Pro-Rata Revenue Sharing Rule

• Pro-Rata Revenue Sharing Rule is a mechanism that allocates revenue proportionally to agents based on their claims or contributions.
• The rule is defined by clear mathematical formulations and axioms such as homogeneity, additivity, and resource monotonicity.
• It is applied in diverse sectors like insurance, music streaming, and sports broadcasting, though its susceptibility to manipulation raises practical challenges.

The pro-rata (proportional) revenue sharing rule is a foundational allocation principle across insurance, digital platforms, sports, auctions, and distributed computing. It stipulates that each participant receives a share of total revenue (or liability or payout) in direct proportion to a well-defined measure of their contribution or claim. This rule is characterized by fundamental axiomatic, statistical, and game-theoretic properties, and appears in both policy-mandated and algorithmic settings.

1. Formal Framework and Mathematical Definition

Let $N = \{1, \dots, n\}$ denote the set of agents, and let $(c_i)_{i\in N}$ be nonnegative claims, weights, or contributions by each agent. Let $R$ be the total revenue to be allocated. The basic pro-rata rule assigns to each $i$ the share

$r_i = R \cdot \frac{c_i}{\sum_{j\in N} c_j}$

This rule applies identically in cases where $c_i$ carries semantic meaning as market share, audience, streams, sales, performance, or asset contributed to a pool.

Weighted versions generalize to

$$r_i = R \cdot \frac{w_i c_i}{\sum_{j\in N} w_j c_j}$$

where $w_i > 0$ are exogenous weights reflecting policy or institutional priorities (Bergantiños et al., 11 Dec 2025, Bergantiños et al., 2024).

In settings with structured “issues” (e.g., per-user, per-match, per-flight), the pro-rata rule appears as a one-stage or two-stage proportional allocation in multi-issue claims problems (Bergantiños et al., 2024, Bergantiños et al., 2023).

2. Axiomatic and Game-Theoretic Characterization

Pro-rata sharing is uniquely determined by a small set of classical axioms:

• Homogeneity (Scale Invariance): Scaling all $c_i$ or $R$ multiplies each $r_i$ accordingly.
• Additivity: For any two revenue pools, sharing each one pro-rata and summing the allocations equals sharing their sum pro-rata.
• Resource Monotonicity: More total revenue yields weakly higher shares for each agent.
• Null/Zero Principle: Agents with zero $c_i$ receive zero share.
• Equal Treatment of Equals: Equal $c_i$ yield equal $r_i$ for any pair of agents (Bergantiños et al., 2023, Bergantiños et al., 11 Dec 2025, Bergantiños et al., 2024, Bergantiños et al., 5 Jan 2026).

For settings with per-issue claims, additivity and scale-invariance extend to multi-issue formulations. In the music streaming and sports broadcasting settings, axiomatic results show that pro-rata is the unique rule satisfying natural consistency and impartiality criteria (Bergantiños et al., 5 Jan 2026, Bergantiños et al., 2023, Bergantiños et al., 2024).

Cooperative game-theoretic analysis connects the pro-rata rule to the Shapley value and the core in particular transferable-utility games. In sports broadcasting, pro-rata coincides with the Shapley value for the convex TU game defined by audience matrices (Bergantiños et al., 5 Jan 2026). For general streaming problems, pro-rata typically lies outside the cooperative game core, in contrast to user-centric rules (Bergantiños et al., 2023).

3. Sector-Specific Implementations

Insurance Marketplaces (ACA Risk Adjustment)

The Affordable Care Act implements a zero-sum, pro-rata transfer scheme across insurance plans:

$\sum_{i=1}^n s_i T_i = 0$

where $s_i$ is plan $i$'s market share and $T_i$ is its transfer as a fraction of premium. This results in mean and variance formulas:

$$\mathbb{E}[T_n] = -\sum_{i=1}^{n-1} \frac{s_i}{s_n} \mu_i;\quad \operatorname{Var}(T_n) = \frac{1}{s_n^2}\left( \sum_{i=1}^{n-1} s_i^2\sigma_i^2 + 2\sum_{i<j}s_is_j\sigma_{ij} \right)$$

Small-share plans are systematically exposed to greater expected transfer magnitudes and extremely high volatility (Li et al., 2017).

Music Streaming and Subscription Platforms

For artist $i$ with total streams $T_i$ and pool of revenue $R$, pro-rata allocates:

$R_i^P = \frac{T_i}{\sum_k T_k} R$

This formulation also appears as GlobalProp in fraud and manipulation-resistance research (Ghosh et al., 6 Nov 2025, Bergantiños et al., 2023). Weighted pro-rata (with $w_i$) is deployed to tune allocations or cap incentives (Yu, 14 Jan 2026).

Airline Revenue Allocation (IATA)

Ticket revenue is split by weighted leg-factors $w_e$:

$$W_i^w(A) = \sum_{j\in M} \frac{\sum_{(e,i)\in f^j} w_e}{\sum_{(e,k)\in f^j} w_e} p^j$$

with weights $w_e$ determined empirically by IATA's MPA-P system (Bergantiños et al., 11 Dec 2025).

Sports Broadcasting

Given an audience matrix $A=(a_{ij})$:

$$s_i = \frac{\alpha_i}{\sum_k \alpha_k} R = \frac{1}{2}\alpha_i$$

where $\alpha_i$ is total audience for club $i$. This is the unique rule satisfying additivity, equal treatment of equals, and null-team axioms (Bergantiños et al., 5 Jan 2026).

Distributed Computing—Edge-Cloud Systems

Ortmann's rule assigns to server $i$:

$r_i = R \cdot \frac{c_i}{\sum_j c_j}$

where $c_i$ is the standalone capacity or revenue potential of server $i$ (Cao et al., 2017).

Auctions with Profit Sharing Contracts

A pro-rata profit-and-loss sharing contract (PLSC) with sharing fraction $\alpha$ provides the seller with

$P_{plsc} = b + \alpha(x - b)$

where $b$ is the auction price and $x$ is realized profit. This increases expected revenue for the seller compared to one-time payment formats (Abhishek et al., 2011).

Claims Problems

Streaming, bankruptcy, and multi-issue claims problems all admit pro-rata (proportional) allocation as the unique solution satisfying homogeneity, additivity, and resource monotonicity (Bergantiños et al., 2024).

Prompt-Based AI Data Markets

Provider $i$ receives $R_i = R \frac{s_i}{S}$, where $s_i$ is the sum of per-prompt engagement scores. Calculation methods include supervised classification and embedding-based similarity models (Zhang, 2023).

4. Strategic, Statistical, and Robustness Properties

The pro-rata rule is susceptible to strategic manipulation where claims or contributions can be gamed:

• