On click-fraud under pro-rata revenue sharing rule
Source: arXiv:2601.09573 · Published 2026-01-14 · By Hao Yu
TL;DR
This paper addresses the strategic click-fraud vulnerability often attributed to the pro-rata revenue sharing rule on music streaming platforms, which allocates artists’ royalties proportional to their total streams across all users. The author develops the first tractable non-cooperative game-theoretic model where artists can invest in undetectable fraud generating fake streams up to a technological limit representing the platform's detection capability. Contrary to common belief, the analysis reveals that pro-rata is "fraud-robust": when fraud technology is weak, honest behavior dominates and a fraud-free, socially efficient equilibrium emerges. When fraud technology is strong, a unique fraud equilibrium exists but aggregate fake streams remain bounded due to diminishing returns caused by stream dilution. Surprisingly, some equilibria with fraud improve egalitarian fairness by redistributing income from top to low-stream artists. To mitigate fraud without fully switching to user-centric rules, a parametric weighted rule interpolating between pro-rata and user-centric is introduced that restores fraud-free equilibria under certain parameters. The paper also critiques Spotify's royalty eligibility threshold as potentially increasing fraud and harming all artists.
Key findings
- Honesty is a strictly dominant strategy under pro-rata if fraud technology λ0 ≤ λ [1 + ξ / (1 - d_min)] where d_min is minimal artist streamshare and ξ is the fraud premium (Theorem 1).
- When fraud technology is strong, i.e., λ0 > λ(1 + ξ / (1 - d_min)), a unique fraud equilibrium emerges with positive fake streams concentrated among low-streamshare artists (Lemma 2, Theorem 2).
- Aggregate fake streams remain bounded, with the total-to-real streams ratio capped by V = (λ0 - λ) / (ξλ) (Proposition 1).
- Fraud reallocates income from high-streamshare artists to low-streamshare artists, sometimes improving egalitarian fairness measured by the minimum utility (Proposition 2, Corollary 3).
- A weighted revenue sharing rule R_pw(α), convex combination of pro-rata (α=1) and user-centric (α=0), can restore fraud-proof equilibria if α ≤ 1 / ((1 - d_min)V) even under strong fraud technology (Theorem 3, Proposition 3).
- Spotify’s eligibility threshold (minimum streams to share revenue) may increase fraud intensity and harm overall artist payoffs rather than deter fraud (Section 6.2).
Threat model
The adversary is a strategic artist who can purchase fraud activity at unit cost δ to generate fake streaming activity that the platform’s detection systems cannot fully identify or remove up to λ0 streams per fake user. Artists aim to maximize royalties by choosing fraud levels. The platform cannot completely distinguish real from fake streams under λ0 but applies cost and detection pressure. The adversary cannot create unlimited fake streams at no cost, and there is full public knowledge of parameters.
Methodology — deep read
Threat Model & Assumptions: The adversaries are music artists who can strategically purchase fraud services producing fake streams that are undetectable up to a limit (fraud technology λ0). Artists know all parameters and choose fraud intensity simultaneously in a non-cooperative game. The platform cannot fully detect or remove fake streams below λ0 per fake user, and fraud costs artists a fee δ per unit.
Data & Model Setup: The model considers a two-sided market with n artists and m users, each user paying a normalized subscription fee of 1. Real streams and user preferences are fixed during a short term. Artists’ real streamshares di are known. The platform allocates revenue fraction β to artists.
Architecture/Algorithm: The main analysis is a game where each artist chooses fraud investment xi ≥ 0 generating λ0 fake streams. Payoffs depend on revenue share under the pro-rata or weighted rule minus fraud cost δxi. The key parameter V = (λ0 - λ)/(ξλ) captures relative fraud advantage versus cost. Utility functions are derived and analyzed for dominant strategy and Nash equilibria.
Training Regime: Not applicable; this is an equilibrium and comparative statics analysis deriving closed-form conditions.
Evaluation Protocol: Theoretical proofs establish conditions for (a) fraud-free equilibria under weak fraud technology, (b) unique fraud equilibria under strong technology, and (c) bounds on aggregate fraud. Fairness comparisons use minimum utility across artists as metric. Parametric weighted rules are analyzed for fraud deterrence. Spotify policy effects are checked by adapting the game to eligibility thresholds.
