mathbb{V}\left[\bar{p}_{1}\right]=p(1-p)$ and collecting all the terms.∎

###### Theorem A.5.

The asymptotic effiency of the sample mean is:

$\lim_{n\to\infty}\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=\lim_{n\to\infty}p\left(1-p\right)\bigg{(}\sum_{i=1}^{n-1}\dfrac{1}{i\left(i+1\right)}\prod_{j=i+1}^{n-1}\bigg{(}1-\dfrac{2\lambda_{j}}{j+1}\bigg{)}\bigg{)}.$

###### Proof.

By re-grouping numerators and denominators in the product of Theorem A.4 and
cancelling out extra terms:

$\displaystyle\mathbb{E}\left[\left(\bar{p}_{n}-p\right)^{2}\right]=p\left(1-p\right)\left(\dfrac{1}{n^{2}}+\sum_{i=1}^{n-1}\dfrac{n+1-2\lambda_{n}}{i\left(i+1\right)n}\prod_{j=i+1}^{n-1}\left(1-\dfrac{2\lambda_{j}}{j+1}\right)\,\right).$

The proof is completed by taking $\lim_{n\to\infty}$ of above expression. ∎

###### Theorem A.6 (Sample mean consistency).

$\sum\lambda_{i}^{2}$ diverging is a necessary condition for the consistency
of $\bar{p}_{n}$ (Equation 4):

(25)
$\forall\epsilon>0:\lim_{n\to\infty}\text{Pr}\left(\left|\bar{p}_{n}-p\right|>\epsilon\right)=0\longrightarrow\lim_{n\to\infty}\sum_{i=1}^{n}\lambda_{i}^{2}=\infty.$

###### Proof.

By Loève (1977, p. 159), consistency of $\bar{p}$ implies its variance
vanishing:
$\forall\epsilon>0:\text{Pr}(|\bar{p}_{n}-p|\geq\epsilon)\geq\mathbb{V}\left[\bar{p}_{n}\right]-\epsilon^{2}$.
Then:
$\forall\epsilon>0:\lim_{n\to\infty}\text{Pr}\left(\left|\bar{p}_{n}-p\right|>\epsilon\right)=0\rightarrow\lim_{n\to\infty}\text{Var}(\bar{p}_{n})=0\rightarrow\forall
i:\lim_{n\to\infty}\prod_{j=i+1}^{n-1}(1-\frac{2\lambda_{j}}{j+1})=0\rightarrow\forall
i:\lim_{n\to\infty}\sum_{j=i+1}^{n-1}\frac{\lambda_{j}}{j+1}=\infty\rightarrow\forall
i:\lim_{n\to\infty}\sqrt{\sum_{j=i+1}^{n-1}\lambda_{j}^{2}}\sqrt{\sum_{j=i+1}^{n-1}\frac{1}{\left(j+1\right)^{2}}}=\infty\rightarrow\lim_{n\to\infty}\sum_{i=1}^{n}\lambda_{i}^{2}=\infty$.
The second step is due to partial products in Theorem A.5, the third by
Leonard (2012, p. 6), the fourth by Cauchy-Schwarz inequality and the fifth by
convergence of p-series with $p=2$. ∎

###### Proposition A.7 (Conditional bias of sample mean).

$\mathbb{E}\left[\bar{p}_{n}-p\mid\bar{p}_{m}\right]=\left(\bar{p}_{m}-p\right)\prod_{i=m+1}^{n}\left(1-\frac{\lambda_{i}}{i}\right)$
where $m<n$.

###### Proof.

By repeatedly applying the induction step in Theorem A.1, replacing
$\mathbb{E}\left[\bar{p}_{n}\right]$ with
$\mathbb{E}\left[\bar{p}_{n}\mid\bar{p}_{m}\right]$ and noting that
$\mathbb{E}\left[\bar{p}_{m}|\bar{p}_{m}\right]=\bar{p}_{m}$:

$\mathbb{E}\left[\bar{p}_{n}-p\,|\,\bar{p}_{m}\right]=\mathbb{E}\left[\bar{p}_{n-1}-p\,|\,\bar{p}_{m}\right]\big{(}1-\frac{\lambda_{n}}{n}\big{)}=\big{(}\bar{p}_{m}-p\big{)}\prod_{i=m+1}^{n}\big{(}1-\frac{\lambda_{i}}{i}\big{)}.\qed$

###### Proposition A.8 (Error bound of $\bar{p}_{n}$).

If $\lambda_{i}=c^{i-1}$ s.t. $c\in\left(0,1\right)$, the asymptotic expected
absolute error of the sample mean is at least
$2p\left(1-p\right)\exp\left(-c\phi\left(c,1,2\right)-c^{2}\phi\left(c^{2},2,2\right)\right)$,
where $\phi(z,s,$
$a)\equiv\sum^{\infty}_{k=0}\frac{z^{k}}{\left(a+k\right)^{s}}$ is Lerch
transcendent and converges to a fixed value $\forall z<1$ (Ferreira et al.,
2017).

