# What will it take to generate fairness-preserving explanations?

Jessica Dai  Sohini Upadhyay  Stephen H. Bach  Himabindu Lakkaraju

###### Abstract

In situations where explanations of black-box models may be useful, the
fairness of the black-box is also often a relevant concern. However, the link
between the fairness of the black-box model and the behavior of explanations
for the black-box is unclear. We focus on explanations applied to tabular
datasets, suggesting that explanations do not necessarily preserve the
fairness properties of the black-box algorithm. In other words, explanation
algorithms can ignore or obscure critical relevant properties, creating
incorrect or misleading explanations. More broadly, we propose future research
directions for evaluating and generating explanations such that they are
informative and relevant from a fairness perspective.

Machine Learning, ICML

##  1 Introduction & Motivation

While fairness and explainability are both generally considered core
components of “responsible” machine learning, surprisingly little work has
explored the two principles in tandem. However, especially in light of common
goals of generating explanations for a black-box models, it is critical that
the explanation itself can be reliably trusted to illustrate important
fairness properties of the black-box. For example, Suresh et al. (2021)’s
framework for characterizing stakeholders in explainable machine learning
provides objectives such as debugging or improving the model, ensuring
regulatory compliance, informing downstream actions, justifying actions based
on algorithm output, and contesting a decision; and specific tasks like
assessing the reliability of a prediction; detecting mistaken or
discriminatory behavior; and understanding the influence of different inputs.
Prior work in this area has outlined similar goals for explanations (Bhatt et
al., 2020a). For obvious reasons, if fairness is a concern related to the
model more broadly, it is also a critical consideration for these tasks and
objectives in the context of explanations. Furthermore, while of course
calculating particular fairness desiderata for the underlying black-box
directly might surface unfairness, the stakeholders who are using an
explainable ML algorithm may not have access to the information needed for
such an analysis; as a result, we might hope that explanations themselves
contain sufficient and accurate information for any stakeholder to confidently
make claims and downstream decisions based on the explanation. This is
especially important given that end-users of explanations may be vulnerable to
overtrusting or being manipulated by explanations (Lakkaraju & Bastani, 2020).

However, current methods for evaluating explanations are designed to be almost
entirely application-agnostic, and therefore do not consider any criteria
related to fairness. While terminology varies across the literature, commonly
used evaluation metrics for explanations include fidelity, the extent to which
a surrogate model generated by an explanation algorithm produces predictions
similar to the black-box, and stability, the extent to which explanations
generated for similar (but non-identical) inputs are similar to one another
(Bhatt et al., 2020b; Yeh et al., ).

A growing portion of the literature points to dangers in focusing solely on
these targets when designing explanation algorithms. Slack et al. (2020b) and
Zhang et al. (2019), for instance, highlight the high degree of inconsistency
of explanations generated by perturbation-based methods under certain
parameter settings—in other words, multiple explanations generated for the
same input may result in wildly different explanations. Kumar et al. (2020)
and Hancox-Li & Kumar (2021), meanwhile, investigate SHAP (Lundberg & Lee,
2017), and find problems from both technical and philosophical perspectives.
Under the framework of fairness specifically, Slack et al. (2020a) and Aïvodji
et al. (2019) illustrate specific ways in which either a black-box algorithm
or an explanation, respectively, may be adversarially constructed such that
the explanation, while having high fidelity (or achieving other desirable
metrics), misleadingly suggests that the black-box model is fair when in
reality it is not. However, adversarial construction may not be necessary for
misleading explanations to occur.

For some baseline intuition as to how fairness and explanations may interact,
consider the following. Explanations are often intended to provide a
digestible approximation of the black-box algorithm’s decision boundary,
whether locally (in the neighborhood of a particular input) or globally (for
all possible inputs). Additionally, fairness concerns arise when there is a
meaningful difference in how two or more demographic groups are distributed or
labelled in the training data, which leads to a meaningful disparity in how
the black-box machine learning algorithm performs on the two groups by
whatever metric one may choose (Corbett-Davies & Goel, 2018). The demographic
information may or may not be used by the black-box. In the case that it is
not used, the explanation will not explicitly encode information about the
sensitive attribute, and an end-user relying on the explanation alone will
have little information about the fairness of the black-box. In the case that
it is used, such as when fairness-constrained learning algorithms or
postprocessing methods are applied, then the black-box may learn a decision
boundary such that the boundaries are different when conditioned on group
membership. However, explanation methods are not designed to approximate such
boundaries. Furthermore, the explanation itself will include information about
the sensitive attribute, such as in the form of a feature importance score; it
is not immediately clear what the proper interpretation of that score should
be. Finally, in either case, the known issue of isolating feature attributions
when features may be correlated with one another (Kumar et al., 2020) is
especially relevant when considering fairness applications.

###  1.1 A simple example

Consider the following scenario where the black-box takes in three features:
group, x0, and x1, where group $\in\\{0,1\\}$, x0 and x1 are continuous, and
x0 is correlated with group membership. For some reason or another (perhaps by
applying a fairness intervention in the training process), the black-box’s
learned decision boundaries are different when conditioned on group
membership: specifically, the black-box predicts 1 for group 0 when x1 $>6$,
and for group 1 when x1 $>5$. Figure 1(a) illustrates this black-box decision
boundary.

(a) Decision boundary of the black-box

(b) Decision boundary learned by LIME

Figure 1: The true decision boundaries of the black-box vs the decision
boundary learned by LIME. The left cluster is the minority group, comprising
27% of the total population.

In the case that both groups constitute $50\%$ of the population, an
explanation method that optimizes for fidelity as measured by performance on
sampled neighbors will approximate the decision boundary at around x1
$>5.5$—an explanation that is simply incorrect for both groups based on what
we know about the black-box. If one group is a minority of the population,
however, the explanation’s approximated decision boundary will be closer to
the majority group’s decision boundary, meaning overall better explanations
for the majority group and overall worse explanations for the minority group.
This is illustrated in Figure 1(b), which visualizes the decision boundary
learned by LIME: note that the learned boundary is much closer to x1 $>5$, the
majority group’s boundary, over all data points, not just the points
corresponding to the majority group. Notably, this is a problem that seems to
arise whenever group-conditional decision boundaries are meaningfully
distinct. The explanations generated here, therefore, may be both misleading
and incorrect.

##  2 Our Framework

We propose a two-part framework for further work in this area: first,
determining what constitutes a mismatch in fairness properties; and second,
generating fairness-preserving explanations.

###  2.1 Diagnosing Fairness Mismatch

First, we provide an initial attempt at outlining what metrics or diagnostic
tests may be useful in detecting a mismatch in fairness; these also serve,
therefore, as potential criteria or definitions for what a fairness-preserving
vs fairness-obscuring explanation may look like. These metrics are broadly
motivated by the principle that if the model is fair, the explanations should
not raise false alarms; similarly, if the model is unfair, the explanations
should not suggest that it is innocuous. In this section we attempt to
pinpoint what exactly it means for an explanation to “raise a false alarm” or
suggest that the model “is innocuous.”

Group fairness. There are a variety of metrics through which models can be
audited or monitored for group fairness: demographic parity focuses primarily
on group-wise outcomes, while other metrics such as equalized odds, equal
opportunity, or predictive parity, reflect some combination of the group-
conditional confusion matrices (Verma & Rubin, 2018).

Let $\mathcal{M}$ represent a metric of group fairness which takes in the
predictions of some model (and potentially information about the true labels);
$f$ represent the black-box; $E_{f}$ be the surrogate model from an
explanation for $f$; and $E_{f}(\vec{x})$ represent evaluating the surrogate
model on some input $\vec{x}$. Then, group fairness is preserved when:

$|\mathcal{M}(f(\vec{x}))-\mathcal{M}(E_{f}(\vec{x}))|\leq\epsilon$

In other words, when, if substituting the black-box model with the
explanation’s surrogate model, the predictions generated result in similar
values of the fairness metric $\mathcal{M}$.

Two obvious issues arise with this initial proposition. First, while this is
straightforward for global explanation methods, many of the most popular
explanation methods like LIME (Ribeiro et al., 2016) or SHAP (Lundberg & Lee,
2017) are local explanation methods, designed to explain specific points: that
is, there is no notion of a global surrogate model from which group fairness
metrics can easily be calculated. Second, only the demographic parity metric
does not require information about the ground-truth labelling of data points;
all other metrics require this information. The question then becomes how to
determine the set of points $x$ on which $\mathcal{M}$ will be calculated for
local explanations. One potential approach is to use the sampled points in the
local neighborhood generated by the explanation method, and calculate
$\mathcal{M}$ on the neighborhood for each of the points in the dataset. Of
course, this approach means that no ground-truth labelling is available for
this set of sampled points, and thus the only metric that can be verified to
match or mismatch in this way is demographic parity.

Counterfactual fairness. In classification, counterfactual fairness and
individual fairness have similar motivations: identifying how the prediction
for a particular input $x$ would change if only the group membership of $x$
was changed (Dwork et al., 2012). Though there is debate about the extent to
which counterfactual or individual fairness is distinct (if not orthogonal)
from group fairness (Lahoti et al., 2019; Binns, 2020), a “fairness-
preserving” explanation should nevertheless capture the counterfactual
behavior of the black-box model. To that end, let $x^{\prime}$ represent the
input $x$ with a changed value for group membership, and $E_{f(x)}$ illustrate
explanations generated for input $x$. Then, counterfactual fairness is
preserved when:

$E_{f(x)}-E_{f(x^{\prime})}\approx f(x)-f(x^{\prime})$

In other words, when the difference between the explanation generated for $x$
and the explanation generated for $x^{\prime}$ follows the difference between
the model’s behavior on $x$ and $x^{\prime}$. This abstraction also raises
open questions about how exactly the similarity should be determined.

Sensitive attribute. The treatment of the sensitive attribute in cases where
it is included in the inputs to the black-box model is also worth additional
attention. In this case, unlike group and counterfactual fairness, we do not
propose a particular normative value of how the sensitive attribute ought to
be treated by the explanation algorithm in relation to the black-box. For
example, the feature importance for the sensitive attribute being 0 does not
necessarily imply that the black-box is not discriminatory: the influence of
the sensitive attribute may have been attributed to another, correlated
feature. Moreover, many algorithms for fair machine learning explicitly use
the sensitive attribute in order to achieve some measure of fairness, such as
the method proposed in Hardt et al. (2016). In this sense, a feature
importance of 0 might even be alarming rather than reassuring. As a result,
future work in this area may include methods which give more meaningful ways
to interpret the influence of the sensitive attribute.

