# Revenue sharing at music streaming platforms††thanks: Financial support from
Ministerio de Ciencia e Innovación, MCIN/AEI/ 10.13039/501100011033, through
grants PID2020-113440GBI00 and PID2020-115011GB-I00, Junta de Andalucía
through grant P18-FR-2933, and Xunta de Galicia through grant ED431B2022/03 is
gratefully acknowledged. We thank audiences at Paris, Istanbul and Oviedo for
helpful comments and suggestions.

Gustavo Bergantiños
Juan D. Moreno-Ternero ECOBAS, Universidade de Vigo, ECOSOT, 36310 Vigo,
EspañaDepartment of Economics, Universidad Pablo de Olavide, 41013 Sevilla,
España<EMAIL_ADDRESS>
###### Abstract

We study the problem of sharing the revenues raised from subscriptions to
music streaming platforms among content providers. We provide direct,
axiomatic and game-theoretical foundations for two focal (and somewhat polar)
methods widely used in practice: pro-rata and user-centric. The former rewards
artists proportionally to their number of total streams. With the latter, each
user’s subscription fee is proportionally divided among the artists streamed
by that user. We also provide foundations for a family of methods compromising
among the previous two, which addresses the rising concern in the music
industry to explore new streaming models that better align the interests of
artists, fans and streaming services.

JEL numbers: D63, C71, L82, O34.

Keywords: Streaming, revenue allocation, axioms, cooperative games, pro-rata,
user-centric.

##  1 Introduction

Steve Jobs unveiled the iPod on October 23, 2001. As he put it himself a few
years later, that event didn’t just change the way we all listen to music, it
changed the entire music industry. And it is difficult to deny such a
statement. Because digital music had started to emerge at the end of the 20th
century, mostly thanks to the advent of peer-to-peer (P2P) technologies. But
successful file-sharing platforms, such as Napster, were under scrutiny by the
music industry, which aggressively pursued stronger copyright enforcement and
regulations (e.g., Bhattacharjee et al., 2007). In that environment, Apple
managed to persuade record companies to sale individual tracks for 99 cents.
Gradually, the industry found a new way to stay profitable and even embraced
new technology advances like streaming. A decade later, Spotify, an early
forerunner music streaming platform, announced a customer base of 1 million
paying subscribers across Europe, and was officially launched in the US.
Alternative services (such as Beats Music, Amazon Music Unlimited and Google
Play Music All-Access, as well as a new Apple Music service) also emerged in
the market. According to Statista, revenue in the Music Streaming market is
projected to reach US$\$25.84$ billions in 2023. The number of users is
expected to amount to $1.1$ billion by
2027.111https://www.statista.com/outlook/dmo/digital-media/digital-
music/music-streaming/worldwide

Streaming platforms generate massive amounts of revenue nowadays. Aleei et al.
(2022) argue that “streaming services generated $\$4.3$ billion in the first
half of 2019 (with $77\%$ of that coming from paid subscriptions)”. Users
typically pay a fixed (monthly) amount to freely access their libraries. A
common practice for platforms is to distribute around $70\%$ of the revenue
received from subscriptions among artists (e.g., Meyn et al., 2023). Platforms
also raise money from other sources (for instance, advertisements) but the
most important source are subscriptions and we shall concentrate on them here.
An ensuing interesting problem is to allocate the corresponding part of those
revenues among participating artists, based on their streaming times. This
will be the object of study in this paper.

We introduce a stylized model, whose ingredients are three: a group of
artists, a group of users and a matrix indicating the streaming times each
user played each artist. Based on this input, a popularity index, which
measures the importance of each artist, is constructed. The reward received by
each artist from the revenues generated in each problem is based on such a
popularity index. The most frequent indices are the so called pro-rata index,
which renders artists rewarded proportionally to the total number of streams
and the so called user-centric index, which renders artists rewarded so that
the revenue generated by each user is divided among the artists listened by
the user proportionally to the total number of streams.222Typically, a user
pays around $\$10$ per month. As 70% of the amount generated by users is
devoted to pay artists, we could just consider in our model each user pays
$\$7$. For ease of exposition, we shall assume that each user pays a
normalized amount of $1$ to have unlimited access to the contents in the
streaming platform, during the period of time (usually, a month). Note also
that, in practice, it is considered that an artist obtained a streaming unit
when a user played a song from that artist for at least 30 seconds (e.g., Meyn
et al., 2023). Thus, the entry of the (streaming) matrix $t_{ij}$ would be the
number of streaming units artist $i$ obtained from user $j$ during one month.
There is, nevertheless, a debate in the industry about how streams should be
computed. Another alternative is a remuneration based on per-second usage
(e.g., Meyn et al., 2023). Our theoretical model is general and allows us to
accommodate other ways of measuring streams. Then, the amount received by each
artist is computed by adding the amounts obtained from each user.

We take several approaches to analyze our model. In the first (axiomatic)
approach, we present axioms that formalize normatively appealing principles
for popularity indices. Some convey structural ideas that reflect operational
features of the index. For instance, additivity says that if we can present a
problem as the sum of smaller problems (such as in, say, multinational
platforms), then the index in the original problem should coincide with the
sum of the indices in the smaller problems. And homogeneity says that the
index should reflect accordingly the cases in which each user has reproduced
content from a given artist a certain times more than content from another
artist. Other axioms (equal individual impact of similar users and equal
global impact of users) model alternative forms of marginalism, i.e., the
impact of extra users in the platform. And we also consider axioms reflecting
concerns for fairmess: reasonable lower bounds and click-fraud-proofness. The
first one states that artists should at least receive the amount paid by the
users that only played content provided by them. The second one states that if
a user changes streaming times, then the amount received by each artist could
not change more than the subscription paid by that user. We explore how the
main indices perform with respect to all these axioms, but our main result in
this axiomatic approach to the problem (Theorem 1) states three
characterization results. First, the combination of additivity, homogeneity
and equal individual impact of similar users characterizes the pro-rata index.
Second, the combination of additivity, homogeneity and equal global impact of
users characterizes the user-centric index. Third, the combination of
additivity and homogeneity characterizes a family of weighted indices that
compromise between the pro-rata index and the user-centric index, upon
assigning each artist the weighted aggregation of streamings, with the weight
depending on the user and her streaming profile.

We then take two other indirect approaches to deal with streaming problems. In
both of them, streaming problems are associated to other problems that have
already been studied in the literature, and we then import well-known
solutions for those problems to solve streaming problems.

First, we associate each streaming problem with a cooperative game with
transferable utility, where the worth of each set of artists is defined as the
amount paid by the users that have only played content from those artists.
Such a game is convex, i.e., the incentives to join a coalition increase as
the coalition grows. It is well known that the core of a convex game is quite
large and we fully characterize it in our Theorem 2. In words, a core
allocation imposes that the amount paid by each user is divided in any way
among the artists listened by this user. Each artist then receives the sum
(over all users) of the corresponding amounts. We show that the pro-rata index
does not always guarantee allocations within the core, whereas the user-
centric index always does so.

Second, we associate each streaming problem with a claims problem, where the
users are identified as the issues in the claims problem. Then, problems can
be solved in two stages. Either focussing on each issue (user) first and
agents (artists) afterwards, or viceversa. We show (Theorem 3) how the indices
we consider for streaming problems can be rationalized as two-stage claims
rules. More precisely, the pro-rata index and the user-centric index can be
rationalized as weighted proportional rules (although there is no bijection
between this family of rules and the weighted indices for streaming problems).
Furthermore, the allcation rules these two indices induce can also be
described as two-stage (claims) rules where we first decide the importance of
each user and then the importance of each artist for each user, which is
computed as the sum over all users.

###  1.1 Related literature

The closest paper to ours is Alaei et al. (2022). They also consider the same
streaming platforms that we consider here, which generate revenues by charging
users a subscription fee for unlimited access to the content and compensate
artists through an allocation rule. They also assume that users are
heterogeneous in both their overall consumption and the distribution of their
consumption over different artists, but they model this by referring to the
probability (per usage) that each user type wants to consume the content of
each artist. In our case, we talk about the number of times a user plays an
artist. But, leaving this minor aspect aside, our models are essentially
equivalent. They are also concerned with the pro-rata and user-centric revenue
allocation methods, but focus on characterizing when these two methods can
sustain a set of artists on the platform, as well as comparing them from both
the platform’s and the artists’ perspectives. In particular, they show that,
despite the cross-subsidization between low- and high-streaming-volume users,
the pro-rata method can be preferred by both the platform and the
artists.333Lei (2023) also obtains that the pro-rata method can outperform the
user-centric method in a simple model with only two artists. More precisely,
they show that artists who are predominantly listened to by users whose
overall streaming volume (consumption) is high receive higher payments with
the pro-rata allocation than with the user-centric allocation. Consequently,
if there is an artist who is extremely popular with users who have high
consumption, the pro-rata rule is preferred from the platform’s viewpoint
(actually, the platform might not even be profitable with the user-centric
rule in this scenario). More generally, the analysis in Alaei et al. (2022)
suggests that “pro-rata should be the artist’s compensation rule of choice if
the users’ behavior points to correlation between artists’ popularity and
consumption among users and/or if the platform might not be able to estimate
heterogeneity of artists’ popularity and consumption among its users”.444Alaei
et al. (2022) also obtain interesting computational results for this problem.
For instance, they show that the platform’s problem of selecting an optimal
portfolio of artists is NP-complete, but develop a polynomial time
approximation scheme for the optimal platform’s profit. We do not consider
computational aspects in this paper.

Our analysis complements Alaei et al. (2022) in several ways. We provide
normative foundations for both methods, as well as game-theoretical and other
indirect approaches to the problem. And we can safely conclude from our
analysis that there exist powerful arguments to prefer the user-centric method
to the pro-rata method. We acknowledge that the pro-rata method made sense
when the technology to divide revenue by individual users’ streams did not
exist. But that is no longer the case and, consequently, a user-centric method
that distributes revenue based on each listener’s individual streams seems to
be more reasonable (based on the arguments we provide in our analysis). But we
are not the first ones suggesting that. Haampland et al. (2022) resort to the
data provided by a leading music streaming platform in France (with 427
million streams coming from more than 140,000 unique users during six months).
They obtain from their empirical analysis that, when compared with the pro-
rata method, the user-centric method renders consumers’ choices better aligned
with revenue sharing, but also that it favors organic streams with respect to
curated streams, it reduces the superstar phenomenon, and it moderates the
potential bias in curated streams. Dimont (2017) also endorses switching from
a pro-rata method to a user-centric method from a legal perspective. Muikku
(2017) finds that as the overall stream count decreases, the revenue
difference between the user-centric and the pro-rata methods increases. The
latter favours artists and tracks, which get the biggest amount of played
streams regardless if they are created by a large number of users with few
plays or a smaller number of users who have played them repeatedly. The former
favours artists with smaller number of streams, especially when the overall
stream count is smaller. However, the results depend on the cumulative effects
of both individual and user groups’ listening habits. Meyn et al. (2023)
investigate the monetary consequences of the switch from the pro-rata method
to the user-centric method. Using individual-level data from 3,326
participants and data from Spotify Web API, they find a substantial
reallocation (mostly shifting revenue from mainstream to niche genres) of
nearly €170 million per year at Spotify, driven by the song length as well as
the payments per listening time.

Beyond the specific issue of sharing the revenue raised from subscriptions to
(online music) platforms, the literature (at the intersection of several
fields such as economics, management, marketing and law) has addressed other
issues regarding the effect that streaming has on the music industry. For
instance, Aguiar and Waldfogel (2018) find that although streaming displaces
sales, it also displaces music piracy. Aguiar and Waldfogel (2021) highlight
the market power of some platforms (especially, Spotify) in the online music
market (via their via platform-operated playlists, which act as promotion
channels). Datta et al., (2018) analyze the effects of the gradual adoption of
streaming services in the music industry. They find that adoption of streaming
leads to very large increases in the quantity and diversity of consumption in
the first months after adoption. And, although the effects attenuate over
time, adopters continue playing substantially more, and more diverse, music.
Jain and Qian (2021) focus on another major source of revenue for platforms;
namely, advertising, which depends on the number of active customers using the
platforms. Their emphasis is on how such sharing incentives are affected by
the nature of competition among various producers, the size of the customer
base, and the type of customers. They show that increased producer competition
can lead to higher compensation for the producers, higher content quality, and
higher producers’ profits. And, more generally, Waldfogel (2017) surveyed the
literature dealing with digitalization and its positive effects in different
art industries (not only music, but also films, literature and television),
whereas Rietveld and Schilling (2020) reviewed the literature (comprising the
last three decades) on platform competition.

