# fraudproof_revenue_division_on_subscription_platforms__0d5ac1d7.pdf Fraud-Proof Revenue Division on Subscription Platforms Abheek Ghosh 1 * Tzeh Yuan Neoh 2 * Nicholas Teh 1 * Giannis Tyrovolas 1 * We study a model of subscription-based platforms where users pay a ﬁxed fee for unlimited access to content, and creators receive a share of the revenue. Existing approaches to detecting fraud predominantly rely on machine learning methods, engaging in an ongoing arms race with bad actors. We explore revenue division mechanisms that inherently disincentivize manipulation. We formalize three types of manipulation-resistance axioms and examine which existing rules satisfy these. We show that a mechanism widely used by streaming platforms, not only fails to prevent fraud, but also makes detecting manipulation computationally intractable. We also introduce a novel rule, SCALEDUSERPROP, that satisﬁes all three manipulation-resistance axioms. Finally, experiments with both real-world and synthetic streaming data support SCALEDUSERPROP as a fairer alternative compared to existing rules. 1. Introduction In September 2024, the FBI criminally charged a musician, Michael Smith, for orchestrating a scheme to fraudulently inﬂate his music streams on platforms such as Amazon Music, Apple Music, Spotify, and You Tube Music and according to court documents, walked away with over US$10 million in royalty payments (United States Attorney s Ofﬁce, 2024). To successfully execute his scheme, he utilized hundreds of thousands of songs created using AI, and built a complicated network of over a thousand bot accounts that artiﬁcially boost streams across these platforms billions of times. Although each stream originated from a bona ﬁde, fee-paying account, the way the platform(s) distributed subscription revenue allowed each bot to generate more in royalties than it cost to maintain its subscription. *Alphabetical ordering 1University of Oxford, UK 2Harvard University, USA. Correspondence to: Tzeh Yuan Neoh , Nicholas Teh . Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s). Subscription platforms have seen signiﬁcant growth in recent years, driven by the rise of internet streaming services such as Spotify, Apple Music, Netﬂix, etc. For instance, the annual revenue of the music streaming industry reached US$27.6 billion in 2023, with signiﬁcant increases over the last ten years (International Federation of the Phonographic Industry, 2024). Under this business model, users pay a ﬁxed subscription fee to enjoy unlimited access to all content on the platform, typically by content creators. The platform then takes a ﬁxed revenue cut and distributes the rest to the creators based on engagement metrics (e.g., play counts or views) and/or speciﬁc agreements between creators and platforms. Despite efforts to curb manipulation, bad actors persist, using bots and click-farms to inﬂate user engagement (Drott, 2020; Snickars & M ahler, 2018). This issue is so signiﬁcant that major music streaming platforms like Amazon Music and Spotify have established an industry advocacy group (Music Fights Fraud Alliance, 2025) to combat such fraud, which is estimated to cost the industry US$300 million annually (Burton, 2021). Additionally, the rise of AI-generated content introduces new challenges platforms are increasingly ﬂooded with synthetic tracks, videos, and live streams designed to exploit engagement-driven algorithms. This AI-generated content often ampliﬁes fraudulent listening activities, making manipulation harder to detect. Current machine learning (ML) approaches to this problem predominantly focus on detecting fraudulent activity using sophisticated algorithms ranging from anomaly detection (Esmaeilzadeh et al., 2022) to unsupervised learning (Mollao glu et al., 2021) and graph neural networks (Li et al., 2021). For instance, music streaming platforms such as Spotify have proprietary models that identify whether a stream is legitimate (using meta-data such as IP location, listening patterns, and other information) and issue ﬁnes if they deem too many streams to be fraudulent (Spotify, 2025). However, as AI continues to evolve, so do the methods used by fraudsters, leading to a continuous arms race. These bad actors increasingly leverage advanced automation tools to make fraudulent activities more sophisticated and harder to detect, challenging the robustness of existing detection frameworks and driving the need for innovative, adaptive solutions (United States Attorney s Ofﬁce, 2024). Fraud-Proof Revenue Division on Subscription Platforms The root of the problem stems from the way revenue is currently distributed to content creators on most subscriptionbased streaming platforms: funds from the royalty pool are allocated proportionally among artists based on their respective percentages of total streams (United States Attorney s Ofﬁce, 2024) we call this rule GLOBALPROP. In this paper, we tackle this problem from a mechanism design perspective, i.e., we mathematically formalize notions of fraud in this setting and investigate the existence of revenue division mechanisms that can inherently disincentivize fraudulent behavior, thereby reducing the industry s reliance on expensive and complex fraud detection methods to combat manipulation. Moreover, if such mechanisms exist, they could complement existing ML-based approaches by providing a foundational layer of fraud resistance. These mechanisms inherently target known forms of fraud, allowing ML systems to focus on adapting to emerging, previously unseen types of fraud that may arise in the future, ensuring continuous improvement in detecting and addressing manipulation. Additionally, many policymakers and academics have also argued against the fairness of GLOBALPROP in favor of an alternative rule USERPROP (which directly allocates a ﬁxed fraction of each user s subscription fee only among the creators of the content the user consumes) from an economic (Meyn et al., 2023; Muikku, 2017), empirical (Moreau et al., 2024), theoretical (Berganti nos & Moreno Ternero, 2025), and legal (Dimont, 2018) perspective. Motivated by these debates, we aim to address fairness considerations in our work as well. Lastly, the primary focus of our work is on fraudulent behavior speciﬁcally related to the creation of fake users (bots) to manipulate engagement metrics. We deliberately do not address the equally prevalent issue of widespread AIgenerated content on these platforms. The legal status of such content can vary, especially since some popular artists openly release their AI-generated voices as (semi-)opensource (Josan, 2024), making its permissibility platformdependent and governed by speciﬁc rules. Nonetheless, our work provides a principled framework for studying similar challenges. As AI continues to evolve and new forms of fraudulent behavior emerge, our approach can be extended to address these issues, provided that appropriate regulatory frameworks are established to guide the platforms. 