Additional considerations. Finally, of note here is the distinction between
evaluating an explanation algorithm itself for how well it preserves fairness
properties in general, and evaluating a given, specific explanation for
whether it is preserving relevant fairness properties once the explanation for
a particular input or model has been generated. These are different tasks—the
first, for example, may be useful for a model developer or engineer in the
process of choosing an explanation method, while the second may be more
relevant to auditing processes once a black-box model (and a corresponding
explanation algorithm) has been deployed. Additional work distinguishing what
approaches or metrics might be comparatively useful in either situation is
warranted; in particular, all of these proposed metrics require a
comparatively high amount of information and access to the black-box, and may
be better-suited towards the first task (evaluating algorithms in general)
rather than the second (auditing individual explanations).

###  2.2 Generating Fairness-Preserving Explanations

As discussed above, algorithms for finding explanations typically focus on
optimizing for metrics such as fidelity and sensitivity. One approach for
generating a fairness-preserving explanation can be similar to early
approaches to fair machine learning algorithms: adding a penalty term in the
objective function for the extent to which the explanation is fairness-
preserving (Kamishima et al., 2012; Zafar et al., 2017). For example, the
original LIME objective function (Ribeiro et al., 2016) is as follows:

$\xi(x)=\arg\min_{g\in G}\mathcal{L}(f,g,\pi_{x})+\Omega(g)$

where $\xi(x)$ is the optimal explanation for input $x$ to model $f$, $G$ is
the class of sparse linear models, $\mathcal{L}$ is a measure of fidelity,
$\pi_{x}$ is a local region around $x$, and $\Omega$ is a measure of
complexity. A modified objective function including a term such as $\psi(f,g)$
measuring the preservation of fairness properties described in Section 2.1,
fits naturally:

$\xi_{\textit{fair}}(x)=\arg\min_{g\in
G}\mathcal{L}(f,g,\pi_{x})+\lambda_{1}\Omega(g)+\lambda_{2}\psi(f,g)$

Here, $\lambda_{1}$ and $\lambda_{2}$ are tuning parameters for the complexity
$\Omega$ and fairness-preservation term $\psi$, respectively.

Figure 2 illustrates the results of using this modified, fairness-preserving
objective function when finding the explanation $\xi$. Here, the dataset used
was COMPAS; the black-box was a three-layer deep neural net; and $\psi$ is
derived from the group fairness equation in Section 2.1. Specifically,
$\psi=|DP(f(x))-DP(E_{f}(x))|$, where $DP$ is the demographic parity metric:
$P(Y=1|S=1)-P(Y=1|S=0)$. In this experiment, the number of perturbations used
to generate the LIME explanation was varied to show the asymptotic fairness
mismatch, as a greater number of perturbations generally results in a higher-
certainty explanation. The fairness mismatch plotted on the y-axis is
calculated exactly in the same way as $\psi$ explained above. Our introduction
of this approach is meant more as a provocation to start the conversation
rather than a full-fledged proposal or argument that this method is
necessarily ideal; however, the results are promising and warrant further
investigation in this direction.

Figure 2: Original vs fairness-preserving LIME algorithms: number of
perturbations used for LIME vs average fairness mismatch over explanations for
all points in the dataset.

##  3 Discussion & Conclusion

In this work, we have given some intuition and preliminary results as to why
it is important to probe the fairness of explanations: not just because of the
often high-stakes and consequential goals for which explanations are used, but
because existing explanation methods focusing on metrics like fidelity may
result in misleading and incorrect explanations even in the absence of an
adversarial actor constructing explicitly discriminatory black-boxes, or
designing explanation methods that explicitly hide discrimination.
Furthermore, fairness can also be viewed as a specific lens on performance for
the model overall. In fact, the phenomenon illustrated in Figure 1 can be
considered to be a performance issue—strictly incorrect decision boundaries,
though the minority group’s decision boundary is much more incorrect—that can
be detected by testing for fairness mismatch as proposed in Section 2. Of
course, the exact behavior in this scenario may be the consequence of LIME’s
choice to focus on sparse linear models, and choosing a more complex
interpretable model class (such as shallow decision trees) may alleviate the
issue.

Nevertheless, an action like this is only made possible by the first step of
diagnosing the fairness mismatch between the black-box and the explanation’s
surrogate model. Thinking about explanations in this context, therefore, also
raises broader questions about the extent to which explanations are in fact
capturing what we want; or, alternatively, ways in which the limitations of
particular explanations or explanation methods may be communicated clearly to
stakeholders and end-users.

In this work, we suggested a framework for evaluating the fairness-preserving
properties of explanations, and proposed one generic approach for producing
fairness-preserving explanations. However, this extended abstract is also
meant to argue for the consideration of evaluation metrics for explanations
more broadly: while fairness was the first angle we considered, there are
undoubtedly additional necessary properties of the model—even privacy, for
example—that explanations should preserve.

## Acknowledgements

We would like to thank the anonymous reviewers for their insightful feedback.
This work is supported in part by the NSF award #IIS-2008461, and Google. The
views expressed are those of the authors and do not reflect the official
policy or position of the funding agencies.

## References

  * Aïvodji et al. (2019) Aïvodji, U., Arai, H., Fortineau, O., Gambs, S., Hara, S., and Tapp, A.  Fairwashing: the risk of rationalization.  In _International Conference on Machine Learning_ , pp. 161–170. PMLR, 2019.
  * Bhatt et al. (2020a) Bhatt, U., Andrus, M., Weller, A., and Xiang, A.  Machine learning explainability for external stakeholders.  _arXiv preprint arXiv:2007.05408_ , 2020a.
  * Bhatt et al. (2020b) Bhatt, U., Weller, A., and Moura, J. M.  Evaluating and aggregating feature-based model explanations.  _arXiv preprint arXiv:2005.00631_ , 2020b.
  * Binns (2020) Binns, R.  On the apparent conflict between individual and group fairness.  In _Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency_ , pp. 514–524, 2020.
  * Corbett-Davies & Goel (2018) Corbett-Davies, S. and Goel, S.  The measure and mismeasure of fairness: A critical review of fair machine learning.  _arXiv preprint arXiv:1808.00023_ , 2018.
  * Dwork et al. (2012) Dwork, C., Hardt, M., Pitassi, T., Reingold, O., and Zemel, R.  Fairness through awareness.  In _Proceedings of the 3rd innovations in theoretical computer science conference_ , pp. 214–226, 2012.
  * Hancox-Li & Kumar (2021) Hancox-Li, L. and Kumar, I. E.  Epistemic values in feature importance methods: Lessons from feminist epistemology.  In _Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency_ , pp. 817–826, 2021.
  * Hardt et al. (2016) Hardt, M., Price, E., and Srebro, N.  Equality of opportunity in supervised learning.  _arXiv preprint arXiv:1610.02413_ , 2016.
  * Kamishima et al. (2012) Kamishima, T., Akaho, S., Asoh, H., and Sakuma, J.  Fairness-aware classifier with prejudice remover regularizer.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , pp. 35–50. Springer, 2012.
  * Kumar et al. (2020) Kumar, I. E., Venkatasubramanian, S., Scheidegger, C., and Friedler, S.  Problems with shapley-value-based explanations as feature importance measures.  In _International Conference on Machine Learning_ , pp. 5491–5500. PMLR, 2020.
  * Lahoti et al. (2019) Lahoti, P., Gummadi, K. P., and Weikum, G.  ifair: Learning individually fair data representations for algorithmic decision making.  In _2019 IEEE 35th International Conference on Data Engineering (ICDE)_ , pp. 1334–1345. IEEE, 2019.
  * Lakkaraju & Bastani (2020) Lakkaraju, H. and Bastani, O.  “how do i fool you?” manipulating user trust via misleading black box explanations.  In _Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society_ , pp. 79–85, 2020.
  * Lundberg & Lee (2017) Lundberg, S. and Lee, S.-I.  A unified approach to interpreting model predictions.  _arXiv preprint arXiv:1705.07874_ , 2017.
  * Ribeiro et al. (2016) Ribeiro, M. T., Singh, S., and Guestrin, C.  “why should i trust you?” explaining the predictions of any classifier.  In _Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining_ , pp. 1135–1144, 2016.
  * Slack et al. (2020a) Slack, D., Hilgard, S., Jia, E., Singh, S., and Lakkaraju, H.  Fooling lime and shap: Adversarial attacks on post hoc explanation methods.  In _Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society_ , pp. 180–186, 2020a.
  * Slack et al. (2020b) Slack, D., Hilgard, S., Singh, S., and Lakkaraju, H.  How much should i trust you? modeling uncertainty of black box explanations.  _arXiv preprint arXiv:2008.05030_ , 2020b.
  * Suresh et al. (2021) Suresh, H., Gomez, S. R., Nam, K. K., and Satyanarayan, A.  Beyond expertise and roles: A framework to characterize the stakeholders of interpretable machine learning and their needs.  In _Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems_ , pp. 1–16, 2021.
  * Verma & Rubin (2018) Verma, S. and Rubin, J.  Fairness definitions explained.  In _2018 ieee/acm international workshop on software fairness (fairware)_ , pp. 1–7. IEEE, 2018.
  * (19) Yeh, C.-K., Hsieh, C.-Y., Suggala, A. S., Inouye, D. I., and Ravikumar, P.  On the (in) fidelity and sensitivity of explanations.
  * Zafar et al. (2017) Zafar, M. B., Valera, I., Rogriguez, M. G., and Gummadi, K. P.  Fairness constraints: Mechanisms for fair classification.  In _Artificial Intelligence and Statistics_ , pp. 962–970. PMLR, 2017.
  * Zhang et al. (2019) Zhang, Y., Song, K., Sun, Y., Tan, S., and Udell, M.  ” why should you trust my explanation?” understanding uncertainty in lime explanations.  _arXiv preprint arXiv:1904.12991_ , 2019.


