# MEASURES OF RELEVANCE TO THE SUCCESS OF STREAMING PLATFORMS††thanks: Juan
Carlos Gonçalves-Dosantos acknowledges the grant PID2021-12403030NB-C31 funded
by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe/EU”. He
also acknowledges financial support from Xunta de Galicia (Grupos de
Referencia Competitiva ED431C-2020/14) and from CITIC that is supported by
Xunta de Galicia (convenio de colaboración entre la Consellería de Cultura,
Educación, Formación Profesional e Universidades y las universidades gallegas
para el refuerzo de los centros de investigaciń del Sistema Universitario de
Galicia (CIGUS)). Ricardo Martínez acknowledges the R&D&I project grant
PID2020-114309GB-I00 funded by MCIN AEI/10.13039/501100011033 and by “ERDF A
way of making Europe/EU”, and he also acknowledges financial support from
Junta de Andalucía under projects FEDER UGR-A-SEJ-14-UGR20 and Grupos PAIDI
SEJ660. Joaquín Sánchez-Soriano acknowledges financial support from the
Generalitat Valenciana under project PROMETEO/2021/063.

Juan Carlos Gonçalves-Dosantos
Universidad Miguel Hernández de Elche  Ricardo Martínez
Universidad de Granada  Joaquín Sánchez-Soriano
Universidad Miguel Hernández de Elche

###### Abstract

Digital streaming platforms, including Twitch, Spotify, Netflix, Disney, and
Kindle, have emerged as one of the main sources of entertainment with
significant growth potential. Many of these platforms distribute royalties
among streamers, artists, producers, or writers based on their impact. In this
paper, we measure the relevance of each of these contributors to the overall
success of the platform, which is information that can play a key role in
revenue allocation. We perform an axiomatic analysis to provide normative
foundations for three relevance metrics: the uniform, the proportional, and
the subscriber-proportional indicators. The last two indicators implement the
so-called pro-rata and user-centric models, which are extensively applied to
distribute revenues in the music streaming market. The axioms we propose
formalize different principles of fairness, stability, and non-manipulability,
and are tailor-made for the streaming context. We complete our analysis with a
case study that measures the influence of the 19 most-followed streamers
worldwide on the Twitch platform.

Keywords: Measure, relevance, proportionality, streaming, axiom

##  1 Introduction

In 2022, Netflix, Amazon Prime Video, Disney+, and HBO MAX – four of the most
successful streaming platforms – reported a joint annual revenue of $103,5
billion. These companies operate so-called over-the-top broadcasting services
by offering to their customers a media library of movies, series, and
documentaries. Viewers must pay a monthly fee to obtain unlimited access to
the catalog. As Table 1 shows, the subscription prices for these platforms are
similar, but the number of subscribers and earned revenue differ
significantly.

Company | Subscription price | Subscribers | Revenue
---|---|---|---
Netflix | $7 | 231 million | $31.6 billion
Amazon Prime Video | $9 | 200 million | $25.2 billion
Disney+ | $8 | 164 million | $7.4 billion
HBO+ | $10 | 95 million | $39.3 billion
Table 1: Fees, subscribers, and revenues of the main streaming platforms.

Two elements are crucial for determining the success (or revenue) of a
platform: the fee and the catalog. Some companies have only two or three shows
each year that subscribers are interested in, but those shows may be massive
hits. Other platforms choose a different model, offering a wider catalog based
on the variety and quantity of content. Be that as it may, it is clear that
not all titles offered by these companies are equally in demand, and the
available titles may be decisive for consumers to decide which platform they
subscribe to. Therefore, if one platform offers content that is more relevant
than others, a worthwhile question to ask is: what is the impact of each
single show on the overall success of a platform? This question can be
answered by applying the theoretical model presented in this paper.

The previous setting is not unique to over-the-top subscription video-on-
demand providers; it can be extended to many similar situations, in which
selling products in a package may be more profitable than selling them
independently (see Adams and Yellen (1976)). Spotify, Twitch, YouTube, and
other web-based services are only a few examples. In all these, subscribers
pay a subscription fee to obtain unlimited access to a set of services, and it
is only natural to wonder what is the individual contribution of each of those
services to the success of the whole package.

In our model, a _platform_ is described by four elements: the set of
_services_ the platform provides (TV shows, artists, streamers, books, etc.),
the set of _subscribers_ who have unlimited access to those services (viewers,
users, readers, etc.), the _subscription price_ paid by each subscriber, and
the _consumption matrix_ that indicates the quantity of each service consumed
by each subscriber.111Our setting is general enough to allow the subscription
price to vary across subscribers. Fixing a common subscription fee is
possible, but it is just a particular case in our model. An _indicator_ is a
measure that determines the relevance of each single service to the success of
the platform, which is the total revenue generated from selling subscriptions.

In this study, we introduce four natural indicators to determine the share of
the platform’s success due to each service. The _uniform indicator_ stipulates
that all services are equally relevant. The _subscriber-uniform indicator_
states that the relevance of each individual subscriber is divided uniformly
among the services that the subscriber consumes. The _proportional indicator_
assigns, to each service, a relevance proportional to its aggregate
consumption. The last proposal, the _subscriber-proportional indicator_ , also
considers a proportional perspective, but applied to each single subscriber.
More precisely, the success of each single subscriber is proportionally
assigned among the services that the subscriber consumes. The relevance of
each service is the lump sum of those proportions across subscribers. In the
music streaming industry, the two most relevant remuneration methods are the
_pro-rate_ and the _user-centric_ mechanisms. The former is based on an
aggregate approach, where the overall revenue from all music subscriptions is
distributed among the artists proportionally to their total number of streams.
In the latter, the calculation is based on the listening habits of each
individual subscriber; the subscription fee she pays is proportionally
distributed, but only among the artists whose music this subscriber consumes.
The _proportional indicator_ and the _subscriber-proportional indicator_
implement the _pro-rate_ and _user-centric_ principles, respectively.

As in Manshadi et al. (2023) and Singal et al. (2022), we follow the axiomatic
methodology to investigate the existence of indicators that satisfy
combinations of properties (called _axioms_) that are suitable for our
setting. The axioms that we propose formalize different principles related to
fairness, stability, and absence of manipulability that an indicator of
relevance should satisfy. The first property is _efficiency_ , which requires
that the sum of the relevance of all services is equal to the total relevance
of the platform. Regarding fairness, _symmetry_ requires that if two services
are equal (i.e., they are equally consumed by all the subscribers), then they
must be equally relevant to the success of the platform. _Strong symmetry_
requires that if two services are consumed by the same subscribers, then both
must be equally relevant. _Nullity service_ states that services not consumed
have no relevance to the platform. With regard to stability, _homogeneity_
states that the relevance must not be affected by the units (thousands or
millions, for example) in which the consumption matrix is expressed.
_Consumption sensitivity_ requires that small mistakes in the measurement of
the consumption do not drastically alter the significance of each service.
_Composition_ states that when data is combined from two subgroups, it is
possible to determine the indicator of the total group by a suitable
composition of the subgroups’ indicators. _Sharing proofness_ and _non-
manipulability_ belong to the third type of principles. The former states that
if subscribers share their subscriptions with friends and relatives, then the
relevance of the services remains unaltered. The latter axiom requires that
the indicator should not be able to be manipulated by merging or splitting
services.

Regarding our findings, we obtain some characterizations of the aforementioned
indicators. More precisely, if we require both _composition_ and _non-
manipulability_ to be satisfied, we must apply the _subscriber-proportional
indicator_. Theorem 2 shows that the _proportional indicator_ is the unique
indicator that fulfills _sharing proofness_ and _non-manipulability_ together.
In Theorem 3, we prove that the combination of _symmetry_ , _homogeneity_ ,
and _consumption sensitivity_ unambiguously yield to the _uniform indicator_.
Finally, Theorem 4 shows that if _strong symmetry_ , _nullity_ , and
_composition_ are demanded, then the _subscriber-uniform indicator_ should be
implemented. As in our setting, the _proportional_ and the _subscriber-
proportional_ reflect the _pro-rate_ and _user-centric_ schemes. Theorems 1
and 2 also reveal characterizations of these two mechanisms. In other words,
based on several principles of fairness, stability, and non-manipulability, we
provide normative foundations for the two most-used revenue allocation methods
in the music streaming industry. On the other hand, all referred indicators
are examined from a game-theoretical perspective. Specifically, we demonstrate
that these indicators coincide with the Shapley value (Shapley (1953)), which
is a prominent solution concept in game theory (Roth (1988), Algaba et al.
(2019)), of some cooperative games that can be defined for the class of
problems analyzed in this paper, which reflect some aspect of how relevance is
obtained. Likewise, an analysis of the indicators and properties is performed
in the particular context of streaming platforms. Next, in the context of
streaming, different aspects related to the information used, the structure of
the indicators, and the properties they satisfy are studied. Moreover, the
impact on relevance measures is analyzed when changes occur in the
visualization matrix and, in this sense, which indicators streamers may
prefer. Likewise, we also analyze the measures of relevance that may be the
most interesting for a platform according to its own business.

We conclude our paper with an illustrative application of the indicators that
we characterize. In particular, we measure the relevance of each of the top 19
most-watched Twitch streamers. This information could play a key role in
determining the distribution of revenues. We highlight and explain the
differences and implications of the four indicators that we analyze in this
work. Based on the obtained results, we conclude that the mere use of either
the number of viewers or exclusive viewers, which is the usual standard in
this industry, is not the most effective approach for determining the impact
of each streamer on the platform.

###  1.1 Related literature

Our work provides a novel approach to determining and valuing the impact of
each single item, service, or contributor to the collective success of a
platform or bundling product. Other papers have also addressed the question of
how to isolate and measure the merit of single agents in the collective
success of online frameworks. Singal et al. (2022) provide an axiomatic
justification of the _counterfactual adjusted Shapley value_ , which measures
the contribution of individual advertiser actions (emails, display ads, search
ads, etc.) to eventual customer acquisition. The authors show that this
Shapley-based metric coincides with an _adjusted unique-uniform_ attribution
scheme. In contrast, the indicators that we characterize in this work are
based on proportionality rather than uniformity. Proportional schemes are
usually perceived by society as fair allocation methods and, therefore, as
less controversial.

The discussion around how to divide the revenue obtained by subscriptions to
music streaming platforms among artists has become particularly relevant in
recent years. In this context, the _pro-rata_ and the _user-centric_ schemes
have emerged as the most prominent and applied methods (see, for example,
Dimont (2017), Page and Safir (2018b), Page and Safir (2018a), Jari (2018),
and Meyn et al. (2023)). Alaei et al. (2022) study the strategic implications
of these two schemes in a model with a two-sided streaming service platform
that generates revenues by charging users a subscription fee for unlimited
access to the content. We contribute to this discussion by providing normative
foundations of the _pro-rata_ and _user-centric_ rules.

López-Navarrete et al. (2019) study the problem of how to allocate the
revenues generated in a Smart TV ecosystem between the provider and the
content producers. They characterize the core of the game associated with the
situation and provide simple formulas for their Shapley and Tijs values, which
belong to the core of the game. López-Navarrete et al. (2023) study how to
allocate the revenues generated in a video platform such as YouTube,
considering the navigation of the users in the platform. For this, they define
dynamic games associated with the problem and provide several allocation
schemes based on the structure of the Shapley value. Our approach differs from
that of López-Navarrete et al. (2019, 2023) since they approach the matter
from a game-theoretical perspective and, in our case, from an axiomatic
perspective. Bergantiños and Moreno-Ternero (2023) also study the problem of
how to share revenue in the streaming industry, with a particular focus on
music streaming services such as Spotify. They develop and characterize
allocation mechanisms based on the _pro-rata_ and the _user-centric_
principles, among others. The axioms we propose in this paper differ from
those presented in Bergantiños and Moreno-Ternero (2023). In particular, the
results obtained by these authors require properties such as _proportionality
on streams_ , _equal importance of similar users_ , or _non-manipulability by
leaving the platform_ , which we do not explore in this work. The definition
of indicator is also different, as we require _efficiency_ while they do not.

A related paper to this, but from the perspective of claim problems and
generalizations, is Ju et al. (2007). The authors describe a general resource
allocation problem where an infinitely divisible good must be distributed
among a set of entities. Each entity is characterized by a vector, with each
value reflecting the entity’s relevance to a specific set of issues. In
contrast to the problem discussed in Ju et al. (2007), where one infinitely
divisible good is allocated, in our model we consider a vector of infinitely
divisible goods, one for each subscriber, the issues in their problem. Where
it depends on the subscription fee of each individual subscriber. In addition
to the proportional rules explored in Ju et al. (2007), our approach allows us
to consider rules that are applied to each subscriber independently. Based on
the normative framework proposed by Ju et al. (2007), which encompasses a wide
range of resource allocation problems, we will introduce several axioms that
are relevant to the context at hand.

Other authors have addressed similar questions from different perspectives in
the literature on allocation and attribution problems. Ginsburgh and Zang
(2003), Bergantiños and Moreno-Ternero (2015) and Martínez and Sánchez-Soriano
(2023) analyze the so-called museum pass problem to distribute the revenue
obtained by selling museum passes that allow entrance into a set of museums.
Martínez and Sánchez-Soriano (2021) develop a theoretical model to measure the
relative relevance of different pathologies to the lethality of a disease in a
society. Additionally, in the context of biology applications, Moretti et al.
(2007), Albino et al. (2008), and Lucchetti et al. (2010) propose indices to
identify the genes that are useful for the diagnosis and prognosis of specific
diseases and cancers, while Martínez and Moreno-Ternero (2022) report that the
weighted averages of the _incidence rate_ , _morbidity rate_ , and _mortality
rate_ are the appropriate methods to evaluate the impact of pandemics. Brander
et al. (2011) offer an account of the importance of the attribution problem in
climate change, and Burger et al. (2020) conduct an in-depth review of the
attribution problem in the context of climate change, both from a technical
perspective and its legal and policy applications. Manshadi et al. (2023)
analyze how governmental and non-profit organizations must face the task of
obtaining equitable and efficient rationing of a social good among individuals
whose needs (as was the case in the recent pandemic) emerge sequentially and
are possibly correlated. Finally, Martínez et al. (2022) characterize several
methods to evaluate the multidimensional proposals of bidders in public
procurement tenders. However, our approach differs from the above mentioned
papers in the problem we study, in the structure of the elements that define
the problem and in several of the properties used to characterize the
solutions presented.

The rest of the paper is organized as follows. In Section 2, we present the
model and introduce four indicators to measure the relevance of services in a
platform. In Section 3, we propose and formalize the normative framework we
consider. We present the normative analysis of the indicators in Section 4,
and we analyze the indicators in the streaming context in Section 5. In
Section 6, we provide an application of our theoretical model. Concluding
remarks are provided in Section 7.

##  2 Mathematical model and relevance indicators

Let $\mathbb{N}$ represent the set of natural numbers, and let $\mathcal{N}$
be the set of all finite and non-empty subsets of $\mathbb{N}$. A platform is
described by a 4-tuple $(N,S,p,C)$, where $N=\\{1,\ldots,|N|\\}\in\mathcal{N}$
($|N|\geq 3$)222Just as Ju et al. (2007) requires this condition for its
results, we also require it for the results presented here. However, it is not
a limiting condition, as the number of services is typically greater than or
equal to 3 on most platforms. is the set of services provided by the platform,
$S=\\{1,\ldots,|S|\\}\in\mathcal{N}$ is the set of subscribers who have
unlimited access to the services in $N$,
$p=(p_{1},\ldots,p_{|S|})\in\mathbb{R}^{|S|}_{++}$ represent the subscription
prices paid by each subscriber, and $C$ is the consumption matrix, each of
whose entries $C_{is}\in\mathbb{R}_{+}$ indicates the quantity of a service
$i$ consumed by subscriber $s$. The success of the platform is the total
revenue generated from selling subscriptions, $\sigma=\|p\|$.333The notation
$\|\cdot\|$ refers to the sum of all absolute values of all entries of a
vector or matrix. We denote by $\mathcal{D}$ the set of all platforms.