1.1. Our Results In this work, we focus on designing manipulation-resistant mechanisms from a computational and axiomatic perspective, setting our research apart from all previous work on this model. Although we build on the standard model for subscription platforms established in prior literature, our key contribution lies in introducing several axioms that aim to capture both resistance to manipulation and maintaining fairness and analyzing these axioms with respect to multiple revenue-division mechanisms three from existing literature and one novel mechanism that we propose. Moreover, we challenge the current status quo rule, GLOB- ALPROP, by demonstrating that detecting suspicious activity under this rule is computationally intractable an important ﬁnding in this context. Since fraud detection (and fraud in general) is highly relevant to the ML community, we believe this result will be of particular interest to researchers and practitioners in the ﬁeld. In Section 2, we establish three fundamental properties that deﬁne the space of mechanisms we consider: anonymity, neutrality, and no free-ridership. The ﬁrst two ensure that payoffs to artists only depend on their engagement with users. In particular, mechanisms cannot distinguish between fraudulent and genuine artists or users. No free-ridership eliminates trivial cases where an artist without engagement receives a non-zero payoff. Next, we formalize three forms of resistance to strategic manipulation. Fraud-proofness prevents adversaries from proﬁtably creating new fraudulent users. Bribery-proofness prevents proﬁtably bribing existing users and is a strengthening of click-fraud-proofness as presented in Berganti nos & Moreno-Ternero (2025). Finally, (strong) Sybil-proofness ensures that artists cannot gain by splitting into multiple identities or merging with others. All three axioms are novel in our setting and are motivated by real-world observations. We also introduce two additional fairness axioms engagement monotonicity and Pigou-Dalton consistency, the latter inspired by an equitability concept in welfare economics. In Section 3, we conduct an axiomatic study (with respect to our proposed concepts) of several rules proposed in the literature so far. Notably, we show that GLOBALPROP fails to satisfy fraud-proofness and bribery-proofness, in contrast to the other two contenders USERPROP and USEREQ. Contributing to existing critiques of GLOBALPROP, we establish a case against GLOBALPROP through a computational lens, and in the context of fraud detection. We show that if a platform uses GLOBALPROP, detecting potentially fraudulent activity is NP-hard. We then analyze the two other existing rules: USERPROP and USEREQ. We study their axiomatic properties and prove that they satisfy our manipulation-resistance axioms, unlike GLOBALPROP. We also demonstrate that portioning rules cataloged in Elkind et al. (2023) fail all the manipulation-resistance axioms we consider. Finally, in Section 4, we propose and study a new rule SCALEDUSERPROP. We show that it has the same axiomatic guarantees as USERPROP but is fairer when measured by the popular pay-per-stream metric. We use this to quantify maximum envy in this setting and empirically Fraud-Proof Revenue Division on Subscription Platforms verify this against existing rules in Section 5. All omitted proofs can be found in the full version of this paper. 1.2. Related Work Our work considers the model proposed and studied by several recent works on (music) streaming platforms.1 Alaei et al. (2022) and Lei (2023) focused on a comparative study between GLOBALPROP and USERPROP. More speciﬁcally, Alaei et al. (2022) focused on providing characterizations of both rules with respect to newly proposed axioms. They were also concerned with which of these two rules could sustain a set of artists proﬁtability on the platform, as well as comparing them from both the platform s and the artists perspectives. Lei (2023) pointed out the shortcomings of USERPROP. They compared the two rules primarily in terms of egalitarian fairness (i.e., the lowest payout among all artists) and efﬁciency (i.e., dominance on quality proﬁle ), but they allow for artists to vary stream quality and thus this concept is not relevant in our model. Berganti nos & Moreno-Ternero (2025) go beyond previous works to consider a family of rules that interpolates between GLOBALPROP and USERPROP, and they provide further characterizations for both rules and their interpolation. Subsequently, Berganti nos & Moreno-Ternero (2024) introduced the Shapley index as a rule for this setting and characterized it using existing and new axioms. Deng et al. (2024) investigate revenue-sharing mechanisms for AI-generated music platforms. Their work centers on the challenge of attributing a new, AI-created track to speciﬁc copyrighted recordings in the training data an attribution problem that underpins royalty allocation in that setting. This challenge is fundamentally distinct from the problems we address. A related stream of work is the museum pass problem, popular in the in the economics literature, and was ﬁrst introduced by Ginsburgh & Zang (2001; 2003). The problem studies the sharing of revenue among museums from the sale of museum passes for a price below the aggregate admission fee of individual member museums (i.e., bundled pricing). B eal & Solal (2010) and