# Predicting Customer Lifetime Value in Free-to-Play Games

Paolo Burelli
IT University of Copenhagen - Tactile Games
<EMAIL_ADDRESS>
###### Abstract

As game companies increasingly embrace a service-oriented business model, the
need for predictive models of players becomes more pressing. Multiple
activities, such as user acquisition, live game operations or game design need
to be supported with information about the choices made by the players and the
choices they could make in the future. This is especially true in the context
of free-to-play games, where the absence of a pay wall and the erratic nature
of the players’ playing and spending behavior make predictions about the
revenue and allocation of budget and resources extremely challenging.

In this chapter we will present an overview of customer lifetime value
modeling across different fields, we will introduce the challenges specific to
free-to-play games across different platforms and genres and we will discuss
the state-of-the-art solutions with practical examples and references to
existing implementations.

##  1 Introduction

Customer lifetime value (CLV or LTV) refers broadly to the revenue that a
company can attribute to one or more customer over the length of their
relationship with the company [55]. The process of predicting the lifetime
value consists in producing one or more monetary values that correspond to the
sum of all the different types of revenues that a specific customer, or a
specific cohort, will generate in the future. The purposes of this prediction
are manifold: for example, having an early estimation of a customer’s
potential value allows more accurate budgeting for future investment;
moreover, monitoring the remaining potential revenue from an established
customer could permit preemptive actions in case of decreased engagement.

Predicting customer lifetime value is a complex challenge and, to date, there
is no single established practice. Furthermore, due to its wide potential
impact in different business aspects, the problem is being researched in
different communities using a plethora of different techniques, varying from
parametric statistical models to deep learning [28, 70]. One of the initial
primary drivers of research in CLV was direct marketing [4] with a focus on
its use in marketing decision problems, such as acquisition or the trade-off
between acquisition and retention costs [6]. Research expanded progressively
to other fields of marketing and customer relationship management, especially
thanks to the pervasiveness of digital technology and the possibility to have
more accurate tracking of the customer relationship [37].

The increased quality and quantity of customer data available for analysis
widened the areas of application of CLV related principles, and helped
accelerating the development of new models for CLV prediction. Initial
analytical models were developed on assumptions about constant and uniform
characteristics of the user behavior (e.g. transaction frequency, margin of
profit). Over time, models have evolved to take into account uncertainty and
variations of user behavior [63] and to be able to draw information from a
wide spectrum of variables [69].

Models have traditionally been based, on transactional data, meaning that the
variables describing the user behavior are based on different characteristics
of the customers transactions with the company (e.g. recency, frequency,
monetary (RFM) [63]). However, with the advent of e-commerce and similar
technologies, the available information about the customer behavior became
increasingly richer, since it started to be possible to track not only the
purchases, but also which objects were observed, how often would the customer
visit the store and many other non-strictly transactional details.

One of the application area with probably the richest amount of user data is
video games; however, until recently, due to its distribution and revenue
models, the need for customer lifetime value prediction was relatively minor.
Traditionally, computer games distribution has relied on a premium pricing
model, in which a game is developed, released and sold for a price. Following
this pricing scheme, the monetization of the customer happens before the
player starts playing the game and its post-purchase behavior does not affect
the customer lifetime value directly [3].

Two main aspects radically changed the relationship between the game
developers and the players/customers: digital distribution platforms and the
free-to-play business model. Digital distribution platforms, such as
Steam111Valve Corporation. Steam. https://store.steampowered.com 2018. or the
App Store222Apple Incorporated. App Store. https://www.apple.com/lae/ios/app-
store 2018, along with game developers’ own on-line services allow tracking of
user behavior beyond a single game, making it more meaningful and necessary to
predict the customer lifetime value across multiple games, either being
purchased on the same store or being linked to the same on-line account.

Free-to-play (F2P/Freemium) games follow a different revenue model than the
traditional one: games are freely available, at no cost, to the players and
the revenue comes from in-game advertisement and the sale of in-game items.
These two revenue streams, contrarily to the classic game revenue model, begin
only after the players/customers have been acquired and vary dramatically
between players. This kind of relationship between the company and the players
resembles in part the relationship between a store and its customer, e.g.
monetary transactions happen at irregular intervals, the transaction value is
not constant and there is no explicit customer churn event.

However, differently from physical or on-line stores, the users are not only
customers but also players, and the data available about their behavior
includes a wide range of features that goes beyond transactions or goods
browsing, such as players progression within the game or players’ skill level.
Within the field of game analytics, the analysis of these features plays a
major role in evaluating the quality and the potential success of a game [19];
furthermore, the behavior has been linked with major business related metrics
such as retention [53] or customer lifetime value [70].

This abundance of accurate and high frequency data, covering different aspects
of the customer experience make free-to-play-games an ideal application area
for research in user modeling and prediction of future customer behavior. In
turn, the development of such models would see an immediate application in the
industry, as they would allow a better optimization of important processes
such as user acquisition and customer relationship management [45].

This, as well as the ever-expanding share of the game industry that leverages
this business model, highlights customer life time value prediction as a key
challenge in games data science and games research. For these reasons, in this
chapter, we will make an attempt to give an overview of the research and
applications of CLV modeling including an in-depth analysis of the works that
have been focused on the F2P field. In the next section, we will outline a
number of foundational concepts that will serve as a basis to the rest of the
chapter. In section 3, we will describe different ways in which CLV can be
used inside and outside of the game industry. In section 4, we will give an
overview of different models used to predict CLV with a more in-depth focus on
models employed in the game industry. In section 5 we will describe a number
of software packages that can be used to perform CLV prediction, and finally,
in section 6, we will outline a number of current and future directions within
the field.

##  2 Definitions and terminology

Pfeifer et al. [55], in their 2004 article, describe a problem present across
the different research works on customer lifetime value: incoherent
terminology. The reasons behind this problem are probably manifold, e.g. the
different research communities; however, in this section we will attempt to
synthesize a number of concepts and terms that are common across the different
fields involved and are foundational to the study of customer lifetime value
prediction in games and beyond.

First of all, it is important to define what is customer lifetime value;
Pfeifer et al. [55] summarize it as: “the present value of the future cash
flows attributed to the customer relationship”. As the authors point out, this
definition makes a couple of important assumptions: first, the value is
associated to the cash flow, which means it is not limited to the inbound flow
of revenue generated by a customer but also to the costs attributed to that
customer; however, the costs considered in this definition are only the ones
directly attributable to a given customer. Second, the term value expresses a
valuation at the current point in time of some future revenues and costs, this
implies some form of discount based on the time in which the future cash flows
are predicted.

In their definition, Pfeifer et al. do not attempt to encompass all possible
definitions of CLV within the literature; instead, they propose one possible
explicit definition of CLV that is terminologically correct and coherent. This
means that other research works on predicting CLV do not necessarily adhere to
the aforementioned definition. Berger and Nasr [4], for instance, do not
consider acquisition cost as part of the lifetime value, considering instead
CLV to be the “maximum profitable acquisition cost” [37]. Sifa et al. [69], on
the other hand, do not take into consideration the present value and use no
factor to discount the future predicted revenue.

One further aspect of this definition that is not shared by the different
research works is the time frame considered for the calculation. Gupta et al.
[28], in their review, find that a number of researchers employ an arbitrary
time horizon for the future prediction [56], while others user an infinite
horizon [25]. While it might appear that employing a fixed time horizon is
just a way to simplify the modeling process, both approaches have their own
merits. Using an infinite horizon does permits to avoid making any assumption
about the maximum length of the customer relationship with the company,
therefore; it allows a more accurate prediction of Pfeifer’s definition of
CLV; however, often for budgeting or other purposes, it is important to know
the return on investment within a given period of time (e.g. 1 year), making a
fixed horizon CLV prediction extremely useful.

Another consideration relative to the temporal aspect of the customer
relationship that has to be taken into account when studying CLV prediction is
whether the relationship is exclusive or it allows the customer to use other
competitors’ services; these two different type of relationship are often
labeled as either lost-for-good or always-a-share [20]. Identifying which type
of relationship is being modeled for CLV prediction is extremely important as
it affects the temporal length of the relationship and the definition of churn
(i.e. when a customer stops using the service).

This is especially true in free-to-play-games where there is no visible churn
event; in this context, the definition of the user state at any given moment
(active or inactive) depends on an ad-hoc formula based one some synthesis of
the players’ actions [61]. Many CLV prediction models include or are built on
top of a churn prediction model, and both the formula and the choice of model
will depend on whether it is possible for the customer to return after a
period of inactivity [54] or whether his or her relationship with the company
is considered finished [27].

Finally, it is important to specify that all of the works cited and described
in this chapter do not consider the effects of the competition with other
services on the customer lifetime value. As pointed out by Gupta et al. [28],
this is the case for most current modeling approaches because of the lack of
data about the competitors.

##  3 Applications of CLV prediction

While marketing serves as the primary application area of customer lifetime
value prediction, a number of other activities, such as customer relationship
management or live game operations, are being increasingly driven by data and
different key performance indicators (KPIs) such as CLV. Schmittlein et al.
[63], in one of the earliest works on customer lifetime value estimation,
motivate their research by stating that:

> ”…the issue is important in at least three settings: monitoring the size and
> growth rate of a firm’s ongoing customer base, evaluating a new product’s
> success based on the pattern of trial and repeat purchases, and targeting a
> subgroup of customers for advertising and promotions.”

They envision that, by knowing which customer are active and what purchases
they make, a manager is able to do more accurate budgeting, a better product
evaluation and target the customers more accurately with re-engagement
initiatives. Similarly to Schmittlein et al., a number of other research works
have tackled the above problems; Table 1 gives and overview of the main
activities for which CLV prediction has been employed.

Each application area includes a number of different activities (e.g. special
offers or advertisement) united by a similar usage of CLV prediction. In the
case of budgeting and finance, CLV predictions is used to predict revenue and
plan budgets for the different activities within the company [63, 18], finding
the balance between acquisition and retention investments [6] or to make a
financial estimation of the company [29]. The second activity, Product
Development, is mentioned only by Schmittlein et al. [63]; however, as we will
describe later in this section, CLV is an important KPI in the evaluation of
the game quality in F2P games and it can be very useful to drive the game
design process.