We denote by $C_{i\cdot}$ the $i$-th row of $C$, which represents the
consumption of service $i$. We also denote by $C_{\cdot s}$ the $s$-th column
of $C$, which represents the consumption of subscriber $s$. Therefore,
$\|C_{i\cdot}\|=\sum_{s\in S}C_{is}$ and $\|C_{\cdot s}\|=\sum_{i\in N}C_{is}$
are the total consumption of service $i\in N$ and subscriber $s\in S$,
respectively. We restrict ourselves to consumption matrices such that
$\|C_{\cdot s}\|>0$ for all $s\in S$, that is, we assume that every subscriber
has paid the subscription fee with the willingness to consume part of the
content that the platform offers. Given a set of services $N^{\prime}\subseteq
N$, or a set of subscribers $S^{\prime}\subseteq S$, the matrices resulting
from removing the rows in $N^{\prime}$ and the columns in $S^{\prime}$ are
denoted by $C_{N\backslash N^{\prime}}$ and $C_{S\backslash S^{\prime}}$,
respectively.

The relevance of each service to the success of the platform is measured using
an indicator $R$, which is a mapping
$R:\mathcal{D}\longrightarrow\mathbb{R}^{|N|}_{+}$ such that

$\sum_{i\in N}R_{i}(N,S,p,C)=\sigma.$

For each $i\in N$, $R_{i}(N,S,p,C)$ indicates the share of the success of the
platform that is due to $i$.

Now, we present some examples of possible indicators. The first one is
straightforward; it attributes equal responsibility to all services.

Uniform indicator. For each $(N,S,p,C)\in\mathcal{D}$ and each $i\in N$,

$R^{U}_{i}(N,S,p,C)=\frac{\sigma}{|N|}.$

The second indicator uniformly distributes the success of each subscriber
among the services that have been consumed, regardless of the consumption
intensity. If subscriber $s\in S$ has not used the service supplied by $i\in
N$, that is, if $C_{is}=0$, then this has no influence on subscriber $s$.

Subscriber-uniform indicator. For each $(N,S,p,C)\in\mathcal{D}$ and each
$i\in N$,

$R^{SU}_{i}(N,S,p,C)=\sum_{s\in S,i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s},$

where $N_{s}=\left\\{j\in N:C_{js}\neq 0\right\\}$.

The next indicator is also natural since it divides the success proportionally
to the consumption of each service. It implements the so-called _pro-rata
rule_ , which is one of the two most prominent remuneration methods in the
literature on the music streaming industry.

Proportional indicator. For each $(N,S,p,C)\in\mathcal{D}$ and each $i\in N$,

$R^{P}_{i}(N,S,p,C)=\frac{\|C_{i\cdot}\|}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma.$

While the previous indicator distributes the success as a whole, the following
measure is based on determining the role of each service in the success of
attracting the interest of each single subscriber. Consider a subscriber $s\in
S$ who has paid the subscription fee $p_{s}$. At this individual level, the
success of the platform is simply $p_{s}$. Now, divide $p_{s}$ among all the
services consumed by $s$ in proportion to the consumption; proceed in the same
way for all subscribers and aggregate across them. This indicator mimics the
principle behind the other focal remuneration method for music streaming:
namely, the _user-centric rule_.

Subscriber-proportional indicator. For each $(N,S,p,C)\in\mathcal{D}$ and each
$i\in N$,

$R^{SP}_{i}(N,S,p,C)=\sum_{s\in S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}.$

In the following example, the relevance of each service for each of the
indicators defined in this section is illustrated.

######  Example 1.

Consider the platform where $N=\\{1,2,3\\}$, $S=\\{1,2,3,4,5,6\\}$,
$p=\left(2,4,\frac{5}{2},2,1,\frac{7}{2}\right)$, and $C$ is given by

$\left(\begin{array}[]{cccccc}0&5&0&1&2&3\\\ 1&1&2&3&6&0\\\ 0&0&0&0&0&0\\\
\end{array}\right)$

Notice that, in this case, the success is $\sigma=15$.The next table shows the
relevance of each service according to the three previous indicators:

| Services
---|---
Indicator | 1 | 2 | 3
$R^{U}$ | 5 | 5 | 5
$R^{SU}$ | 7 | 8 | 0
$R^{P}$ | $\frac{105}{16}$ | $\frac{135}{16}$ | 0
$R^{SP}$ | $\frac{13}{2}$ | $\frac{17}{2}$ | 0

##  3 Normative framework

The axiomatic approach to analyzing allocation mechanisms is one of the common
approaches for determining what is most appropriate, depending on the
properties that are considered most relevant to the problem or context in
question. In this paper, we consider some basic axioms of the literature taken
from the normative framework provided in Ju et al. (2007) for a wide range of
resource allocation problems. Some of these axioms have been conveniently
adapted to the context of the problem studied in this paper, while others are
new. The first axiom requires that the sum of the success of all services is
equal to the total success of the platform. We impose this axiom in the very
definition of indicator, as follows:

Efficiency. For each $(N,S,p,C)$, $\sum_{i\in N}R_{i}(N,S,p,C)=\sigma$.

The following two axioms are minimum requirements of impartiality when
considering different levels of information. They both require symmetric
services to be treated symmetrically, but they differ on when two services
should be considered as such. Thus, the first property states that two
services are symmetric if each subscriber consumes one of them, as long as she
does so with the other service, regardless of the consumption intensity.
_Strong symmetry_ requires that two symmetric services have the same relevance
to the success of the platform.

Strong Symmetry. For each $(N,S,p,C)$ and each pair $\\{i,j\\}\subseteq N$, if
$C_{is}>0\Leftrightarrow C_{js}>0$ for all $s\in S$, then
$R_{i}(N,S,p,C)=R_{j}(N,S,p,C)$.

As an alternative to the previous formulation, the next axiom considers that
two services are symmetric if they are equally consumed by all subscribers. In
such a case, both services must be equally relevant.

Symmetry. For each $(N,S,p,C)$ and each pair $\\{i,j\\}\subseteq N$, if
$C_{is}=C_{js}$ for all $s\in S$, then $R_{i}(N,S,p,C)=R_{j}(N,S,p,C)$.

Obviously, _strong symmetry_ implies _symmetry_. The distinction between these
two properties is particularly relevant in this setting, since some indicators
may satisfy the milder axiom of not the stronger one. This is a peculiarity of
our model. In alternative contexts (for example, in museum pass problems
(Ginsburgh and Zang (2003), Martínez and Sánchez-Soriano (2023)) or in risk
factor problems in epidemiology (Martínez and Sánchez-Soriano (2021), Martínez
and Moreno-Ternero (2022)), among others.) _strong symmetry_ becomes the same
as the property of _symmetry_.

The following axiom states that if a service is not consumed by any
subscriber, then it has no relevance within the platform, which means that its
relevance is equal to zero:

Nullity. For each $(N,S,p,C)\in\mathcal{D}$, and each $i\in N$ such that
$C_{is}=0$ for all $s\in S$, then $R_{i}(N,S,p,C)=0$.

The next axiom states that changing units or magnitudes does not impact the
relevance of services in determining their success on the platform. In simpler
terms, multiplying the consumption matrix by a positive number does not affect
the relevance of the services.

Homogeneity. For each $(N,S,p,C)$ and each $\lambda\in\mathbb{R}_{++}$,
$R(N,S,p,\lambda C)=R(N,S,p,C)$.

Suppose that there are two disjoint groups of subscribers $S$ and $S^{\prime}$
(e.g., subscribers based in two regions of a country). Now consider a larger
society resulting from combining $S$ and $S^{\prime}$. The question is how to
recalculate the indicator for $S\cup S^{\prime}$ from the indicators for $S$
and $S^{\prime}$. _Composition_ states that the relevance to the success of
each service in the large population, $S\cup S^{\prime}$, is the sum of the
relevance in $S$ and $S^{\prime}$.444This property is related to the
additivity or decomposability into parts of a problem, which is also common in
the resource allocation literature (see, for example, Ju et al. (2007);
Bergantiños and Moreno-Ternero (2015); Algaba et al. (2019b); Martínez and
Moreno-Ternero (2022); López-Navarrete et al. (2023); Martínez and Sánchez-
Soriano (2021, 2023)).

Composition. For each
$(N,S,p,C),(N,S^{\prime},p^{\prime},C^{\prime})\in\mathcal{D}$ such that
$S\cap S^{\prime}=\emptyset$,

$R(N,S\cup S^{\prime},p\oplus p^{\prime},C\oplus
C^{\prime})=R(N,S,p,C)+R(N,S^{\prime},p^{\prime},C^{\prime}),$

where $C\oplus C^{\prime}$ is the matrix resulting from concatenating $C$ and
$C^{\prime}$ by rows, and $p\oplus p^{\prime}$ is the vector resulting from
concatenating $p$ and $p^{\prime}$.

The upcoming axiom, which we will term _consumption sensitivity_ states that
minor alterations in the consumption matrix of the problem do not result in
significant changes in the relevance of the services.

Consumption sensitivity. For each $\delta\geq 0$, if
$\|C-C^{\prime}\|<\delta$, then
$\|R(N,S,p,C)-R(N,S,p,C^{\prime})\|<\varepsilon(\delta)$, so that when
$\delta$ goes to $0$, then $\varepsilon(\delta)$ also goes to $0$.

This property implies that if we have two successions of problems
$\\{(N,S,p,C^{n})\\}$ and $\\{(N,S,p,C^{\prime n})\\}$, such that for every
$\delta>0$, there exists $n_{0}$ such that $\forall n\geq n_{0}$,
$\|C^{n}-C^{\prime n}\|<\delta$, then $\|R(N,S,p,C^{n})-R(N,S,p,C^{\prime
n})\|<\varepsilon(\delta)$. Therefore, as $\|C^{n}-C^{\prime n}\|$ goes to
$0$, $\|R(N,S,p,C^{n})-R(N,S,p,C^{\prime n})\|$ also goes to $0$. Notice that
consumption sensitivity differs from the _continuity_ requirement that is used
in the resource allocation literature. In fact, the proportional indicator,
for example, satisfies continuity but violates consumption sensitivity (see
Appendix A).

Furthermore, this property should not be confused with the continuity property
that is usually used in the resource allocation literature. The continuity
property states that if there is a sequence of problems convergent to a given
one, then the associated sequence of allocations converges to the allocation
of the limit problem.

It is common for subscribers to streaming platforms to share their
subscriptions among friends and relatives. They split the subscription price
so that each one can access the platforms services. Sharing proofness requires
the indicator not to be altered by this type of behavior.

Sharing proofness. For each $(N,S,p,C)\in\mathcal{D}$, each non-empty
$S^{\prime}\subseteq S$, and each $s\in S^{\prime}$, where
$p^{\prime}_{s}=\sum_{t\in S^{\prime}}p_{t}$ and $C^{\prime}_{is}=\sum_{t\in
S^{\prime}}C_{it}$ for all $i\in N$, then

$R(N,S,p,C)=R\left(N,\\{s\\}\cup S\backslash
S^{\prime},(p^{\prime}_{s},p_{S\backslash S^{\prime}}),(C^{\prime}_{\cdot
s},C_{S\backslash S^{\prime}})\right),$

where $(C^{\prime}_{\cdot s},C_{S\backslash S^{\prime}})$ is the matrix whose
entries are equal to $(C^{\prime}_{\cdot s},C_{S\backslash
S^{\prime}})_{is}=C^{\prime}_{is}$ and $(C^{\prime}_{\cdot s},C_{S\backslash
S^{\prime}})_{it}=C_{it}$ for all $t\in S\backslash S^{\prime}$, all $i\in N$
and $s\in S^{\prime}$.

Our next axiom states that the relevance to success cannot be manipulated by
merging or splitting services. In other words, this implies that no group of
services can enhance its success on the platform by consolidating their
consumption, and similarly no individual service can increase its success by
creating additional services and dividing its consumption among them. This
axiom was introduced by O’Neill (1982) in the context of bankruptcy problems
and recurrently used in resource allocation problems. For instance, see Ju et
al. (2007).

Non-manipulability. For each $(N,S,p,C)\in\mathcal{D}$, each non-empty
$N^{\prime}\subseteq N$, and each $i\in N^{\prime}$, where
$C^{\prime}_{is}=\sum_{j\in N^{\prime}}C_{js}$ for all $s\in S$, then

$R_{i}\left(\\{i\\}\cup N\backslash
N^{\prime},S,p,(C^{\prime}_{i\cdot},C_{N\backslash
N^{\prime}})\right)=\sum_{j\in N^{\prime}}R_{j}(N,S,p,C),$

where $(C^{\prime}_{i\cdot},C_{N\backslash N^{\prime}})$ is the matrix whose
entries are equal to $(C^{\prime}_{i\cdot},C_{N\backslash
N^{\prime}})_{is}=C^{\prime}_{is}$, and $(C^{\prime}_{i\cdot},C_{N\backslash
N^{\prime}})_{js}=C_{js}$ for all $s\in S$, all $j\in N\backslash N^{\prime}$,
and $i\in N^{\prime}$.

Another way to analyze allocation mechanisms is from the perspective of game
theory. To do so, associated with each problem, a game is defined that
describes in some way what you want to distribute, in this case, the relevance
to platform success. Once we have the game, we analyze whether the allocation
mechanism coincides with any known solution from game theory, usually the
Shapley value, as this is perhaps one of the most relevant (Shapley (1953),
Roth (1988), Algaba et al. (2019)). For the same problem, there may be
different ways of defining a cooperative game associated with it, depending on
the approach or idea you have about how a coalition generates value, in our
case relevance. Next, various cooperative games are introduced and an analysis
is performed to identify which of the proposed indicators – if any – the
Shapley value of these games coincides with. Before analyzing the possible
associated cooperative games, the mathematical expression of the Shapley value
(Shapley (1953)) is presented below for completeness. A cooperative game is
defined by a pair $(N,v)$, where $N$ is a finite set of players and $v$ is a
function from the set of all possible subsets of $N$ to $\mathbb{R}$, such
that $v(\varnothing)=0$, and the Shapley value of the game is given by

$Sh_{i}(N,v)=\sum_{S\subset
N\backslash\\{i\\}}\frac{|S|\cdot(|N|-|S|-1)}{|N|!}\left(v(S\cup\\{i\\})-v(S)\right),\quad
i\in N.$

Given a problem $(N,S,p,C)\in\mathcal{D}$, a cooperative game $(N,v)$ can be
associated with it, where the set of players $N$ is equal to the set of
services, and $v$ is the characteristic function that measures the relevance
of coalition $S$ on the success of the platform. In all games that we
introduce, the set of players remains the same; the only difference among them
is the characteristic function. First, if we consider a kind of “essentiality”
property of the services, which means that if a service leaves the platform,
all subscribers who view on that service will cease to subscribe, then the
characteristic function of the associated cooperative game is defined as
$v^{SU}(R)=\displaystyle\sum_{s\in N_{R}}p_{s}$ where $N_{R}=\\{s\in
S:C_{js}=0\>\forall j\in N\backslash R\\}$ for all $R\subseteq N$. Following
Ginsburgh and Zang (2003), it is easy to check that the Shapley value of this
game coincides with the _subscriber-uniform indicator_.

From a different perspective, if we consider a kind of “inessentiality”
property of the services, which means that all subscribers are subscribed to
the platform regardless of the services and the time they view them, then we
can define the characteristic function of the associated cooperative game as
$v^{U}(R)=\sigma$ for all $R\subseteq N$. In this new game, since the game
remains constant for all coalitions, it is easy to observe that the Shapley
value of this game coincides with the _uniform indicator_.