Our work also relates to the industrial organization literature dealing with
bundling, which can be traced back to Adams and Yellen (1976). Bundling
products is typically an effective mechamism to increase revenue with respect
to selling products independently. Examples abounded in real life even before
the advent of digital content platforms (e.g., Ginsburgh and Zang, 2003;
Bergantiños and Moreno-Ternero, 2015). Online bundling might nevertheless have
a different nature, as it often occurs in a distribution channel where a
downstream firm (retailer) adopts a bundling strategy involving products made
by separate and multiple upstream firms (manufacturers). This sort of practice
not only applies to the platforms offering entertainment products, such as the
ones we analyze here, but also to other information goods such as online
services (e.g., Bhargava, 2013).555Geng et al. (2005) point out how consumers’
average value for consuming a stream of information goods (such as streaming
music, or other online entertainment) declines with the number consumed and
provide basic guidelines for optimal bundling marketing strategies in such a
case. The complex relationship between the independent price of each product
and the bundled price renders the problem of sharing the revenue from periodic
charges to unlimited streaming among the participating agents a complex one
too. Nevertheless, we believe our results could shed light on that problem.

Our paper (especially, Section 3) is also connected to the sizable axiomatic
literature on resource allocation or, more generally, the axiomatics of
economic design (e.g., Thomson, 2023). A variety of axioms formalizing
principles with ethical or operational appeal for resource allocation have
been introduced in economic theory for more than half a century. Instances are
axioms formalizing no-envy, impartiality, priority, or solidarity (e.g.,
Foley, 1967; Moreno-Ternero and Roemer, 2006; Thomson, 2023). They have
contributed to our understanding of normative issues concerning the allocation
of goods and services. Our paper extends the scope of this literature in order
to deal with a special and somewhat new type of goods and services that have
arisen in the current digital era. But the characterizations we provide in
Section 3 are also reminiscent of other recent characterizations in the
axiomatic literature on resource allocation. For instance, Bergantiños and
Moreno-Ternero (2020) analyze the problem of sharing the (collectively raised)
revenues from sportscast.666Dietzenbacher and Kondratev (2023) study the
related problem of allocating a prize endowment among competitors, when their
ranking is known. The input in their case is the audience matrix indicating
the number of viewers of each game between two clubs participating in a given
competition (thus, similar to the input of the streaming problems we analyze
in this paper). And they also characterize two focal rules to share those
(collectively raised) revenues.777Families of rules compromising among the two
rules have also been considered in the literature (e.g., Bergantiños and
Moreno-Ternero, 2021, 2022). Singal et al. (2022) also develop an axiomatic
framework to study the related problem of attribution in online advertising;
namely, assessing the contribution of individual advertiser actions to
eventual conversion.888Both Bergantiños and Moreno-Ternero (2020) and Singal
et al. (2022) also explore the game-theoretical approach to solve their
problems, as we do in this paper. Somewhat related, Flores-Szwagrzak and
Treibich (2020) introduce an innovative productivity index that disentangles
individual from collaborative productivity. This index defines an individual’s
productivity score as the sum of her credit over all the projects she has
contributed to. Credit on each project is allocated proportionally to the
score of each teammate and the scores are thus determined endogenously and
simultaneously (solving a fixed point problem). Martinez and Moreno-Ternero
(2022) characterize a family of pandemic performance indicators arising from a
weighted average of the incidence rate, morbidity rate and mortality rate.
Previously, Hougaard and Moulin (2014) studied how to share the cost of
finitely many public goods (items) among users with different needs. They
characterized a family of cost ratios based on simple liability indices, one
for each agent and for each item, measuring the relative worth of this item
across agents, and generating cost allocation rules additive in costs. And,
more recently, Gudmundsson et al. (2023) also take the axiomatic approach to
analyze the problem of sharing sequentially triggered losses. They
characterize a class of fixed-fraction rules, which strike a balance between
incentives for accident prevention and fairness to assign liabilities.
Finally, Gonçalves-Dosantos et al. (2023) also take the axiomatic approach
(but not the game-theoretical, or another indirect approach) to study general
content streaming platforms (such as, for instance, Netflix). As music
streaming platforms have some specific characteristics that are not
necessarily extended to general content streaming platforms, most of the
axioms considered in both papers are of a different nature. Characterization
results are thus genuinely different.999As a matter of fact, in all of our
characterizations we use axioms that do not appear in Gonçalves-Dosantos et
al. (2023), whereas in all of the characterizations they provide, they use
some axioms that do not appear in our paper.

To conclude with the introduction, we mention that our paper is also related
to the literature on cooperative game theory. There is a tradition of
analyzing problems involving agents’ cooperation with a game-theoretical
approach. Classical instances are the so-called airport problems (e.g.,
Littlechild and Owen, 1973), in which the cost of a runway has to be shared
among different types of airplanes, bankruptcy problems from the Talmud (e.g.,
Aumann and Maschler, 1985), telecommunications problems and the rerouting of
international telephone calls (e.g., van den Nouweland et al., 1996), or
modern transportation systems (e.g., Krajewska et al., 2008). One of the
approaches we take in this paper (Section 4) is precisely this one. The
(cooperative) game we associate to streaming problems is convex and, thus, it
has a large core encompassing allocations guaranteeing participation
constraints. We fully characterize such a core and show that the pro-rata
method can generate allocations that violate these constraints. We also show
that the resulting axiom of core-selection is key to characterize the user-
centric method. The combination of these two results is another powerful
argument we provide to support the user-centric method with respect to the
pro-rata method.

The game-theoretical approach is an indirect way of solving streaming
problems. Another indirect approach suggests to solve them via associating a
claims problem (rather than a cooperative game). The problem of adjudicating
conflicting claims (in short, claims problem) models a basic situation in
which an endowment is allocated among agents who have claims on it, and the
available amount is not enough to fully honor all claims. This is a classic
problem that can be traced back to ancient sources, such as Aristotle and the
Talmud, although is formal treatment is somewhat recent (e.g., O’Neill, 1982;
Thomson, 2019). Ju et al. (2007) generalize these problems to account for
multiple issues.101010Csóka and Herings (2018, 2021) also studied recently a
different generalization of claims problems to deal with financial networks,
as pioneered by Eisenberg and Noe (2001). See also Calleja and Llerena (2023).
We show that some of the two-stage rules from generalized claims problems can
rationalize some of the allocation rules we consider to solve streaming
problems.

The rest of the paper is organized as follows. In Section 2, we introduce our
model and main concepts for our analysis of streaming problems. In Section 3,
we present the axiomatic approach to our problems including the main
characterization results we obtain. In Section 4, we explore the game-
theoretical approach to our problem. In Section 5, we explore another indirect
approach to solve our problems based on claims problems. Section 6 concludes.
Most of the proofs have been postponed to an appendix.

##  2 Streaming problems

Let $N=\left\\{1,...,n\right\\}$ denote a set of artists and
$M=\left\\{1,...,m\right\\}$ a set of users. For each pair $i\in N,j\in M$,
let $t_{ij}$ denote the times user $j$ played (via streaming) contents
uploaded by artist $i$ in the platform, briefly streams, during a certain
period of time (e.g., month). Let $t=\left(t_{ij}\right)_{i\in N,j\in M}$
denote the corresponding matrix encompassing all playing times. A streaming
problem is $P=\left(N,M,t\right)$.111111As mentioned at the Introduction, and
for ease of notation, we normalize the amount paid by each user to 1. Thus,
the amount to be divided among artists in a problem $\left(N,M,t\right)$ is
just $m$, the number of users. The set of problems so defined is denoted by
$\mathcal{P}$. For each $j\in M,$ we denote by $t^{-j}$ the matrix obtained
from $t$ by removing the column corresponding to user $j$.

For each artist $i\in N$,

$T_{i}\left(N,M,t\right)=\sum_{j\in M}t_{ij},$

denotes the total times $i$ was played. Likewise, for each user $j\in M$

$T^{j}\left(N,M,t\right)=\sum_{i\in N}t_{ij},$

denotes the total times $j$ played content.

We define the set of fans of each artist as the set of users who have played
content from the artist at least once. Formally, $F:N\rightarrow M$ is such
that for each $i\in N,$

$F_{i}\left(N,M,t\right)=\left\\{j\in M:t_{ij}>0\right\\}.$

Similarly, we define the list of artists of a user as those from which the
user has played content at least once. Formally, $L:M\rightarrow N$ such that
for each $j\in M,$

$L^{j}\left(N,M,t\right)=\left\\{i\in N:t_{ij}>0\right\\}.$

The profile of user $j$ is defined as the streaming vector associated to such
a user. Namely,

$t_{.j}\left(N,M,t\right)=\left(t_{ij}\right)_{i\in N}.$

When no confusion arises we write $T_{i}$ instead of
$T_{i}\left(N,M,t\right)$, $T^{j}$ instead of $T^{j}\left(N,M,t\right),$
$F_{i}$ instead of $F_{i}\left(N,M,t\right),$ $L^{j}$ instead of
$L^{j}\left(N,M,t\right),$ and $t_{.j}$ instead of $t_{.j}\left(N,M,t\right).$

We illustrate our model with a basic example that will surface throughout our
ensuing analysis. Assume two users ($a,b$) join a platform to listen to their
favorite artists (1,2). Assume artist $1$ is listened by user $a$, whereas
artist $2$ is listened by user $b$. This situation can be included in our
theoretical model as follows.

###### Example 1

Let $N=\left\\{1,2\right\\},$ $M=\left\\{a,b\right\\},$ and

$t=\left(\begin{array}[]{cc}10&0\\\ 0&90\end{array}\right).$

A popularity index $\left(I\right)$ for streaming problems is a mapping that
measures the importance of each artist in each problem. Formally,
$I:\mathcal{P}\to\mathbb{R}_{+}^{N}$ and, for each pair $i,j\in N$,
$I_{i}\left(N,M,t\right)\geq I_{j}\left(N,M,t\right)$ if and only if $i$ is at
least as important as $j$ at problem $\left(N,M,t\right)$. We assume that
$\sum\limits_{i\in N}I_{i}\left(N,M,t\right)>0$.

The reward received by each artist $i\in N$ from the revenues generated in
each problem ($m$ because the amount paid by each user has been normalized to
1) is based on the importance of that artist in that problem. Formally,

$R_{i}^{I}\left(N,M,t\right)=\frac{I_{i}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,M,t\right)}m.$

Note that any positive linear transformation of a given index generates the
same allocation of rewards. Formally, for each $\lambda>0$ and each index $I$,
$R^{\lambda I}\equiv R^{I}$. Thus, in what follows, we shall slightly abuse
language to identify an index with all its positive linear transformations
too.

The index used in most of the platforms is the so called pro-rata index, which
simply measures importance by the total number of streams. Formally, for each
problem $\left(N,M,t\right)\in\mathcal{P}$ and each artist $i\in N,$

$P_{i}\left(N,M,t\right)=T_{i}=\sum_{j\in M}t_{ij}.$

Thus, the amount received by each artist $i\in N$ under $P$ is

$R^{P}_{i}\left(N,M,t\right)=\frac{T_{i}}{\sum\limits_{j\in N}T_{j}}m.$

Another basic index is the so called user-centric index. All users have the
same importance (normalized to 1). The importance of each user is divided
among the artists listened by this user proportionally to the total number of
streams. Then, the importance of each artist is the sum, over all users, of
the importance of the artist given by each user. Formally, for each problem
$\left(N,M,t\right)$ and each artist $i\in N,$

$U_{i}\left(N,M,t\right)=\sum_{j\in M}\frac{t_{ij}}{T^{j}}.$

As

$\sum_{i\in N}\sum_{j\in M}\frac{t_{ij}}{T^{j}}=\sum_{j\in M}\sum_{i\in
N}\frac{t_{ij}}{T^{j}}=\sum_{j\in M}1=m,$

it follows that the amount received by each artist $i\in N$ from this index is
precisely

$R_{i}^{U}\left(N,M,t\right)=U_{i}\left(N,M,t\right).$

In Example 1, we then have the following:

| $R_{i}^{P}\left(N,M,t\right)$ | $R_{i}^{U}\left(N,M,t\right)$ | $T_{i}$
---|---|---|---
1 | 0.2 | 1 | 10
2 | 1.8 | 1 | 90

That is, pro-rata gives to artist 2 much more than to artist 1, whereas the
user-centric gives the same to both artists (which sounds more reasonable in
this example). This might illustrate why some platforms are moving from pro-
rata to user-centric.121212For instance, SoundCloud did so in 2021 (e.g.,
Ingham, 2021) whereas Deezer started an online campaign promoting user-centric
(e.g., Deezer, 2019). Muikku (2017), Alaei et al. (2022), Haampland et al.
(2022), and Meyn et al. (2023), among others, have compared the two indices on
different grounds.