###  3.1 Customer Acquisition

Customer Acquisition encompasses all activities aimed at getting new customers
to start using the service or buying the products offered. As mentioned by
Berger et al. [4, 5], having an estimation of the customers’ lifetime value
can help deciding more accurately how much to spend on a given promotional
campaign and how much margin for profitability there could be in a new market
segment; Dwyer [20] further develops this concepts by identifying in CLV the
ceiling of the customer acquisition spend. Having an early indication of
whether the costs of acquisition are matched by the CLVs, can help deciding
whether a certain promotional initiative should be continued or stopped [45,
69, 70].

###  3.2 Customer Retention

Customer retention and segmentation initiatives are often intertwined and not
completely distinguishable; in this article, we describe customer
retention/loyalty activities as any activity explicitly aimed at prolonging
the lifetime of a customer, while, with customer targeting/segmentation, we
identify all activities aimed at identifying different homogeneous groups
within the customer base that can be targeted with a custom user experience.
Such a customer experience could be aimed at prolonging the customer lifetime;
however, for the purpose of this categorization, we define this as a secondary
objective.

Application area | Article
---|---
Budgeting and Finance | [63, 6, 18, 29]
Product Development | [63]
Customer Acquisition | [20, 4, 5, 45, 69, 70]
Customer Retention/Loyalty | [4, 50, 56, 44, 59, 58]
| [48, 67, 11, 61, 70]
Customer Targeting/Segmentation | [50, 56, 75, 74, 35, 30, 67, 42, 40]
Other | [1]
Table 1: Applications of CLV prediction in different business activities

Latchowetz et al., in their study on the impact of season ticket holders on
the NBA franchises revenue [44], argue that a customer should not be evaluated
solely on the revenue that she or he generates within a season. At the time of
their estimation, an average season ticket holder had a lifetime value of more
than eighty thousand US dollars; therefore, the authors advice the
entertainment industry to acknowledge the importance of using customer
lifetime value and developing long term retention strategies for their
customers.

Berger and Nasr [4] discuss how knowing the customer lifetime value can help
determining the effects of adopting different marketing strategies for
retention and acquisition and how to balance the spending between the two
activities. The authors point out also that any acquisition strategy might
impact also the customers’ retention; therefore, it is hard to see the two
activities independently and both their budgets must by aligned to the
predicted customer lifetime value. Mulhern extends further Berger and Nasr’s
idea about the strategic use of CLV prediction by seeing it as a primary
indicator to driving resources allocation in the marketing mix [50].

Rosset at al.[59, 58] extend the idea of using CLV prediction to drive
retention activities and expand it by investigating how to estimate the impact
of retention efforts to the CLV. They demonstrate an application of their
approach in the context of a retention campaign for a group of potential
churners. They show how, by having a model able to predict CLV in different
configurations of the service, it is possible to compare the current estimated
lifetime value of a group of potential churners with different projected
lifetime values given a number of different incentives to improve the cohort’s
retention; thus, taking a more informed decision on the correct initiative to
be taken.

The predictions used for these decisions, as any other prediction of future
events, have some degree for uncertainty in the form of prediction error or
bias; taking this uncertainty into account can be also important in decision
making. Malhouse and Blattberg [48] study the impact of uncertainty in the
management of the customer relationship helping to understand not only whether
engaging into a retention activity can be profitable but also how often it
could be.

###  3.3 Customer Segmentation

Most of the research works presented in this section make the assumption that
a higher retention (i.e. a longer lifetime for the customers) will have a
positive impact on the company’s revenue in the form of an increased CLV;
however, while this is generally true, Reinartz and Kumar [56] found that this
assumption does not always howl and different customer segments have different
spending patterns, which means that, in a number of cases, prolonging the
customer’s lifetime would no yield any more revenue. For this reason, they
suggest that, the company should not necessarily pursuit long-term customer
relationship, but it should customize the relationship based on prediction on
future customer lifetime value and retention.

Segmenting the target audience and personalizing marketing activities is an
established practice [17]; however, the predominant guiding factors used to
define the segments and the most appropriate actions have, for long time, been
based on demographics, questionnaires and interviews [47]. The advent of large
scale data collection and analytics, in many industries, has dramatically
changed many of the common practices in marketing research, allowing
practitioners to access more detailed information from wider audiences at
lowers costs.

Mulhern [50], for example, proposes customer lifetime value as a possible
segmentation criterion beside usage volume and brand loyalty: he suggests that
one simple segmentation could divide the users in customers worth keeping, as
their CLV is profitable, and customers that are not worth the retention cost.
Verhoef and Donkers [75] propose a similar segmentation to design personalized
insurance offers.

Venkatesan and Kumar [74] pose the question whether using predicted CLV for
segmentation con yield better results than more established KPIs such as
previous-period customer revenue or past customer value. To compare the
effectiveness of the different metrics, they rank the customers based on each
metric and segment the customers based on the raking; the results show the
segmentation based on CLV to be superior to the other segmentation approaches.

Similar strategies to the ones mentioned so far have been applied to a number
of different industries such as telecommunications [35], banking [30],
retailing [67, 42] or beauty [40]. In the last few years, a similar trend has
emerged in the gaming industry and, especially in the mobile game and free-to-
play industry, all the aforementioned marketing practices are established and
widely undertaken [66].

###  3.4 Free-To-Play Games

Figure 1: Three screen grabs from the iOS version of Bee Brilliant Blast by
Tactile Games depicting, from left to right, the phone’s store interface, a
typical puzzle level and the in-game store interface, in which the customer
can purchase different amounts of in-game currency.

Free-To-Play video games are games than can be played without an upfront
payment and potentially completely free of charge; players can, within the
game, purchase some virtual goods and pay to enable some features. While this
type of games was originally mostly distributed through on-line social media
platforms, the business model is currently vastly diffused on many different
platforms, such as mobile phones, home consoles and personal computers [2].

Figure 1 shows an example of a mobile free-to-play game; as it is possible to
see on the first screen grab to the left, the game is freely available on the
phone’s store, however, the store specifies that the game contains the
possibility to perform in-app purchases. The last image to the right shows the
in-game store that allows the customer to purchase different packs of in-game
currency, which can be used throughout the game to unlock different features.
Using an intermediary currency, while a very common practice, is not defining
characteristic of free-to-play games, and some games allow customer to enable
directly in-game features directly using a real currency.

The second main source of revenue in free-to-play games comes from
advertisement: a large portion of free-to-play game display to the players ads
in different formats (e.g. videos or interstitials). These ads are provided by
external services that act as mediators between the advertiser and the
publisher. The game (i.e. the publisher) received ads from the ad provider
through an API and shows them to the player, the ad provider will in return
pay the game developer depending on the number of ads being displayed, the
number of times the ads are clicked or the number of times the application
being advertised is being installed.

Both aforementioned revenue streams – in-app purchase and advertisement – have
a non-contractual nature, as the player can freely choose if and when to make
a purchase or click on and advertisement campaign; furthermore, a conversion
rate (i.e. the percentage of players making an in-app purchase) well below 10%
is considered normal in the industry [51], making it especially challenging to
predict the customers’ lifetime value. At the same activities such as customer
acquisition and retention rely on accurate CLV predictions for accurate
targeting and budgeting.

As of November 2018, there are around 816000 games available only on the Apple
App Store333https://www.pocketgamer.biz/metrics/app-store/categories/;
therefore, targeting the right customers and accurately measure their
potential impact in the company’s revenue is particularly important in
customer acquisition in the free-to-play games market, given the extremely
high number of games competing for the same users. A significant part of the
customer acquisition efforts in the industry is executed through performance
marketing campaigns: in these campaigns, he advertiser competes with other
advertisers to show ads the potential new customers, often times through some
form of auctioning system. Customer lifetime value predictions can be used as
a benchmark to evaluate the profit margin of these campaigns as both the
potential revenue coming from the new customers and their cost of acquisition
vary greatly depending on the segmentation chosen for the campaign, the market
context and the quality of the advertisement creative content [70]. Even
beyond performance marketing, an accurate CLV prediction is very important to
be able to budget expensive and higher risk marketing efforts such as
television advertisement.

Furthermore, the F2P games market is characterized by very low retention
number – e.g. the best android games have a day 30 retention rate below
5%444https://www.forbes.com/sites/johnkoetsier/2017/10/20/the-very-best-
android-game-has-just-4-5-user-retention-after-30-days/ – making any effort to
improve such low number a priority for most companies. Within this context,
CLV prediction can be used, as previously described, to target different
segments of users with special offers; However, it is also possible to
personalize the game experience at large by adapting, for instance, the flow
of the game progression or by providing custom events. Harrison and Roberts
show an example of such adaptation aimed to improving retention in a digital
version of Scrabble [32].

Given the importance of CLV prediction in the free-to-play and other
industries, a number of algorithms have been developed over time , in the next
two section we will describe the most significant ones and analyses a number
of software packages that can be used for CL prediction.

##  4 Models

Berger and Nasr are the first researchers that attempted a categorization of
CLV prediction models [4]. In this early review of the field, all methods
mentioned are aimed at calculating an average customer lifetime value for the
entirety of the company’s customer base or a given cohort. In a latter review
[43], Kumar et al. group these approaches under the label ”average LTV
approach”, while Jain and Singh [37] use the term ”basic structural model”. In
this chapter we adopt the definition used by Kumar et al. and we include a
number of other practices commonly used in the free-to-play industry that
approach LTV prediction in a similar fashion.

The methods for CLV prediction discussed in this article are dividend in two
further categories: customer history models and machine learning models. The
first category contains all methods that attempt to predict either a single
customer’s or a cohort’s lifetime value based on some mathematical projection
if the past behavior of the given customer or cohort. The second category
includes all algorithms which make use of some learning algorithm to build a
computational model of the customer behavior, which is then used to make
predictions.

###  4.1 Average Models

The aim of the models discussed in this section is to give an estimation of
the future discounted cash flow for a group of customers or for the whole user
base. The common aspect between these models is that they make a number of
assumptions: they assume a constant (or otherwise known) margin of profit over
time, they do not consider the stochastic nature of the purchase behavior of
the customers and they assume that the customer behavior is uniform across the
estimated cohort.

The most basic model proposed by Berger and Nasr [4] assumes that the customer
retention rate and the costs of retention are constant over time and both
costs and revenues happen periodically at a constant rate, within these
conditions the formula for CLV is as follows:

$CLV=GC*\sum_{\\_}{i=0}^{n}\frac{r^{i}}{(1+d)^{i}}-M*\sum_{\\_}{i=0}^{n}\frac{r^{i-1}}{(1+d)^{i-0.5}}$
(1)

Where CG is the customer’s expected gross contribution margin per period, M is
the promotion costs per customer per period, n is the prediction horizon
expressed in number of periods, r is the retention between one period and the
next, and d is the discount rate.