The two previous games are extreme cases in terms of considering the
“essentiality” of a service in the success of the platform. However, they can
also be considered intermediate cases when the subscription prices paid by
consumers are taken as divisible. Thus, we consider the other three
cooperative games, which take into account both the consumption matrix and the
fact that the subscription prices are divisible. In the first one, we consider
that the subscription prices can be divided among the services that
subscribers consume, regardless of the time consumed for each service. In this
case, the characteristic function $v$ is defined as
$v^{SU}(R)=\displaystyle\sum_{s\in S}\frac{|supp(C_{Rs})|}{|supp(C_{\cdot
s})|}p_{s}$ for all $R\subseteq N$, where $C_{Rs}=\left(C_{is}\right)_{i\in
R}$ and for any $x\in\mathbb{R}^{n}$, $n\in\mathbb{N}_{++}$,
$supp(x)=\\{x_{i}:x_{i}>0\\}$. It is easy to see that for all $R\subseteq N$
$v^{SU}(R)=\displaystyle\sum_{i\in R}v^{SU}(i)$, the Shapley value coincides
with _subscriber-uniform indicator_. Finally, when – in addition – the time
consumed by the subscribers of each service is taken into account, then two
other games can be defined. These two cooperative games can be seen as more ad
hoc, but they still enable us to obtain our indicators using the Shapley
value. The characteristic function of the first of them is defined as
$v^{SP}(R)=\displaystyle\sum_{i\in R}\sum_{s\in S}\frac{C_{is}}{C_{\cdot
s}}p_{s}$ for all $R\subseteq N$; in other words, the relative weight of the
coalition $R$ in the price paid by each subscriber. It is easy to see that for
all $R\subseteq N$ $v^{SP}(R)=\displaystyle\sum_{i\in R}v^{SP}(i)$, and then
the Shapley value coincides with the _subscriber-proportional indicator_. The
characteristic function of the second and last game is defined as
$v^{P}(R)=\displaystyle\sum_{i\in R}\frac{\|C_{i\cdot}\|}{\sum{j\in
N}\|C_{j\cdot}\|}\sigma$ for all $R\subseteq N$; in other words, the relative
weight of the coalition in the overall success. Once again, it is easy to see
that for all $R\subseteq N$ $v^{P}(R)=\displaystyle\sum_{i\in R}v^{P}(i)$, and
then the Shapley value coincides with the _proportional indicator_.

Therefore, all the indicators that are introduced coincide with the Shapley
value of some cooperative game whose characteristic function measures how each
subset of services contributes to the success of the platform.

##  4 Normative analysis

In this section, we present our main results, which are characterizations of
the four indicators presented in Section 2. These characterizations are given
in terms of the axioms introduced in the previous section and, in some way,
show what properties are associated with each of the indicators proposed to
measure the relevance of the services to platform success. The next theorem
states that the _subscriber-proportional indicator_ is the unique indicator
that fulfills both _composition_ and _non-manipulability_. Based on the
connection between the _subscriber-proportional indicator_ and the _user-
centric principle_ , Theorem 1 also provides instrumental and behavioral
justifications for the application of the latter, given that it provides
insights into its calculation and the prevention of strategic behaviors to
manipulate the measure of the success.

######  Theorem 1.

An indicator satisfies composition and non-manipulability if and only if it is
the subscriber-proportional indicator.

###### Proof.

We start by showing that the subscriber-proportional indicator satisfies the
axioms in the statement.

  * •

Composition. Let
$(N,S,p,C),(N,S^{\prime},p^{\prime},C^{\prime})\in\mathcal{D}$ such that
$S\cap S^{\prime}=\emptyset$. Let $i\in N$. We have that

$\displaystyle R_{i}^{SP}(N,S\cup S^{\prime},p\oplus p^{\prime},C\oplus
C^{\prime})$ $\displaystyle=\sum_{s\in S\cup S^{\prime}}\dfrac{(C\oplus
C^{\prime})_{is}}{\|(C\oplus C^{\prime})_{\cdot s}\|}(p\oplus p^{\prime})_{s}$
$\displaystyle=\sum_{s\in S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}+\sum_{s\in
S^{\prime}}\dfrac{C^{\prime}_{is}}{\|C^{\prime}_{\cdot s}\|}p^{\prime}_{s}$
$\displaystyle=R_{i}^{SP}(N,S,p,C)+R_{i}^{SP}(N,S^{\prime},p^{\prime},C^{\prime}).$

  * •

Non-manipulability. Let $(N,S,p,C)\in\mathcal{D}$, and let
$N^{\prime}\subseteq N$ and $i\in N^{\prime}$ such that
$C^{\prime}_{is}=\sum_{j\in N^{\prime}}C_{js}$ for all $s\in S$. Then

$\displaystyle R^{SP}_{i}\left(\\{i\\}\cup N\backslash
N^{\prime},S,p,(C^{\prime}_{i\cdot},C_{N\backslash N^{\prime}})\right)$
$\displaystyle=\sum_{s\in S}\dfrac{C^{\prime}_{is}}{C^{\prime}_{is}+\sum_{k\in
N\backslash N^{\prime}}C_{ks}}p_{s}$ $\displaystyle=\sum_{s\in
S}\dfrac{\sum_{j\in N^{\prime}}C_{js}}{\sum_{j\in N^{\prime}}C_{js}+\sum_{k\in
N\backslash N^{\prime}}C_{ks}}p_{s}$ $\displaystyle=\sum_{s\in
S}\dfrac{\sum_{j\in N^{\prime}}C_{js}}{\sum_{j\in N}C_{js}}p_{s}$
$\displaystyle=\sum_{j\in N^{\prime}}\sum_{s\in S}\dfrac{C_{js}}{\|C_{\cdot
s\|}}p_{s}$ $\displaystyle=\sum_{j\in N^{\prime}}R^{SP}_{j}(N,S,p,C).$

Now, we prove the converse. Let $R$ be an indicator that fulfills composition
and non-manipulability. Let $(N,S,p,C)\in\mathcal{D}$. Suppose that $S$ is a
singleton (i.e., there is only one subscriber $S=\\{s\\}$), and let $p^{(s)}$
and $C^{(s)}=\left(C_{1s}^{(s)},\ldots,C_{|N|s}^{(s)}\right)$ denote the
corresponding subscription price and consumption matrix of this platform,
respectively. Since $R$ satisfies _non-manipulability_ , we can apply Theorem
9 from Ju et al. (2007)555It is easy to check that when $|S|=1$, our model
coincides with the resource allocation model proposed by Ju et al. (2007). to
obtain that, for each $i\in N$,

$R_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=\frac{C^{(s)}_{is}}{\sum_{i\in
N}C^{(s)}_{is}}p^{(s)}=R^{SP}_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right).$

Now, suppose that $S$ is such that $|S|\geq 2$. Notice that

$p=p^{(1)}\oplus\ldots\oplus p^{(|S|)}\text{ and }C=C^{(1)}\oplus\ldots\oplus
C^{(|S|)},$

where each pair $p^{(s)}$ and $C^{(s)}$ corresponds to the subscription price
and the consumption matrix with only one subscriber. Since $R$ satisfies
_composition_ , it follows that, for each $i\in N$,

$R_{i}\left(N,S,p,C\right)=\sum_{s\in
S}R_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=\sum_{s\in
S}R^{SP}_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=R^{SP}_{i}\left(N,S,p,C\right).$

∎

As the next remark shows, Theorem 1 is tight, and both axioms are necessary
for the characterization.

######  Remark 1.

The axioms of Theorem 1 are independent.

  * (a)

The uniform indicator satisfies composition, but not non-manipulability.

  * (b)

The proportional indicator satisfies non-manipulability, but not composition.

Our second characterization shows that if we require _sharing proofness_ and
_non-manipulability_ to be satisfied, then the relevance of each service must
be measured using the _proportional indicator_. As mentioned in the
introduction, in our setting, the _proportional indicator_ implements the
_pro-rate principle_. Therefore, Theorem 2 can also be interpreted as a way to
justify, from a normative perspective, the application of this principle in
the music streaming industry.

######  Theorem 2.

An indicator satisfies sharing proofness and non-manipulability if and only if
it is the proportional indicator.

###### Proof.

We start by showing that the proportional indicator satisfies the axioms in
the statement.

  * •

Sharing proofness. Let $(N,S,p,C)\in\mathcal{D}$. Let $S^{\prime}\subseteq S$
and $s\in S^{\prime}$ such that $p^{\prime}_{s}=\sum_{t\in S^{\prime}}p_{t}$
and $C^{\prime}_{is}=\sum_{t\in S^{\prime}}C_{it}$ for all $i\in N$ and
$C^{\prime}_{is}=C_{is}$ for all $i\in N$ and $s\in S\backslash S^{\prime}$.
Then

$\displaystyle R^{P}_{i}\left(N,\\{s\\}\cup S\backslash
S^{\prime},(p^{\prime}_{s},p_{S\backslash S^{\prime}}),C^{\prime}\right)$
$\displaystyle=\dfrac{\|C^{\prime}_{i\cdot}\|}{\sum_{j\in
N}\|C^{\prime}_{j\cdot}\|}\sigma$ $\displaystyle=\dfrac{\sum_{t\in\\{s\\}\cup
S\backslash S^{\prime}}C^{\prime}_{it}}{\sum_{j\in N}\sum_{t\in\\{s\\}\cup
S\backslash S^{\prime}}C^{\prime}_{jt}}\sigma$
$\displaystyle=\dfrac{C^{\prime}_{is}+\sum_{t\in S\backslash
S^{\prime}}C^{\prime}_{it}}{\sum_{j\in N}\left(C^{\prime}_{js}+\sum_{t\in
S\backslash S^{\prime}}C^{\prime}_{jt}\right)}\sigma$
$\displaystyle=\dfrac{\sum_{t\in S^{\prime}}C_{it}+\sum_{t\in S\backslash
S^{\prime}}C_{it}}{\sum_{j\in N}\left(\sum_{t\in S^{\prime}}C_{jt}+\sum_{t\in
S\backslash S^{\prime}}C_{jt}\right)}\sigma$ $\displaystyle=\dfrac{\sum_{t\in
S}C_{it}}{\sum_{j\in N}\sum_{t\in S}C_{jt}}\sigma$
$\displaystyle=R^{P}_{i}(N,S,p,C).$

  * •

Non-manipulability. Let $(N,S,p,C)\in\mathcal{D}$. Let $N^{\prime}\subseteq N$
and $i\in N^{\prime}$ such that $C^{\prime}_{is}=\sum_{j\in N^{\prime}}C_{js}$
for all $s\in S$. Then

$\displaystyle R^{P}_{i}\left(\\{i\\}\cup N\backslash
N^{\prime},S,p,(C^{\prime}_{i\cdot},C_{N\backslash N^{\prime}})\right)$
$\displaystyle=\dfrac{\|C^{\prime}_{i\cdot}\|}{\|C^{\prime}_{i\cdot}\|+\sum_{k\in
N\backslash N^{\prime}}\|C_{k\cdot}\|}\sigma$ $\displaystyle=\dfrac{\sum_{j\in
N^{\prime}}\|C_{j\cdot}\|}{\sum_{j\in N^{\prime}}\|C_{j\cdot}\|+\sum_{k\in
N\backslash N^{\prime}}\|C_{k\cdot}\|}\sigma$ $\displaystyle=\sum_{j\in
N^{\prime}}\dfrac{\|C_{j\cdot}\|}{\sum_{k\in N}\|C_{k\cdot}\|}\sigma$
$\displaystyle=\sum_{j\in N^{\prime}}R^{P}_{j}(N,S,p,C).$

Now, we prove the converse. Let $R$ be an indicator that fulfills sharing
proofness and non-manipulability. Let $(N,S,p,C)\in\mathcal{D}$. Now, consider
the platform $(N,\\{1\\},p^{(1)},C^{(1)})$ where
$p^{(1)}=p_{1}+\ldots+p_{|S|}$ and $C^{(1)}_{i}=\sum_{s\in S}C_{is}$ for all
$i\in N$. _Sharing proofness_ requires that

$R(N,S,p,C)=R\left(N,\\{1\\},p^{(1)},C^{(1)}\right).$

Notice that in $\left(N,\\{1\\},p^{(1)},C^{(1)}\right)$ there is only one
subscriber. Applying Theorem 9 in Ju et al. (2007), _non-manipulability_
implies that for each $i\in N$,

$R_{i}\left(N,\\{1\\},p^{(1)},C^{(1)}\right)=\frac{C^{(1)}_{i}}{\|C^{(1)}\|}\sigma=\frac{\|C_{i\cdot}\|}{\|C\|}\sigma.$

Therefore, for each $i\in N$, we conclude that

$R_{i}(N,S,p,C)=\frac{\|C_{i\cdot}\|}{\|C\|}\sigma=R^{P}_{i}(N,S,p,C).$

∎

######  Remark 2.

The axioms of Theorem 2 are independent.

  * (a)

The uniform indicator satisfies sharing proofness, but not non-manipulability.

  * (b)

The subscriber-proportional indicator satisfies non-manipulability, but not
sharing proofness.

Our third characterization is given in Theorem 3. This result states that the
combination of _symmetry_ , _homogeneity_ , and _consumption sensitivity_
unambiguously yield to the application of the _uniform indicator_.

######  Theorem 3.

An indicator satisfies symmetry, homogeneity, and consumption sensitivity if
and only if it is the uniform indicator.

###### Proof.

We start by showing that the uniform indicator satisfies the axioms in the
statement.

  * •

Symmetry. Let $(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that
$C_{is}=C_{js}$ for all $s\in S$. It follows that

$\displaystyle R_{i}^{U}(N,S,p,C)$
$\displaystyle=\frac{\sigma}{|N|}=R_{j}^{U}(N,S,p,C).$

  * •

Homogeneity. Let $(N,S,p,C)\in\mathcal{D}$ and $\lambda\in\mathbb{R}_{++}$.
Let $i\in N$. It follows that

$\displaystyle R_{i}^{U}(N,S,p,C)$
$\displaystyle=\frac{\sigma}{|N|}=R_{i}^{U}(N,S,p,\lambda C).$

  * •

Consumption sensitivity. Since $R_{i}^{U}(N,S,p,C)=\frac{\sigma}{|N|}$ for all
$i\in N$, it is easy to see that small changes in the consumption matrix do
not alter the relevance of the services given by the uniform indicator.

Next, we prove the converse. Let $R$ be an indicator that satisfies symmetry,
homogeneity, and consumption sensitivity, and $R\neq R^{U}$. Therefore, there
exists a problem $(N,S,p,C)$ such that $R(N,S,p,C)\neq R^{U}(N,S,p,C)$.

For an arbitrary $n\in\mathbb{N}_{++}$, let $C^{0n}$ be the consumption matrix
for which all coordinates are exactly equal to $\frac{1}{n}$. On the one hand,
by _symmetry_ we know that $R_{i}(N,S,p,C^{0n})=\frac{\sigma}{|N|}$ for all
$i\in N$ and for all $n\in\mathbb{N}_{++}$. On the other hand, by
_homogeneity_ , $R(N,S,p,\frac{1}{n}C)=R(N,S,p,C)$ for all
$n\in\mathbb{N}_{++}$.

Now, it is straightforward to prove that $\|C^{0n}-\frac{1}{n}C\|$ goes to $0$
when $n$ goes to infinity, but since $R(N,S,p,C)\neq R^{U}(N,S,p,C)$, it
follows that
$\|R(N,S,p,C^{0n})-R(N,S,p,\frac{1}{n}C)\|=\|R^{U}(N,S,p,C^{0n})-R(N,S,p,\frac{1}{n}C)\|=k>>0$
for all $n\in\mathbb{N}_{++}$. This contradicts the fact that $R$ satisfies
consumption sensitivity.

∎

######  Remark 3.

The axioms of Theorem 3 are independent.