There is also room for alternative payment schemes beyond the previous two.
For instance, Meyn et al.., (2022) write “remuneration based on quality
ratings, or a combination of user-centric and pro-rata remuneration”. And,
also, “in addition to pro-rata and user-centric, the distribution parameters
(e.g., the unit of distribution) must be investigated further”. Partly
accounting for this motivation, we now introduce a family of indices that
allow us to consider alternative payment schemes compromising between the
previous two.

A weight system is a function
$\omega:M\times\mathbb{N}_{+}^{N}\rightarrow\mathbb{R}$ such that for each
$j\in M$ and each $x\in\mathbb{N}_{+}^{N},$ $\omega\left(j,x\right)>0.$ For
each weight system $\omega$, its weighted index $I^{\omega}$ is defined so
that for each $\left(N,M,t\right)$ and each $i\in N,$

$I_{i}^{\omega}\left(N,M,t\right)=\sum_{j\in M}w\left(j,t._{j}\right)t_{ij}.$

The index of each artist is obtained as the sum, over all users, of the
streams of the user weighted by a factor that depends on the user (allowing,
for instance, for popular users to have more importance than unknown users)
and the streaming profile of the user (allowing, for instance, for more active
users to have more importance than less active users). The pro-rata index is
the weighted index associated with $\omega\left(j,x\right)=1$ for each $j\in
M$, and each $x\in\mathbb{N}_{+}^{N}\ $(namely, the users and the profile do
not matter) and the user-centric index is the weighted index associated with
$\omega\left(j,x\right)=\frac{1}{\sum\limits_{i\in N}x_{i}}$ for each $j\in
M$, and each $x\in\mathbb{R}_{+}^{N}$ (namely, the user does not matter but
the profile matters).

Notice that under pro-rata the importance of each user to all artists, namely
$\sum\limits_{i\in N}w\left(j,t._{j}\right)t_{ij}=T^{j},$ is proportional to
his/her total streams. Under user-centric all users have the same importance,
namely $\sum\limits_{i\in N}w\left(j,t._{j}\right)t_{ij}=1$. We can consider
weight systems reflecting that all users have a minimum importance and also
that users with more streamings contribute more, but with an upper bound. For
instance, for each $j\in M$,

$w\left(j,x\right)=\left\\{\begin{tabular}[]{ll}$\frac{1}{\sum\limits_{i=1}^{n}x_{i}}$&if
$\sum\limits_{i=1}^{n}x_{i}\leq\alpha$,\\\ $\frac{1}{\alpha}$&if
$\alpha<\sum\limits_{i=1}^{n}x_{i}\leq\beta$,\\\
$\frac{\beta}{\alpha\sum\limits_{i=1}^{n}x_{i}}$&if
$\sum\limits_{i=1}^{n}x_{i}>\beta$.\end{tabular}\right.$

Notice that $\omega\left(j,x\right)$ does not depend on $j.$ Thus, we only
distinguish users through the streaming times. More precisely, all users with
a number of streams below $\alpha$ have the same importance as in the user-
centric index. Namely, for each $j\in M$,

$\sum_{i\in N}w\left(j,t_{.j}\right)t_{ij}=1.$

Users with a number of streams between $\alpha$ and $\beta$ have a
proportional importance to the streams.Namely, for each $j\in M$,

$\sum_{i\in N}w\left(j,t_{.j}\right)t_{ij}=\frac{T^{j}}{\alpha}.$

Finally, users with a number of streams above $\beta$ have the same
importance, given by the thresholds $\alpha$ and $\beta$. Namely, for each
$j\in M$,

$\sum_{i\in N}w\left(j,t_{.j}\right)t_{ij}=\frac{\beta}{\alpha}.$

We now revisit Example 1, adding a new user ($c$) with $5$ streams for artist
$1$ and $35$ streams for artist $2$. Besides, let $\alpha=20$ and $\beta=60.$
Then, the weigthed index $I^{\alpha,\beta}$ defined as above is

$\displaystyle I_{1}^{\alpha,\beta}\left(N,M,t\right)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{10}\right)10+\left(\frac{60}{20\ast
90}\right)0+\left(\frac{1}{20}\right)5=1.25\text{ and }$ $\displaystyle
I_{2}^{\alpha,\beta}\left(N,M,t\right)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{10}\right)0+\left(\frac{60}{20\ast
90}\right)90+\left(\frac{1}{20}\right)35=4.75.$

In this case, $P\left(N,M,t\right)=\left(15,125\right)$ whereas
$U\left(N,M,t\right)=\left(1.125,1.875\right)$. Then, the rewards induced for
artists are

| $R_{i}^{P}\left(N,M,t\right)$ | $R_{i}^{U}\left(N,M,t\right)$ | $R_{i}^{I^{\alpha,\beta}}\left(N,M,t\right)$
---|---|---|---
1 | 0.3 | 1.1 | 0.6
2 | 2.7 | 1.9 | 2.4

We observe that $I^{\alpha,\beta}$ yields an allocation in between those the
pro-rata and user-centric yield.

##  3 An axiomatic approach

In this section, we take the axiomatic approach to solve streaming problems.
That is, we formalize axioms of indices that model principles with normative
(ethical or operational) appeal. Some of them will echo the concern that
artists are paid fairly.131313Haamland et al., (2022) provide an interesting
discussion about this issue. Some others are inspired from related discussions
in the music industry or from the literature on resource allocation.

To introduce the first axiom, which is standard in axiomatic work, assume that
each user has played artist $i$ a certain times more than artist $i^{\prime}.$
Then, the index should preserve that ratio. Formally,

Homogeneity. For each $\left(N,M,t\right)\in\mathcal{P}$, each pair
$i,i^{\prime}\in N,$ and each $\lambda\geq 0$ such that $t_{ij}=\lambda
t_{i^{\prime}j}$ for all $j\in M,$

$I_{i}\left(N,M,t\right)=\lambda I_{i^{\prime}}\left(N,M,t\right).$

The second axiom is also a standard axiom in resource allocation and says that
if we can divide a problem in the sum of two smaller problems, then the
solution to the original problem should be the sum of the solutions in the two
smaller problems. The intuition in streaming problems is the following.
Suppose that a platform operates in several countries. Then, we can reward
artists in two ways. First, we consider all countries in the same market and
we allocate artists according with the streams in all countries. Second, we
consider each country as a different market. Thus, each artist receives an
allocation in each country according with the streams in this country. The
total allocation to an artist is the sum over all countries. Additivity says
that both ways should coincide. Formally,

Additivity. For each trio
$\left(N,M^{1},t^{1}\right),\left(N,M^{2},t^{2}\right),\left(N,M,t\right)\in\mathcal{P}$
such that $M=M^{1}\cup M^{2}$, $t_{ij}=t_{ij}^{1}$ when $j\in M^{1}$ and
$t_{ij}=t_{ij}^{2}$ when $j\in M^{2},$

$I\left(N,M,t\right)=I\left(N,M^{1},t^{1}\right)+I\left(N,M^{2},t^{2}\right).$

We now consider two alternative forms of modeling the impact of extra users.

Assume that two users (say, $j$ and $j^{\prime})$ have the same streams on
artist $i.$ We can then consider that both users are similar for artist $i$.
Then, both users should have the same impact over this artist.141414A
rationale for this axiom is provided by Page and Safir (2018a), who write
“some listeners may find it inequitable that a given track be awarded
significantly different per-stream values”.  Formally,

Equal individual impact of similar users. For each
$\left(N,M,t\right)\in\mathcal{P}$, each $i\in N,$ and each pair
$j,j^{\prime}\in M$ such that $t_{ij}=t_{ij^{\prime}}$,

$I_{i}\left(N,M\backslash\left\\{j\right\\},t^{-j}\right)=I_{i}\left(N,M\backslash\left\\{j^{\prime}\right\\},t^{-j^{\prime}}\right).$

It is also often argued that as all users pay the same, all users should have
the same impact on the index. We formalize this idea as follows:

Equal global impact of users.  For each $\left(N,M,t\right)\in\mathcal{P}$ and
each pair $j,j^{\prime}\in M,$

$\sum_{i\in
N}I_{i}\left(N,M\backslash\left\\{j\right\\},t^{-j}\right)=\sum_{i\in
N}I_{i}\left(N,M\backslash\left\\{j^{\prime}\right\\},t^{-j^{\prime}}\right).$

The next result states the characterizations we obtain combining the previous
axioms.

###### Theorem 1

The following statements hold:

$\left(a\right)$ An index satisfies homogeneity, and additivity if and only if
it is a weighted index.

$\left(b\right)$ An index satisfies homogeneity, additivity, and equal
individual impact of similar users if and only if it is the pro-rata index.

$\left(c\right)$ An index satisfies homogeneity, additivity, and equal global
impact of users if and only if it is the user-centric index.

Theorem 1, whose proof is postponed to the appendix, provides normative
grounds for the indices considered above. Statement $\left(a\right)$ provides
the characterization of the family of indices satisfying the two basic axioms.
Pro-rata and user-centric arise from the family by imposing an additional
axiom. Equal individual impact of similar users for the former and equal
global impact of users for the latter. Both axioms reflect different ways to
address users and streams. The former focuses on streams while stating,
roughly speaking, that all streams are equally important, independently of who
has produced the streams. The latter focuses on users while stating, roughly
speaking, that all users are equally important, independently of the streams
of each user. Thus, we can see pro-rata and user-centric as two somewhat polar
weighted indices. The first one “equalizes” streams, whereas the second one
“equalizes” users.

We conclude this section complementing the axiomatic analysis from Theorem 1
upon formalizing two axioms that capture ideas that have recently received
attention in the literature on the music industry. We shall see that, although
Theorem 1 puts both the user-centric index and the pro-rata index on a similar
foot, the following analysis (with respect to the new two axioms) favors the
user-centric index.

Haampland et al. (2022) write the following: “such a payment system can be
perceived to be quite unfair since individual users’ payments are not aligned
with their actual musical preferences.” Similarly, Meyn et al. (2023) argue
that users could “prefer that their payment only goes towards the content they
use”. Thus, we consider an axiom that says that given a set of users $C$ and
the set of artists $A$ listened by those users, the amount received by the
artists in $A$ should be, at least, the amount paid by users in $C.$ Formally,

Reasonable lower bound. For each $\left(N,M,t\right)\in\mathcal{P}$ and each
$C\subset M$, let $L^{C}=\bigcup\limits_{j\in C}L^{j}.$ Then,

$\sum_{i\in L^{C}}\frac{I_{i}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,M,t\right)}m\geq\left|C\right|.$

On another matter, Meyn et al. (2023) write the following: “some artists have
asked their listeners to play their music silently while asleep to generate a
larger share of remuneration”. Dimont (2017) calls this phenomenon click fraud
and discuss it giving some real examples. We first notice that it is not
possible to know if some user is making click fraud or not (namely, if he/she
is listening the song or not when playing it). Thus, it seems reasonable that
if some song is played, the artists should be rewarded for it. Nevertheless,
it should not be rewarded too much. For instance, it does not seem reasonable
that if a user pays $\$10$ and decides to listen to an artist many times, then
the amount received by this artist increases more than $\$50$. Otherwise,
artists would have incentives to create fictitious users listening artist’s
songs all the day. We formalize this axiom by saying that if a user changes
his/her streaming times, then the amount received by each artist could not
change more that the subscription paid by the user. Formally,

Click-fraud-proofness. Let
$\left(N,M,t\right),\left(N,M,t^{\prime}\right)\in\mathcal{P}$ and $j\in M$ be
such that $t_{ij^{\prime}}=t_{ij^{\prime}}^{\prime}$ for all $j^{\prime}\in
M\backslash\left\\{j\right\\}$ and $i\in N$. Then, for all $i\in N,$

$\left|\frac{I_{i}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,M,t\right)}m-\frac{I_{i}\left(N,M,t^{\prime}\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,M,t^{\prime}\right)}m\right|\leq 1.$

The user-centric index satisfies the previous two axioms, whereas pro-rata
satisfies none of them (see the appendix for the details).