Ginsburgh & Zang (2001; 2003) studied the problem as a coalitional game, whereas Casas M endez et al. (2011) and Est evez-Fern andez et al. (2012) studied the problem as a bankruptcy game. Wang (2011) studied the dual version of the problem the museum cost sharing problem. All of the works above (including several more recent works which look at the Shapley value as a rule (Berganti nos & Moreno-Ternero, 2015; 2016)) es- 1However, we note that this model is also applicable to many other content subscription platforms (e.g., education, art, etc.). sentially conduct an axiomatic study of popular rules in their respective games modeled, but adapted to this new setting. We refer the reader to the Casas-M endez et al. (2014) for a survey on earlier works on this area. From 2001 to 2014, works on the topic cumulatively studied more than 30 axioms, with broadly two kinds of manipulation-resistant axioms one based on ticket prices and the other based on reported number of visitors . However, we note that the museum pass problem is fundamentally different from our problem, and thus the way axioms (and rules) are conceptualized would also naturally be distinct. This distinction is particularly apparent when it comes to concepts relating to manipulation. Our work also contributes to the broader literature on applying computational and algorithmic methods to address incentive-related challenges in online economic systems and platforms. For example, manipulation issues have been studied in the contexts of online advertising markets (Golrezaei et al., 2021a; Kanoria & Nazerzadeh, 2014), recommendation systems (Eilat & Rosenfeld, 2023; Yao et al., 2023), and e-commerce platforms (Golrezaei et al., 2021b; He et al., 2022; Mayzlin et al., 2014). 2. Model and Axioms For each positive integer k, let [k] := {1, . . . , k}. Let N = [n] be the set of users and C = [m] be the set of artists. Suppose that an adversary controls a set of fake users b N N and a set of fake artists b C C; let bn = | b N|. For each i 2 N and j 2 C, let wij 0 denote the number of interactions user i has with artist j.2 For each user i 2 N, we assume that P j2C wij > 0, i.e., the user has some non-zero interactions.3 Let wi = (wi1, . . . , wim) for each i 2 N. The engagement proﬁle is w = (w1, . . . , wn). Without loss of generality, we assume that the subscription fee for each user is 1 unit. Then, the total subscription fee collected from the users is n. As assumed in the prior works on this topic, and as observed in the real-world, we assume that the platform takes a cut of (1 )n and distributes the remaining n to the artists, for some 2 (0, 1]. A problem instance I = (N, C, w) is deﬁned by the set of users N, the set of artists C, and the engagement proﬁle w. A payment rule (or simply rule) is a function φ that maps each instance I to an m-valued vector (φI(1), . . . , φI(m)), where φI(j) is the payment to artist j 2 C. To simplify notation, for a subset of artists S C, we use φI(S) to 2This is typically deﬁned as a stream (on music streaming platforms like Spotify), whereby a user plays a track for a minimum duration, or a view (on video streaming platforms like You Tube Live) when a user joins and stays for a minimum amount of time. 3Note that in many of our proofs, we can without loss of generality assume that weights are rational numbers. Fraud-Proof Revenue Division on Subscription Platforms denote the sum of the payments to the artists in the set S: φI(S) = P 2.1. Preliminary Axioms We begin by introducing three fundamental fairness properties that any reasonable revenue division mechanism in our setting should satisfy. We will then provide a rationale for the necessity of these axioms in our setting. The ﬁrst axiom anonymity prescribes that the rule cannot distinguish between real and fake users. Deﬁnition 2.1 (Anonymity). A rule φ is anonymous if permuting the labels of the users does not affect the payoffs of the artists. Formally, rule φ is anonymous if for all instances I = (N, C, w) and I0 = (N, C, w0) and all permutations σ : N ! N, if wi = w0 σ(i) for all users i 2 N, then for all artists j 2 C, φI(j) = φI0(j). The second axiom neutrality is similar in nature to anonymity, but for artists. It prescribes that the rule cannot distinguish between real and fake artists. Deﬁnition 2.2 (Neutrality). A rule φ is neutral if permuting the labels of the artists permutes their payoffs. Formally, rule φ is neutral if for all instances I = (N, C, w) and I0 = (N, C, w0) and all permutations σ : C ! C, if wij = w0 iσ(j) for all users i 2 N and artists j 2 C, then for all artists j 2 C, φI(j) = φI0(σ(j)). In our setting, it is crucial to consider only rules that are anonymous and neutral. In practice, given the number of users/artists, it is virtually impossible to detect all fake users/artists, even with existing fraud detection techniques, as noted in our introduction. This inability to reliably distinguish between real and fake users or artists underscores the importance of addressing the questions we aim to answer. Finally, the last fundamental axiom we consider is the notion of no free-ridership. Intuitively, this means that artists who receive no user engagement should not receive any payment. Deﬁnition 2.3 (No free-ridership). A rule φ satisﬁes no free-ridership if, for any instance I = (N, C, w) and artist j 2 C where P i2N wij = 0, then φI(j) = 0. This axiom rules out trivial rules that allocate payments disregarding user engagement (e.g., giving equal payment to each artist irrespective of user engagement) and are, therefore, resistant to strategic manipulation. 