  * (a)

The proportional indicator satisfies symmetry and homogeneity, but not
consumption sensitivity. To see this, consider a situation with three services
and only one subscriber. Let us consider the following two successions of
consumption matrices:

$C^{1n}=\left(\begin{array}[]{c}\frac{1}{n}\\\ \frac{1}{n}\\\ \frac{1}{n}\\\
\end{array}\right)\text{ and }C^{2n}=\left(\begin{array}[]{c}0\\\
\frac{1}{n}\\\ \frac{2}{n}\\\ \end{array}\right).$

It is obvious that $\|C^{1n}-C^{2n}\|$ goes to $0$ when $n$ goes to infinity,
but
$\|R^{P}(N,S,p,C^{1n})-R^{P}(N,S,p,C^{2n})\|=\|(\frac{p}{3},\frac{p}{3},\frac{p}{3})-(0,\frac{p}{3},\frac{2p}{3})\|=\frac{2p}{3}>>0$
for all $n$.

  * (b)

Let $R^{1}$ be defined as follows. For each $i\in N$,

$R^{1}_{i}(N,S,p,C)=\frac{\sigma}{|N|}-\displaystyle\min\left\\{\frac{1}{\|C_{i\cdot}\|},\|C_{i\cdot}\|\right\\}+\frac{\displaystyle\sum_{j\in
N}\min\left\\{\frac{1}{\|C_{j\cdot}\|},\|C_{j\cdot}\|\right\\}}{|N|}.$

The indicator $R^{1}$ satisfies symmetry and consumption sensitivity, but not
homogeneity.

  * (c)

Let $R^{2}$ be defined as follows. For each $i\in N$,

$R^{2}_{i}(N,S,p,C)=\begin{cases}\dfrac{2\sigma}{|N|}&\text{if
}i=1\\\\[11.38092pt]
\dfrac{(|N|-2)\sigma}{|N|(|N|-1)}&\text{otherwise.}\end{cases}$

The indicator $R^{2}$ satisfies homogeneity and consumption sensitivity, but
not symmetry.

The final characterization shows that if _composition_ , _nullity_ , and
_strong symmetry_ must be satisfied for an indicator, then the relevance of
each service must be measured using the _subscriber-uniform indicator_.

######  Theorem 4.

An indicator satisfies composition, nullity, and strong symmetry if and only
if it is the subscriber-uniform indicator.

###### Proof.

We start by showing that the _subscriber-proportional indicator_ satisfies the
axioms in the statement.

  * •

Composition. Let
$(N,S,p,C),(N,S^{\prime},p^{\prime},C^{\prime})\in\mathcal{D}$ such that
$S\cap S^{\prime}=\emptyset$. Let $i\in N$. It follows that

$\displaystyle R_{i}^{SU}(N,S\cup S^{\prime},p\oplus p^{\prime},C\oplus
C^{\prime})$ $\displaystyle=\sum_{s\in S\cup S^{\prime},i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}(p\oplus p^{\prime})_{s}$
$\displaystyle=\sum_{s\in S,i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s}+\sum_{s\in S^{\prime},i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p^{\prime}_{s}$
$\displaystyle=R_{i}^{SU}(N,S,p,C)+R_{i}^{SP}(N,S^{\prime},p^{\prime},C^{\prime}).$

  * •

Nullity. Let $(N,S,p,C)\in\mathcal{D}$ and $i\in N$ such that $C_{is}=0$ for
all $s\in S$, then $i\notin N_{s}=\\{j\in N:C_{js}\neq 0\\}$ for all $s\in S$
and

$\displaystyle R_{i}^{SU}(N,S,p,C)$ $\displaystyle=\sum_{s\in S,i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s}=0.$

  * •

Strong symmetry. Let $(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that
$C_{is}\cdot C_{js}\neq 0$ or $C_{is}=C_{js}=0$ for all $s\in S$. Then $i\in
N_{s}$ if and only if $j\in N_{s}$ for all $s\in S$, and it follows that

$\displaystyle R_{i}^{SU}(N,S,p,C)$ $\displaystyle=\sum_{s\in S,i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s}=\sum_{s\in S,j\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s}=R_{j}^{SU}(N,S,p,C).$

Now, we prove the converse. Let $R$ be an indicator that fulfills
_composition_ , _nullity_ , and _strong symmetry_. Let
$(N,S,p,C)\in\mathcal{D}$. Suppose that $S$ is a singleton (i.e., there is
only one subscriber $S=\\{s\\}$), and let $p^{(s)}$ and
$C^{(s)}=\left(C_{1s}^{(s)},\ldots,C_{|N|s}^{(s)}\right)$ be the corresponding
subscription price and consumption matrix of this platform, respectively.
Since $R$ satisfies _nullity_ , it follows that

$R_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=0,$

for all $i\in N$ such that $C^{(s)}_{is}=0$. Additionally, since $R$ satisfies
_strong symmetry_ , it follows that

$R_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=\frac{p^{(s)}}{|N_{s}|},$

for all $i\in N$ such that $C^{(s)}_{is}>0$, where $N_{s}=\\{j\in
N:C^{(s)}_{js}>0\\}$.

Now, suppose that $S$ is such that $|S|\geq 2$. Notice that

$p=p^{(1)}\oplus\ldots\oplus p^{(|S|)}\text{ and }C=C^{(1)}\oplus\ldots\oplus
C^{(|S|)},$

where each pair $p^{(s)}$ and $C^{(s)}$ corresponds to the subscription price
and the consumption matrix with only one subscriber. Since $R$ satisfies
_composition_ , it follows that for each $i\in N$,

$R_{i}\left(N,S,p,C\right)=\sum_{s\in
S}R_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=\sum_{s\in
S}R^{SU}_{i}\left(N,\\{s\\},p^{(s)},C^{(s)}\right)=R^{SU}_{i}\left(N,S,p,C\right).$

∎

######  Remark 4.

The axioms of Theorem 4 are independent.

  * (a)

The subscriber-proportional indicator satisfies composition and nullity, but
not strong symmetry. To see this, consider a situation with three services,
only one subscriber, $p_{1}=1$, and the following consumption matrix:

$C^{1n}=\left(\begin{array}[]{c}2\\\ 1\\\ 0\\\ \end{array}\right)$

For the strong symmetry axiom, the success of service one and two should be
equal. However,
$R^{SP}_{1}(N,S,p,C)=\frac{2}{3}\neq\frac{1}{3}=R^{SP}_{2}(N,S,p,C)$.

  * (b)

The uniform indicator satisfies composition and strong symmetry, but not
nullity. Consider the example of the previous item. Service three is not
viewed by the subscriber, but $R^{SP}_{3}(N,S,p,C)=\frac{1}{3}$.

  * (c)

Let $R^{1}$ be defined as follows. For each $i\in N$,

$R^{1}_{i}(N,S,p,C)=\begin{cases}\dfrac{\sigma}{|M|}&\text{if
}M\neq\emptyset\\\\[11.38092pt]
R^{P}_{i}(N,S,p,C)&\text{otherwise,}\end{cases}$

where $M\subseteq N$ such that if $i\in M$, there exists at least one service
$j\in M$, $C_{is}\cdot C_{js}\neq 0$ or $C_{is}=C_{js}=0$ for all $s\in S$.

The indicator $R^{1}$ satisfies nullity and strong symmetry, but not
composition.

Table 2 summarizes our theoretical findings. As we can observe, the four
indicators satisfy several interesting properties, even though not all
properties are required for their characterizations. The _uniform indicator_
fulfills all the axioms except _non-manipulability_ and _nullity_. The
_subscriber-uniform indicator_ satisfies all the axioms except _consumption
sensitivity_ , _sharing proofness_ , and _non-manipulabity_ , and it is
characterized by _composition_ , _strong symmetry_ , and _nullity_. _Sharing
proofness_ and _non manipulability_ unambiguously yield to the _proportional
indicator_ , but they also satisfy _symmetry_ , _homogeneity_ , and _nullity_.
The _subscriber-proportional indicator_ is characterized by _composition_ and
_non-manipulability_ , but it also fulfills _symmetry_ , _homogeneity_ , and
_nullity_. From the standpoint of the axiom, three axioms – (_efficiency_ ,
_symmetry_ , and _homogeneity_) – are satisfied by all indicators in the
paper. Moreover, all axioms except _consumption sensitivity_ are satisfied by
at least two of the four indicators.

Property | $R^{U}$ | $R^{SU}$ | $R^{P}$ | $R^{SP}$
---|---|---|---|---
Efficiency | Y | Y | Y | Y
Symmetry | Y∗ | Y | Y | Y
Strong symmetry | Y | Y∗ | N | N
Homogeneity | Y∗ | Y | Y | Y
Consumption sensitivity | Y∗ | N | N | N
Composition | Y | Y∗ | N | Y∗
Sharing proofness | Y | N | Y∗ | N
Non-manipulability | N | N | Y∗ | Y∗
Nullity | N | Y∗ | Y | Y
Table 2: Summary of theoretical findings. Y means that the indicator in the
column does satisfy the axiom in the row, N means that the indicator does not
satisfy the axiom, and Y∗ means that the axiom is used to characterize the
indicator.

For proofs and counterexamples of all properties for all studied relevance
indicators that have not been proven in Theorems 1 to 4, see Appendix A.

##  5 Analysis of the indicators in the streaming context

In this section, we analyze relevance indicators in the context of streaming
platforms. To do this, different aspects related to the information elements
of the problem, the structure of the indicators, and the properties that they
satisfy are considered. Likewise, we analyze which indicators are most
attractive to the interests of the platform.

Given a platform described by $\left(N,S,p,C\right)$, it is possible to
distinguish four levels of the use of the information contained in the problem
description. The simplest level, which we denote by $\varnothing$, would
correspond to using only the number of streaming creators (streaming channels
or simply streamers) on the platform as information. In this case, both the
number of subscribers and their streamer consumption times are considered
irrelevant and, therefore, the only important consideration is how many
streamers are offered on the platform and how much is earned from
subscriptions. At the second level, we consider two possibilities. On the one
hand, we consider the streamers, the total time they are watched for, and the
total subscription revenue as relevant information. In this case, the number
of subscribers and the time that each of them watches the streamers is not
taken into account. Therefore, the only relevant consideration is how much
time each streamer is watched and how much is earned. We denote this level of
information by $\mathcal{C}$. On the other hand, consider subscribers, the
streamers they watch, and their subscription fees as relevant information. In
this case, the consumption times of each subscriber of each of the streamers
are not taken into account. Therefore, the only important consideration is how
many subscribers watch each streamer and how much they pay. We denote this
level of information by $\mathcal{S}$. The richest level in terms of
information, which we denote by $\mathcal{C+S}$, would correspond to using all
the information related to subscribers, the streamers they watch, their
watching times, and their subscription fees. Therefore, in this case, all the
information available in the problem description is used. All these levels of
information are depicted in Figure 1.

$\varnothing$$\mathcal{C}$$\mathcal{S}$$\mathcal{C+S}$ Figure 1: Levels of
information on a platform.

Below we relate each of the levels of information described in Figure 1 with
the relevance indicators presented in Section 2 and the properties introduced
in Section 3. Of the nine properties presented in the normative framework, two
of them – _efficiency_ and _homogeneity_ – are present at all levels of
information; they do not provide differentiating elements in the study of the
relationship between the properties and the levels of information. These two
properties are related to the earnings obtained by the platform through the
payment of subscription fees, which are a common element in all levels of
information considered. On the other hand, the properties of _composition_ and
_non-manipulability_ are associated with two different paths of information
aggregation. The path of _non-manipulability_ takes us to the information
levels $\mathcal{C}$ and $\mathcal{C+S}$, which have in common that the time
watching the streamers is a fundamental piece of information. In this sense,
the fact of considering the watching times of the streamers would be related
to the non-possibility of manipulating the relevance measures by the streamers
themselves. The _composition_ path leads us to the information levels
$\mathcal{S}$ and $\mathcal{C+S}$ that have in common that what individual
subscribers watch on the platform is an essential piece of information. In
this case, considering what each subscriber sees on the platform is related to
the possibility of decomposing the problem of measuring relevance into smaller
parts. The _nullity_ is related to all levels of information that take into
account, in one way or another, subscribers’ behavior on the platform;
therefore, this property is not associated with the information level
$\varnothing$, but with the other three levels of information. Figure 2
depicts the properties and levels of information discussed here.

$\varnothing$$\mathcal{C}$$\mathcal{S}$$\mathcal{C+S}$Composition pathNon-
manipulability pathNullity region Figure 2: Levels of information and
properties.

The relationship of the four remaining properties with the different levels of
information is as follows. The property of _consumption sensitivity_ implies
that in the environment of null values, the information regarding which
subscribers and how long they watch the streamers becomes almost irrelevant,
as long as the subscription rates that they pay remain the same. Therefore,
this property would be associated with the information level $\varnothing$. In
this way, this property becomes the antagonist of _nullity_ , which refers to
null viewing time values; however, knowing which subscribers they are and
which streamers are not watched, then which subscribers and how long they
watch the streamers, is relevant. The symmetry property is associated with the
information level $\mathcal{C+S}$ since its very definition requires
information about which streamers subscribers watch and for how long. However,
the _strong symmetry_ property is only related to the streamers that
subscribers watch, but without going into the detail of how long they watch
them; therefore, this property is associated with the information level
$\mathcal{S}$. It is easy to check that _symmetry_ implies _strong symmetry_ ,
but the converse is not true. The last property proposed in the normative
framework is _sharing proofness_. In this property, the only relevant element
is the time that each streamer is watched, regardless of who and how many
subscribers watch them, then it is related to the information level
$\mathcal{C}$.

As has been done with properties, it is also possible to relate relevance
indicators to information levels. First, since the _uniform indicator_
exclusively relies on the number of services to distribute the platform’s
success, this indicator would be related to the level of information
$\varnothing$. Second, the _proportional indicator_ only considers the total
viewing time of each streamer to measure their relevance to platform success;
in this way, this indicator would be related to the level of information
$\mathcal{C}$. Third, the _subscriber-uniform indicator_ uses information
regarding which subscribers watch the streamers, but not for how long they
watch them, meaning that this indicator would be related to the level of
information $\mathcal{S}$. Finally, in the definition of the _subscriber-
proportional indicator_ , both the subscribers who watch the streamers and the
time they watch them are taken into consideration; therefore, this indicator
would be related to the level of information $\mathcal{C+S}$. In this way,
through the levels of information, the indicators are also related to the
different properties. At this point, it is worth saying that there may be
indicators that satisfy properties related to other levels of information
other than the one that the indicator is related to, as is the case in this
context (see Table 2). The reason for this circumstance is largely due to the
structure of the indicators themselves. For example, all of the presented
indicators satisfy the _symmetry_ property. In the case of the _uniform
indicator_ , it satisfies the _symmetry_ property by the simple fact that all
streamers are treated identically regardless of other elements. The
_proportional indicator_ satisfies the _symmetry_ property as a consequence of
the fact that if all the times of two symmetric streamers are added together,
their sum will obviously be equal and the indicator will treat them the same
by its own definition. Finally, the _subscriber-uniform indicator_ satisfies
the _symmetry_ property for the simple fact that if two streamers are
symmetrical, then they will be watched by the same subscribers.

The following example illustrates when a streamer can prefer one indicator
over another. This is a toy example with two subscribers, in which three types
of streamers are considered: one watched by one subscriber and not watched by
the other, another watched equally by both subscribers, and another not
watched by either subscriber. In particular, let $(N,S,p,C)\in\mathcal{D}$,
where $|N|=3$ and $|S|=2$; then the subscription prices $p=(p_{1},p_{2})$ and
the consumption matrix is equal to

$\left(\begin{array}[]{cc}M&0\\\ 1&1\\\ 0&0\\\ \end{array}\right),$

where $M\in\mathbb{R}_{++}$.