##  4 A game-theoretical approach

A standard approach to solve resource allocation problems is through
cooperative game theory. That is, given a resource allocation problem we
associate a cooperative game, we compute a cooperative solution (for instance
the core), and finally we compute the allocation induced by the cooperative
solution in the original problem. We shall follow that approach in this
section to solve our streaming problems.

A cooperative game with transferable utility, briefly a TU game, is a pair
$\left(N,v\right)$, where $N$ denotes a set of agents and
$v:2^{N}\rightarrow\mathbb{R}$ satisfies $v\left(\varnothing\right)=0.$ The
core of a cooperative game is defined as the set of feasible payoff vectors,
for which no coalition can improve upon. Formally, for each game
$\left(N,v\right)$,

$C\left(N,v\right)=\left\\{x=(x_{i})_{i\in N}\text{: }\sum_{i\in
N}x_{i}=v\left(N\right)\text{ and }\sum_{i\in S}x_{i}\geq v(S)\text{ for each
}S\subset N\right\\}.$

We now associate with each streaming problem $\left(N,M,t\right)$ a $TU$ game
$\left(N,v_{\left(N,M,t\right)}\right)$ where the set of agents of the
cooperative game is the set of artists. Given $S\subset N$ we define
$v_{\left(N,M,t\right)}\left(S\right)$ as the amount paid by the users that
have only listened to artists in $S$.151515The definition of
$\left(N,v_{\left(N,M,t\right)}\right)$ is similar to the one used in the so-
called museum pass problem (e.g., Ginsburg and Zang, 2003; Bergantiños and
Moreno-Ternero, 2015). Formally,

$v_{\left(N,M,t\right)}\left(S\right)=\left|\left\\{j\in M:L^{j}\subset
S\right\\}\right|.$

When no confusion arises we write $v$ instead of $v_{\left(N,M,t\right)}$.
Note that this game is based on a pessimistic premise. To wit, it is
reasonable to assume that if artists do not consider themselves to be well
paid, they might leave the platform.161616Kanye West, a highly popular artist,
recently launched his own streaming service (e.g., Meyn et al., 2023). The
issue is to estimate the revenues seceding artists might obtain after leaving
the platform (and creating a new one). Our estimation is that just all users
that listened only to the seceding artists will leave the platform to join the
new one promoted by them. We acknowledge that this could be a quite
pessimistic stance. For instance, consider a user whose $99\%$ of streams are
devoted to artists within the seceding group ($S$) and only $1\%$ to other
artists. It seems plausible to conjecture that this user would leave the
platform, whereas our assumption on $v(S)$ does not consider it.

It is not difficult to show that $v$ is a convex game, i.e., the incentives to
join a coalition increase as the coalition grows (or, more formally, its
characteristic function is supermodular). It is well known that the core of a
convex game is quite large. The next result actually characterizes all the
allocations within the core of this game. In words, they must satisfy the
following: the amount paid by each user (which we have normalized to 1) is
divided in any way among the artists listened by this user. Each artist then
receives the sum (over all users) of the corresponding amounts. Formally, for
each $\left(N,M,t\right)$ we define the following set of allocations:

$A\left(N,M,t\right)=\left\\{\begin{array}[]{c}x\in\mathbb{R}^{N}:x=\sum\limits_{j\in
M}x^{j}\text{ where for each }j\in M\\\ x^{j}\in\mathbb{R}^{N},\\\
x_{i}^{j}=0\text{ for each }i\in N\backslash L^{j}\text{,}\\\ x_{i}^{j}\geq
0\text{ for each }i\in L^{j}\text{ and}\\\ \sum\limits_{i\in
N}x_{i}^{j}=1.\end{array}\right\\}.$

###### Theorem 2

For each $\left(N,M,t\right)\in\mathcal{P}$,

$C\left(N,v\right)=A\left(N,M,t\right).$

The proof of Theorem 2 can be found in the appendix.

We note that the rewards induced by the pro-rata index could be outside the
core. To do so, consider for instance Example 1 and $S=\left\\{1\right\\}.$
Then, $R_{1}^{P}\left(N,M,t\right)=\frac{20}{100}<1=v\left(S\right)$.

On the other hand, by definition, the user-centric index yields for each
problem $\left(N,M,t\right)$ an allocation that belongs to
$A\left(N,M,t\right)$. Thus, by Theorem 2, the rewards induced by the user-
centric index always belong to the core. More generally, we introduce the
axiom stating that the rewards generated by an index should always lie within
the core of the associated cooperative game. Formally,

Core selection. For each $\left(N,M,t\right)\in\mathcal{P}$,
$R^{I}\left(N,M,t\right)\in C\left(N,v\right)$.

As the next result states, core selection actually characterizes the user-
centric index, when combined with the axioms of homogeneity and additivity
(already introduced in the previous section), provided we restrict to the
following relevant subdomain of streaming problems.

More precisely, let $\mathcal{P}^{\ast}$ be the set of all problems (with at
least three users) where no user has played content from all the artists.
Namely,

$\mathcal{P}^{\ast}=\left\\{\left(N,M,t\right)\in\mathcal{P}:\left|M\right|\geq
3\text{ and }L^{j}\left(N,M,t\right)\neq N\text{ for all }j\in M\right\\}.$

###### Theorem 3

In the domain $\mathcal{P}^{\ast}$, an index satisfies homogeneity,
additivity, and core selection if and only if it is the user-centric index.

The proof of Theorem 3 can be found in the appendix. Note that
$\mathcal{P}^{\ast}$ is not a very restrictive domain for real-life
platforms.171717Obviously, all platforms have at least three users. And a user
can play at most $89280$ streams per month (assuming 30 seconds to qualify as
a valid streaming, and the user is somehow playing 24/7 those shortest valid
streamings), which is typically below the number of artists platforms have in
their catalogue. Nevertheless, one could extend Theorem 3 to the full domain
$\mathcal{P}$ upon simply adding an axiom of independence of null artists,
stating that removing an artist with no streamings does not change the value
of the index for the remaining artists.

Based on the analysis in this section, we can safely state that the game-
theoretical approach favors the user-centric index with respect to the pro-
rata index.

##  5 A claims approach

In this section, we consider another (indirect) approach to solve streaming
problems, based on claims problems. Claims problems refer to an amount of a
homogeneous and infinitely divisible good (e.g., money) to be divided among a
set of agents, who have claims on the good. This is certainly the case of the
canonical and well-known bankruptcy problem.181818Although this problem can be
traced back to Aristotle and the Talmud, the seminal contribution is O’Neill
(1982). Thomson (2019) is an excellent survey with an extensive treatment of
the sizable literature emanating from that seminal contribution. Formally, a
bankruptcy problem is a triple $\left(N,c,E\right)$ where $N$ is the set of
agents, $c\in\mathbb{R}_{+}^{N}$ is the claims vector and $E\in\mathbb{R}_{+}$
is the amount to be divided. It is assumed that $\sum\limits_{i\in N}c_{i}\geq
E$. A rule is a function $R$ assigning to each bankruptcy problem
$\left(N,c,E\right)$ a vector $R\left(N,c,E\right)\in\mathbb{R}^{N}$ such that
for each $i\in N$, $0\leq R_{i}\left(N,c,E\right)\leq c_{i}$ and
$\sum\limits_{i\in N}R_{i}\left(N,c,E\right)=E$. Some popular rules are the
proportional rule, which yields awards proportionally to claims, and the
constrained equal awards rule, which equalizes the amount received by each
agent as much as possible. Formally, for each $(N,c,E)$ and each $i\in N,$

$P_{i}(N,c,E)=\frac{c_{i}}{\sum\limits_{i^{\prime}\in
N}c_{i^{\prime}}}\,E,\text{ and }$ $CEA_{i}(N,c,E)=\min\\{\lambda,c_{i}\\}$

where $\lambda$ satisfies $\sum\limits_{i\in N}\min\\{\lambda,c_{i}\\}=E$.

We now associate with each streaming problem $\left(N,M,t\right)$ a bankruptcy
problem $\left(N,c\left(N,M,t\right),E_{\left(N,M,t\right)}\right)$ where
$c_{i}\left(N,M,t\right)=T_{i}$, and $E\left(N,M,t\right)=m.$ Notice that
$R_{i}^{P}\left(N,M,t\right),$ the amount received by artist $i$ under the
pro-rata index, coincides with $P_{i}\left(N,c,m\right),$ the amount received
by agent $i$ in the associated bankruptcy problem under the proportional rule.
Besides, $U_{i}\left(N,M,t\right),$ the amount received by artist $i$ under
the user-centric index, can not be computed through $\left(N,T,m\right)$
because it depends on numbers ($t_{ij})$ that do not appear in
$\left(N,T,m\right).$ We now present an extension of bankruptcy problems,
following Ju et al., (2007), that allows us to use the values $t_{ij}$.

A claims problem is a tuple $\left(N,K,c,E\right)$ where $N$ is a set of
agents, $K$ is a set of issues, $c=(c_{ij})_{i\in N,j\in K}\ $where for all
$i\in N$ and $j\in K,$ $c_{ij}\geq 0$ denotes the characteristic of agent $i$
on issue $j$, and $E\in\mathbb{R}_{+}$ is the amount of a homogeneous and
infinitely divisible good to be divided. A rule is a function $R$ assigning to
each claims problem $\left(N,K,c,E\right)$ a vector
$R\left(N,K,c,E\right)\in\mathbb{R}^{N}.$

We now associate with each streaming problem $\left(N,M,t\right)$ a claims
problem
$\left(N,K\left(N,M,t\right),c\left(N,M,t\right),E\left(N,M,t\right)\right)$
where $K\left(N,M,t\right)=M,$ $c\left(N,M,t\right)=t$, and
$E\left(N,M,t\right)=m.$

Ju et al., (2007) characterize several families of rules.191919The do so
mostly thanks to the axiom of reallocation-proofness, which does not have a
parallel in this paper. One of the families is formally defined as follows.

For each claims problem $\left(N,K,c,E\right)$ and each $j\in K$, let
$c_{.j}=(c_{ij})_{i\in N}$ and $C^{j}=\sum\limits_{i\in N}c_{ij}.$ A weight
function is a function
$\omega:\mathbb{R}_{+}^{K}\times\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}^{K}$,
which assigns a probability distribution $\omega(x,y)$ (namely, $0\leq
w_{j}\left(x,y\right)\leq 1$ for all $j\in K$ and $\sum\limits_{j\in
K}w_{j}\left(x,y\right)=1)$. The weighted proportional rule associated to
$\omega$ assigns, for each problem $\left(N,K,c,E\right)$ and each $i\in N$
the amount

$P_{i}^{\omega}\left(N,K,c,E\right)=\sum_{j\in
K}\frac{c_{ij}}{C^{j}}\omega_{j}(\left(C^{j}\right)_{j\in K},E)E.$

In words, rule $P^{\omega}$ first applies the proportional rule to each
single-dimensional sub-problem $\left(N,\left\\{j\right\\},c_{.j},E\right)$
and then it takes the weighted average of the solutions to the sub-problems
according to the vector of weights $\omega_{j}(\left(C^{j}\right)_{j\in
K},E)$.