Further iterations of this model presented by Berger and Nasr [4], by Jain and
Singh [37] and by Kumar et al. [43], allow more sophisticated ways to express
non constant retention rates and margins of profit and to include acquisition
costs in the calculation.

In their article about the relationship between the length of service and CLV,
Rosset et al. [58] formulate an extended average model that employs a Kaplan-
Meier [39] estimator to calculate the duration of the relationship between the
company and the average customer .

A similar approach is currently among the basic models used in the free-to-
play industry for CLV prediction [65]: after selecting a specific function to
express the retention curve of a given cohort, the function is fit on the
training data and the resulting customer lifetime value is calculated as
follows:

$CLV=\sum_{\\_}{i=0}^{n}ARPDAU*ret(i)$ (2)

Where $ARPDAU$ stands for average revenue per daily active user and $ret(i)$
is the value of the chosen retention function at the $i^{th}$ day. This method
for CLV prediction, while quite simplistic and not necessarily extremely
accurate, has the advantage of being easily readable and based on established
business KPIs such as APRDAU and retention.

Another example of simplistic but widely adopted model is described by Runge
[60]: instead of basing the projected revenue on retention and APRDAU, the
method fits a function to approximate the average monetization curve, which is
then used to project the customer lifetime value for a player based the amount
of money the she spent at the present day. If we consider $rev(n)$ as the
revenue produced by a customer up the $n^{th}$ day of the customer
relationship and $mon(n)$ as the average fraction of the revenue that is
produced by day $n$ the formula for CLV prediction is as follows:

$CLV=rev(n)/mon(n)$ (3)

The major advantage of this model is its simplicity and, similarly to the
retention-based model, its readability. However, with such a model it is not
possible to predict the CLV for customers that have not yet produced any
revenue – i.e. the projection would always result 0; moreover, in a context
such as free-to-play games, in which the amount of paying user is very small,
the variably of the behavior between different paying user has a large impact
on the prediction, and such variability is completely disregarded by the
model.

###  4.2 Customer History Models

The aforementioned methods based on retention and monetization curves are
partly based on historical customer data as, in both methods, a function
representing the average retention or monetization behaviors is fit on some
past customer data and then used for prediction. These methods however do not
allow to use any particular user’s track record to personalize the prediction,
hence the average nature of the methods.

Based on the customer categorization by Jackson [36] that divides the
customers in two types: always-a-share and lost-for-good, Dwyer [20] argues
that an average retention model is not sufficient to model CLV for customers
belonging to the always-a-share category. For this type of customers, he
proposed a migration model based on the customers’ recency of purchase.

Always-a-share customers differ from lost-for-good customers in that the
latter have a long-term commitment with the company and switching to a
competitor is costly, while the first ones can have simultaneous relationships
with multiple companies competing for the same product/service. Typical
examples of lost-for-good customers are bank customers or tenants in a rental
property. Whether free-to-play game players can be considered belonging to the
first or second category is an open question as there is no monetary barrier
that stops a player to switch to another game; however, to a certain extent,
the longer the time a player invests in a game, the less likely is that the
player will be stopping playing that game.

In his customer migration model, Dwyer uses purchase recency to estimate the
customer’s probability of performing a purchase in the next period and the
potential value of such a purchase. In the training phase, based on past
customers’ data, the method builds a model of purchase probabilities and
values that depend on the length of the period of inactivity. These recency
cells are then used as a lookup table to predict, depending on each customer’s
past purchase behavior data, her purchases in the next time unit. Dwyer’s
model allows to leverage the customer’s past behavior for a personalized CLV
prediction and considers the probabilistic nature of the customers purchase
behavior.

####  4.2.1 Recency, Frequency and Monetary Value

The main limitation of Dwyer’s methods is that the probability of a purchase
in a given period is based only in the recency of the last purchase, which is
not necessarily true for all businesses. Especially in free-to-play game,
different types of purchase will affect differently the future purchases, for
example, a player purchasing a large package of virtual currency recently
might be less likely to make purchase as the amount of resources acquired will
allow her to play longer without a need for any help within the game.

The concept of recency used by Dwyer is an established metric in direct
marketing and alongside purchase frequency and average purchase monetary value
– commonly known together as RFM – have been long used in the field to predict
customer behavior [28]. Hughes describes a method for customer quality
estimation based on these three variables: the current customer based of the
company is sorted along these three variables each categories into 5
quantiles, creating therefore $5*5*5$ groups, these groups are then used to
score the different customers and target them with specific offers [34].

Inspired by Hughes’s work, Shih and Liu propose a method based on RFM and CLV
clustering to rank the customer according to their profitability [68]. As a
first step the method relies on a group of expert evaluation to identify the
relative importance of the recency, frequency and monetary variables using
analytical hierarchical processing. The customers are than clustered based on
the RFM space and the resulting clusters are ranked through a simple weighted
sum of the three normalized variables.

The aforementioned methods are not designed to produce a numerical predictor
of CLV but only to score the customers by their potential profitability.
Furthermore, these methods are limited in predicting the customer behavior
only for one future time period [26]. Finally, these methods disregard the
fact that customers’ pas behavior is, often times, the result of past company
activities.

Fader et al. describe a model for CLV prediction, using RFM as input variables
based on the e Pareto/NBD framework [63], which overcomes some of the
aforementioned limitations [26].

####  4.2.2 Pareto/NBD

The Pareto/NBD model [63] aims at predicting individual customer’s purchase
behavior based on their past purchase patterns. More specifically, for each
customer, based on the recency ($t$) and frequency ($X$) of the purchases and
the length of the relationship between the customer and the company ($T$), the
model estimates the expected future number of purchases. The model is based on
five assumptions:

  * •

while active, a customer makes purchases following a Poisson process with rate
$\lambda$;

  * •

the purchase rate $\lambda$ differs between customers and is distributed
according to a gamma distribution across the customer base;

  * •

the duration of the active period of the customer is exponentially distributed
with a death rate $\mu$;

  * •

the death rate $\mu$ differs between customers and is distributed according to
a gamma distribution across the customer base;

  * •

the death rate and the purchase rates are independent.

Based in these assumptions, the authors find that the customers “deaths” for a
sample follow a Pareto distribution of the second kind [38], while the number
of purchases made by an active customer follow a negative binomial
distribution [22] – hence the name Pareto/NBD. The two distributions are
controlled by four parameters ($r$ and $\alpha$ for NBD and $s$ and $\beta$
fro Pareto);

Schmittlein suggests two approaches to estimate the parameters based on the
customers’ past behavior: maximum likelihood and fitting observed moments; in
case a model in needed without prior data, Schmittlein discusses also the
possibility to handpick the parameters based on management judgments. Once the
parameters have been identified, each customer’s future number of purchases is
predicted using the two aforementioned distributions and the customer’s $X$,
$t$ and $T$.

One of the main issues with the Pareto/NBD is it’s computational complexity
[25] as the estimation requires multiple evaluations of the Gauss
Hypergeometric Function; this problem is mitigated by the modified BG/NBD
model by Fader et al. [25], which models the customer activity using a beta-
geometric model that is easier to implement efficiently.

Both aforementioned models are able to predict the future number of purchases
for a given customer and they can, for instance, be used to count expected
active users ate a give point in time, however these methods do not model in
any way the value of each purchase, therefore, they can’t directly predict the
customer lifetime value.

Reinartz and Kumar, in their studies on profitable lifetime duration [56, 57],
employ Schmittlein’s model to be able to calculate the number of time periods
in which a customer will perform a purchase. The authors transform the
continuous probability that a customer is active into a dichotomous measure of
whether the customer is active or inactive it a given point in time. Given a
probability threshold, it is possible to identify a customer’s future date of
churn, which, combined with the first date of the customer relationship, gives
an expected customer relationship duration. This duration, expressed as number
of periods $n$, is used to calculate that customer’s lifetime value using the
formula defined by Berger and Nasr [4] and described in Equation 1.

Schmittlein and Peterson[64] propose an extension of the original Pareto/NBD
in which the future monetary value of each transaction is samples from a
normal distributed around the mean monetary value of a cohort. However, Fader
et al. [26] observe that, in the data they analyzed, there are large
differences between mean, mode and median, indicating that the distribution of
the monetary values is highly skewed. Therefore, the authors propose to model
the average transaction value using an adapted version of the gamma-gamma
model proposed by Colombo and Jiang [15].

One of the main strengths and, at the same, the main limitations of the
Pareto/NBD or BG/NBD models is that they rely on a small number of purely
transactional data. This is a strength in that it makes the model general
enough to be easily applied in different context; however, the resulting model
will always risk to be sub-optimal as it disregards potentially relevant
information. Sing et al. address this limitation and propose an estimation
framework that can flexibly incorporate multiple statistical distributions and
consider a number of covariates such as age or gender [72].

Glady et al. [27] address another important limitation of the aforementioned
models: they all assume independence between the frequency of transactions and
the profit per transaction. The authors of the study demonstrate that such
assumption does not hold in multiple real-world data sets and propose a
modified Pareto/Dependent model that performs better than Pareto/NBD in such
circumstances.

###  4.3 Markov Chain Models

An alternative approach to customer lifetime value prediction is proposed for
the first time by Pfeifer and Carraway [54]: the authors suggest to model the
customer relationship as a Markov Chain Model (MCM), in which the different
states of the model represent different conditions of the relationship between
the customer and the company in terms of transactions and customer activity,
and the transition probabilities between states represent the probability of a
customer to move from one condition to the other one – e.g. for a customer to
make a purchase or to churn.

Figure 2: Simple customer relationship Markov Chain Model. States r1 to r4
represent different value of recency from 1 to 4.

Figure 2 shows an example of a simple Markov Chain Model representing the
possible states of the relationship between a customer and the company and the
valid transitions between the states. In the depicted case, the states are
based on the recency of the last purchase (e.g. r1 corresponds to recency
equal to 1) and there are only 5 valid stages before churn. However, as in the
examples reported by Pfeifer and Carraway, in a real-world scenario, the
states are based on a multitude of factors and there are many more valid
transitions each with their own probability.