In this scenario, we have the following three types of streamers: Streamer 1
is viewed exclusively by the first subscriber, with a variable consumption
quantity; Streamer 2 is viewed by the first subscriber for a single unit of
time (we can normalize this unit for homogeneity with respect to what this
subscriber watches of the first service), and it is exclusively viewed by the
second subscriber; and Streamer 3, on the other hand, remains entirely
unwatched by any subscriber. The following table shows the relevance of each
streamer according to the four indicators:

| Streamers
---|---
Indicator | 1 | 2 | 3
$R^{U}$ | $\frac{(p_{1}+p_{2})}{3}$ | $\frac{(p_{1}+p_{2})}{3}$ | $\frac{(p_{1}+p_{2})}{3}$
$R^{SU}$ | $\frac{p_{1}}{2}$ | $\frac{p_{1}}{2}+p_{2}$ | 0
$R^{P}$ | $\frac{M}{M+2}(p_{1}+p_{2})$ | $\frac{2}{M+2}(p_{1}+p_{2})$ | 0
$R^{SP}$ | $\frac{M}{M+1}p_{1}$ | $\frac{1}{M+1}p_{1}+p_{2}$ | 0

It is obvious that a streamer that is not viewed (or hardly watched) by any
subscriber will prefer the _uniform indicator_. Since the rest of the
indicators satisfy the axiom of _nullity_ , it would be assigned null
relevance in any other case. For the other two streamers, we have the
following cases :

  * •

$p_{1}>2p_{2}$ then $R^{SU}_{1}>R^{U}_{1}$.

  * •

$p_{1}<2p_{2}$ then $R^{SU}_{1}<R^{U}_{1}$.

  * •

For all values $R^{SU}_{2}>R^{U}_{2}$.

Since Streamer 1 is viewed exclusively by the first subscriber, it would
prefer to measure its relevance using the _subscriber-uniform indicator_
rather than the _uniform indicator_ , when the first subscriber’s subscription
is more than twice as large as the second subscriber’s. Streamer 2 will always
prefer the _subscriber-uniform indicator_ over the _uniform indicator_ for any
$p_{1}$ and $p_{2}$.

On the other hand, we can also analyze when a streamer prefers the
_proportional indicator_ over the _uniform indicator_ :

  * •

$M>1$ then $R^{P}_{1}>R^{U}_{1}$.

  * •

$M<1$ then $R^{P}_{1}<R^{U}_{1}$.

Note that Streamer 1 prefers the _proportional indicator_ over the _uniform
indicator_ when its viewing time is greater than the viewing time of Streamer
2.

  * •

$M<4$ then $R^{P}_{2}>R^{U}_{2}$.

  * •

$M>4$ then $R^{P}_{2}<R^{U}_{2}$.

However, Streamer 2 will prefer the _proportional indicator_ to measure its
relevance when the viewing time of Streamer 1 is not excessively greater (in
this example, four times) than the viewing time of Streamer 2. Otherwise,
Streamer 2 will prefer the _uniform indicator_ , since if it is much greater,
Streamer 2 would be very close to the situation of Streamer 3.

We now explore when the _subscriber-proportional indicator_ is preferred to
the _uniform indicator_ by Streamers 1 and 2.

  * •

$p_{1}\leq\frac{p_{2}}{2}$ then $R^{SP}_{1}<R^{U}_{1}$.

  * •

$p_{1}>\frac{p_{2}}{2}$

    * –

$M<\frac{p_{1}+p_{2}}{2p_{1}-p_{2}}$ then $R^{SP}_{1}<R^{U}_{1}$.

    * –

$M>\frac{p_{1}+p_{2}}{2p_{1}-p_{2}}$ then $R^{SP}_{1}>R^{U}_{1}$.

If the first subscriber’s subscription is half or less than that of the second
subscriber, Streamer 1 prefers the _uniform indicator_ over the _subscriber-
proportional indicator_. This is because, with the _subscriber-proportional
indicator_ , Streamer 1 does not benefit from this subscription since the
second subscriber does not view Service 1. Therefore, the distribution of
subscription values plays a crucial role in deciding between the indicators.
If the first subscriber’s subscription is greater than that of the second, the
choice of indicator will depend on the viewing time that this streamer
receives; for a sufficiently high value, the chosen indicator will be the
_subscriber-proportional indicator_.

  * •

$p_{1}\leq 2p_{2}$ then $R^{SP}_{2}>R^{U}_{1}$.

  * •

$p_{1}>2p_{2}$

    * –

$M<\frac{2(p_{1}+p_{2})}{p_{1}-2p_{2}}$ then $R^{SP}_{2}>R^{U}_{2}$.

    * –

$M>\frac{2(p_{1}+p_{2})}{p_{1}-2p_{2}}$ then $R^{SP}_{2}<R^{U}_{2}$.

If either the first subscriber’s subscription price is less than double that
of the second, or the viewing time for Streamer 1 is below the threshold
$\frac{2(p_{1}+p_{2})}{p_{1}-2p_{2}}$, then Streamer 2 will opt for the
_subscriber-proportional indicator_ over the _uniform_ one. This preference is
due to the fact that Streamer 2 receives the entire subscription $p_{2}$,
while also receiving a proportional share of $p_{1}$.

Next, we analyze when the _subscriber-uniform indicator_ is preferred to the
_proportional indicator_ by Streamers 1 and 2.

  * •

$M>\frac{2p_{1}}{p_{1}+2p_{2}}$ then $R^{P}_{1}>R^{SU}_{1}$ and
$R^{P}_{2}<R^{SU}_{2}$.

  * •

$M<\frac{2p_{1}}{p_{1}+2p_{2}}$ then $R^{P}_{1}<R^{SU}_{1}$ and
$R^{P}_{2}>R^{SU}_{2}$.

When the viewing time for Streamer 1 exceeds a certain threshold, denoted as
$\frac{2p_{1}}{p_{1}+2p_{2}}$, Streamer 1 will opt for the _proportional
indicator_ , while Streamer 2 will choose the _subscriber-uniform indicator_.
This decision is driven by the fact that the _proportional indicator_
considers the total viewing time, whereas the _subscriber-uniform indicator_
accounts for the number of streamers consumed by each subscriber. In this
case, the second subscriber exclusively consumes Streamer 2.

We now study when the _subscriber-proportional indicator_ is preferred to the
_subscriber-uniform indicator_ by Streamers 1 and 2.

  * •

$M>1$ then $R^{SP}_{1}>R^{SU}_{1}$ and $R^{SP}_{2}<R^{SU}_{2}$.

  * •

$M<1$ then $R^{SP}_{1}<R^{SU}_{1}$ and $R^{SP}_{2}>R^{SU}_{2}$.

When the viewing time for Streamer 1 exceeds the viewing time for Streamer 2,
Streamer 1 prefers the _subscriber-proportional indicator_ , while Streamer 2
opts for the _subscriber-uniform indicator_. This preference arises because
both streamers compete for the first subscriber’s subscription. With a higher
viewing time, Streamer 1 receives a larger share of this subscription. As a
result, Streamer 2 prefers an equal distribution, which is achieved through
the _subscriber-uniform indicator_.

Finally, we study when the _proportional indicator_ is preferred to the
_subscriber-proportional indicator_ by Streamers 1 and 2.

  * •

$p_{1}<p_{2}$ then $R^{P}_{1}>R^{SP}_{1}$ and $R^{P}_{2}<R^{SP}_{2}$.

  * •

$p_{1}>p_{2}$

    * –

$0<M<\frac{p_{1}-p_{2}}{p_{2}}$ then $R^{P}_{1}<R^{SP}_{1}$ and
$R^{P}_{2}>R^{SP}_{2}$.

    * –

$M>\frac{p_{1}-p_{2}}{p_{2}}$ then $R^{P}_{1}>R^{SP}_{1}$ and
$R^{P}_{2}<R^{SP}_{2}$.

If the first subscriber’s subscription price is lower than that of the second
subscriber, Streamer 1 will prefer the _proportional indicator_ over the
_subscriber-proportional indicator_. This preference is driven by the fact
that the _proportional indicator_ allows Streamer 1 to receive a share of the
second subscriber’s subscription fee, whereas the _subscriber-proportional
indicator_ does not. Conversely, Streamer 2 will opt for the _subscriber-
proportional indicator_ , as it will receive the entirety of the second
subscriber’s subscription fee. If the first subscriber’s subscription price is
higher, the choice of indicator will depend on the viewing time of Streamer 1.

The following lemmas are formulated for general problems that establish
conditions under which a streamer will prefer one indicator over another.
Proofs of all lemmas are available in Appendix B.

######  Lemma 1.

Let $(N,S,p,C)\in\mathcal{D}$. For a streamer $i\in N$ such that

$\displaystyle\sum_{s\in S,i\in
N_{s}}\frac{\left(|N|-|N_{s}|\right)}{|N_{s}|}p_{s}\geq\sum_{s\in S,i\notin
N_{s}}p_{s}$

then $R_{i}^{SU}(N,S,p,C)\geq R_{i}^{U}(N,S,p,C)$.

######  Lemma 2.

Let $(N,S,p,C)\in\mathcal{D}$. For a streamer $i\in N$ such that $C_{is}>0$
for all $s\in S$, then $R_{i}^{SU}(N,S,p,C)\geq R_{i}^{U}(N,S,p,C)$.

Lemma 2 asserts that when a streamer is viewed by all subscribers, it will
favor the _subscriber-uniform indicator_ over the _uniform indicator_ to
measure its influence on the platform.

######  Lemma 3.

Let $(N,S,p,C)\in\mathcal{D}$. For a streamer $i\in N$ such that
$\|C_{i\cdot}\|\geq\frac{\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{|N|-1}$, then $R_{i}^{P}(N,S,p,C)\geq
R_{i}^{U}(N,S,p,C)$.

Lemma 3 states that if a streamer is viewed for more than the average viewing
time of all other streamers, it will favor the _proportional indicator_ over
the _uniform indicator_ to assess its influence within the platform.

######  Lemma 4.

Let $(N,S,p,C)\in\mathcal{D}$. For a streamer $i\in N$ such that
$C_{is}\geq\frac{\sum_{j\in N\backslash\\{i\\}}C_{js}}{|N|-1}$ for all $s\in
S$ then $R_{i}^{SP}(N,S,p,C)\geq R_{i}^{U}(N,S,p,C)$.

Lemma 4 states that if a streamer is viewed for more than the average viewing
time of all other streamers by each subscriber, then the streamer will prefer
the _subscriber-proportional indicator_ over the _uniform indicator_ to
measure its success on the platform.

######  Lemma 5.

Let $(N,S,p,C)\in\mathcal{D}$. For a streamer $i\in N$ such that
$C_{is}\geq\frac{\sum_{j\in N_{s}\backslash\\{i\\}}C_{js}}{|N_{s}|-1}$, where
$N_{s}=\left\\{j\in N:C_{js}\neq 0\right\\}$ for all $s\in S$ with $i\in
N_{s}$, then $R_{i}^{SP}(N,S,p,C)\geq R_{i}^{SU}(N,S,p,C)$.

Lemma 5 asserts that when a streamer accumulates more viewing time from each
subscriber than the average viewing time on other streamers seen by the same
subscriber, the streamer prefers the _subscriber-proportional indicator_ over
the _subscriber-uniform indicator_.

Let us now see how the _sharing proofness_ and _non-manipulability_ axioms
affect the distribution of each indicator. To do this, let us consider the
previous example again and assume that subscriber one comprises two
individuals who share the subscription equally. Both have viewed Streamer 1
for the same duration, but only one of them has watched Streamer 2. Therefore,
we have the following viewing time matrix:

$\left(\begin{array}[]{ccc}\frac{M}{2}&\frac{M}{2}&0\\\ 1&0&1\\\ 0&0&0\\\
\end{array}\right).$

This situation is directly related to the _sharing proofness_ axiom. Let us
now explore how it influences the distribution of each indicator:

| Streamers
---|---
Indicator | 1 | 2 | 3
$R^{U}$ | $\frac{(p_{1}+p_{2})}{3}$ | $\frac{(p_{1}+p_{2})}{3}$ | $\frac{(p_{1}+p_{2})}{3}$
$R^{SU}$ | $\frac{3p_{1}}{4}$ | $\frac{p_{1}}{4}+p_{2}$ | 0
$R^{P}$ | $\frac{M}{M+2}(p_{1}+p_{2})$ | $\frac{2}{M+2}(p_{1}+p_{2})$ | 0
$R^{SP}$ | $\frac{M+1}{M+2}p_{1}$ | $\frac{1}{M+2}p_{1}+p_{2}$ | 0

The _uniform_ and _proportional indicators_ satisfy the aforementioned axiom,
so that the distribution obtained is maintained compared to the first
scenario. On the other hand, the _subscriber-uniform_ and _subscriber-
proportional indicators_ attributed greater relevance to Streamer 1. This is
because in the first scenario, the subscription was uniformly or
proportionally divided between Streamers 1 and 2 based on the viewing time. In
this new scenario, half of the subscription is allocated exclusively to
Streamer 1, as it is viewed solely by one of the subscribers, while the
remaining half is divided either uniformly or proportionally.

In the latter scenario, we will assume that Streamer 1 is divided into two
such that the sum of their viewing times remains constant. The viewing time
matrix is as follows:

$\left(\begin{array}[]{cc}\frac{M}{2}&0\\\ \frac{M}{2}&0\\\ 1&1\\\ 0&0\\\
\end{array}\right).$

This new scenario is linked to the _non-manipulability_ axiom. The
distributions obtained with the four indicators are as follows:

| Services
---|---
Indicator | $1^{1}$ | $1^{2}$ | 2 | 3
$R^{U}$ | $\frac{(p_{1}+p_{2})}{4}$ | $\frac{(p_{1}+p_{2})}{4}$ | $\frac{(p_{1}+p_{2})}{4}$ | $\frac{(p_{1}+p_{2})}{4}$
$R^{SU}$ | $\frac{p_{1}}{3}$ | $\frac{p_{1}}{3}$ |  $\frac{p_{1}}{3}$ \+ $p_{2}$ | 0
$R^{P}$ | $\frac{\frac{M}{2}}{M+2}(p_{1}+p_{2})$ | $\frac{\frac{M}{2}}{M+2}(p_{1}+p_{2})$ | $\frac{2}{M+2}(p_{1}+p_{2})$ | 0
$R^{SP}$ | $\frac{\frac{M}{2}}{M+2}p_{1}$ | $\frac{\frac{M}{2}}{M+2}p_{1}$ | $\frac{2}{M+2}p_{1}+p_{2}$ | 0

As seen previously, the _proportional_ and _subscriber-proportional
indicators_ satisfy the _non-manipulability_ axiom; therefore, their relevance
remains the same as in the first scenario. On the other hand, as the number of
streamers increases, both the _uniform_ and _subscriber-uniform indicators_
assign lower relevance to each streamer in the new scenario. Nevertheless, the
total relevance achieved by Streamer 1 surpasses that of the initial scenario.
The following lemma establishes this result for any problem:

######  Lemma 6.

Let $(N,S,p,C),(N^{\prime},S,p,C^{\prime})\in\mathcal{D}$. Let $i\in N$, and
assume
$N^{\prime}=N\backslash\\{i\\}\cup\\{i^{\prime}_{1},\ldots,i^{\prime}_{m}\\}$,
with $m\geq 2$. Suppose $C_{is}=\sum_{l=1}^{m}C^{\prime}_{i^{\prime}_{l}s}$
and $C_{js}=C^{\prime}_{js}$ for all $j\in N\backslash\\{i\\}$ and $s\in S$.
Then, $\sum_{l=1}^{m}R^{U}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{U}_{i}(N,S,p,C)$ and
$\sum_{l=1}^{m}R^{SU}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{SU}_{i}(N,S,p,C)$.

Lemma 6 states that if a streamer is divided into several parts by
distributing its viewing time among them, it will yield greater relevance with
the _uniform_ and _subscriber-uniform indicators_.

Now, let us examine how the relevance of each streamer within the platform is
affected when the viewing time of one of the streamers increases or decreases.
First, since the _uniform indicator_ solely depends on the number of
streamers, the relevance of each streamer, under this indicator, remains
constant regardless of changes in the viewing time matrix.