Calleja et al., (2005) introduce multi-issue allocation situations, which are
a particular case of claims problems.202020Actually the tuple
$\left(N,K,c,E\right)$ is defined as in claims problems but some additional
constraints on $c$, $E$ and the definition of a rule are added. As our results
are not affected by that, we avoit the details. Bergantiños et al., (2010,
2011, 2018) consider two-stage rules for claims problems where in the first
stage the endowment is divided among the issues and in the second stage the
amount assigned to each issue is divided among the agents. The final amount
received by each agent is the sum over all issues.

Formally, let $\psi$ and $\phi$ be two bankruptcy rules. The two-stage rule
$R^{\psi,\phi}$ is the claims rule obtained from the following two-stage
procedure:

  1. 1.

First stage. We consider the bankruptcy problem among the issues
$(K,c^{K},E)$, where $c^{K}=(c_{j}^{K})_{j\in K}$ and for each $j\in K$,
$c_{j}^{K}=\sum\limits_{i\in N}c_{ji}$ . We compute $\psi(K,c^{K},E).$

  2. 2.

Second stage. For each $j\in K$, we consider the bankruptcy problem
$(N,\psi_{j}\left(K,E,c^{K}\right),c_{.j})$. We compute
$\phi(N,c_{.j},\psi_{j}\left(K,c^{K},E\right))$.

Thus, for each $i\in N$,

$R_{i}^{\psi,\phi}(N,K,C,E)=\sum_{j\in
K}\phi_{i}\left(N,c_{.j},\psi_{j}\left(K,c^{K},E\right)\right).$

Moreno-Ternero (2009) and Bergantiños et al., (2010) study the two-stage rule
where the proportional rule is used in both stages (namely, $\psi=\phi=P).$
Bergantiños et al. (2011) study the two-stage rule where the constrained equal
awards rule is used in both stages (namely, $\psi=\phi=CEA).$ Bergantiños et
al. (2018) study the two-stage rule where the constrained equal awards rule is
used in the first stage and the proportional rule in the second stage (namely,
$\psi=CEA$ and $\phi=P).$ But all those papers take the axiomatic approach
inspired on the literature of bankruptcy problems, whch is unrelated to the
axiomatic study of this paper.

We obtain the following relationships between the pro-rata index and the user-
centric index and some of the rules from the literature on claims problems.

###### Theorem 4

Let $\left(N,M,t\right)$ be a streaming problem and $\left(N,K,c,E\right)$ be
the associated claims problem. Then,

$\left(a\right)$ There exist weight functions $\omega^{P}$ and $\omega^{U}$
such that $R^{P}\left(N,M,t\right)=P^{\omega^{P}}\left(N,K,c,E\right)$ and
$R^{U}\left(N,M,t\right)=P^{\omega^{U}}(N,K,c,E).$

$\left(b\right)$ $R^{P}\left(N,M,t\right)=R^{P,P}\left(N,K,c,E\right).$

$\left(c\right)$ $U\left(N,M,t\right)=R^{CEA,P}\left(N,K,c,E\right).$

The proof of Theorem 4 can be found in the appendix.

From Theorem $\ref{claims results}\left(a\right)$ we obtain that the pro-rata
index and the user-centric index can be rationalized as weighted proportional
rules. We have seen in Theorem 1 that they are weighted indices. Thus, one
might naturally conjecture whether there is a bijection between weighted
proportional rules (for claims problems) and weighted indices (for streaming
problems). The answer is not. We can indeed find weighted proportional rules
that can not be obtained through a weighted index and weighted indices such
that the induced rule to allocate awards in streaming problems is not a
weighted proportional rule.

Theorem $\ref{claims results}\left(b\right)$ and $\ref{claims
results}\left(c\right)$ allow us to consider both indices from another
perspective. As the statements indicate, the allcation rules they induce (for
streaming problems) can actually be described as two-stage (bankruptcy) rules
where we first decide the importance of each user an then the importance of
each artist for each user, which is computed as the sum over all users. The
pro-rata and user-centric indices measure the importance of each artist in
different ways. For the latter, all users have the same importance, whereas
for the former the importance of each user is proportional to the user’s
streams. They then measure the importance of each artist for each user in the
same way; namely, proportionally to the artists’ streams.

To conclude with this section, we reiterate that Theorem 4 states that both
indices could be rationalized as a combination of two well-known rules from
the literature on bankruptcy problems. Besides, both of them can also be seen
as members of the same family of weighted proportional rules. Thus, we can
conclude that (in contrast with the previous sections) the analysis in this
section does not favor one of the indices over the other.

##  6 Conclusion

We have analyzed in this paper the problem of sharing the revenues raised from
subscription fees to music platforms among participating artists. Our analysis
has highlighted two central methods (pro-rata and user-centric) which can
actually be seen as focal (and somewhat polar) members of a family of methods
which evaluate artists by the weighted aggregation of users’ streaming
choices. The weight assigned to each user might actually depend on the user
herself and her whole streaming profile. We therefore provide a solid common
ground for both methods, in the form of the characterization result for the
whole family. We, nevertheless, provide additional (normative, as well as
game-theoretical) arguments to favor the user-centric method with respect to
the pro-rata method. To wit, we show that the former satisfies two natural and
appealing axioms (reasonable lower bound and click-fraud proofness) that the
second violates. They are somewhat connected to a feature the second exhibits
(whereas the first does not); namely, cross-subsidization between high- and
low-streaming-volume users. Furthermore, the former satisfies core-selection,
while the latter does not (which implies that it does not guarantee
allocations preventing incentives for artist to leave the platform)

To complement the above analysis, we simply stress that two aspects are
relevant when it comes to measuring the importance of artists (within a
platform). On the one hand, the users playing content from the artist. On the
other hand, the streaming times the artist achieved. Although both aspects are
related, they may differ largely. For instance, two artists may have been
listened by the same number of users but with very different streaming times.
We believe the pro-rata index manages well the streaming times but not the
number of users. The user-centric index manages well both the streaming times
and the number of users.

Our analysis should contribute to the debate between the pro-rata and user-
centric methods in the music industry. Nevertheless, the discussion in that
industry nowadays goes beyond the debate between those two methods. For
instance, the French streaming service Deezer claims to be pioneer in fair
payments to artist, “being a main advocate for a re-evaluation of music
streaming’s economic model”. In March 2023, Deezer announced an initiative
with Universal Music Group, the world leader in music-based entertainment, to
explore new streaming models that better align the interests of artists, fans
and streaming services. Using deep data analysis, this partnership aims to
improve the fairness of the current streaming model in various ways, whether
by helping artists monetize their music better or by eliminating issues within
the current system. This initiative will not prioritize just the most-streamed
artists on the platform, but will level the playing field for artists at every
stage of their career and benefit the wider music community as a
whole.212121See https://www.deezer-blog.com/how-much-does-deezer-pay-artists/
We believe that some of these goals might be achieved with other members of
the more general family of weigthed indices we characterize in this paper. As
we mentioned above, weigthed indices are precisely constructed on the premise
that each artist is assigned the weighted aggregation of streamings, accross
users, with the weight depending on the user and her streaming profile.

##  7 Appendix

### Proof of Theorem 1

$\left(a\right)$ We first prove that each weighted index satisfies the two
axioms.

For each $\omega,$ $I^{\omega}$ satisfies homogeneity. Let
$\left(N,M,t\right)\in\mathcal{P}$, $i,i^{\prime}\in N,$ and
$\lambda\in\mathbb{N}_{+}$ such that $t_{ij}=\lambda t_{i^{\prime}j}$ for all
$j\in M,$

$I_{i}^{\omega}\left(N,M,t\right)=\sum_{j\in
M}\omega\left(j,t._{j}\right)t_{ij}=\sum_{j\in
M}\omega\left(j,t._{j}\right)\lambda t_{i^{\prime}j}=\lambda
I_{i^{\prime}}^{\omega}\left(N,M,t\right).$

For each $\omega,$ $I^{\omega}$ satisfies additivity. Let
$\left(N,M^{1},t^{1}\right),\left(N,M^{2},t^{2}\right),\left(N,M,t\right)\in\mathcal{P}$
such that $M=M^{1}\cup M^{2}$, $t_{ij}=t_{ij}^{1}$ when $j\in M^{1}$ and
$t_{ij}=t_{ij}^{2}$ when $j\in M^{2}.$ For each $i\in N$,

$\displaystyle
I_{i}^{\omega}\left(N,M^{1},t^{1}\right)+I_{i}^{\omega}\left(N,M^{2},t^{2}\right)$
$\displaystyle=$ $\displaystyle\sum_{j\in
M^{1}}\omega\left(j,t_{.j}^{1}\right)t_{ij}^{1}+\sum_{j\in
M^{2}}\omega\left(j,t_{.j}^{2}\right)t_{ij}^{2}$ $\displaystyle=$
$\displaystyle\sum_{j\in
M}\omega\left(j,t_{.j}\right)t_{ij}=I_{i}^{\omega}\left(N,M,t\right).$

Conversely, we now prove that if an index $I$ satisfies the two axioms, then
there exists a weight system for streams $\omega$ such that $I=I^{\omega}.$

Let $\left(N,M,t\right)\in\mathcal{P}$. By additivity, for each $i\in N,$

$I_{i}\left(N,M,t\right)=\sum_{j\in
M}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Let $j\in M.$ By homogeneity, for each $i\in N$ such that $t_{ij}=0$,
$I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=0.$

Recall the basic assumption in our model that
$L^{j}\left(N,\left\\{j\right\\},t_{.j}\right)\neq\varnothing$. Then, let
$i\in L^{j}\left(N,\left\\{j\right\\},t_{.j}\right).$ By homogeneity, for each
$i^{\prime}\in$ $L^{j}\left(N,\left\\{j\right\\},t_{.j}\right),$

$I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)=\frac{t_{i^{\prime}j}}{t_{ij}}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Then,

$\displaystyle\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)$ $\displaystyle=$
$\displaystyle\sum_{i^{\prime}\in
L^{j}\left(N,\left\\{j\right\\},t\right)}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)$
$\displaystyle=$ $\displaystyle\sum_{i^{\prime}\in
L^{j}\left(N,\left\\{j\right\\},t\right)}\frac{t_{i^{\prime}j}}{t_{ij}}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)$
$\displaystyle=$
$\displaystyle\frac{T^{j}\left(N,\left\\{j\right\\},t_{.j}\right)}{t_{ij}}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Hence,

$I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=\frac{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)}{T^{j}\left(N,\left\\{j\right\\},t_{.j}\right)}t_{ij}.$

Taking $\omega\left(j,t_{.j}\right)=\frac{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)}{T^{j}\left(N,\left\\{j\right\\},t_{.j}\right)}$
we have that

$I_{i}\left(N,M,t\right)=\sum_{j\in
M}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=\sum_{j\in
M}\omega\left(j,t_{.j}\right)t_{ij}=I_{i}^{\omega}\left(N,M,t\right).$

$\left(b\right)$ As the pro-rata index is a weighted index, we know it
satisfies homogeneity and additivity. As for equal individual impact of
similar users, let $\left(N,M,t\right)\in\mathcal{P}$, $i\in N$ and
$j,j^{\prime}\in M$ such that $t_{ij}=t_{ij^{\prime}}$. Then,

$\displaystyle P_{i}\left(N,M\backslash\left\\{j\right\\},t^{-j}\right)$
$\displaystyle=$ $\displaystyle T_{i}\left(N,M,t^{-j}\right)-t_{ij}$
$\displaystyle=$ $\displaystyle
T_{i}\left(N,M,t^{-j^{\prime}}\right)-t_{ij^{\prime}}$ $\displaystyle=$
$\displaystyle
P_{i}\left(N,M\backslash\left\\{j^{\prime}\right\\},t^{-j^{\prime}}\right).$

Let now $I$ be an index satisfying the three axioms in the second statement.
By the proof of the previous statement, we know that $I=I^{\omega}$ for some
weight system $\omega.$