While Pfeifer and Carraway show how it is possible to calculate CLV using a
MCM representation of the customer relationship with a given set of states and
transaction probabilities, Etzion et al. focus on the process of learning the
states and the transition from past customer data [24]. The process requires
first to identify the variables determining the states of the model, to define
their ranges and discretize them.

At this point the transition probabilities between the states can be
calculated through the following three steps:

  1. 1.

initialize a transition matrix – i.e. a square matrix that contains the
probability of transitioning between one state and another one – with zeros in
all cells;

  2. 2.

for each customer performing a transition between a state $i$ and a state $j$,
the matrix cells identified by $ij$ is incremented.

  3. 3.

at the end of the processing, each line of the matrix is normalized between 0
and 1 using a min-max normalization.

The resulting matrix can be used, as suggested by Pfeifer and Carraway [54] to
calculate CLV for a customer in any given state.

Ching et al. [12] employ the same approach to build multiple transition
matrices that describe the customer behaviors in different market conditions –
e.g. with or without a promotion – finding that the transition matrices differ
and so does the CLV and the customer retention. The authors further showcase
how to use stochastic dynamic programming to estimate the optimal promotion
strategy that optimizes CLV given the previously found transition matrices and
a set of constraint on the promotions budget.

###  4.4 Supervised Learning Models

The general principle at the core of all supervised learning algorithms is to
learn a function between some example input and output data – e.g. past
customer behavior and their recorded lifetime value. This example data is
usually called training data, and the function resulting from this training
phase – the model – is used to predict the target variable – e.g. CLV – on
newly collected data. This procedure is not dissimilar to some of the MCM
based approaches previously described [24]; however, the models we present in
this section, differ from MCM models in that they trade explanatory power for
the possibility to capture more complex behaviors using some form of
computational/black-box model.

This means that the resulting models are less useful to inform the business
about specific customers’ preferences, but they can potentially be more
accurate and be deployed, for instance, for marketing automation. It is
important to note that this is not a hard distinction and many models
described in this and the previous sections include a mix of explanatory and
black-box models.

An example of a model incorporating different learning algorithms is the one
proposed by Haenlein et al. [30]. The authors describe an approach to CLV
prediction that uses a combination of CART analysis [8] and MCM: first, the
customers of a retail bank are divided into age groups and, for each age
group, a regression tree model is built to predict the profit for a customer
within that group; second, the transition probabilities between the groups are
modeled as a Markov Chain so that the model is able to follow the transitions
of the customers throughout their lifetime. The resulting lifetime value is
calculated as a discounted sum of each of the CLV predicted in the possible
customer states weighted by the transition probabilities.

Cheng at al. [11] approach CLV prediction in a similar way but they extend the
model by building a different Markov Chain for each period the customer
relationship; each chain includes a number of states depending of the activity
level of the customers in that specific period. The total CLV for a customer
is similarly calculated by summing the predicted CLVs over the different
states and periods weighted by the transition probabilities and the survival
probability of each period. The models used to predict the profit contribution
of a customer and a given state is modeled using an artificial neural network,
while the survival probability in each period is calculated using a logistic
regression model.

Within the context of free-to-play games, supervised learning methods have
been employed in a number of studies. Compared to the previously mentioned
approaches, supervised learning offers two key advantages that make them
particularly interesting within this industry: the ability to leverage a wide
variety of data about the customer behavior and the ability to make
predictions about CLV for players that are not yet customers – i.e. that have
not yet made a single purchase.

The first aspect is important as in the freemium games industry, customers are
also players and the data describing the gaming behavior of players/customers
is much richer and at a much higher frequency compared to their purchase
behavior data. Furthermore, there are a number of studies that show how past
in-game behavior is a predictor of future in-game behavior [46], of player
preferences [9] and of the probability of making a purchase within the game
[31].

The second aspect is partially related to the first one: since the predictions
can be based also on non-purchase related data, it is potentially possible to
perform predictions also for players that have not yet made a purchase and
that might never be making one. As described in Section 3, most of free-to-
play players do not make any in-game purchase and generate little to no
revenue; it is therefore important for a model that predicts CLV in such
context to be able to discern which customers will be making a purchase before
the first one is made.

In one of the first studies on the application of supervised learning for CLV
prediction in freemium games, Lange et al. present a comparison of the
performance of a series of linear and non-linear models in the prediction of
CLV [45]. The dataset used for the study contains in-game events from the
first 7 days of gameplay of about 38 million players; the input variables
include data about player spending, gameplay, game progression, social
interactions, success metrics and game settings preferences. The target
variable is the amount of revenue generated by each player after 180 days.

The results of the study show that an ensemble of two artificial neural
networks using the absolute-differences technique over-performed the other
methods in terms of mean square error in a 10-fold cross-validated comparison.

A similar study is presented by Sifa et al. [69] in their article on purchase
behavior prediction in mobile free-to-play games. Similarly to Lange et al.,
the authors present a comparative study of multiple supervised learning
algorithms; however, the objective of the prediction is slightly different as
the authors argue that CLV prediction is a combination of three prediction
tasks: predicting whether player will ever make a purchase, predict the number
of purchases and predict the value of each purchase. The results of the study
show that, for the classification task, the models having the best performance
are tree-based models – e.g. random forest and decision tree classifiers – and
that it is extremely valuable to re-sample the dataset in to avoid as the
players performing purchases are extremely under represented. For this
purpose, the authors employ the SMOTE-NC [10] method to over-sample the
“converted” users data points.

In a latter article, Sifa et al. present a study that attempts to predict CLV
in one single step [70], more in line with the approach presented by Lange et
al. [45]. The study evaluates the performance of a deep neural network in
predicting day 360 cumulative undiscounted revenue per player, and it compares
the results against a number of algorithms such as random forest and linear
regression. The results of the comparison show that the deep neural network
out-performs the other regression algorithms in terms or normalized root mean
square error.

Furthermore, the authors demonstrate how to adapt SMOTE-NC to re-sample the
dataset and reduce the imbalance between customers that made a purchase and
customers that did not. This process requires a number of adjustments of the
standard implementation of SMOTE-NC, as the original algorithms is design for
classification problems, while a direct prediction of CLV is a regression
problem.

##  5 Software Packages

In Section 4, we gave an overview of the current state-of-the art in CLV
prediction mostly from an academic perspective, listing and explaining a
number of scientific articles describing the main approaches to solve the
prediction problem. However, most of the algorithms presented so far have been
translated, to different degrees, in software packages that can be applied to
easily perform CLV predictions on new datasets.

In this article, we identify three main packages that can be either used
directly to predict CLV or that can be used to construct a model for CLV
prediction. The software packages are Lifetimes [16] and BTYD [21] that
implement some of the algorithms presented in Section 4.2.2, and scikit-learn
[49] a popular machine learning package for the Python programming language
that can be used to implement the models presented in Section 4.4. Alternative
packages can be used to develop supervised learning models – e.g. Keras [14]
or XGBoost [13] – however, most of them are compatible with scikit-learn,
therefore we chose this package as the representative in this article.

To build and operate with Markov Chain Models, there are a large number of
different alternatives for both the Python and the R language – e.g. the DTMC
pack [73] and PyMC3 [62] – however, these packages are designed to offer
functionalities that go far beyond the code needed to implement the models
presented in Section 4.3, which can be easily implemented using standard
algebra functionalities. For this reason, we chose not cover any MCM package
in this Section.

###  5.1 Lifetimes/BTYD

The Lifetimes [16] and BTYD [21] packages are both software implementations of
the Pareto/NBD, BG/NBD and Gamma-Gamma models; they allow to learn the
parameters of the various distributions from past purchases data and to
predict future purchases numbers and the purchases values for new customers.
In this section we will give sample of how to train and produce predictions
with Lifetimes; however, the procedure is very similar for the BTYD software
package.

The main components of the Lifetimes package are the classes BetaGeoFitter and
GammaGammaFitter. The first class (BetaGeoFitter) contains all the logic
necessary to fit a BG/NBD model from transactional data in the format of
purchase recency and frequency, and current age of the customer in terms of
customer relationship duration. It is also possible to alternatively fit a
Pareto/NBD model using the ParetoNBDFitter class instead.

Given a dataset with correctly formatted inputs, the code to fit the
BetaGeoFitter looks as follows:

⬇

from lifetimes import BetaGeoFitter

bgf = BetaGeoFitter(penalizer_coef=0.0)

bgf.fit(data[’frequency’], data[’recency’], data[’T’])

The package provides a function named summarydatafromtransactiondata that
allows to produce correctly formatted data for the algorithm from a list of
transactions.

The second class (GammaGammaFitter) provides all the methods necessary to fit
a Gamma-Gamma model and can be used, in combination with the BetaGeoFitter, to
predict the customer lifetime value. The Gamma-Gamma model is also trained on
transactional data containing the frequency and the total monetary value of
the purchases made by each customer.

The code to fit the Gamma-Gamma model looks as follows:

⬇

from lifetimes import GammaGammaFitter

returning_customers_summary = data[data[’frequency’]>0]

ggf = GammaGammaFitter(penalizer_coef = 0)

ggf.fit(returning_customers_summary[’frequency’],

returning_customers_summary[’monetary_value’])

And the code to perform CLV predictions on new data using the previously fit
models looks like this:

⬇

ggf.customer_lifetime_value(

bgf,

new_data[’frequency’],

new_data[’recency’],

new_data[’T’],

new_data[’monetary_value’],

time=180,

discount_rate=0.01

The time and discount rate depend on the parameters of the specific prediction
to be performed.

Further documentation and tutorials about the two packages can be found on
their project web
pages555https://github.com/CamDavidsonPilon/lifetimes666https://cran.r-project.org/web/packages/BTYD/index.html,
and both packages are currently available for download and install
respectively on the Python Package Index777https://pypi.org and The
Comprehensive R Archive Network888https://cran.r-project.org.

###  5.2 scikit-learn

Scikit-learn [49] is a library for the Python programming language that
provides functionalists to perform supervised and unsupervised learning tasks.
The supervised learning part of the library implements a wide variety of
algorithms ranging from linear models to tree-based methods for both
classification and regression. In this section, we will show how to use
scikit-learn to build a regression models based on the Random Forest algorithm
similar to the one presented in [45] and [70].