2.2. Axioms for Preventing Strategic Manipulation We start by formalizing the fraud alleged in the indictment mentioned in the introduction. Intuitively, no adversary should be able to create fake users ( b N), pay their subscrip- tion fee, and earn a proﬁt from her own fake artists ( b C).4 Rules that make such fraud impossible are fraud-proof. Deﬁnition 2.4 (Fraud-proofness). A rule φ is fraud-proof if the following holds: For any two instances I = (N \ b N, C, w) and I0 = (N, C, w0) with wi = w0 i for all i 2 N \ b N, and any b C C, we have that φI0( b C) φI( b C) bn. A rule φ is single-user fraud-proof if bn = 1. Our deﬁnition of fraud-proofness considers only an adversary s proﬁt from creating fake users, not fake artists. This means an adversary can introduce fake artists to earn proﬁts without using fake users. However, without fake users, any fake artist must attract engagement from real users to proﬁt (by the no free-ridership assumption). Whether this practice violates a platform s rules is a separate issue beyond our scope we focus on the extra proﬁt an adversary can gain by adding fake users, assuming a ﬁxed set of artists (which may include fake ones). Next, we show that single-user fraud-proofness is equivalent to (multi-user) fraud-proofness, simplifying how one can reason about fraud-proofness. Proposition 2.5. A rule φ is fraud-proof if and only if it is single-user fraud-proof. Another form of manipulation is bribery. Bribery is particularly relevant in scenarios where the platform imposes substantially stringent access requirements, making creating fake users signiﬁcantly more challenging. However, under such conditions, artists may resort to colluding with and bribing users offering to pay the subscription fees of the users to manipulate their engagement proﬁles. This practice is commonly observed in streaming farms, the streaming equivalent of click farms in advertising (Drott, 2020). We call resistance to such bribery as bribery-proofness. Deﬁnition 2.6 (Bribery-proofness). A rule φ is briberyproof if the following holds: For any two instances I = (N, C, w) and I0 = (N, C, w0) with wi 6= w0 i for exactly k users, and any b C C, we have that φI0( b C) φI( b C) k. A rule φ is single-user bribery-proof if k = 1. Similarly to fraud-proofness, multi-user bribery-proofness and single-user bribery-proofness are equivalent. Proposition 2.7. A rule is bribery-proof if and only if it is single-user bribery-proof. We note that (single-user) bribery-proofness substantially strengthens the axiom of click-fraud-proofness proposed in Berganti nos & Moreno-Ternero (2025). Click-fraudproofness requires that a single user altering their engagement cannot alter the payoff of any artist by more than 4Note that we do not impose any constraints on the listening behavior or engagement proﬁles of these fake users. Fraud-Proof Revenue Division on Subscription Platforms 1. Formally, for all j, |φI0(j) φI(j)| 1. Singleuser bribery-proofness requires that for all subsets of artists b C C, |φI0( b C) φI( b C)| 1.5 Bribery-proofness implies click-fraud-proofness and protects from multiple artists colluding. Fraud-proofness and bribery-proofness capture resilience to two different kinds of manipulation. Despite being similar, we show that the axioms are not equivalent. Recall that is the fraction of each user s subscription fee that is allocated to the artists, with the remaining portion retained by the platform as a ﬁxed cut. Theorem 2.8. Consider some rule φ. Then: (i) If = 1 and φ is fraud-proof, it is also bribery-proof; (ii) For 2 (0, 1], there exists a rule that is bribery-proof but not fraud-proof, even when m = 2; (iii) For < 1, there exists a rule that is fraud-proof but not bribery-proof, even when m = 2. The last pair of axioms that we consider Sybil-proofness6 and its strong counterpart addresses a different form of strategic manipulation compared to the two earlier concepts. Intuitively, these axioms are designed to prevent any artist(s) from splitting or merging to gain an unfair advantage and fraudulently increasing their revenue share, thus also ensuring that all artists are treated fairly based on their actual level of user engagement. Deﬁnition 2.9 (Sybil-proofness). A rule φ is Sybil-proof if the following holds: For any two instances I = (N, C, w) and I0 = (N, C0, w0) whereby C C0, if for every subset of artists C C such that (i) wij = w0 ij for all i 2 N, j 2 C ; and j2C\C wij = P ij for all i 2 N, then we must have that φI(C \ C ) = φI0(C0 \ C ). We can deﬁne a stronger notion of Sybil-proofness by relaxing (i) and (ii), deﬁned as follows. Note that strong Sybil-proofness implies Sybil-proofness. Deﬁnition 2.10 (Strong Sybil-proofness). A rule φ is strongly Sybil-proof if the following holds: For any two instances I = (N, C, w) and I0 = (N, C0, w0) whereby C C0, if for any subset of artists C C such that i2N wij = P ij for all j 2 C ; and 5Note that by Proposition 2.7, it sufﬁces to only consider singleuser bribery-proofness. 6The name is inspired by the concept of a Sybil attack in computer networks. j2C\C wij = P then we must have that φI(C \ C ) = φI0(C0 \ C ). We will show later that GLOBALPROP is the only neutral rule satisfying strong Sybil-proofness (Theorem 3.2), hence also motivating our study of (the weaker) Sybil-proofness. 2.3. Fairness Axioms Next, we consider two fairness properties engagement monotonicity and Pigou-Dalton consistency. Intuitively, if an artist s engagement increases while every other artists engagement does not increase, this artist s payoff should not decrease this aligns with basic economic principles. It would be fundamentally unfair for a creator s rising popularity to result in a lower payoff. We formalize this fairness property as follows. Deﬁnition 2.11 (Engagement monotonicity). A rule φ is engagement monotone if the following holds: For any two instances I = (N, C, w) and I0 = (N, C, w0), if there exists a j 2 C such that ij for all i 2 N; and (ii) wij w0 ij for all i 2 N and j 2 C \ {j }, then we must have that φI(j ) φI0(j ). Next, the Pigou-Dalton principle (Pigou, 1920; Dalton, 1920), is a fundamental fairness notion from welfare eco- nomics and often referenced in collective decision-making (Moulin, 2003) it states that among similar outcomes, the equitable one should be picked. We interpret this principle in our setting: all other things being equal, an artist who is more uniformly enjoyed should receive weakly more payoff from an equally popular but polarizing artist. Deﬁnition 2.12 (Pigou-Dalton consistency). A rule φ is Pigou-Dalton consistent if the following holds: For any two instances I = (N, C, w) and I0 = (N, C, w0), if there exists some i, i0 2 N and j 2 C such that ij = wij δ (where δ > 0 and wij δ > 0); i0j = wi0j + δ and w0 (iii) wkj0 = w0 kj0 for all k 2 N and j0 2 C \ {j}, and wkj = w0 kj for all k 2 N \ {i, i0}. then we must have that φI(j) φI0(j). 