Second, let us see what happens with the _subscriber-uniform indicator_. Let
$(N,S,p,C)\in\mathcal{D}$. Given $s\in S$ and $i\in N$, suppose $C^{\prime}$
such that $C^{\prime}_{is}>C_{is}$ and
$C^{\prime}_{js^{\prime}}=C_{js^{\prime}}$ for all $j\in N$ and $s^{\prime}\in
S\backslash\\{s\\}$. Now, we have the following two cases:

  1. 1.

If $C_{is}=0$, then
$R^{SU}_{i}(N,S,p,C^{\prime})=R^{SU}_{i}(N,S,p,C)+\frac{p_{s}}{|N_{s}|+1}$ and
$R^{SU}_{j}(N,S,p,C^{\prime})=R^{SU}_{j}(N,S,p,C)-\frac{p_{s}}{(|N_{s}|+1)|N_{s}|}$,
where $N_{s}=\left\\{j\in N:C_{js}\neq 0\right\\}$.

  2. 2.

If $C_{is}>0$, then $R^{SU}_{i}(N,S,p,C^{\prime})=R^{SU}_{i}(N,S,p,C)$ and
$R^{SU}_{j}(N,S,p,C^{\prime})=R^{SU}_{j}(N,S,p,C)$, where $N_{s}=\left\\{j\in
N:C_{js}\neq 0\right\\}$.

Therefore, under the _subscriber-uniform indicator_ , streamers will not
actively pursue additional views from subscribers who have already watched
them, as it will not enhance their relevance on the success on the platform.
Instead, the emphasis should be on subscribers who have not yet viewed the
streamer, as this will yield a positive outcome. Since the allocated relevance
is inversely proportional to the number of streamers viewed by a subscriber,
the focus should be on those subscribers who watch the fewest services for the
greatest improvements.

Third, we analyze the _proportional indicator_. The mathematical formula for
this indicator is given by

$R^{P}_{i}(N,S,p,C)=\frac{\|C_{i\cdot}\|}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma.$

Then, taking the partial derivative of the previous expression with respect to
$C_{is}$, we obtain that

$\frac{\partial R^{P}_{i}(N,S,p,C)}{\partial
C_{is}}=\dfrac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\displaystyle\left(\sum_{j\in
N}\|C_{j\cdot}\|\right)^{2}}\sigma.$

Therefore, the improvement in the relevance of streamer $i$ when a subscriber
increases the viewing time of that streamer is inversely proportional to the
square of the total viewing time and directly proportional to the total
viewing time of the rest of the streamers on the platform. Note that if
instead of examining the viewing time of a streamer by a subscriber, we
examined the total viewing time of a streamer, the result would be identical.
On the other hand, if we rewrite the previous expression as

$\frac{\partial R^{P}_{i}(N,S,p,C)}{\partial
C_{is}}=\left(\dfrac{1}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}-\dfrac{\|C_{i\cdot}\|}{\displaystyle(\sum_{j\in
N}\|C_{j\cdot}\|)^{2}}\right)\sigma,$

then we observe that the improvement in streamer relevance is less than the
income per unit of viewing time when the viewing time of a streamer increases.
Furthermore, we observe that this improvement will be less the greater the
viewing time that the streamer initially had.

Fourth, we now study the changes in the _subscriber-proportional indicator_.
The mathematical expression for this indicator is given by

$R^{SP}_{i}(N,S,p,C)=\sum_{s\in S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}.$

Then, taking the partial derivative of the previous expression with respect to
$C_{is}$, it follows that

$\frac{\partial R^{SP}_{i}(N,S,p,C)}{\partial
C_{is}}=\dfrac{\displaystyle\sum_{j\in N\backslash\\{i\\}}C_{js}}{\|C_{\cdot
s}\|^{2}}p_{s}.$

Therefore, we obtain a result similar to that of the _proportional indicator_.
In this case, the improvement in the relevance of streamer $i$ when a
subscriber increases the viewing time of that streamer is inversely
proportional to the square of the total viewing time of that particular
subscriber and directly proportional to the total viewing time of the rest of
the streamers on the platform, again of that particular subscriber. As in the
case of the _proportional indicator_ , if we rewrite the previous expression
as

$\frac{\partial R^{SP}_{i}(N,S,p,C)}{\partial
C_{is}}=\left(\dfrac{1}{\displaystyle\|C_{\cdot
s}\|}-\dfrac{C_{is}}{\displaystyle\|C_{\cdot s}\|^{2}}\right)p_{s},$

then we observe the following: that the improvement of the relevance of a
streamer is less than the subscription fee per unit of viewing time when the
viewing time of that streamer by the subscriber increases; and that this
improvement will be less the greater the viewing time that the streamer
initially had for that subscriber. In essence, the interpretation is analogous
to that obtained for the _proportional indicator_ , but considering only the
subscription rate of the particular subscriber.

Finally, we analyze which indicator would be chosen (or preferred) by the
platform. To do this, we must consider how a platform’s success is measured.
It can be assessed that success is determined by the number of subscriptions;
the more subscribers a platform has, the higher its profits. The indicators
that are directly linked to the number of subscribers are the _subscriber-
uniform indicator_ and the _subscriber-proportional indicator_. The
_subscriber-uniform indicator_ does not consider the viewing time from each
subscriber. Consequently, each streamer constantly aims to attract new
subscribers, regardless of the viewing time. This leads to the addition of new
subscribers to the platform, which in turn increases the platform’s success
and profits. On the other hand, when utilizing the _subscriber-proportional
indicator_ , streamers seek to attract new subscribers, although to a lesser
degree compared to the _subscriber-uniform indicator_. Under this new metric,
streamers also strive to retain these new subscribers, with the goal of
prolonging their viewing time. Let us now contemplate a scenario in which the
number of subscribers remains fixed, and the platform derives benefits (for
instance, through advertising revenue) from the time that subscribers spend
viewing streamers. In such a situation, the chosen indicators for the platform
would be the _proportional indicator_ and the _subscriber-proportional
indicator_. Although the _subscriber-proportional indicator_ aims to achieve a
balance between the number of subscribers and the time they spend viewing the
streamers, the _proportional indicator_ strives directly to increase the
viewing time of subscribers. Therefore, the latter indicator could lead to
greater benefits for the platform in this new situation. In an average
scenario where the platform aims to boost its success and profits (for
instance, through advertising revenue) by increasing both the number of
subscribers and the time they spend viewing the streamers, the platform should
prioritize the _subscriber-proportional indicator_.

##  6 A case study: Twitch

In the previous sections, we characterized and analyzed the _uniform_ ,
_subscriber-uniform_ , _proportional_ , and _subscriber-proportional_
indicators in the context of streaming platforms. In this section, we propose
an application of these four indicators to measure the relevance of creators
on the Twitch streaming platform Twitch.666https://www.twitch.tv/

We screened the 19 most-watched streamers worldwide during a three-week period
from 23 December 2022 to 15 January 2023. In particular, we collected data on
the viewers (identified through their nicknames) who were watching each
streamer every streaming hour. In terms of the theoretical model introduced in
Section 2, the services $N$ are the streamers, the subscribers $S$ are the
viewers, and the consumption $C_{is}$ is the time each viewer spent watching
the online content provided by each streamer. The subscription price is
assumed to be fixed (and normalized to 1).777Information on the individual
subscription fee that each viewer pays is not public. However, the platform
has access to this information, which it can easily use to compute the
distribution of the revenue that corresponds to each of the indicators.

The success of the Twitch platform is $\sigma=5,164,910$, which coincides with
the number of viewers. Our database contains approximately five million unique
users who watched one or more of the streamers. For the analyzed period, Table
3 shows the country, streaming language, number of viewers, overall share of
viewers, exclusive viewers (those who exclusively consume the content of one
streamer), and the percentage they represent out of the total amount of
exclusive viewers. In terms of the number of users, AuronPlay, Ibai, and XQc
are the three most relevant streamers, significantly outperforming the others.
In terms of exclusive viewers, AuronPlay and XQc perform relatively well, but
Ibai drops significantly.

[Insert Table 3 about here.]

|  |  | Viewers | Exclusive viewers
---|---|---|---|---
Streamer | Country | Language | Number | % | Number | %
Adin Ross | United States | English | 522425 | 6.97 | 279905 | 7.49
AuronPlay | Spain | Spanish | 758108 | 10.11 | 302403 | 8.09
Casimito | Brazil | Portuguese | 224771 | 3.00 | 143199 | 3.83
ElSpreen | Argentina | Spanish | 354882 | 4.73 | 160886 | 4.30
elXokas | Spain | Spanish | 237442 | 3.17 | 57847 | 1.55
Fextralife | United States | English | 254907 | 3.40 | 228050 | 6.10
Gaules | Brazil | Portuguese | 239381 | 3.19 | 162742 | 4.35
HasanAbi | United States | English | 372542 | 4.97 | 238815 | 6.39
Ibai | Spain | Spanish | 738317 | 9.85 | 237515 | 6.35
IlloJuan | Spain | Spanish | 476256 | 6.35 | 157036 | 4.20
Juansguarnizo | Mexico | Spanish | 400161 | 5.34 | 128466 | 3.44
Kai Cenat | United States | English | 501301 | 6.69 | 270001 | 7.22
Loltyler1 | United States | English | 299832 | 4.00 | 201687 | 5.40
Loud$\\_$coringa | Brazil | Portuguese | 279470 | 3.73 | 236475 | 6.33
Roshtein | Malta | English | 4451 | 0.06 | 2782 | 0.07
Rubius | Spain | Spanish | 378760 | 5.05 | 115037 | 3.08
Tarik | United States | English | 410166 | 5.47 | 299966 | 8.02
TheGrefg | Spain | Spanish | 318345 | 4.25 | 92534 | 2.48
XQc | Canada | English | 725958 | 9.68 | 422948 | 11.31
|  |  | 7497475 | 100 | 3738294 | 100
Table 3: Viewers and exclusive viewers.

Beyond the mere number of users or exclusive users, what is the specific
impact of each streamer on the success of the Twitch? Table 4 illustrates the
application of the _uniform_ , _subscriber-uniform_ , _proportional_ , and
_subscriber-proportional_ indicators to our case study. For each metric, we
indicate both the value and the percentage of success that is due to each
streamer. Not surprisingly, for the _uniform indicator_ they are all equally
relevant. Although this indicator may be seen as an uninformative measure, it
serves as a useful benchmark for comparisons since, by definition, it
coincides with the average share. The _subscriber-uniform indicator_ assigns
the success of each individual viewer among the streamers whose content the
subscriber viewed. Under this indicator, exclusive viewers have a significant
impact on the distribution of success to each streamer. The subscription fee
is fully assigned to that specific streamer. For the remaining viewers,
success is distributed uniformly among all the streamers that they viewed.
However, exclusive viewers do not completely determine the success rate. For
example, Tarik is the third streamer with the most exclusive viewers, yet he
is the sixth most successful streamer by this indicator.

The _proportional indicator_ (which implements the _pro-rata_ principle in
Alaei et al. (2022)) distributes the success proportionally to the overall
consumption of each streamer. Notice that according to $R^{P}$, only a few
streamers are above average (Adin Ross, AuronPlay, Ibai, IlloJuan, and XQc).
Comparing the percentage of viewers with the _proportional indicator_ , we
observe that the impact of Adin Ross is substantially higher than one might
expect by focusing only on the users. His relevance increases in excess of 51%
to the $R^{P}$ indicator. IlloJuan, for example, is another streamer whose
influence is undervalued. The other side of the coin is that the impact of
TheGrefg is 35% lower if we consider the _proportional indicator_.

The _subscriber-proportional indicator_ (which captures the idea behind the
_user-centric_ principle in Alaei et al. (2022)) assigns the success of each
single viewer among the streamers whose content she has consumed. Therefore,
by definition, the impact of an exclusive user is completely attributed to the
unique streamer she watches. It is for this reason that the number of
exclusive viewers has a higher impact in the _subscriber-proportional
indicator_ than in the _proportional indicator_. However, $R^{SP}$ and
exclusive viewers are not completely correlated. For instance, Ibai is the
seventh streamer in terms of the most exclusive viewers (6.35%), but he rises
to the third position for the _subscriber-proportional indicator_ (8.32%),
with an increase of a 31%. This is because, in addition to his exclusive
followers, his content is consumed by many other users.

We also observe several differences by comparing the _proportional_ and
_subscriber-proportional_ indicators. The most significant change is the
relevance of the streamer IlloJuan. He represents 10.15% of the total success
of the platform for the former, but only the 5.80% for the latter. This is due
to the fact that IlloJuan is one of the most watched streamers, relative to
the total number of hours viewed, but the number of exclusive viewers
(approximately $33\%$) is substantially lower than in other streamers with
higher benefits. In contrast, Gaules is more relevant according to the
_subscriber-proportional indicator_ compared to the _proportional indicator_.
The reason is that Gaules does not have many viewing hours compared to the
total number of hours viewed by the users. However, this is a Portuguese
streamer, while the others are Spanish or English, so he has a significant
number of exclusive viewers, around $68\%$.

[Insert Table 4 about here.]

| $R^{U}$ | $R^{SU}$ | $R^{P}$ | $R^{SP}$
---|---|---|---|---
Streamer | Value | % | Value | % | Value | % | Value | %
Adin Ross | 271837.4 | 5.26 | 384314.07 | 7.44 | 545333.40 | 10.56 | 399648.12 | 7.74
AuronPlay | 271837.4 | 5.26 | 467563.86 | 9.05 | 448855.80 | 8.69 | 462333.00 | 8.95
Casimito | 271837.4 | 5.26 | 181757.52 | 3.52 | 219242.70 | 4.24 | 193421.99 | 3.74
ElSpreen | 271837.4 | 5.26 | 228803.63 | 4.43 | 189751.10 | 3.67 | 228175.61 | 4.42
elXokas | 271837.4 | 5.26 | 117556.95 | 2.28 | 106974.10 | 2.07 | 100438.95 | 1.94
Fextralife | 271837.4 | 5.26 | 239775.26 | 4.64 | 172821.40 | 3.35 | 238549.53 | 4.62
Gaules | 271837.4 | 5.26 | 198698.27 | 3.85 | 105343.10 | 2.04 | 185148.12 | 3.58
HasanAbi | 271837.4 | 5.26 | 294519.76 | 5.70 | 191565.80 | 3.71 | 277481.56 | 5.37
Ibai | 271837.4 | 5.26 | 417746.54 | 8.09 | 466978.00 | 9.04 | 429475.25 | 8.32
IlloJuan | 271837.4 | 5.26 | 269565.17 | 5.22 | 524492.80 | 10.15 | 299687.49 | 5.80
Juansguarnizo | 271837.4 | 5.26 | 223595.42 | 4.33 | 255009.10 | 4.94 | 222511.11 | 4.31
Kai Cenat | 271837.4 | 5.26 | 369516.65 | 7.15 | 266601.80 | 5.16 | 353599.40 | 6.85
Loltyler1 | 271837.4 | 5.26 | 243739.64 | 4.72 | 188135.50 | 3.64 | 244417.87 | 4.73
Loud$\\_$coringa | 271837.4 | 5.26 | 256230.97 | 4.96 | 252097.30 | 4.88 | 258610.57 | 5.00
Roshtein | 271837.4 | 5.26 | 3409.45 | 0.07 | 5992.60 | 0.11 | 3537.18 | 0.06
Rubius | 271837.4 | 5.26 | 204089.17 | 3.95 | 253755.40 | 4.91 | 198031.44 | 3.83
Tarik | 271837.4 | 5.26 | 346841.69 | 6.72 | 219131.10 | 4.24 | 351297.07 | 6.80
TheGrefg | 271837.4 | 5.26 | 166702.99 | 3.23 | 141868.60 | 2.75 | 160103.74 | 3.10
XQc | 271837.4 | 5.26 | 550482.98 | 10.66 | 610960.50 | 11.83 | 558442.00 | 10.81
Table 4: Relevance of streamers according to the proposed indicators.

At this point, a question that arises is how the relevance of each streamer
can be translated into revenue allocation. One way to do this is to distribute
the part of the total revenue from subscription fees that the platform
dedicates to rewarding streamers in proportion to the relevance of the
platform’s success that the indicators attribute to them. Note that this form
of distribution will not modify in any way the structure of the relevance
indicators presented, since the percentages obtained in Table 4 have been
obtained using these indicators. In this sense, the comparative analysis
carried out in the previous paragraphs on the indicators would also be valid
for the allocation of revenue among streamers in proportion to these
indicators.

##  7 Concluding remarks

In recent years, streaming services have risen significantly in popularity. In
return for a subscription, consumers have unlimited access to the catalog of
services that platforms offers, such as movies, TV shows, music, and books.
Evidently, not all services are equally relevant for attracting users’
interest to pay for a subscription. In this paper, we have presented a model
to measure the relevance of each service with respect to the global success of
the platform. We have addressed this problem using an axiomatic approach, and
we have provided normative foundations for four basic indicators: the
_uniform_ , _subscriber-uniform_ , _proportional_ , and _subscriber-
proportional_ indicators. These four metrics are characterized by means of
several axioms, which represent different principles of fairness and
stability. Table 4 summarizes the properties fulfilled by each indicator and
the ones that are necessary for the characterization. Moreover, different
issues related to information, structure of the indicators, and axioms have
been analyzed to establish their importance in the context of streaming
platforms. This analysis also includes an understanding of which relevance
indicators may be preferred by streamers and the marginal impact on a
streamer’s relevance when viewing time increases (or decreases). We have also
discussed the question of which indicators are the most interesting for the
platform.

In the particular context of music streaming platforms and the distribution of
revenues among artists, two mechanisms have emerged as focal allocation rules:
the _pro-rata_ and the _user-centric_ rules (see Meyn et al. (2023)). In our
model, the _pro-rata_ and _user-centric_ schemes coincide with the
_proportional_ and _subscriber-proportional_ indicators, respectively.
Therefore, Theorems 1 and 2 also reveal characterizations of these two
allocation rules. In other words, based on principles of fairness, stability,
and non-manipulability, we provide normative foundations for the two most
prominent revenue allocation mechanisms in the music streaming market.

To illustrate the performance of the four indicators, we have analyzed the
streaming platform Twitch. Over a three-week period, we recorded the number of
viewers and viewing times of the 19 most-followed Twitch streamers worldwide.
Based on the normative approach presented in Section 4, we have concluded that
the mere number of viewers is not a suitable metric to capture the impact of
each streamer. We have also found that exclusive viewers (i.e., users who
consume the content of a unique streamer) have a higher influence in the
_subscriber-proportional indicator_ than in the _proportional indicator_.

We acknowledge that there are still some issues that deserve a deeper
analysis. This paper focuses on measuring the influence of each service from
the point of view of consumption. However, there are cases where the
perspective of the production should also be considered. For instance,
considering TV series, the viewers’ reaction is important, but the production
cost cannot be ignored. In this respect, a natural extension of the proposed
model is to define indicators that consider both parts of the market. The
literature on resource allocation and attribution problems is quite extensive,
and other metrics and axioms can also be explored.

## References

  * Adams and Yellen (1976) Adams, W. J. and Yellen, J. L. (1976).  Commodity bundling and the burden of monopoly.  _The Quarterly Journal of Economics_ , 90:475–497.
  * Alaei et al. (2022) Alaei, S., Makhdoumi, A., Malekian, A., and Pekec̆, S. (2022).  Revenue-Sharing Allocation Strategies for Two-Sided Media Platforms: Pro-Rata vs. User-Centric.  _Management Science_ , 68:8699–8721.
  * Albino et al. (2008) Albino, D., Scaruffi, P., Moretti, S., Coco, S., Truini, M., Cristofano, C. D., Cavazzana, A., Stigliani, S., Bonassi, S., and Tonini, G. P. (2008).  Identification of low intratumoral gene expression heterogeneity in neuroblastic tumors by genome-wide expression analysis and game theory.  _Cancer_ , 113:1412–1422.
  * Algaba et al. (2019b) Algaba, E., Fragnelli, V., Llorca, N., and Sánchez-Soriano, J. (2019).  Horizontal cooperation in a multimodal public transport system: The profit allocation problem.  _European Journal of Operational Research_ , 275:659–665..
  * Algaba et al. (2019) Algaba, E., Fragnelli, V. and Sánchez-Soriano, J. (2022).  Handbook of the Shapley value.  _CRC Press_.
  * Bergantiños and Moreno-Ternero (2015) Bergantiños, G. and Moreno-Ternero, J. D. (2015).  The axiomatic approach to the problem of sharing the revenue from museum passes.  _Games and Economic Behavior_ , 89:78–92.
  * Bergantiños and Moreno-Ternero (2023) ——— (2023).  An axiomatic approach to the revenue sharing in the streaming industry.  _22nd annual SAET Conference Paris, France_.
  * Brander et al. (2011) Brander, K., Bruno, J., Hobday, A., and Schoeman, D. (2011).  The value of attribution.  _Nature Climate Change_ , 1:70–71.
  * Burger et al. (2020) Burger, M., Wentz, J., and Horton, R. (2020).  The law and science of climate change attribution.  _Columbia Journal of Environmental Law_ , 5:57–240.
  * Dimont (2017) Dimont, J. (2017).  Royalty inequity: Why music streaming services should switch to a per-subscriber model.  _CU Hastings Law Journal_ , 69:675–700.
  * Ginsburgh and Zang (2003) Ginsburgh, V. and Zang, I. (2003).  The museum pass game and its value.  _Games and Economic Behavior_ , 43:322–325.
  * Jari (2018) Jari, M. (2018).  Pro-rata and user-centric distribution models: A comparative study.  Technical report.
  * Ju et al. (2007) Ju, B. G., Miyagawa, E., and Sakai, T. (2007).  Non-manipulable division rules in claim problems and generalizations.  _Journal of Economic Theory_ , 132:1–16.
  * López-Navarrete et al. (2019) López-Navarrete, F., Sánchez-Soriano, J., and Bonastre, O. M. (2019).  Allocating revenues in a Smart TV ecosystem.  _International Transactions in Operational Research_ , 26:1611–1632.
  * López-Navarrete et al. (2023) ——— (2023).  Dynamic generation and attribution of revenues in a video platform.  ArXiv, 4859841 [math.OC].
  * Lucchetti et al. (2010) Lucchetti, R., Moretti, S., Patrone, F., and Radrizzani, P. (2010).  The Shapley and Banzhaf values in microarray games.  _Computers & Operations Research_, 37:1406–1412.
  * Manshadi et al. (2023) Manshadi, V., Niazadeh, R., and Rodilitz, S. (2023).  Fair dynamic rationing.  _Management Science_ , forthcoming.
  * Martínez and Moreno-Ternero (2022) Martínez, R. and Moreno-Ternero, J. D. (2022).  An axiomatic approach towards pandemic performance indicators.  _Economic Modelling_ , 116:105983.
  * Martínez and Sánchez-Soriano (2021) Martínez, R. and Sánchez-Soriano, J. (2021).  Mathematical indices for the influence of risk factors on the lethality of a disease.  _Journal of Mathematical Biology_ , 83:74.
  * Martínez and Sánchez-Soriano (2023) Martínez, R. and Sánchez-Soriano, J. (2023).  Order preservation with dummies in the museum pass problem.  _Annals of Operations Research_ , forthcoming.
  * Martínez et al. (2022) Martínez, R., Sánchez-Soriano, J., and Llorca, N. (2022).  Assessments in public procurement procedures.  _Omega_ , 111:102660.
  * Meyn et al. (2023) Meyn, J., Kandziora, M., Albers, S., and Clement, M. (2023).  Consequences of platforms’ remuneration models for digital content: initial evidence and a research agenda for streaming services.  _Journal of the Academy of Marketing Science_ , 51:114–131.
  * Moretti et al. (2007) Moretti, S., Patrone, F., and Bonassi, S. (2007).  The class of microarray games and the relevance index for genes.  _TOP_ , 15:256–280.
  * O’Neill (1982) O’Neill, B. (1982).  A problem of rights arbitration from the Talmud.  _Mathematical social sciences_ , 2:345–371.
  * Page and Safir (2018a) Page, W. and Safir, D. (2018a).  Money in, money out: lessons from cmos in allocating and distributing licensing revenue.  Technical report.
  * Page and Safir (2018b) ——— (2018b).  User-centric revisited: The unintended consequences ofroyalty distribution.  Technical report.
  * Roth (1988) Roth, A. E. (1988).  The Shapley value: essays in honor of Lloyd S. Shapley.  _Cambridge University Press_.
  * Shapley (1953) Shapley, L.S. (1953).  A value for n-person games.  _Contributions to the Theory of Games_ , 2:307–317.
  * Singal et al. (2022) Singal, R., Besbes, O., Desir, A., Goyal, V., and Iyengar, G. (2022).  Shapley meets uniform: An axiomatic framework for attribution in online advertising.  _Management Science_ , 68:7457–7479.

## Appendix A

Proof of properties in Table 2.

The uniform indicator satisfies _strong symmetry_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that $C_{is}\cdot C_{js}\neq 0$
or $C_{is}=C_{js}=0$ for all $s\in S$. Then we have
$R^{U}_{i}(N,S,p,C)=\frac{\sigma}{n}=R^{U}_{j}(N,S,p,C)$.

The uniform indicator satisfies _composition_. Let
$(N,S,p,C),(N,S^{\prime},p^{\prime},C^{\prime})\in\mathcal{D}$ such that
$S\cap S^{\prime}=\emptyset$. Let $i\in N$. Then we have that

$\displaystyle R_{i}^{U}(N,S\cup S^{\prime},p\oplus p^{\prime},C\oplus
C^{\prime})$ $\displaystyle=\sum_{s\in S\cup S^{\prime}}\dfrac{(p\oplus
p^{\prime})_{s}}{|N|}$ $\displaystyle=\sum_{s\in
S}\dfrac{p_{s}}{|N|}+\sum_{s\in S^{\prime}}\dfrac{p^{\prime}_{s}}{|N|}$
$\displaystyle=R_{i}^{U}(N,S,p,C)+R_{i}^{U}(N,S^{\prime},p^{\prime},C^{\prime}).$

The uniform indicator satisfies _sharing proofness_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $S^{\prime}\subseteq S$ and $s\in S^{\prime}$
such that $p^{\prime}_{s}=\sum_{t\in S^{\prime}}p_{t}$ and
$C^{\prime}_{is}=\sum_{t\in S^{\prime}}C_{it}$ for all $i\in N$ and
$C^{\prime}_{is}=C_{is}$ for all $i\in N$ and $s\in S\backslash S^{\prime}$.
Then

$\displaystyle R^{U}_{i}\left(N,\\{s\\}\cup S\backslash
S^{\prime},(p^{\prime}_{s},p_{S\backslash S^{\prime}}),C^{\prime}\right)$
$\displaystyle=\dfrac{p^{\prime}_{s}+\sum_{t\in S\backslash
S^{\prime}}p_{t}}{|N|}$ $\displaystyle=\dfrac{\sum_{t\in
S^{\prime}}p_{t}+\sum_{t\in S\backslash S^{\prime}}p_{t}}{|N|}$
$\displaystyle=\dfrac{\sum_{t\in S}p_{t}}{|N|}$
$\displaystyle=R^{U}_{i}(N,S,p,C).$

The uniform indicator does not satisfy _non-manipulabity_. Consider a platform
where $N=\\{1,2,3\\}$, $S=\\{1\\}$, $p_{1}=1$, and $C$ is given by

$\left(\begin{array}[]{cccccc}0\\\ 1\\\ 2.5\\\ \end{array}\right).$

The uniform indicator is equal to
$R^{U}_{i}(N,S,p,C)=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)$. Now,
consider $N^{\prime}=\\{1,2\\}$ and $C^{\prime}_{11}=C_{11}+C_{12}=1$. Then

$R_{1}\left(\\{1\\}\cup N\backslash
N^{\prime},S,p,(C^{\prime}_{1\cdot},C_{N\backslash
N^{\prime}})\right)=\frac{1}{2}\neq\sum_{j\in
N^{\prime}}R_{j}(N,S,p,C)=\frac{2}{3}.$

The uniform indicator does not satisfy _nullity_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i\in N$ such that $C_{is}=0$ for all $s\in S$.
Then

$R^{U}_{i}(N,S,p,C)=\frac{\sigma}{|N|}\neq 0.$

The subscriber-uniform indicator satisfies _symmetry_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that $C_{is}=C_{js}$ for all
$s\in S$. Then $i\in N_{s}$ if and only if $j\in N_{s}$. Therefore, we have
$R^{SU}_{i}(N,S,p,C)=R^{SU}_{j}(N,S,p,C)$.

The subscriber-uniform satisfies _homogeneity_. Let $(N,S,p,C)\in\mathcal{D}$
and $\lambda\in\mathbb{R}_{++}$. Let $i\in N$. We have that

$\displaystyle R_{i}^{SU}(N,S,p,\lambda C)$ $\displaystyle=\sum_{s\in S,i\in
N_{s}}\dfrac{1}{\left|N_{s}\right|}p_{s}=R_{i}^{P}(N,S,p,C),$

where $N_{s}=\left\\{j\in N:C_{js}\neq 0\right\\}$.

The subscriber-uniform indicator does not satisfy _consumption sensitivity_.
Let us consider the following two successions of consumption matrices:

$C^{1n}=\left(\begin{array}[]{c}\frac{1}{n}\\\ \frac{1}{n}\\\ \frac{1}{n}\\\
\end{array}\right)\text{ and }C^{2n}=\left(\begin{array}[]{c}0\\\
\frac{1}{n}\\\ \frac{2}{n}\\\ \end{array}\right).$

It is obvious that $\|C^{1n}-C^{2n}\|$ goes to $0$ when $n$ goes to infinity,
but
$\|R^{SU}(N,S,p,C^{1n})-R^{SU}(N,S,p,C^{2n})\|=\|(\frac{p}{3},\frac{p}{3},\frac{p}{3})-(0,\frac{p}{2},\frac{p}{2})\|=\frac{2p}{3}>>0$
for all $n$.

The subscriber-uniform indicator satisfies _composition_. Let
$(N,S,p,C),(N,S^{\prime},p^{\prime},C^{\prime})\in\mathcal{D}$ such that
$S\cap S^{\prime}=\emptyset$. Let $i\in N$. We have that

$\displaystyle R_{i}^{SU}(N,S\cup S^{\prime},p\oplus p^{\prime},C\oplus
C^{\prime})$ $\displaystyle=\sum_{s\in S\cup
S^{\prime}}\dfrac{1}{\left|N_{s}\right|}(p\oplus p^{\prime})_{s}$
$\displaystyle=\sum_{s\in S}\dfrac{1}{\left|N_{s}\right|}p_{s}+\sum_{s\in
S^{\prime}}\dfrac{1}{\left|N_{s}\right|}p^{\prime}_{s}$
$\displaystyle=R_{i}^{SU}(N,S,p,C)+R_{i}^{SU}(N,S^{\prime},p^{\prime},C^{\prime}).$

The subscriber-uniform indicator does not satisfy _sharing proofness_.
Consider a platform where $N=\\{1,2,3\\}$, $S=\\{1,2\\}$, $p=(1,1)$, and $C$
is given by

$\left(\begin{array}[]{cccccc}0&2\\\ 1&0\\\ 1&0\\\ \end{array}\right).$

Then $R^{SU}(N,S,p,C)=\left(1,\frac{1}{2},\frac{1}{2}\right)$. Let
$S^{\prime}=\\{1,2\\}\subseteq S$ and $s=1\in S^{\prime}$ such that
$p^{\prime}_{s}=\sum_{t\in S^{\prime}}p_{t}$ and $C^{\prime}_{is}=\sum_{t\in
S^{\prime}}C_{it}$ for all $i\in N$. Then
$R^{SP}(N,S^{\prime},p^{\prime},C^{\prime})=\left(\frac{2}{3},\frac{2}{3},\frac{2}{3}\right)\neq
R^{SP}(N,S,p,C)$.