Random Forest [7] is an ensemble algorithms that combines a number of tree
predictors trained on different portions of the data, the output of the model
is generally either the mode or the mean of the outputs of the different tree
predictors depending on whether the task is a classification or a regression
task. Scikit-learn implements this class of algorithms though the
RandomForestRegressor and RandomForestClassifier classes; for the purpose of
this article we will show how to use the RandomForestRegressor to predict CLV
through regression.

Given a dataset similar to the one used in [70] with one line per player, the
code to fit the RandomForestRegressor looks as follows:

⬇

from sklearn.ensemble import RandomForestRegressor

X = data[’number_of_sessions’, ’number_of_rounds’,

’number_of_days’, ’number_of_purchases’,

’total_purchase_amount’]

y = data[”day_360_CLV”]

model = RandomForestRegressor(n_estimators=100)

model.fit(X, y)

For readability, the number of features selected as inputs is reduced in
comparison to [70]; furthermore, the number of trees in the ensemble has been
chosen without any specific motivation as no precise description is included
in the article. Both the size of the ensemble and maximum depth the trees, as
well as the other parameters of the ensemble, are problem specific and should
be selected either algorithmically or through extensive experimentation.

The code necessary to perform CLV prediction using the previously fit model
looks as follows:

⬇

X = new_data[’number_of_sessions’, ’number_of_rounds’,

’number_of_days’, ’number_of_purchases’,

’total_purchase_amount’]

y = model.predict(X)

The resulting array $y$ will contain the predicted customer lifetime values.
The sci-kit learn library includes also a number of functionalities to
estimate the predictions error and to perform different forms of validation
and testing – e.g. k-folds cross-validation. More information can be found on
the project’s documentation page999https://scikit-
learn.org/stable/documentation.html.

##  6 Conclusions and Future Directions

In this chapter we have described the applications of customer lifetime value
prediction in the free-to-play games and other industries and we attempted to
give a comprehensive description of the different methods that are currently
used to predict CLV. We have identified a number of activities that can
benefit from an accurate prediction ranging from customer acquisition to
market segmentation and we have described how the need for such prediction is
even more important in free-to-play games due to the characteristics of the
marker and the customer relationship – e.g. incredibly high competition in
user acquisition and very low number of paying customers compared to the
player base.

In Section 4, we have identifies within the literature four main group of
methods for the prediction: average based, Pareto/NBD and derivatives, Markov
Chain Models and Supervised Learning Models. The methods that are currently
dominant within the free-to-play games industry are either average based or
some form of Pareto/NBD. However, we can see an emerging trend in recent years
of more studies being published on the application of different supervised
learning algorithms to CLV prediction.

These methods have a number of advantages over classical statistical methods
in this context as they make no assumption of the distributions of the input
data and easily allow multiple co-variates to be included in the model.
Furthermore, in an industry with such a low rate of conversion to paying
users, being able to predict revenue from a player that has not yet made any
purchase has the potential to be very useful, for instance, for early
estimation of the profitability of a player acquisition campaign.

Based on the analysis presented and in light of the recent improvements within
the field of machine learning, we close the article by proposing a number of
possible interesting future directions for customer lifetime value prediction
research.

##### Deep Learning

One group of supervised learning algorithms that has only minimally been
explored in one of the most recent articles [70] is Deep Neural Networks. The
work by Sifa et al. don’t give an in-depth description of the deep multi-layer
perceptron used; however, in the absence of other information, we can assume
that the model was based on a standard fully-connected feed-forward neural
network with multiple hidden layers. Given the results achieved with a
relatively simple architecture, it would be interesting to test more advanced
deep learning techniques such as auto-encoders for feature extraction [76] or
deep convolutional neural networks [41].

##### Time Series

All of the models presented in this chapter act on some form of summary data
representing the behavior of customers/players up to a given point in time.
While some of these summary representations do include some representation of
the temporal aspect of the behaviors – e.g. the recency and frequency features
in RFM based models – these cannot capture any particular sequences of
purchases or any sequence of in-game events that is connected to the
probability that a player will perform a purchase or not. One possible
approach to leverage this kind of information is to treat player behavior data
as time series and perform some form of time-series regression or
classification to predict CLV or a purchase event. Within aforementioned deep
learning field algorithms such as Long-short term memory networks [33] or the
more recent temporal convolution networks [23].

##### Transfer Learning and Lifelong Learning

Game developers in the mobile free-to-play market are challenged with the
increasing need to manage a multitude of games live at the same time. These
games have often very long lifetimes and, throughout their lifetimes, the game
is adjusted and evolved and the player base changes dues to the game evolution
and the changes in the customer acquisition initiatives. This adds a new
dimension of the problem of CLV prediction as models need to be constantly
updated and new models need to be built quickly for new version of the games
and for new games. Within this scenario, research in cross-game player
behavior analysis [52], transfer learning [77] and life-long machine learning
[71], and their application to CLV prediction will become increasingly
important.