We now prove that $\omega\left(j,t_{.j}\right)$ does not depend on $j$ and
$t_{.j}.$ Let $j\in M$ and $i\in
L^{j}\left(N,\left\\{j\right\\},t_{.j}\right).$ For each $x>0$ consider the
problem $\left(N,\left\\{j^{\prime}\right\\},t^{x}\right)$ such that
$t_{ij^{\prime}}=x$ for all $i\in N.$ By homogeneity, for each pair
$i,i^{\prime}\in N,$
$I_{i}\left(N,\left\\{j^{\prime}\right\\},t^{x}\right)=I_{i^{\prime}}\left(N,\left\\{j^{\prime}\right\\},t^{x}\right).$
Let $\left(N,M^{\prime},t^{\prime}\right)\in\mathcal{P}$ be such that
$M^{\prime}=\left\\{j,j^{\prime}\right\\}$ and for all $i^{\prime}\in N,$
$t_{i^{\prime}j}^{\prime}=t_{i^{\prime}j}$ and
$t_{i^{\prime}j^{\prime}}^{\prime}=t_{i^{\prime}j^{\prime}}^{t_{ij}}.$

By equal individual impact of similar users,

$I_{i}\left(N,M^{\prime}\backslash\left\\{j\right\\},t^{-j}\right)=I_{i}\left(N,M^{\prime}\backslash\left\\{j^{\prime}\right\\},t^{-j^{\prime}}\right).$

Then,

$\displaystyle\omega\left(j,t_{.j}\right)t_{ij}$ $\displaystyle=$
$\displaystyle
I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=I_{i}\left(N,M^{\prime}\backslash\left\\{j^{\prime}\right\\},t^{-j^{\prime}}\right)$
$\displaystyle=$ $\displaystyle
I_{i}\left(N,M^{\prime}\backslash\left\\{j\right\\},t^{-j}\right)=I_{i}\left(N,\left\\{j^{\prime}\right\\},t^{t_{ij}}\right)$
$\displaystyle=$
$\displaystyle\omega\left(j^{\prime},t^{t_{ij}}\right)t_{ij}.$

Hence, $\omega\left(j,t_{.j}\right)=\omega\left(j^{\prime},t^{t_{ij}}\right).$

Let $t^{\ast}\in\mathbb{N}_{+}^{N}$ be such that $t_{ij}^{\ast}=t_{ij}$ and
$t_{i^{\prime}j}^{\ast}=1$ when $i^{\prime}\neq j.$ Similarly to
$\omega\left(j,t_{.j}\right)$, we can argue that
$\omega\left(j,t^{\ast}\right)=\omega\left(j^{\prime},t^{t_{ij}}\right).$

Take $\left(N,\left\\{j\right\\},t^{\ast}\right)\in\mathcal{P}$ and
$i^{\prime}\neq i.$ Similarly to $\omega\left(j,t_{.j}\right)$ with
$i^{\prime}$ instead of $i$ we can argue that
$\omega\left(j,t^{\ast}\right)=\omega\left(j^{\prime},t^{1}\right).$ Thus, for
each $j,j^{\prime}\in M$ and each $t_{.j}$ we have that
$\omega\left(j,t_{.j}\right)=\omega\left(j^{\prime},t^{1}\right).$ Then,
$\omega\left(j,t_{.j}\right)$ does not depend on $j$ and $t_{.j}$ and we can
define

$\lambda=\omega\left(j,t_{.j}\right).$

Now, for each $\left(N,M,t\right)$ and each $i\in N,$

$I_{i}\left(N,M,t\right)=I_{i}^{\omega}\left(N,M,t\right)=\sum_{j\in M}\lambda
t_{ij}=\lambda T_{i}=\lambda P_{i}\left(N,M,t\right).$

Then, $I$ is a (positive) linear transformation of $P$ and hence it coincides
with $P$ (recall that we define indices up to a positive linear
transformation, because the resulting ones generate the same allocation of
rewards among artists).

$\left(3\right)$ As the user-centric index is a weighted index, we know it
satisfies homogeneity and additivity. As for equal global impact of users, let
$\left(N,M,t\right)\in\mathcal{P}$ and $j\in M,$

$\sum_{i\in
N}U_{i}\left(N,M\backslash\left\\{j\right\\},t^{-j}\right)=\sum_{i\in
N}\sum_{k\in M\backslash\left\\{j\right\\}}\frac{t_{ik}}{T^{k}}=\sum_{k\in
M\backslash\left\\{j\right\\}}\sum_{i\in N}\frac{t_{ik}}{T^{k}}=\sum_{k\in
M\backslash\left\\{j\right\\}}1=m-1.$

Thus, $\sum\limits_{i\in
N}U_{i}\left(N,M\backslash\left\\{j\right\\},t^{-j}\right)$ does not depend on
$j\in M$ and hence the user-centric index satisfies equal global impact of
users.

Let now $I$ be an index satisfying the axioms in the third statement. By the
proof of the first statement, for each $\left(N,M,t\right)\in\mathcal{P}$ and
each $i\in N,$

$I_{i}\left(N,M,t\right)=\sum_{j\in M}\omega\left(j,t_{.j}\right)t_{ij},$

where

$\omega\left(j,t_{.j}\right)=\frac{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)}{T^{j}\left(N,\left\\{j\right\\},t_{.j}\right)}$

Let $j,j^{\prime}\in M$ and $M^{\ast}=\left\\{j,j^{\prime}\right\\}.$ By equal
global impact of users,

$\sum_{i\in
N}I_{i}\left(N,M^{\ast}\backslash\left\\{j\right\\},t_{.j^{\prime}}\right)=\sum_{i\in
N}I_{i}\left(N,M^{\ast}\backslash\left\\{j^{\prime}\right\\},t_{.j}\right).$

Thus,

$\sum_{i\in
N}I_{i}\left(N,\left\\{j^{\prime}\right\\},t_{.j^{\prime}}\right)=\sum_{i\in
N}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Then, $\sum\limits_{i\in N}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)$ does
not depend on $j$ and $t_{.j}$ and we can define

$\lambda=\sum_{i\in N}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

As $T^{j}\left(N,\left\\{j\right\\},t_{.j}\right)=T^{j}\left(N,M,t\right)$, it
follows that

$I_{i}\left(N,M,t\right)=\sum_{j\in
M}\frac{\lambda}{T^{j}\left(N,M,t\right)}t_{ij}=\lambda
U_{i}\left(N,M,t\right).$

Then, $I$ is a (positive) linear transformation of $U$ and hence it coincides
with $U$.

### Independence of the axioms of Theorem 1

We now show that all the axioms used in the previous characterizations are
independent.

$\left(a\right)$ Let $I^{1}$ be the uniform index that assigns to each artist
a constant number. Namely, for each $\left(N,M,t\right)$ and each $i\in N$,
$I_{i}^{1}\left(N,M,t\right)=1$. $I^{1}$ satisfies additivity but not
homogeneity.

Let $I^{2}$ be defined as follows. For each $\left(N,M,t\right)$ and each
$i\in N,$

$I_{i}^{2}\left(N,M,t\right)=\sum_{j=1}^{m}\frac{t_{ij}+T_{i}}{T^{j}+\sum\limits_{i^{\prime}\in
N}T_{i^{\prime}}}.$

$I^{2}$ satisfies homogeneity but not additivity.

$\left(b\right)$ Let $I^{3}$ be defined as follows. For each
$\left(N,M,t\right)$ and each $i\in N,$

$I_{i}^{3}\left(N,M,t\right)=\sum_{j=1}^{m}t_{ij}^{2}.$

$I^{3}$ satisfies additivity and equal individual impact of similar users but
not homogeneity.

$I^{2}$, defined above, satisfies homogeneity and equal individual impact of
similar users, but not additivity.

The user-centric index satisfies homogeneity and additivity but not equal
individual impact of similar users.

$\left(c\right)$ $I^{1}$, defined above, satisfies additivity and equal global
impact of users, but not homogeneity.

Let $I^{4}$ be defined as follows. For each $\left(N,M,t\right)$ and each
$i\in N,$

$I_{i}^{4}\left(N,M,t\right)=\frac{T_{i}}{\sum\limits_{i^{\prime}\in
N}T_{i^{\prime}}}m.$

$I^{4}$ satisfies homogeneity and equal global impact of users, but not
additivity.

The pro-rata index satisfies homogeneity and additivity, but not equal global
impact of users.

### Reasonable lower bound and click-fraud-proofness

The user-centric index satisfies reasonable lower bound. Let
$\left(N,M,t\right)\in\mathcal{P}$, $C\subset M$ and
$L^{C}=\bigcup\limits_{j\in C}L^{j}.$ Then,

$\displaystyle\sum_{i\in
L^{C}}\frac{U_{i}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}U_{i^{\prime}}\left(N,M,t\right)}m$ $\displaystyle=$
$\displaystyle\sum_{i\in L^{C}}U_{i}\left(N,M,t\right)=\sum_{i\in
L^{C}}\sum_{j\in M}\frac{t_{ij}}{T^{j}}$ $\displaystyle\geq$
$\displaystyle\sum_{i\in L^{C}}\sum_{j\in C}\frac{t_{ij}}{T^{j}}=\sum_{j\in
C}\sum_{i\in L^{C}}\frac{t_{ij}}{T^{j}}$ $\displaystyle=$
$\displaystyle\sum_{j\in C}1=\left|C\right|.$

The pro-rata index does not satisfy it. To show that, consider Example 1 and
let $C=\left\\{a\right\\}.$ Then $L^{a}=\left\\{1\right\\}$ but

$\frac{P_{1}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}P_{i^{\prime}}\left(N,M,t\right)}2=\frac{20}{100}<1.$

The user-centric index satisfies click-fraud-proofness. To show this, let
$\left(N,M,t\right)$, $\left(N,M,t^{\prime}\right)$, and $j\in M$ such that
$t_{ij^{\prime}}=t_{ij^{\prime}}^{\prime}$ for all $j^{\prime}\in
M\backslash\left\\{j\right\\}.$ Then, for all $i\in N,$

$\displaystyle\left|\frac{U_{i}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}U_{i^{\prime}}\left(N,M,t\right)}m-\frac{U_{i}\left(N,M,t^{\prime}\right)}{\sum\limits_{i^{\prime}\in
N}U_{i^{\prime}}\left(N,M,t^{\prime}\right)}m\right|$ $\displaystyle=$
$\displaystyle\left|U_{i}\left(N,M,t\right)-U_{i}\left(N,M,t^{\prime}\right)\right|$
$\displaystyle=$
$\displaystyle\left|\frac{t_{ij}}{T^{j}\left(N,M,t\right)}-\frac{t_{ij}^{\prime}}{T^{j}\left(N,M,t^{\prime}\right)}\right|$
$\displaystyle\leq$ $\displaystyle 1,$

where the last inequality is true because
$\frac{t_{ij}}{T^{j}\left(N,M,t\right)},\frac{t_{ij}^{\prime}}{T^{j}\left(N,M,t^{\prime}\right)}\in\left[0,1\right]$.

The pro-rata index, however, does not satisfy it. To show that, consider
Example 1. If we let

$t^{\prime}=\left(\begin{array}[]{cc}10&0\\\ 0&2\end{array}\right),$

we obtain

$\left|\frac{P_{2}\left(N,M,t\right)}{\sum\limits_{i^{\prime}\in
N}P_{i^{\prime}}\left(N,M,t\right)}m-\frac{P_{2}\left(N,M^{{}^{\prime}},t^{\prime}\right)}{\sum\limits_{i^{\prime}\in
N}P_{i^{\prime}}\left(N,M^{\prime},t^{\prime}\right)}m\right|=\left|\frac{90}{100}2-\frac{2}{12}2\right|=\frac{22}{15}>1.$

### Proof of Theorem 2

We prove Theorem 2 using a classical result from the literature on cooperative
game theory. We first introduce some notation. We then explain the classical
result. Finally, we apply it to our game $\left(N,v\right).$

The unanimity game $u_{R}$ associated with the nonempty coalition $R\subset N$
is defined as $u_{R}\left(S\right)=1$ when $R\subset S$ and
$u_{R}\left(S\right)=0$ otherwise.

Harsanyi (1959) proved that for every $TU$ game $\left(N,v\right)$, there
exist unique weights $\Delta_{v}\left(R\right)\in\mathbb{R}$ such that
$v=\sum\limits_{\varnothing\neq R\subset N}\Delta_{v}\left(R\right)u_{R}.$ The
weights $\Delta_{v}\left(R\right)$ are called the (Harsanyi) dividends of $v.$

Given a $TU$ game $v,$ the Harsanyi set $H(N,v)$ (Vasil’ev 1978, 1981) is
defined as the set of allocations obtained by distributing the dividend of any
coalition $R$ in any way among the agents in $R.$ We now introduce it
formally.222222The presentation we consider here follows the one in van den
Brink et al., (2014).