Reproducibility: Full mathematical proofs and formulas are provided. No code or empirical datasets are used, as this is a theoretical economics paper. The model is transparent and tractable with closed-form characterizations. The author acknowledges dependence on parameters that platforms can estimate.
Example End-to-End: For given artist streamshares di, subscription fee β, fraud cost δ, average real streams λ, and detection threshold λ0, one computes ξ and V. If fraud technology is weak (λ0 below threshold), all artists choose no fraud xi=0 and receive payoffs proportional to di. If above threshold, equilibrium analyses solve a fixed point problem yielding the set Nd of dishonest artists (low streaming ones) who choose positive fraud, ensuring equalized capped total streams on cheating artists and bounded aggregate fake streams. The model shows how increased fake streams reduce marginal benefit, limiting runaway fraud.
Technical innovations
- Modeling strategic click-fraud by music artists as a non-cooperative game with heterogeneous streamshares and bounded fraud technology.
- Proof that pro-rata revenue sharing is robust to click-fraud under a threshold of fraud detection capability, yielding an efficient equilibrium.
- Characterization of a unique bounded fraud equilibrium when fraud detection is weak, with fraud concentrated among low-streamshare artists.
- Introduction of a parametric weighted revenue sharing rule interpolating between pro-rata and user-centric to restore fraud-proofness under stronger fraud technology.
Baselines vs proposed
- Pure pro-rata rule: fraud-free equilibrium exists if λ0 ≤ λ(1 + ξ/(1-d_min)) vs weighted rule with α ≤ 1/((1-d_min)V) extends fraud-free equilibrium to stronger fraud technology.
- Spotify’s royalty eligibility threshold policy: may increase aggregate fake streams and lower all artists’ utilities vs baseline with no threshold.
Limitations
- The model abstracts away from dynamic, long-run participation or quality choices by users and artists, focusing on short-run strategic manipulation.
- No empirical validation or parameter estimation with real platform data is provided; applicability depends on estimating λ0, δ, β in practice.
- Assumes full information, i.e., all artists know all parameters; does not analyze asymmetric information or adaptive adversaries.
- The analysis does not consider sophisticated detection improvements over time beyond a fixed λ0 parameter.
- Spotify policy examination is stylized and may not capture all real-world nuances or indirect effects of eligibility thresholds.
- No explicit modeling of multi-artist fraud collusion or coordinated attacks, only individual strategic choices.
Open questions / follow-ons
- How would dynamic, repeated interaction between artists and adaptive platform detection alter fraud incentives and equilibrium?
- What is the impact of asymmetric information where artists have private fraud capabilities or costs unknown to others?
- Can more sophisticated joint fraud strategies or collusion among artists break the bounded fraud equilibrium?
- How does incorporating user engagement and quality-choice behavior affect fraud incentives beyond short-run streaming?
Why it matters for bot defense
This work provides theoretical validation that a classically vilified revenue sharing rule—pro-rata—is more resilient to click-fraud than previously assumed under realistic fraud detection constraints. Bot-defense or CAPTCHA practitioners focused on streaming platforms can draw insight into the economic incentives and natural fraud bounds arising from revenue dilution, helping calibrate detection thresholds and fraud cost parameters. The parametric weighted rule offers a concrete mechanism design approach to balance efficiency and fraud robustness without fully abandoning pro-rata's simplicity. Understanding that fraud aggregates remain bounded and concentrated among marginal artists informs prioritization of detection efforts and anti-bot defenses that focus on subtle but limited attack vectors rather than unbounded fraudulent explosions. Lastly, the analysis of platform-level policy (like Spotify eligibility cutoffs) cautions that poorly chosen thresholds can increase fraud incentives, a principle generalizable when engineering CAPTCHA or bot mitigation steps that impact revenue or access eligibility.
Cite
@article{arxiv2601_09573,
title={ On click-fraud under pro-rata revenue sharing rule },
author={ Hao Yu },
journal={arXiv preprint arXiv:2601.09573},
year={ 2026 },
url={https://arxiv.org/abs/2601.09573}
}