## References

  * [1] Harsha Aeron, Tarun Bhaskar, Ramasubramanian Sundararajan, Ashwani Kumar, and Janakiraman Moorthy.  A metric for customer lifetime value of credit card customers.  Journal of Database Marketing & Customer Strategy Management, 15(3):153–168, jun 2008.
  * [2] Kati Alha, Elina Koskinen, Janne Paavilainen, Juho Hamari, and Jani Kinnunen.  Free-to-Play Games : Professionals’ Perspectives.  Nordic DiGRA, pages 1–14, 2014.
  * [3] Charles Baden-Fuller and Stefan Haefliger.  Business Models and Technological Innovation.  Long Range Planning, 46(6):419–426, 2013.
  * [4] Paul D. Berger and Nada I. Nasr.  Customer lifetime value: Marketing models and applications.  Journal of Interactive Marketing, 12(1):17–30, 1998.
  * [5] Paul D. Berger, Bruce Weinberg, and Richard C. Hanna.  Customer lifetime value determination and strategic implications for a cruise-ship company.  Journal of Database Marketing & Customer Strategy Management, 11(1):40–52, sep 2003.
  * [6] Robert C Blattberg and John Deighton.  Manage Marketing by the Customer Euqity.  Harvard business review, 74(4):136, 1996.
  * [7] Leo Breiman.  Random forests.  Machine Learning, 2001.
  * [8] Leo Breiman.  Classification And Regression Trees.  Routledge, oct 2017.
  * [9] Paolo Burelli and Georgios N. Yannakakis.  Adaptive Virtual Camera Control Trough Player Modelling.  User Modelling and User-Adapted Interaction, 2015.
  * [10] N. V. Chawla, K. W. Bowyer, L. O. Hall, and W. P. Kegelmeyer.  SMOTE: Synthetic Minority Over-sampling Technique.  Journal of Artificial Intelligence Research, 16:321–357, jun 2002\.
  * [11] C.-J. Cheng, S.W. Chiu, C.-B. Cheng, and J.-Y. Wu.  Customer lifetime value prediction by a Markov chain based data mining model: Application to an auto repair and maintenance company in Taiwan.  Scientia Iranica, 19(3):849–855, jun 2012.
  * [12] W. K. Ching, M. K. Ng, K. K. Wong, and E. Altman.  Customer lifetime value: Stochastic optimization approach.  Journal of the Operational Research Society, 55(8):860–868, 2004\.
  * [13] Hyunsu Cho.  XGBoost Python Package, 2018.
  * [14] Francois Chollet.  Keras: Deep Learning for humans, 2018.
  * [15] Richard Colombo and Weina Jiang.  A stochastic RFM model.  Journal of Interactive Marketing, 13(3):2–12, 1999.
  * [16] Cam Davidson-Pilon.  Lifetimes: Measure customer lifetime value in Python, 2018.
  * [17] Peter R. Dickson and James L. Ginter.  Market Segmentation, Product Differentiation, and Marketing Strategy.  Journal of Marketing, 1987.
  * [18] Bas Donkers, Peter C. Verhoef, and Martijn de Jong.  Predicting Customer Lifetime Value in Multi-Service Industries.  Technical report, Erasmus Research Institute of Management (ERIM), 2003\.
  * [19] Anders Drachen, Magy Seif El-Nasr, and Alessandro Canossa.  Game Analytics – The Basics.  In Game Analytics, pages 13–40. 2013.
  * [20] F Robert Dwyer.  Customer Lifetime Valuation to Support Marketing Decision Making.  Journal of Direct Marketing, 11(4):6–13, 1997.
  * [21] Lukasz Dziurzynski, Edward Wadsworth, and Daniel McCarthy.  BTYD: Implementing Buy ’Til You Die Models, 2014.
  * [22] A. S. C. Ehrenberg.  Repeat Buying.  North-Holland, 1972.
  * [23] Maha Elbayad, Laurent Besacier, and Jakob Verbeek.  Pervasive Attention: 2D Convolutional Neural Networks for Sequence-to-Sequence Prediction.  In Conference on Computational Natural Language Learning, Brussels, Belgium, 2018.
  * [24] O. Etzion, A. Fisher, and S. Wasserkrug.  e-CLV: a modelling approach for customer lifetime evaluation in e-commerce domains, with an application and case study for online auctions.  In IEEE International Conference on e-Technology, e-Commerce and e-Service, 2004. EEE ’04. 2004, pages 149–156. IEEE, 2004.
  * [25] Peter S Fader, Bruce G S Hardie, Ka Lok Lee, and Peter S Fader.  Counting Your Customers the Easy Way : An Alternative to the Pareto / NBD Model.  Marketing Science, 24(2):275–284, 2005.
  * [26] Peter S. Fader, Bruce G.S. Hardie, and Ka Lok Lee.  RFM and CLV: Using Iso-Value Curves for Customer Base Analysis.  Journal of Marketing Research, 42(4):415–430, nov 2005.
  * [27] Nicolas Glady, Bart Baesens, and Christophe Croux.  A Modified Pareto / NBD Approach for Predicting Customer Lifetime Value.  Expert Systems with Applications, 36(2, Part 1):1–26, 2009.
  * [28] Sunil Gupta, Dominique M. Hanssens, Bruce G. S. Hardie, Wiliam Kahn, V. Kumar, Nathaniel Lin, N. Ravishanker, S. Sriram, Nalini Ravishanker S. Sriram, V. Kumar, and Sunil Gupta.  Modeling Customer Lifetime Value.  Journal of Service Research, 9(2):139–155, 2006.
  * [29] Sunil Gupta and Donald R. Lehmann.  Customer Lifetime Value and Firm Valuation.  Journal of Relationship Marketing, 5(2-3):87–110, oct 2006.
  * [30] Michael Haenlein, Andreas M. Kaplan, and Anemone J. Beeser.  A Model to Determine Customer Lifetime Value in a Retail Banking Context.  European Management Journal, 25(3):221–234, jun 2007.
  * [31] Nicolai Hanner and Ruediger Zarnekow.  Purchasing behavior in free to play games: Concepts and empirical validation.  In Proceedings of the Annual Hawaii International Conference on System Sciences, volume 2015-March, pages 3326–3335, 2015.
  * [32] Brent Harrison and David L. Roberts.  Analytics-Driven Dynamic Game Adaption for Player Retention in a 2-Dimensional Adventure Game.  IEEE Conference on Computatonal Intelligence and Games, CIG, (Aiide):23–29, 2013.
  * [33] Sepp Hochreiter and Jürgen Schmidhuber.  Long Short-Term Memory.  Neural Computation, 9(8):1735–1780, nov 1997.
  * [34] Arthur M. Hughes.  Strategic Database Marketing: The Masterplan for Starting and Managing a Profitable Customer-Based Marketing Program.  McGraw-Hill, 2000.
  * [35] Hyunseok Hwang, Taesoo Jung, and Euiho Suh.  An LTV model and customer segmentation based on customer value: a case study on the wireless telecommunication industry.  Expert Systems with Applications, 26(2):181–188, feb 2004.
  * [36] Barbara Jackson.  Winning and keeping industrial customers.  Lexington Books, 1985.
  * [37] Dipak Jain and Siddhartha S. Singh.  Customer lifetime value research in marketing: A review and future directions.  Journal of Interactive Marketing, 16(2):34–46, jan 2002.
  * [38] Norman L. Johnson and Samuel Kotz.  Continuous Univariate Distributions, volume 1.  John Wiley & Sons, 1970.
  * [39] E. L. Kaplan and Paul Meier.  Nonparametric Estimation from Incomplete Observations.  Journal of the American Statistical Association, 1958.
  * [40] Mahboubeh Khajvand, Kiyana Zolfaghar, Sarah Ashoori, and Somayeh Alizadeh.  Estimating customer lifetime value based on RFM analysis of customer purchase behavior: Case study.  Procedia Computer Science, 3(January):57–63, 2011.
  * [41] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, Lake Tahoe, NV, USA, 2012. Neural Information Processing Systems Foundation.
  * [42] V. Kumar.  A Customer Lifetime Value-Based Approach to Marketing in the Multichannel, Multimedia Retailing Environment.  Journal of Interactive Marketing, 24(2):71–85, may 2010.
  * [43] V. Kumar, Girish Ramani, and Timothy Bohling.  Customer lifetime value approaches and best practice applications.  Journal of Interactive Marketing, 18(3):60–72, 2004.
  * [44] Tony Lachowetz, Mark McDonald, William Sutton, and John Clark.  The National Basketball Association: Application of Customer Lifetime Value.  Sport Marketing Quarterly, 10(3):181, 2001.
  * [45] Sascha Lange, Michael Lenz, and Martin Riedmiller.  Case Study : Behavioral Prediction of Future Revenues in Freemium Games.  In Workshop New Challenges in Neural Computation, pages 26–33, 2014\.
  * [46] Tobias Mahlmann, Anders Drachen, Julian Togelius, Alessandro Canossa, and Georgios N. Yannakakis.  Predicting player behavior in Tomb Raider: Underworld.  In IEEE Conference on Computational Intelligence and Games, pages 178–185, Copenhagen, Denmark, 2010. IEEE.
  * [47] Naresh K. Malhotra.  Review of marketing research, 2008.
  * [48] Edward C. Malthouse and Robert C. Blattberg.  Can we predict customer lifetime value?  Journal of Interactive Marketing, 19(1):2–16, jan 2005.
  * [49] Andreas Mueller.  scikit-learn: A set of python modules for machine learning and data mining, 2018.
  * [50] Francis Mulhern.  Customer Profitability Analysis: Measurement, Concentration and Research Directions.  Journal of Interactive Marketing, 13(1):25–40, 1999.
  * [51] David B. Nieborg.  Crushing Candy: The Free-to-Play Game in Its Connective Commodity Form.  Social Media and Society, 1(2), 2015.
  * [52] Héctor Perez Martínez, Maurizio Garbarino, Georgios N. Yannakakis, Hector Perez Martinez, Maurizio Garbarino, and Georgios N. Yannakakis.  Generic physiological features as predictors of player experience.  In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), volume 6974 LNCS, pages 267–276, Memphis, 2011.
  * [53] Africa Perianez, Alain Saas, Anna Guitart, and Colin Magne.  Churn Prediction in Mobile Social Games: Towards a Complete Assessment Using Survival Ensembles.  In 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), pages 564–573, 2016.
  * [54] Phillip E. Pfeifer and Robert L. Carraway.  Modeling customer relationships as Markov chains.  Journal of Interactive Marketing, 14(2):43–55, jan 2000.
  * [55] Phillip E Pfeifer, Mark E Haskins, and Robert M Conroy.  Customer Lifetime Value, Customer Profitability, and the Treatment of Acquisition Spending.  Journal of Managerial Issues, 17(1):25, 2004.
  * [56] Werner J. Reinartz and V. Kumar.  On the Profitability of Long-Life Customers in a Noncontractual Setting: An Empirical Investigation and Implications for Marketing.  Journal of Marketing, 64(4):17–35, oct 2000.
  * [57] Werner J. Reinartz and V. Kumar.  The Impact of Customer Relationship Characteristics on Profitable Lifetime Duration.  Journal of Marketing, 67(1):77–99, jan 2003.
  * [58] Saharon Rosset, Einat Neumann, U R I Eick, and Nurit Vatnik.  Customer Lifetime Value Models for Decision Support.  Data Mining and Knowledge Discovery, 7(3):321–339, 2003.
  * [59] Saharon Rosset, Einat Neumann, Uri Eick, Nurit Vatnik, and Yizhak Idan.  Customer lifetime value modeling and its use for customer retention planning.  Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining - KDD ’02, page 332, 2002.
  * [60] Julian Runge.  The Golden Curve: Determining Player Value in Freemium Apps - https://gameanalytics.com/blog/golden-curve-determining-player-value-freemium-apps.html (Accessed: 2018-11-08), 2014.
  * [61] Julian Runge, Peng Gao, Florent Garcin, and Boi Faltings.  Churn prediction for high-value players in casual social games.  In 2014 IEEE Conference on Computational Intelligence and Games, pages 1–8. IEEE, aug 2014.
  * [62] John Salvatier, Thomas V Wiecki, and Christopher Fonnesbeck.  Probabilistic programming in Python using {PyMC}3.  {PeerJ} Computer Science, 2:e55, apr 2016.
  * [63] David C. Schmittlein, Donald G. Morrison, and Richard Colombo.  Counting Your Customers : Who Are They and What Will They Do Next?  Management Science, 33(1):1–24, 1987.
  * [64] David C Schmittlein and Robert A Peterson.  Customer Base Analysis : An Industrial Purchase Process Application.  Marketing Science, 1994.
  * [65] Eric Benjamin Seufert.  Two Methods for Modeling LTV with a Spreadsheet - https://www.slideshare.net/EricSeufert/ltv-spreadsheet-models-eric-seufert (Accessed: 2018-11-08), 2013.
  * [66] Venkatesh Shankar and Sridhar Balasubramanian.  Mobile Marketing: A Synthesis and Prognosis.  Journal of Interactive Marketing, 23(2):118–129, 2009.
  * [67] Cc Shen and Hm Chuang.  A study on the applications of data mining techniques to enhance customer lifetime value.  WSEAS Transactions on Information Science and Applications, 6(2):319–328, 2009.
  * [68] Ya-Yueh Shih and Chung-Yuan Liu.  A method for customer lifetime value ranking — Combining the analytic hierarchy process and clustering analysis.  Journal of Database Marketing & Customer Strategy Management, 11(2):159–172, dec 2003.
  * [69] Rafet Sifa, Fabian Hadiji, Julian Runge, Anders Drachen, Kristian Kersting, and Christian Bauckhage.  Predicting Purchase Decisions in Mobile Free-to-Play Games.  In Proceedings, The Eleventh AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment (AIIDE-15), pages 79–85, 2015\.
  * [70] Rafet Sifa, Julian Runge, Christian Bauckhage, and Daniel Klapper.  Customer Lifetime Value Prediction in Non-Contractual Freemium Settings: Chasing High-Value Users Using Deep Neural Networks and SMOTE.  In Data, Text and Web Mining for Business Analytics, volume 9, pages 923–932, 2018.
  * [71] Daniel L Silver, Qiang Yang, and Lianghao Li.  Lifelong Machine Learning Systems : Beyond Learning Algorithms.  AAAI Spring Symposium Series, (Solomonoff 1989):49–55, 2013.
  * [72] Siddharth S. Singh, Sharad Borle, and Dipak C. Jain.  A Generalized Framework for Estimating Customer Lifetime Value When Customer Lifetimes are Not Observed.  SSRN Electronic Journal, pages 1–32, 2007.
  * [73] Giorgio Alfredo Spedicato.  Discrete Time Markov Chains with R.  The R Journal, 2017.
  * [74] Rajkumar Venkatesan and V Kumar.  A Customer Lifetime Value Framework for Customer Selection and Resource Allocation Strategy.  Journal of Marketing, 68(4):106–125, oct 2004.
  * [75] Peter C. Verhoef and Bas Donkers.  Predicting Customer Potential Value: an Application in the Insurance Industry.  Technical Report 0, Erasmus Research Institute of Management, 2001.
  * [76] Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol.  Extracting and composing robust features with denoising autoencoders.  In Proceedings of the 25th international conference on Machine learning - ICML ’08, number April, pages 1096–1103, New York, New York, USA, 2008. ACM Press.
  * [77] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson.  How transferable are features in deep neural networks?  In Advances in Neural Information Processing Systems, volume 27, pages 1–9, 2014.