###### Proof.

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\left|\bar{p}_{n}-p\right|\right]\geq\lim_{n\to\infty}\mathbb{E}\left[\left|\mathbb{E}\left[\bar{p}_{n}-p|r_{1}\right]\right|\right]=2p\left(1-p\right)$
$\displaystyle\cdot\exp\left(\sum_{i=2}^{\infty}\ln\left(1-\dfrac{\lambda_{i}}{i}\right)\right)\geq
2p\left(1-p\right)\exp\bigg{(}\sum_{i=2}^{\infty}\Big{(}-\dfrac{\lambda_{i}}{i}-\dfrac{\lambda_{i}^{2}}{i^{2}}\Big{)}\bigg{)}$
$\displaystyle=2p\left(1-p\right)\exp\Bigg{(}-c\sum^{\infty}_{i=0}\dfrac{c^{i}}{\left(2+i\right)}-c^{2}\sum^{\infty}_{i=0}\dfrac{c^{2i}}{\left(2+i\right)^{2}}\Bigg{)}.$

The first step is due to the law of total expectation and Jensen’s inequality
for conditional expectation. The second is by Proposition A.7 with $m=1$. The
third is due to a known inequality $\forall x>-0.68:\ln\left(1+x\right)\geq
x-x^{2}$ (Kozma, 2021). ∎

###### Theorem A.9.

Each $\hat{r}_{i}$ (and thus also $\hat{p}_{n}$) is unbiased for all
$\hat{\lambda}_{i}$.

###### Proof.

By Theorem A.1, for any choice of $i$ and $\hat{\lambda}_{i}$:

$\mathbb{E}\left[\hat{r}_{i}\mid\hat{\lambda}_{i}\right]=\mathbb{E}\bigg{[}\dfrac{r_{i}-(1-\hat{\lambda}_{i})\bar{p}_{i-1}}{\hat{\lambda}_{i}}\bigg{]}=\dfrac{\mathbb{E}\left[r_{i}-\bar{p}_{i-1}\right]}{\hat{\lambda}_{i}}+\mathbb{E}\left[\bar{p}_{i-1}\right]=p.\qed$

###### Theorem A.10 (Conditional bias of
$\hat{r}_{i}\mid\hat{\lambda}_{i}=\lambda_{i}$).

If $\hat{\lambda}_{i}=\lambda_{i}$, the affine mean estimator $\hat{r}_{i}$ is
an unbiased estimator of $p$ when conditioned on $\hat{r}_{j}$ or
$\bar{p}_{j}$ s.t. $\forall j<i$.

###### Proof.

For any function of previous ratings $f_{i}=f(r_{1},\ldots,r_{i-1})$:

$\displaystyle\mathbb{E}\left[\hat{r}_{i}\mid f_{i},\hat{\lambda}_{i}\right]$
$\displaystyle=\mathbb{E}\bigg{[}\dfrac{\lambda_{i}p+(1-\lambda_{i})\bar{p}_{i-1}-(1-\hat{\lambda}_{i})\bar{p}_{i-1}}{\hat{\lambda}_{i}}\mid
f_{i}\bigg{]}$ (26)
$\displaystyle=\dfrac{\lambda_{i}}{\hat{\lambda}_{i}}p+\left(\dfrac{\hat{\lambda}_{i}-\lambda_{i}}{\hat{\lambda}_{i}}\right)\mathbb{E}\left[\bar{p}_{i-1}\mid
f_{i}\right]=p.$

The proof is completed by setting $f_{i}=\hat{r}_{j}$ or $f_{i}=\bar{p}_{j}$.
∎

###### Corollary A.11 (Conditional bias of
$\hat{r}_{i}\mid\hat{\lambda}_{i}\neq\lambda_{i}$).

Affine estimator $\hat{r}_{i}$ is not unbiased when conditioned on a function
of previous interactions as the last equality of Equation 26 may not hold.

###### Lemma A.12 (Covariance of affine estimators).

$\hat{r}_{i}$ and $\hat{r}_{j<i}$ are uncorrelated as long as
$\hat{\lambda}_{i}=\lambda_{i}$.