We say that $p=\left(p^{R}\right)_{\varnothing\neq R\subset N}$ is a sharing
system if for each $\varnothing\neq R\subset N$ we have that
$p^{R}\in\mathbb{R}^{N},$ $p_{i}^{R}\geq 0$ for all $i\in R,$ $p_{i}=0$ for
all $i\in N\backslash R,$ and $\sum\limits_{i\in N}p_{i}^{R}=1.$ Each sharing
system $p$ induces the allocation $y^{p}$ where for each $i\in N$,

$y_{i}^{p}=\sum_{i\in R}\Delta_{v}\left(R\right)p_{i}^{R}.$

The Harsanyi set is defined as

$H\left(N,v\right)=\left\\{y^{p}:p\text{ is a sharing system}\right\\}.$

Vasil’ev (1981) proved that if all dividends of $\left(N,v\right)$ are
nonnegative, $C\left(N,v\right)=H\left(N,v\right)$.

It is straightforward to check that given a streaming problem
$\left(N,M,t\right)$,

$v=\sum_{\varnothing\neq R\subset N}\left|j\in M:L^{j}=R\right|u_{R}.$

Then, for each $\varnothing\neq R\subset N,$
$\Delta_{v}\left(R\right)=\left|j\in M:L^{j}=R\right|\geq 0.$ Thus, for our
result, it suffices to prove that $H\left(N,v\right)=A\left(N,M,t\right).$

We first prove that $H\left(N,v\right)\subset A\left(N,M,t\right).$ Let $p$ be
a sharing system. We must prove that $y^{p}\in A\left(N,M,t\right).$ We define
$\left(x^{j}\right)_{j\in M}$ and $x=\sum\limits_{j\in M}x^{j}\in
A\left(N,M,t\right)$ as follows. Given $j\in M$ we take $x^{j}=p^{L^{j}}.$
Because of the definition of a sharing system we have that $x^{j}$ satisfies
the condition of $A\left(N,M,t\right).$ Besides, for each $i\in N$

$x_{i}=\sum_{j\in M}x_{i}^{j}=\sum_{j\in M:i\in L^{j}}p_{i}^{L^{j}}=\sum_{i\in
R\subset N}\left|j\in M:L^{j}=R\right|p_{i}^{R}=y_{i}^{p}.$

We now prove that $A\left(N,M,t\right)\subset H\left(N,v\right).$ Let
$\left(x^{j}\right)_{j\in M}$ and $x=\sum\limits_{j\in M}x^{j}\in
A\left(N,M,t\right).$ We define $p$ as follows. For each $R\subset N$ and
$i\in N,$

$p_{i}^{R}=\left\\{\begin{tabular}[]{ll}$\frac{\sum\limits_{j\in
M,L^{j}=R}x_{i}^{j}}{\left|j\in M,L^{j}=R\right|}$&$\text{when }\left|j\in
M,L^{j}=R\right|>0$\\\ $\frac{1}{\left|N\right|}$&when $\left|j\in
M,L^{j}=R\right|=0.$\end{tabular}\right.$

We prove that $p$ is a sharing system. Let $R\subset N.$ Since $x_{i}^{j}\geq
0$ for all $i\in N$ and $j\in M,$ $p_{i}^{R}\geq 0$ when $i\in R.$ Since
$x_{i}^{j}=0$ when $i\in N\backslash L^{j},$ $p_{i}^{R}=0$ when $i\in
N\backslash R.$ If $\left|j\in M,L^{j}=R\right|=0,$ obviously
$\sum\limits_{i\in N}p_{i}^{R}=1.$ Assume that $\left|j\in
M,L^{j}=R\right|>0.$ Now,

$\sum\limits_{i\in N}p_{i}^{R}=\sum\limits_{i\in N}\frac{\sum\limits_{j\in
M,L^{j}=R}x_{i}^{j}}{\left|j\in M,L^{j}=R\right|}=\sum\limits_{j\in
M,L^{j}=R}\frac{\sum\limits_{i\in N}x_{i}^{j}}{\left|j\in
M,L^{j}=R\right|}=\sum\limits_{j\in M,L^{j}=R}\frac{1}{\left|j\in
M,L^{j}=R\right|}=1.$

Besides, for each $i\in N,$

$\displaystyle y_{i}^{p}$ $\displaystyle=$ $\displaystyle\sum_{i\in R\subset
N}\left|j\in M:L^{j}=R\right|p_{i}^{R}$ $\displaystyle=$
$\displaystyle\sum_{i\in R\subset N}\left|j\in
M:L^{j}=R\right|\frac{\sum\limits_{j\in M,L^{j}=R}x_{i}^{j}}{\left|j\in
M,L^{j}=R\right|}$ $\displaystyle=$ $\displaystyle\sum_{i\in R\subset
N}\sum\limits_{j\in M,L^{j}=R}x_{i}^{j}=\sum\limits_{j\in M}\sum_{i\in
L_{j}}x_{i}^{j}=\sum\limits_{j\in M}x_{i}^{j}$ $\displaystyle=$ $\displaystyle
x_{i}.$

### Proof of Theorem 3

We have already seen that the user-centric index satisfies homogeneity,
additivity, and core selection.

Conversely, let $I$ be an index satisfying the axioms. Let
$\left(N,M,t\right)\in\mathcal{P}^{\ast}$ and $j\in M.$ We consider the
problem $\left(N,\left\\{j\right\\},t_{.j}\right)\in\mathcal{P}^{\ast}$. By
core selection, taking $S=$ $L^{j}\left(N,\left\\{j\right\\},t_{.j}\right)\neq
N$, we have

$\sum_{i\in
S}\frac{I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)}\geq
v_{\left(N,\left\\{j\right\\},t_{.j}\right)}\left(S\right)=1,$

which implies

$\sum_{i\in
L^{j}\left(N,\left\\{j\right\\},x\right)}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)\geq\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right).$

Thus,

$\sum_{i\in N\backslash
L^{j}\left(N,\left\\{j\right\\},x\right)}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)\leq
0.$

And, therefore,

$I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=0\text{ for each }i\in
N\backslash L^{j}\left(N,\left\\{j\right\\},t_{.j}\right).$ (1)

Now, let $i,i^{\prime}\in L^{j}\left(N,\left\\{j\right\\},t_{.j}\right).$ By
homogeneity,

$I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)=\frac{t_{i^{\prime}j}}{t_{ij}}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Then, for each $i\in L^{j}\left(N,\left\\{j\right\\},t_{.j}\right)$,

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)=\frac{\sum\limits_{i^{\prime}\in
N}t_{i^{\prime}j}}{t_{ij}}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

Hence,

$I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=\left[\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)\right]\frac{t_{ij}}{\sum\limits_{i^{\prime}\in
N}t_{i^{\prime}j}}=\left[\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right)\right]U_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

We now prove the following claim.

Claim. For each pair $j^{1},j^{2}\in M$, such that $j^{1}\neq j^{2}$ and
$L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\cap
L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)=\varnothing$ we have

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

Suppose that the claim is false. Then, without loss of generality, assume that

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)<\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

Let $\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)\in\mathcal{P}^{\ast}$
where $\hat{t}$ denotes the restriction of $t$ to
$\left\\{j^{1},j^{2}\right\\}.$ By core selection, taking $S=$
$L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right),$

$1=v_{\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)}\left(S\right)\leq\sum_{i\in
S}\frac{I_{i}\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)}2$

By additivity,

$\displaystyle\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)$
$\displaystyle=$ $\displaystyle\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)+\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)\text{ and }$
$\displaystyle\sum_{i\in
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}I_{i}\left(N,\left\\{j^{1},j^{2}\right\\},\hat{t}\right)$
$\displaystyle=$ $\displaystyle\sum_{i\in
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}I_{i}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)+\sum_{i\in
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}I_{i}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

By $\left(\ref{0 en N - Lj}\right),$

$\displaystyle\sum_{i\in
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}I_{i}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)$
$\displaystyle=$ $\displaystyle\sum_{i\in
N}I_{i}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\text{ and }$
$\displaystyle\sum_{i\in
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}I_{i}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)$
$\displaystyle=$ $\displaystyle 0.$

Then,

$\displaystyle 1$ $\displaystyle\leq$ $\displaystyle\frac{\sum\limits_{i\in
N}I_{i}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)+\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)}2$
$\displaystyle<$ $\displaystyle\frac{\sum\limits_{i\in
N}I_{i}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}{\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)+\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)}2=1,$

which is a contradiction. Hence, the claim holds.

We now prove that, for each pair $j^{1},j^{2}\in M$, with $j^{1}\neq j^{2}$

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

By the claim, the result holds when
$L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\cap
L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)=\varnothing.$ Assume
then otherwise. We consider two cases.

  1. 1.

$L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\supset
L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)$.232323The case
$L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\subset
L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)$ is similar and,
thus, we omit it.

As $\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\in\mathcal{P}^{\ast}$, it
follows that $S=N\setminus
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\neq\varnothing.$ Let
$j^{3}\in M\backslash\left\\{j^{1},j^{2}\right\\}$ and $t_{.j^{3}}^{S}$ be
such that

$t_{ij^{3}}^{S}=\left\\{\begin{tabular}[]{ll}$1$&for each $i\in S$,\\\ $0$&for
each $i\in N\setminus S$.\end{tabular}\right.$

Then, by the Claim above,

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{3}\right\\},t_{.j^{3}}^{S}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

  2. 2.

$S=L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\setminus
L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)\neq\varnothing$ and
$S^{\prime}=L^{j^{2}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right)\setminus
L^{j^{1}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)\neq\varnothing.$ Let
$t_{.j^{1}}^{S}$ and $t_{.j^{2}}^{S^{\prime}}$ be defined as in Case 1. Then,
by the Claim above,

$\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}^{S^{\prime}}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{1}\right\\},t_{.j^{1}}^{S}\right)=\sum_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j^{2}\right\\},t_{.j^{2}}\right).$

To conclude the proof, we define $p=\sum\limits_{i^{\prime}\in
N}I_{i^{\prime}}\left(N,\left\\{j\right\\},t_{.j}\right).$ Based on the above,
$p$ is well defined because it neither depends on $j$, nor on $t_{.j}.$ Then,

$I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=pU_{i}\left(N,\left\\{j\right\\},t_{.j}\right).$

By additivity,

$I_{i}\left(N,M,t\right)=\sum_{j\in
M}I_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=\sum_{j\in
M}pU_{i}\left(N,\left\\{j\right\\},t_{.j}\right)=pU_{i}\left(N,M,t\right).$

Thus, $I$ is also the user-centric index, as desired.

### Independence of the axioms of Theorem 3

We now show that all the axioms used in the previous characterization are
independent.

Let $I^{5}$ be defined as follows. For each $\left(N,M,t\right)$ and each
$i\in N,$

$I_{i}^{5}\left(N,M,t\right)=\sum_{j\in M:i\in
L^{j}\left(N,\left\\{j\right\\},t_{.j}\right)}\frac{1}{\left|L^{j}\left(N,\left\\{j\right\\},t_{.j}\right)\right|}.$

$I^{5}$ satisfies all axioms but homogeneity.

$I^{2}$, defined above, satisfies all axioms but additivity.

The pro-rata satisfies all axioms but core selection.

### Proof of Theorem 4

$\left(a\right)$ We consider the weight function $\omega^{P}$ given by

$\omega_{j}^{P}\left(\left(C^{j^{\prime}}\right)_{j^{\prime}\in
K},E\right)=\frac{C^{j}}{\sum\limits_{j^{\prime}\in K}C^{j^{\prime}}}.$

For each $i\in N,$

$P_{i}^{\omega^{P}}\left(N,K,c,E\right)=\sum_{j\in
M}\frac{t_{ij}}{T^{j}}\frac{T^{j}}{\sum\limits_{j^{\prime}\in
K}T^{j^{\prime}}}m=\frac{\sum\limits_{j\in
M}t_{ij}}{\sum\limits_{j^{\prime}\in
K}T^{j^{\prime}}}m=\frac{T_{i}}{\sum\limits_{i^{\prime}\in
N}T_{i^{\prime}}}m=R_{i}^{P}\left(N,M,t\right).$

Thus, $R^{P}$ is the weighted proportional rule associated to $\omega^{P}$.