3. Existing Mechanisms In this section, we formally deﬁne the three existing mechanisms proposed in the literature, and study which axioms Fraud-Proof Revenue Division on Subscription Platforms Axioms / Rules GP UP UEQ SCUP Fraud-proofness 7 3 3 3 Bribery-proofness 7 3 3 3 Sybil-proofness 3 3 7 3 Strong Sybil-proofness 3 7 7 7 Engagement monotonicity 3 3 3 3 Pigou-Dalton consistency 3 7 3 7 Table 1. Axiomatic properties of the revenue division mechanisms. GP is GLOBALPROP, UP is USERPROP, UEQ is USEREQ, and SCUP is SCALEDUSERPROP. they satisfy. We summarize our results in Table 1. At the end of the section, we also include a reference to a discussion on how our model generalizes portioning rules. The rules we consider in this and the next section trivially satisfy anonymity and neutrality. Therefore, among the three preliminary axioms introduced in Section 2.1, we will only formally prove the satisfaction of no free-ridership. 3.1. GLOBALPROP: The Status Quo GLOBALPROP distributes the payoff to each artist proportionally to the artist s share of total engagement. For example, if there are 500 users, and an artist gets 25% of the total user engagement in the platform, then the artist correspondingly receives a payment of 0.25 500 = 125 under GLOBALPROP. According to court documents (United States Attorney s Ofﬁce, 2024), this is the rule that major streaming platforms use.7 Given an instance I = (N, C, w) and for each j 2 C, the payment rule GLOBALPROP is deﬁned as follows. i2N wij0 n. It is easy to observe that users with higher engagement exert a disproportionate inﬂuence on revenue distribution. Given this, it is not surprising that this rule fails to satisfy both fraud-proofness and bribery-proofness. Theorem 3.1. GLOBALPROP satisﬁes strong Sybilproofness, but fails fraud-proofness and bribery-proofness. Moreover, strong Sybil-proofness uniquely characterizes GLOBALPROP, given our neutrality assumption. Theorem 3.2. GLOBALPROP is the only neutral rule satisfying strong Sybil-proofness. GLOBALPROP also satisﬁes our fairness axioms. 7It is also sometimes known as the pro-rata rule. Theorem 3.3. GLOBALPROP satisﬁes no free-ridership, engagement monotonicity, and Pigou-Dalton consistency. A Case Against GLOBALPROP: The Computational Intractability of Fraud Detection. We have shown that GLOBALPROP is not fraud-proof. One might hope that artists beneﬁting from fraud could be easily identiﬁed and removed. Unfortunately, detecting the artists who gain the most from fraudulent activity is computationally intractable. Importantly, a user who streams music extensively is not inherently suspicious some people naturally listen to music for most of their waking hours. Thus, instead of targeting individual active users, we should focus on identifying artists who may be used as vehicles for fraud by an adversary.8 Deﬁnition 3.4 (Potentially Suspicious Proﬁts). Given a set of artists U C, their potentially suspicious proﬁt (PSP) from GLOBALPROP is their maximum marginal proﬁts from a set of users V , less the cost of creating these users: PSP(U) = max (n |V |) |V | Thus, our objective of identifying suspicious artists can be framed as ﬁnding a set of artists U C such that PSP(U) is high. However, the choice of |U| is crucial. If we restrict U to a single artist (|U| = 1), an adversary can easily evade detection by distributing fake users listening activity across multiple fraudulent artists. On the other hand, if we impose no constraint on |U|, we risk identifying a set of legitimate artists with dedicated fan bases. Also, while an adversary can create multiple fake artists, doing so incurs administrative overhead such as setting up identiﬁcation and banking details which makes the creation of an arbitrarily large number of fake artists impractical in many circumstances. Therefore, we deﬁne the problem of ﬁnding suspicious artists as ﬁnding the set U C of size at most k artists that maximize PSP(U). However, we show that this problem is computationally intractable, with the following result. Theorem 3.5. Given an instance I = (N, C, w) and parameters k |C| and γ > 0, it is NP-hard to determine if there exists a U C such that |U| k and PSP(U) γ. 3.2. User-Additive Rules At the opposite extreme from GLOBALPROP are rules where each user s subscription fee is distributed solely based on their individual engagement proﬁle. Under these rules, an 8Our objective is to identify fraudulent artists as a means of detecting suspicious interactions between fake users and fake artists. Fraud-Proof Revenue Division on Subscription Platforms artist s total payoff is simply the sum of the amounts they would receive from each user in a single-user setting. We refer to this class of rules as user-additive.9 Deﬁnition 3.6 (User-additive rules). For each instance I = (N, C, w), deﬁne instances Ii = ({i}, C, wi) for each i 2 N. Then, a rule φ is user-additive if for all instances I and artists j 2 C, φI(j) = P i2N φIi(j). We then show the following. Proposition 3.7. A user-additive rule is fraud-proof and bribery-proof. We focus on two user-additive rules that have been discussed in the existing literature: USERPROP and USEREQ. Under USERPROP, an fraction of each user s subscription fee is allocated to the artists proportional to the user s engagement. For example, if a user listens to three artists the ﬁrst artist 50% of the time and the other two artists 25% each then under USERPROP, the artists will receive payments of /2, /4, and /4 from this user, respectively. The total payment of an artist is the sum of such payments from each user. Given an instance I = (N, C, w) and for each j 2 C, the payment rule USERPROP is deﬁned as follows. j02C wij0 . We show that it satisﬁes all of the manipulation-resistant axioms (excluding strong Sybil-proofness) and engagement monotonicity, but fails Pigou-Dalton consistency. Theorem 3.8. USERPROP is fraud-proof, bribery-proof, and Sybil-proof, but fails strong Sybil-proofness. Theorem 3.9. USERPROP satisﬁes no free-ridership and engagement monotonicity, but fails