The proportional indicator satisfies _symmetry_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that $C_{is}=C_{js}$ for all
$s\in S$. We have that

$\displaystyle R_{i}^{P}(N,S,p,C)$
$\displaystyle=\frac{\|C_{i\cdot}\|}{\displaystyle\sum_{t\in
N}\|C_{t\cdot}\|}\sigma=\frac{\sum_{s\in S}C_{is}}{\displaystyle\sum_{t\in
N}\|C_{t\cdot}\|}\sigma=\frac{\sum_{s\in S}C_{js}}{\displaystyle\sum_{t\in
N}\|C_{t\cdot}\|}\sigma=\frac{\|C_{j\cdot}\|}{\displaystyle\sum_{t\in
N}\|C_{t\cdot}\|}\sigma=R_{j}^{P}(N,S,p,C).$

The proportional indicator does not satisfy _strong symmetry_. Consider a
platform where $N=\\{1,2,3\\}$, $S=\\{1\\}$, $p_{1}$, where $C$ is given by

$\left(\begin{array}[]{cccccc}0\\\ 1\\\ 2\\\ \end{array}\right).$

Then $C_{21}\cdot C_{32}\neq 0$ and
$R_{2}^{P}(N,S,p,C)=\frac{p_{1}}{3}\neq\frac{2p_{1}}{3}=R_{3}^{P}(N,S,p,C)$.

The proportional indicator satisfies _homogeneity_. Let
$(N,S,p,C)\in\mathcal{D}$ and $\lambda\in\mathbb{R}_{++}$. Let $i\in N$. We
have that

$\displaystyle R_{i}^{P}(N,S,p,\lambda C)$ $\displaystyle=\frac{\|\lambda
C_{i\cdot}\|}{\displaystyle\sum_{j\in N}\|\lambda
C_{j\cdot}\|}\sigma=\frac{\lambda\|C_{i\cdot}\|}{\displaystyle\sum_{j\in
N}\lambda\|C_{j\cdot}\|}\sigma=R_{i}^{P}(N,S,p,C).$

The proportional indicator does not satisfy _composition_. Consider a platform
where $N=\\{1,2,3\\}$, $S=\\{1\\}$, $S^{\prime}=\\{2\\}$, $p_{1}=1$, where
$C^{1}$ and $C^{2}$ are given by

$\left(\begin{array}[]{cccccc}0\\\ 1\\\ 1\\\ \end{array}\right),$
$\left(\begin{array}[]{cccccc}2\\\ 1\\\ 1\\\ \end{array}\right).$

Then $R^{P}(N,\\{1\\},p_{1},C^{1})=\left(0,\frac{1}{2},\frac{1}{2}\right)$ and
$R^{P}(N,\\{2\\},p_{2},C^{2})=\left(\frac{1}{2},\frac{1}{4},\frac{1}{4}\right)$.
By another way, $R^{P}(N,\\{1\\}\cup\\{2\\},p_{1}\oplus p_{2},C^{1}\oplus
C^{2})=\left(\frac{2}{3},\frac{2}{3},\frac{2}{3}\right)\neq\left(\frac{1}{2},\frac{3}{4},\frac{3}{4}\right)=R^{P}(N,\\{1\\},p_{1},C^{1})+R^{P}(N,\\{2\\},p_{2},C^{2})$.

The proportional indicator satisfies _nullity_. Let $(N,S,p,C)\in\mathcal{D}$.
Let $i\in N$ such that $C_{is}=0$ for all $s\in S$. Then

$R^{P}_{i}(N,S,p,C)=\frac{\|C_{i\cdot}\|}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma=\frac{\sum_{s\in S}C_{is}}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma=0.$

The subscriber-proportional indicator satisfies _symmetry_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i,j\in N$ such that $C_{is}=C_{js}$ for all
$s\in S$. We have that

$\displaystyle R_{i}^{SP}(N,S,p,C)$ $\displaystyle=\sum_{s\in
S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}=\sum_{s\in S}\dfrac{C_{js}}{\|C_{\cdot
s}\|}p_{s}=R_{j}^{SP}(N,S,p,C).$

The subscriber-proportional indicator does not satisfy _strong symmetry_.
Consider a platform where $N=\\{1,2,3\\}$, $S=\\{1\\}$, $p_{1}$, where $C$ is
given by

$\left(\begin{array}[]{cccccc}0\\\ 1\\\ 2\\\ \end{array}\right).$

Then $C_{21}\cdot C_{32}\neq 0$ and
$R_{2}^{SP}(N,S,p,C)=\frac{p_{1}}{3}\neq\frac{2p_{1}}{3}=R_{3}^{SP}(N,S,p,C)$.

The subscriber-proportional indicator satisfies _homogeneity_. Let
$(N,S,p,C)\in\mathcal{D}$ and $\lambda\in\mathbb{R}_{++}$. Let $i\in N$. We
have that

$\displaystyle R_{i}^{SP}(N,S,p,\lambda C)$ $\displaystyle=\sum_{s\in
S}\dfrac{\lambda C_{is}}{\|\lambda C_{\cdot s}\|}p_{s}=\sum_{s\in
S}\dfrac{\lambda C_{is}}{\lambda\|C_{\cdot s}\|}p_{s}=R_{i}^{SP}(N,S,p,C).$

The subscriber-proportional indicator does not satisfy _consumption
sensitivity_. Let us consider the following two successions of consumption
matrices:

$C^{1n}=\left(\begin{array}[]{c}\frac{1}{n}\\\ \frac{1}{n}\\\ \frac{1}{n}\\\
\end{array}\right)\text{ and }C^{2n}=\left(\begin{array}[]{c}0\\\
\frac{1}{n}\\\ \frac{2}{n}\\\ \end{array}\right).$

It is obvious that $\|C^{1n}-C^{2n}\|$ goes to $0$ when $n$ goes to infinity,
but
$\|R^{SP}(N,S,p,C^{1n})-R^{SP}(N,S,p,C^{2n})\|=\|(\frac{p}{3},\frac{p}{3},\frac{p}{3})-(0,\frac{p}{3},\frac{2p}{3})\|=\frac{2p}{3}>>0$
for all $n$.

The subscriber-proportional indicator does not satisfy _sharing proofness_.
Consider a platform where $N=\\{1,2,3\\}$, $S=\\{1,2\\}$, $p=(1,1)$, and $C$
is given by

$\left(\begin{array}[]{cccccc}0&2\\\ 1&1\\\ 1&1\\\ \end{array}\right)$

Then, $R^{SP}(N,S,p,C)=\left(\frac{1}{2},\frac{3}{4},\frac{3}{4}\right)$. Let
$S^{\prime}=\\{1,2\\}\subseteq S$ and $s=1\in S^{\prime}$ such that
$p^{\prime}_{s}=\sum_{t\in S^{\prime}}p_{t}$ and $C^{\prime}_{is}=\sum_{t\in
S^{\prime}}C_{it}$ for all $i\in N$, then
$R^{SP}(N,S^{\prime},p^{\prime},C^{\prime})=\left(\frac{2}{3},\frac{2}{3},\frac{2}{3}\right)\neq
R^{SP}(N,S,p,C)$.

The subscriber-proportional indicator satisfies _nullity_. Let
$(N,S,p,C)\in\mathcal{D}$. Let $i\in N$ such that $C_{is}=0$ for all $s\in S$,
then

$R^{SP}_{i}(N,S,p,C)=\sum_{s\in S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}=0.$

##  8 Appendix B

Proof of Lemma 1.

###### Proof.

Let $(N,S,p,C)\in\mathcal{D}$, and suppose we have a streamer $i\in N$ such
that

$\displaystyle\sum_{s\in S,i\in
N_{s}}\frac{\left(|N|-|N_{s}|\right)}{|N_{s}|}p_{s}\geq\sum_{s\in S,i\notin
N_{s}}p_{s}.$

Then we have that

$\displaystyle R_{i}^{U}(N,S,p,C)$ $\displaystyle=\sum_{s\in
S}\frac{1}{|N|}p_{s}=\sum_{s\in S,i\in N_{s}}\frac{1}{|N|}p_{s}+\sum_{s\in
S,i\notin N_{s}}\frac{1}{|N|}p_{s}$ $\displaystyle\leq\sum_{s\in S,i\in
N_{s}}\frac{1}{|N|}p_{s}+\sum_{s\in S,i\in
N_{s}}\frac{\left(|N|-|N_{s}|\right)}{|N||N_{s}|}p_{s}$
$\displaystyle=\sum_{s\in S,i\in
N_{s}}\frac{\left(|N_{s}|+|N|-|N_{s}|\right)}{|N||N_{s}|}p_{s}$
$\displaystyle=\sum_{s\in S,i\in
N_{s}}\frac{1}{|N_{s}|}p_{s}=R_{i}^{SU}(N,S,p,C).$

∎

Proof of Lemma 2.

###### Proof.

Let $(N,S,p,C)\in\mathcal{D}$, and suppose we have a streamer $i\in N$ such
that $C_{is}>0$ for all $s\in S$. Then we have that

$\displaystyle R_{i}^{SU}(N,S,p,C)$ $\displaystyle=\sum_{s\in S,i\in
N_{s}}\frac{1}{|N_{s}|}p_{s}=\sum_{s\in S}\frac{1}{|N_{s}|}p_{s}\geq\sum_{s\in
S}\frac{1}{|N|}p_{s}=R_{i}^{SU}(N,S,p,C).$

∎

Proof of Lemma 3.

###### Proof.

Let $(N,S,p,C)\in\mathcal{D}$, and suppose we have a streamer $i\in N$ such
that $\|C_{i\cdot}\|\geq\frac{\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{|N|-1}$. Then we have that

$\displaystyle R^{P}_{i}(N,S,p,C)$
$\displaystyle=\frac{\|C_{i\cdot}\|}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma\geq\frac{\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{|N|-1}}{\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma=\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\left(|N|-1\right)\displaystyle\sum_{j\in
N}\|C_{j\cdot}\|}\sigma$ $\displaystyle=\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\left(|N|-1\right)\displaystyle\left(\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|+\|C_{i\cdot}\|\right)}\sigma$
$\displaystyle\geq\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\left(|N|-1\right)\displaystyle\left(\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|+\frac{\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{|N|-1}\right)}\sigma$
$\displaystyle=\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\left(|N|-1\right)\displaystyle\left(\frac{|N|}{|N|-1}\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|\right)}\sigma$
$\displaystyle=\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}{\displaystyle|N|\sum_{j\in
N\backslash\\{i\\}}\|C_{j\cdot}\|}\sigma=\frac{\sigma}{|N|}=R_{i}^{U}(N,S,p,C).$

∎

Proof of Lemma 4.

###### Proof.

Let $(N,S,p,C)\in\mathcal{D}$, and suppose we have a streamer $i\in N$ such
that $C_{is}\geq\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}}{|N|-1}$ for all $s\in S$. Then we have that

$\displaystyle R^{SP}_{i}(N,S,p,C)$ $\displaystyle=\sum_{s\in
S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}=\sum_{s\in
S}\dfrac{C_{is}}{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}+C_{is}}p_{s}$ $\displaystyle\geq\sum_{s\in
S}\dfrac{\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}}{|N|-1}}{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}+\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}}{|N|-1}}p_{s}$ $\displaystyle=\sum_{s\in
S}\dfrac{\frac{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}}{|N|-1}}{\frac{|N|}{|N|-1}\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}}p_{s}=\sum_{s\in
S}\frac{1}{|N|}p_{s}=R_{i}^{SU}(N,S,p,C).$

∎

Proof of Lemma 5.

###### Proof.

Let $(N,S,p,C)\in\mathcal{D}$, and suppose we have a streamer $i\in N$ such
that $C_{is}\geq\frac{\sum_{j\in N_{s}\backslash\\{i\\}}C_{js}}{|N_{s}|-1}$
for all $s\in S$ with $i\in N_{s}$. Then we have that

$\displaystyle R^{SP}_{i}(N,S,p,C)$ $\displaystyle=\sum_{s\in
S}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}=\sum_{s\in S,i\in
N_{s}}\dfrac{C_{is}}{\|C_{\cdot s}\|}p_{s}=\sum_{s\in S,i\in
N_{s}}\dfrac{C_{is}}{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}+C_{is}}p_{s}$ $\displaystyle\geq\sum_{s\in S,i\in
N_{s}}\dfrac{\frac{\displaystyle\sum_{j\in
N_{s}\backslash\\{i\\}}C_{js}}{|N_{s}|-1}}{\displaystyle\sum_{j\in
N\backslash\\{i\\}}C_{js}+\frac{\displaystyle\sum_{j\in
N_{s}\backslash\\{i\\}}C_{js}}{|N_{s}|-1}}p_{s}$ $\displaystyle=\sum_{s\in
S,i\in N_{s}}\dfrac{\frac{\displaystyle\sum_{j\in
N_{s}\backslash\\{i\\}}C_{js}}{|N_{s}|-1}}{\frac{|N_{s}|}{|N_{s}|-1}\displaystyle\sum_{j\in
N_{s}\backslash\\{i\\}}C_{js}}p_{s}=\sum_{s\in S,i\in
N_{s}}\frac{1}{|N_{s}|}p_{s}=R_{i}^{SU}(N,S,p,C).$

∎

Proof of Lemma 6.

###### Proof.

Let $(N,S,p,C),(N^{\prime},S,p,C^{\prime})\in\mathcal{D}$, let $i\in N$ where
$N^{\prime}=N\backslash\\{i\\}\cup\\{i^{\prime}_{1},\ldots,i^{\prime}_{m}\\}$
with $m\geq 2$ and $C_{is}=\sum_{l=1}^{m}C^{\prime}_{i^{\prime}_{l}s}$ and
$C_{js}=C^{\prime}_{js}$ for all $j\in N\backslash\\{i\\}$ and $s\in S$. Let
us first see that
$\sum_{l=1}^{m}R^{U}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{U}_{i}(N,S,p,C)$. We have that

$\displaystyle\sum_{l=1}^{m}R^{U}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{U}_{i}(N,S,p,C)\Leftrightarrow\frac{m\sigma}{|N|-1+m}\geq\frac{\sigma}{|N|}\Leftrightarrow|N|m\geq|N|+m-1\Leftrightarrow|N|\geq
1.$

Now, let us see that
$\sum_{l=1}^{m}R^{SU}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{SU}_{i}(N,S,p,C)$. We have that

$\displaystyle\sum_{l=1}^{m}R^{SU}_{i^{\prime}_{l}}(N^{\prime},S,p,C^{\prime})\geq
R^{U}_{i}(N,S,p,C)\Leftrightarrow\sum_{l=1}^{m}\sum_{s\in S,i^{\prime}_{l}\in
N^{\prime}_{s}}\frac{1}{|N^{\prime}_{s}|}p_{s}\geq\sum_{s\in S,i\in
N_{s}}\frac{1}{|N_{s}|}p_{s},$

where $N_{s}=\\{j\in N:C_{js}>0\\}$ and $N^{\prime}_{s}=\\{j\in
N^{\prime}:C^{\prime}_{js}>0\\}$ for all $s\in S$. Then it is sufficient to
prove that if $i\in N_{s}$, then there exists at least one $i^{\prime}_{l}\in
N^{\prime}_{s}$ for all $s\in S$. Since
$C_{is}=\sum_{l=1}^{m}C^{\prime}_{i^{\prime}_{l}s}$ for all $s\in S$ if
$C_{is}>0$ then $C^{\prime}_{i^{\prime}_{l}s}>0$ for some
$l\in\\{1,\ldots,m\\}$, and the proof is concluded. ∎