We now take the weight function $\omega^{U}$ given by

$\omega_{j}^{U}\left(\left(C^{j^{\prime}}\right)_{j^{\prime}\in
K},E\right)=\frac{1}{\left|K\right|}.$

For each $i\in N,$

$P_{i}^{\omega^{U}}\left(N,K,c,E\right)=\sum_{j\in
M}\frac{t_{ij}}{T^{j}}\frac{1}{m}m=\sum_{j\in
M}\frac{t_{ij}}{T^{j}}=U_{i}\left(N,M,t\right).$

Thus $U$ is the weighted proportional rule associated to $\omega^{U}$.

$\left(b\right)$ Given $i\in N,$ we compute $R_{i}^{P,P}\left(N,K,c,E\right)$

First stage. For each $j\in K,$

$P_{j}(K,c^{K},E)=\frac{T^{j}}{\sum\limits_{j^{\prime}\in K}T^{j^{\prime}}}m.$

Second stage.

$P_{i}(N,c_{.j},P_{j}\left(K,c^{K},E\right))=\frac{t_{ij}}{T^{j}}\frac{T^{j}}{\sum\limits_{j^{\prime}\in
M}T^{j^{\prime}}}m=\frac{t_{ij}}{\sum\limits_{j^{\prime}\in
M}T^{j^{\prime}}}m.$

Thus, for each $i\in N$,

$\displaystyle R_{i}^{P,P}(R,N,E,C)$ $\displaystyle=$ $\displaystyle\sum_{j\in
K}P_{i}(N,c_{.j},P_{j}\left(K,c^{K},E\right))$ $\displaystyle=$
$\displaystyle\sum_{j\in M}\frac{t_{ij}}{\sum\limits_{j^{\prime}\in
M}T^{j^{\prime}}}m=\frac{T_{i}}{\sum\limits_{i^{\prime}\in
N}T_{i^{\prime}}}m=R_{i}^{P}\left(N,M,t\right).$

$\left(c\right)$ Given $i\in N,$ we compute
$R_{i}^{CEA,P}\left(N,K,c,E\right).$

First stage. For each $j\in K,$
$CEA_{j}(K,c^{K},E)=\min\left\\{\lambda,T^{j}\right\\}$ where
$\sum\limits_{j\in K}\min\left\\{\lambda,T^{j}\right\\}=E.$ It is
straightforward to check that $\lambda=1.$ Hence,

$CEA_{j}(K,c^{K},E)=1.$

Second stage.

$P_{i}(N,c_{.j},CEA_{j}(K,c^{K},E))=\frac{t_{ij}}{T^{j}}.$

Thus, for each $i\in N$,

$\displaystyle R_{i}^{CEA,P}(R,N,E,C)$ $\displaystyle=$
$\displaystyle\sum_{j\in K}P_{i}(N,c_{.j},CEA_{j}(K,c^{K},E)))$
$\displaystyle=$ $\displaystyle\sum_{j\in
M}\frac{c_{ij}}{T^{j}}=U_{i}\left(N,M,t\right).$

## References

  * [1] Adams, W.J., Yellen, J.L., 1973. Commodity bundling and the burden of monopoly. Quarterly Journal of Economics 90, 475-98.
  * [2] Aguiar, L., Waldfogel, J., 2018. As streaming reaches flood stage, does it stimulate or depress music sales? International Journal of Industrial Organization 57, 278-307.
  * [3] Aguiar, L., Waldfogel, J., 2021. Platforms, power, and promotion: Evidence from Spotify playlists. The Journal of Industrial Economics 69, 653-691.
  * [4] Aumann R.J., Maschler M., 1985. Game theoretic analysis of a bankruptcy problem from the Talmud. Journal of Economic Theory 36, 195-213.
  * [5] Alaei S, Makhdoumi A, Malekian A, Pekeč S., 2022. Revenue-Sharing Allocation Strategies for Two-Sided Media Platforms: Pro-Rata vs. User-Centric. Management Science 68, 8699-8721.
  * [6] Bhargava, H.K., 2013. Mixed bundling of two independently valued goods. Management Science 59, 2170-2185.
  * [7] Bhattacharjee, S., Gopal, R.D., Lertwachara, K., Marsden, J.R., Telang, R., 2007. The effect of digital sharing technologies on music markets: A survival analysis of albums on ranking charts. Management Science 53, 1359-1374.
  * [8] Bergantiños G, Lorenzo, L. Lorenzo-Freire, S., 2010. A characterization of the proportional rule in multi issue allocation situations. Operations Research Letters 38, 17-19.
  * [9] Bergantiños G, Lorenzo, L. Lorenzo-Freire, S., 2011. New characterizations of the constrained equal awards rule in multi-issue allocation situations. Mathematical Methods of Operations Research 74, 311-325.
  * [10] Bergantiños G, Chamorro, J.M. Lorenzo, L. Lorenzo-Freire, S., 2018\. Mixed rules in multi-issue allocation situations. Naval Research Logistics 65, 66-77.
  * [11] Bergantiños G, Moreno-Ternero JD., 2015. The axiomatic approach to the problem of sharing the revenue from museum passes. Games and Economic Behavior 89, 78-92.
  * [12] Bergantiños, G., Moreno-Ternero, J.D., 2020. Sharing the revenues from broadcasting sport events. Management Science 66, 2417-2431.
  * [13] Bergantiños, G., Moreno-Ternero, J.D., 2021. Compromising to share the revenues from broadcasting sports leagues. Journal of Economic Behavior and Organization 183, 57-74.
  * [14] Bergantiños, G., Moreno-Ternero, J.D., 2022. On the axiomatic approach to sharing the revenues from broadcasting sports leagues. Social Choice and Welfare 58, 321-347.
  * [15] van den Brink, R., van der Laan, G., Vasilev, V., (2014), Constrained core solutions for totally positive games with ordered players. International Journal of Game Theory 43, 351-368.
  * [16] Calleja, P., Borm, P., and Hendrickx, R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730-747
  * [17] Calleja, P., Llerena, F., 2023. Proportional clearing mechanisms in financial systems: an axiomatic approach. Journal of Mathematical Economics. Forthcoming.
  * [18] Csóka, P., Herings, P.J.J., 2018. Decentralized clearing in financial networks. Management Science 64, 4471-4965.
  * [19] Csóka, P., Herings, P.J.J., 2021. An axiomatization of the proportional rule in financial networks. Management Science 67, 2799-2812.
  * [20] Deezer., 2019. Deezer wants artists to be paid fairly - By adopting a User-Centric Payment System (UCPS), Retrieved December 20, 2019 from https:// www.deezer.com/ucps.
  * [21] Dietzenbacher, B.J., Kondratev, A.Y., 2023. Fair and consistent prize allocation in competitions. Management Science 69, 3319-3339.
  * [22] Dimont J., 2017. Royalty inequity: Why music streaming services should switch to a per-subscriber model. Hastings Law J. 69:675.
  * [23] Eisenberg, L., Noe, T.H., 2001. Systemic risk in financial systems. Management Science 47, 236-249.
  * [24] Flores-Szwagrzak, K., Treibich, R., 2020. Teamwork and Individual Productivity. Management Science 66, 2523-2544.
  * [25] Geng, X., Stinchcombe, M.B., Whinston, A.B., 2005. Bundling information goods of decreasing value. Management Science 51, 662-667.
  * [26] Ginsburgh V., Zang I., 2003. The museum pass game and its value. Games and Economic Behavior 43, 322-325.
  * [27] Gonçalves-Dosantos, J., Martinez, R., Sánchez-Soriano, J., 2023\. Measures of relevance in the success of streaming platforms. Mimeo
  * [28] Gudmundsson, J., Hougaard, J.L., Ko, C.Y., 2023. Sharing sequentially triggered losses: Automated conflict resolution through smart contracts. Management Science. Forthcoming.
  * [29] Haampland O, Johannessen R, Moreau F, Wikstrom P., 2022. Fairness and royalty payment systems on music streaming platforms. Mimeo.
  * [30] Harsanyi J., 1959. A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton University Press, Princeton, pp 325-355.
  * [31] Hougaard, J.L., Moulin, H., 2014. Sharing the cost of redundant items. Games and Economic Behavior 87, 339-352.
  * [32] Ingham T., 2021. SoundCloud revolutionizes streaming music payouts, launching new royalties system. Music Business Worldwide. https://www.musicbusinessworldwide.com/soundcloud-isabout-to-revolutionize-streaming-payouts-launching-user-centric-royalties-for-100000-indieartists/
  * [33] Jain, S., Qian, K., 2021. Compensating online content producers: A theoretical analysis. Management Science 67, 7075-7090.
  * [34] Ju, B-G., Miyagawa, E. and Sakai, T., 2007. Non-Manipulable Division Rules in Claim Problems and Generalizations. Journal of Economic Theory 132, 1-26.
  * [35] Krajewska, M., Kopfer, H., Laporte, G., Ropke, S., Zaccour, G., 2008\. Horizontal cooperation among freight carriers: request allocation and profit sharing. Journal of the Operational Research Society 59, 1483-1491.
  * [36] Littlechild, S., Owen. G., 1973. A simple expression for the Shapley value in a special case. Management Science 20, 370-372.
  * [37] Lei, X., 2023. Pro-rata vs User-centric in the music streaming industry. Economics Letters 226, 111111.
  * [38] Martinez, R., Moreno-Ternero, J.D., 2022. An axiomatic approach towards pandemic performance indicators. Economic Modelling 116, 105983.
  * [39] Meyn J, Kandziora M, Albers S, 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.
  * [40] Moreno-Ternero, J., 2009. The proportional rule for multi-issue bankruptcy problems. Economics Bulletin 29, 483-490.
  * [41] Moreno-Ternero, J., Roemer, J., 2006. Impartiality, priority and solidarity in the theory of justice. Econometrica 74, 1419-1427.
  * [42] Muiku J., 2017. Pro rata and user centric distribution models: a comparative study. Digital Media Finland.
  * [43] van den Nouweland, A., Borm, P., van Golstein Brouwers, W., Groot Bruinderink, R., Tijs, S., 1996. A Game Theoretic Approach to Problems in Telecommunication. Management Science 42, 294-303.
  * [44] O’Neill, B., 1982. A problem of rights arbitration from the Talmud. Mathematical Social Sciences 2, 345-371.
  * [45] Page W, Safir D., 2018a. Money in, money out: Lessons from CMOS in allocating and distributing licensing revenue. Technical report.
  * [46] Page W, Safir D., 2018b. User-centric revisited: The unintended consequences of royalty distribution. Technical report, Spotify, Stockholm, Sweden.
  * [47] Rietveld, J., Schilling, M.A., 2021. Platform competition: A systematic and interdisciplinary review of the literature. Journal of Management 47, 1528-1563.
  * [48] Shapley L., 1953. A value for n-person games, in Contributions to the Theory of Games II (Annals of Mathematics Studies 28), ed. by H.W. Kuhn and A.W. Tucker, Princeton: Princeton University Press, 307-317.
  * [49] Singal, R., Besbes, O., Desir, A., Goyal, V., Iyengar, G., 2022. Shapley Meets Uniform: An Axiomatic Framework for Attribution in Online Advertising. Management Science 68, 7457-7479.
  * [50] Thomson W., 2019. _How to divide when there isn’t enough: from Aristotle, the Talmud, and Maimonides to the axiomatics of resource allocation_ , Econometric Society Monograph.
  * [51] Thomson, W., 2023. _The Axiomatics of Economic Design_ , Vol. 1\. Studies in Choice and Welfare. Springer, Cham.
  * [52] Vasilev, 1978. Support function of the core of a convex game. Optimizacija Vyp 21, 30-35 (in Russian).
  * [53] Vasilev VA, 1981. On a class of imputations in cooperative games. Sov Math Dokl 23, 53-57.
  * [54] Waldfogel, J., 2017. How digitization has created a golden age of music, movies, books, and television. Journal of Economic Perspectives 31, 195-214.