Pigou-Dalton consistency. Next, we consider the USEREQ rule, ﬁrst studied in Berganti nos & Moreno-Ternero (2024). They established the equivalence between USEREQ and the Shapley value, a fundamental measure in cooperative game theory that ensures a fair distribution of payoffs among players based on their contributions (Shapley, 1953). Now, given an instance I = (N, C, w), for each i 2 N and j 2 C, let 1wij>0 be the indicator function that returns the value 1 if wij > 0, and 0 otherwise. In USEREQ, an fraction of each user s subscription fee is distributed equally among the artists with strictly positive engagement from the user. For example, if a user listens to only three artists 80%, 19%, and 1% of the time, respectively and does not 9This term is distinct from user-centric, which is sometimes used in the literature to refer to USERPROP. listen to other artists, then these three artists each receives a payment of /3 from this user, and the remaining artists do not receive any payment from the user. The total payment to an artist is the sum of such payments from each user. Given an instance I = (N, C, w) and for each j 2 C, the payment rule USEREQ is deﬁned as follows. 1wij>0 |{j0 2 C : wij0 > 0}| . USEREQ has similar guarantees as USERPROP, with the difference being that it fails Sybil-proofness, but satisﬁes Pigou-Dalton consistency. Theorem 3.10. USEREQ is fraud-proof and bribery-proof, but fails Sybil-proofness. Theorem 3.11. USEREQ satisﬁes no free-ridership, engagement monotonicity, and Pigou-Dalton consistency. A Generalization of Portioning We also make an important observation: our model can be viewed as a generalization of portioning under cardinal preferences (Elkind et al., 2023; Freeman et al., 2021), where each agent subjectively divides a contiguous resource (such as time or money) among a given set of candidates, and the goal is to aggregate these preferences to obtain one (fair) division. This is similar to our model if we let agents be users, candidates be artists, and preferences be interactions.10 However, portioning rules require that the engagement of each user is normalized (i.e., sums to 1). We can then generate rules for our setting by normalizing each wi and applying a portioning rule to the instance. There are eight portioning rules cataloged in Elkind et al. (2023). One of them is equivalent to USERPROP, but the other seven fail fraud-proofness, bribery-proofness and Sybil-proofness. We present these rules and prove the results in the full version of this paper. 4. SCALEDUSERPROP: A Fairer Mechanism The three rules we considered above are conceptually distinct: GLOBALPROP allows dedicated fans to exert a disproportionate inﬂuence on revenue distribution, but this also creates opportunities for fraud by fabricating users who may appear as dedicated fans. In contrast, USERPROP is often viewed by policymakers as a more desirable alternative to GLOBALPROP. However, USERPROP is not necessarily fairer (Lei, 2023), and user-additive rules in general may fail to meaningfully reward artists for increasing the engage- 10We note that this analogy requires imposing rational number constraints on preferences, as assumed in the preliminaries. Fraud-Proof Revenue Division on Subscription Platforms ment within their existing fanbase. To better understand differences in payment fairness, it is useful to examine the pay-per-stream metric (Dimont, 2018; Meyn et al., 2023). Given an instance I and an artist j, let the artist pay-per-stream (PPS) for rule φ be PPS(φ, I, j) = φI(j) P i2N wij . Using this, we deﬁne the maxi- mum envy (ME) of I as ME(φ, I) = maxj2C PPS(φ,I,j) minj02C PPS(φ,I,j0). This ratio quantiﬁes the disparity in PPS between the highest-paid and lowest-paid artists, providing a measure of the maximum envy in revenue distribution. Then, we obtain the following result, which essentially implies that any fraud-proof or bribery-proof rule has the potential to be extremely unfair (unbounded maximum envy). Proposition 4.1. For all 2 (0, 1] and rules φ, if there exists k 2 R such that for all instances I, ME(φ, I) k, then φ fails fraud-proofness and bribery-proofness. However, not all such rules may perform equally bad on this front we will analyze this later through experiments (in Section 5), with a slight variant of the ME deﬁnition. Given this, we attempt to achieve a compromise by designing a rule that has the same axiomatic guarantees as USERPROP, while offering empirically (in Section 5) stronger fairness guarantees than USERPROP and USEREQ. SCALEDUSERPROP works by having the platform take a disproportionate amount of commission from low-engagement users. The platform then runs USERPROP on the remaining subscription fees. It is deﬁned as follows. SCALEDUSERPROP Given an instance I = (N, C, w), let γ be a constant such that P = n. Then, for each j 2 C, the payment rule SCALEDUSERPROP is deﬁned as follows. wij0, 1) wij P Note that when = 1, we have min(γ P j02C wij0, 1) = 1 for all i 2 N, making SCALEDUSERPROP equivalent to USERPROP. For < 1, if no user s engagement exceeds 1 times the average engagement, then SCALEDUSERPROP is equivalent to GLOBALPROP, which we show below. Theorem 4.2. Fix an instance I = (N, C, w). If P j2C wij 1 n j2C wij for all i 2 N, then SCALEDUSERPROP is equivalent to GLOBALPROP. Thus, SCALEDUSERPROP can be viewed as a variant of GLOBALPROP that limits the inﬂuence of users who have engagement signiﬁcantly above average. We then show that SCALEDUSERPROP has exactly the same axiomatic guarantees as USERPROP, with the following results. Theorem 4.3. SCALEDUSERPROP satisﬁes fraudproofness, bribery-proofness, and Sybil-proofness, but fails strong Sybil-proofness. Theorem 4.4. SCALEDUSERPROP satisﬁes no freeridership, engagement monotonicity, but fails Pigou-Dalton consistency. 5. Experiments To complement our theoretical analysis, we conduct experiments to evaluate our fraud-proof (and briberyproof) mechanisms USERPROP, USEREQ, SCALE- DUSERPROP using both synthetic and real-world datasets. Motivated by our deﬁnition of maximum envy in Proposition 4.1, for each rule, we analyze the top and bottom few users based on their pay-per-stream (PPS) relative to GLOBALPROP s PPS, as the revenue share ( ) varies.11 Notably, only SCALEDUSERPROP is inﬂuenced non-linearly by changes in (the other rules scale linearly with ). Consequently, the pay-per-stream values for the other three rules remain constant across different values of . Synthetic datasets We generate synthetic problem instances involving 10, 000 users and 1, 000 artists. For each user, we ﬁrst determine the number of artists they interact with by drawing a value uniformly at random from the range [1, 100]. Based on this value, we randomly select the corresponding number of artists from the pool of 1, 000. For each chosen artist, the number of times the user streams their music is sampled from a Poisson distribution with λ = 1. We repeat the experiments 100 times. Real-world datasets We utilize data from the Music Listening Histories Dataset (Vigliensoni & Fujinaga, 2017), that contains the listening history of approximately 583, 000 users, 439, 000 artists, and a cumulative total of 27 billion listening events (i.e., user-artist interactions).12 Discussion On real-world data, SCALEDUSERPROP emerges as fairest mechanism among those considered, especially for values of not close to 1; whereas USEREQ, which treats avid and casual listeners equally, is the least fair. SCALEDUSERPROP signiﬁcantly reduces the top 100 11Note that in Proposition 4.1, maximum envy is deﬁned with respect to the single top and bottom user, which differs from the metric used in this section. In our experiments, we chose to report metrics for the top and bottom few users rather than just the single best and worst, as we believe this provides a more robust assessment mitigating the impact of potential outliers that may disproportionately affect the extremes. However, our deﬁnition and theoretical results would easily extend to top and bottom few users, making it consistent with that used for the experiments. 12Our code is accessible at https://github.com/ nicteh/Fraud-Proof-Revenue-Division. Fraud-Proof Revenue Division on Subscription Platforms (a) Real data, top 100 artists PPS relative to GP (b) Real data, bottom 100 artists PPS relative to GP (c) Synthetic data, top 10 artists PPS relative to GP (d) Synthetic data, bottom 10 artists PPS relative to GP Figure 1. Overview of graphs from real and synthetic data. (a) and (b) show results for real data, while (c) and (d) show results for synthetic data. GP is short for GLOBALPROP. artists PPS even for > 0.9, but it only gradually increases the bottom 100 PPS as decreases. To understand this outcome, we ﬁrst observe that artists with high PPS typically attract infrequent listeners, while those with low PPS tend to have a more dedicated, avid fanbase. We also observe that under SCALEDUSERPROP, each stream from a user contributes min(γ, 1 P j2C wij ), whereas under USERPROP, it contributes P j2C wij . For avid lis- teners with high P j2C wij, a stream under SCALEDUSERPROP is worth 1 times its value under USERPROP. Conversely, for infrequent listeners, SCALEDUSERPROP caps a stream s worth at γ, while under USERPROP, it can reach up to in the extreme case where P j2C wij = 1. On synthetic data, SCALEDUSERPROP remains the fairest mechanism as decreases. However, in contrast to the real-world data, we observe two key differences: (1) the top and bottom PPS are much closer in magnitude, and (2) USERPROP and USEREQ perform nearly identically. These differences can be partly attributed to the way synthetic instances are generated. While our model accounts for users with varying streaming frequencies, it does not capture the real-world tendency of certain artists to attract predominantly avid or infrequent listeners. 6. Conclusion In this work, we formalized three types of manipulation by fraudulent agents in subscription-based platforms, motivated by a real-world multi-million dollar fraud case. We show that the axioms we introduced are not equivalent and study the rules that satisfy them. GLOBALPROP, which is used by streaming platforms, does not satisfy fraud-proofness or bribery-proofness. However, we show that USERPROP and USEREQ do. We introduce a novel rule, SCALEDUSERPROP. It is as strong in resisting manipulation as USERPROP and incentivizes artists to increase their overall engagement similarly to GLOBALPROP. Our empirical study on real and synthetic data of fraud-proof rules support SCALEDUSERPROP is a fairer fraud-proof alternative to other rules A natural follow-up direction would be to study a freemium model, by incorporating users who have to watch advertisements to gain access to content on the platform, and have been adopted by streaming platforms such as You Tube and Spotify, among others. Revenue division in this context would have different considerations and call for more appropriate axioms to be deﬁned. Machine learning approaches have been adopted here as well (Goli et al., 2024); it would be interesting to explore these questions from a mechanism design perspective. Impact Statement This paper presents work whose goal is to advance the theory of machine learning. There are many potential societal consequences of our work, none which we feel must be speciﬁcally highlighted here. Alaei, S., Makhdoumi, A., Malekian, A., and Pekeˇc, S. Revenue-sharing allocation strategies for two-sided media platforms: Pro-rata vs. user-centric. Management Science, 68(12):8699 8721, 2022. Berganti nos, G. and Moreno-Ternero, J. D. The axiomatic approach to the problem of sharing the revenue from museum passes. Games and Economic Behavior, 89: 78 92, 2015. Berganti nos, G. and Moreno-Ternero, J. D. A new rule for the problem of sharing the revenue from museum passes. Operations Research Letters, 44(2):208 211, 2016. Berganti nos, G. and Moreno-Ternero, J. D. The shapley index for music streaming platforms. ar Xiv preprint ar Xiv:2411.07166, 2024. Berganti nos, G. and Moreno-Ternero, J. D. Revenue shar- ing at music streaming platforms. Management Science, Forthcoming, 2025. Burton, C. S. Playola and fraud on digital music plat- forms: Why legislative action is required to save the mu- Fraud-Proof Revenue Division on Subscription Platforms sic streaming market. Journal of Business & Technology Law, 16(2):387 435, 2021. B eal, S. and Solal, P. R egles d allocation pour les pro- grammes de pass culturel. Revue Economique, 61:1099 1109, 2010. Casas-M endez, B., Fragnelli, V., and Garc ıa-Jurado, I. Weighted bankruptcy rules and the museum pass problem. European Journal of Operational Research, 215(1): 161 168, 2011. Casas-M endez, B., Fragnelli, V., and Garc ıa-Jurado, I. A survey of allocation rules for the museum pass problem. Journal of Cultural Economics, 38:191 205, 2014. Chlamt aˇc, E., Dinitz, M., and Makarychev, Y. Minimizing the union: Tight approximations for small set bipartite vertex expansion. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 881 899, 2017. Dalton, H. The measurement of the inequality of incomes. The Economic Journal, 30(119):348 361, 1920. Deng, J., Zhang, S., and Ma, J. Computational copyright: Towards a royalty model for music generative AI. ar Xiv preprint ar Xiv:2312.06646, 2024. Dimont, J. Royalty inequity: Why music streaming services should switch to a per-subscriber model. Hastings Law Journal, 69(2):675 700, 2018. Drott, E. Fake streams, listening bots, and click farms: Counterfeiting attention in the streaming music economy. American Music, 38(2):153 175, 2020. Eilat, I. and Rosenfeld, N. Performative recommendation: Diversifying content via strategic incentives. In Proceedings of the 40th International Conference on Machine Learning (ICML), pp. 9082 9103, 2023. Elkind, E., Suksompong, W., and Teh, N. Settling the score: Portioning with cardinal preferences. In Proceedings of the 26th European Conference in Artiﬁcial Intelligence (ECAI), pp. 621 628, 2023. Esmaeilzadeh, S., Salajegheh, N., Ziai, A., and Boote, J. Abuse and fraud detection in streaming services using heuristic-aware machine learning. ar Xiv preprint 2203.02124, 2022. Est evez-Fern andez, A., Borm, P., and Hamers, H. A note on passepartout problems. International Game Theory Review, 14(2):1250013, 2012. Freeman, R., Pennock, D., Peters, D., and Wortman Vaughan, J. Truthful aggregation of budget proposals. Journal of Economic Theory, 193:105234, 2021. Ginsburgh, V. and Zang, I. Sharing the income of a museum pass program. Museum Management and Curatorship, 19(4):371 383, 2001. Ginsburgh, V. and Zang, I. The museum pass game and its value. Games and Economic Behavior, 43(2):322 325, 2003. Goli, A., Reiley, D. H., and Zhang, H. Personalizing ad load to optimize subscription and ad revenues: Product strategies constructed from experiments on pandora. Marketing Science, 44(2):327 352, 2024. Golrezaei, N., Javanmard, A., and Mirrokni, V. Dynamic incentive-aware learning: Robust pricing in contextual auctions. Operations Research, 69(1):297 314, 2021a. Golrezaei, N., Manshadi, V., Schneider, J., and Sekar, S. Learning product rankings robust to fake users. In Proceedings of the 22nd ACM Conference on Economics and Computation (EC), pp. 560 561, 2021b. He, S., Hollenbeck, B., and Proserpio, D. The market for fake reviews. Marketing Science, 41(5):896 921, 2022. International Federation of the Phonographic Indus- try. Global music report, 2024. URL https:// www.ifpi.org/wp-content/uploads/2024/ 04/GMR 2024 State of the Industry.pdf. Josan, H. H. S. AI and deepfake voice cloning: Innovation, copyright and artists rights. Digital Policy Hub Working Paper, 2024. Kanoria, Y. and Nazerzadeh, H. Dynamic reserve prices for repeated auctions: Learning from bids. In Proceedings of the 10th International Conference on Web and Internet Economics (WINE), pp. 232, 2014. Lei, X. Pro-rata vs user-centric in the music streaming industry. Economics Letters, 226:111111, 2023. Li, Z., Wang, H., Zhang, P., Hui, P., Huang, J., Liao, J., Zhang, J., and Bu, J. Live-streaming fraud detection: A heterogeneous graph neural network approach. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD), pp. 3670 3678, 2021. Mayzlin, D., Dover, Y., and Chevalier, J. Promotional re- views: An empirical investigation of online review manipulation. American Economic Review, 104(8):2421 2455, 2014. Meyn, J., Kandziora, M., Albers, S., and Clement, M. Con- sequences 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, 2023. Fraud-Proof Revenue Division on Subscription Platforms Mollao glu, A., Baltao glu, G., C akrr, E., and Aktas , M. S. Fraud detection on streaming customer behavior data with unsupervised learning methods. In Proceedings of the 2021 International Conference on Electrical, Communication, and Computer Engineering (ICECCE), pp. 1 6, 2021. Moreau, F., Wikstr om, P., Haampland, O., and Johannessen, R. Alternative payment models in the music streaming market: A comparative approach based on stream-level data. Information Economics and Policy, 68:101103, 2024. Moulin, H. Fair Division and Collective Welfare. MIT Press, 2003. Muikku, J. Pro rata and user centric distribution models: A comparative study. Digital Media Finland, 2017. Music Fights Fraud Alliance. Music ﬁghts fraud alliance, 2025. URL https:// www.musicfightsfraud.com/. Pigou, A. C. The Economics of Welfare. Macmillan and Co., London, 1920. Shapley, L. S. A value for n-person games. In Kuhn, H. and Tucker, A. (eds.), Contributions to the Theory of Games, Volume II, pp. 307 317. Princeton University Press, 1953. Snickars, P. and M ahler, R. Spotibot turing testing spotify. Digital Humanities Quarterly, 12(1), 2018. Spotify. Royalties, 2025. URL https: //support.spotify.com/us/artists/ article/royalties/. United States Attorney s Ofﬁce, S. D. o. N. Y. North carolina musician charged with music streaming fraud aided by artiﬁcial intelligence, 2024. URL https://www.justice.gov/usaosdny/pr/north-carolina-musiciancharged-music-streaming-fraud-aidedartificial-intelligence. Vigliensoni, G. and Fujinaga, I. The music listening his- tories dataset. In Proceedings of the 18th International Society for Music Information Retrieval Conference (ISMIR), pp. 96 102, 2017. Wang, Y. A museum cost sharing problem. American Journal of Operations Research, 1(2):51 56, 2011. Yao, F., Li, C., Nekipelov, D., Wang, H., and Xu, H. How bad is top-k recommendation under competing content creators? In Proceedings of the 40th International Conference on Machine Learning (ICML), pp. 39674 39701, 2023.