###### Proof.

By Theorem A.10 and the law of total expectation:

$\displaystyle\mathbb{E}\left[\hat{r}_{i}\hat{r}_{j}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\hat{r}_{i}\hat{r}_{j}\mid\hat{r}_{j}\right]\right]=\mathbb{E}\left[\hat{r}_{j}\mathbb{E}\left[\hat{r}_{i}\mid\hat{r}_{j}\right]\right]$
(27)
$\displaystyle=\mathbb{E}\left[\hat{r}_{j}\mathbb{E}\left[\hat{r}_{i}\right]\right]=\mathbb{E}\left[\hat{r}_{j}\right]\mathbb{E}\left[\hat{r}_{i}\right].\qed$

###### Theorem A.13.

The variance of affine estimator in Equation 15 is:

$\mathbb{V}\left[\hat{p}_{n}\right]=\frac{1}{\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}}\sum_{i=1}^{n}\frac{\omega_{i}^{2}}{\lambda_{i}^{2}}\left(p\left(1-p\right)-\left(1-\lambda_{i}\right)^{2}\mathbb{V}\left[\bar{p}_{i-1}\right]\right).$

###### Proof.

By Lemmas A.12, A.3 and A.2:

$\displaystyle\mathbb{V}\left[\hat{p}_{n}\right]=\dfrac{1}{\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}}\left(\sum_{i=1}^{n}\omega_{i}^{2}\mathbb{V}\left[\hat{r}_{i}\right]+2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\omega_{i}\omega_{j}\mathbb{C}\text{ov}\left[\hat{r}_{i},\hat{r}_{j}\right]\right)$
$\displaystyle=\dfrac{1}{\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}}\sum_{i=1}^{n}\dfrac{\omega_{i}^{2}}{\lambda_{i}^{2}}\big{(}\mathbb{V}\left[r_{i}\right]+\left(1-\lambda_{i}\right)^{2}\mathbb{V}\left[\bar{p}_{i-1}\right]-2\left(1-\lambda_{i}\right)$
(28)
$\displaystyle\cdot\mathbb{C}\text{ov}\left[r_{i},\bar{p}_{i-1}\right]\big{)}=\left(\sum_{i=1}^{n}\omega_{i}\right)^{-2}\sum_{i=1}^{n}\dfrac{\omega_{i}^{2}}{\lambda_{i}^{2}}\left(p-p^{2}-\left(1-\lambda_{i}\right)^{2}\mathbb{V}\left[\bar{p}_{i-1}\right]\right).\qed$

###### Theorem A.14 (Convergence of affine estimator).

When $\forall\hat{\lambda}_{i}=\lambda_{i}$,
$\hat{p}_{n}=\frac{\sum_{i=1}^{n}\omega_{i}\hat{r}_{i}}{\sum_{i=1}^{n}\omega_{i}}$
converges within $\epsilon$ of the true value $p$ with probability of at least
$1-\alpha$ if
$\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}\geq\frac{1}{2\epsilon^{2}}\log{\frac{2}{\alpha}}\sum_{i=1}^{n}\big{(}\frac{\omega_{i}}{\lambda_{i}}\big{)}^{2}$.

###### Proof.

Define $X_{n}$ as $\sum_{i=1}^{n}\omega_{i}\left(\hat{r}_{i}-p\right)$,
$X_{0}=0$. Then $X_{n}$ is a martingale as $\mathbb{E}\left[X_{n}\mid
X_{0},X_{1},...,X_{n-1}\right]=X_{n-1}+\mathbb{E}\left[w_{i}\left(\hat{r}_{i}-p\right)\mid\bar{p}_{n-1}\right]=X_{n-1}$.
For each $n:X_{n}-X_{n-1}=\omega_{n}\left(\hat{r}_{n}-p\right)$, whose values
for $r_{n}=0$ and $r_{n}=1$ are separated by
$d_{n}=\frac{\omega_{n}}{\lambda_{n}}$. By Azuma’s inequality:

(29)
$\begin{split}\text{Pr}\left(\left|\dfrac{\sum_{i=1}^{n}\omega_{i}\left(\hat{r}_{i}-p\right)}{\sum_{i=1}^{n}\omega_{i}}\right|>\epsilon\right)&=\text{Pr}\bigg{(}|X_{n}-X_{0}|>\epsilon\sum_{i=1}^{n}\omega_{i}\bigg{)}\\\
&\leq
2\exp\left(\dfrac{-2\epsilon^{2}\left(\sum_{i=1}^{n}\omega_{i}\right)^{2}}{\sum_{i=1}^{n}d_{i}^{2}}\right).\end{split}$

The proof is completed by taking the logarithm of both sides and
redistributing the terms. ∎

###### Corollary A.15 (Convergence of $\bar{p}_{n}$ for specific
$\omega_{i}$).

By Theorem A.14, when $\forall{}i:\hat{\lambda}_{i}=\lambda_{i}$,
$\omega_{i}=1$, $\hat{p}_{n}$ is a consistent estimator of $p$ if
$\sum_{i=1}^{n}\frac{1}{\lambda_{i}^{2}}$ grows slower than $n^{2}$.
Similarly, when $\forall\hat{\lambda}_{i}=\lambda_{i}=\omega_{i}$,
$\hat{p}_{n}$ is a consistent estimator of $p$ if
$\left(\sum_{i=1}^{n}\lambda_{i}\right)^{2}$ grows quicker than $n$.

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  *[RecSys]: Recommender Systems
  *[CTR]: Click-Through Rate
  *[IPS]: Inverse Propensity Scoring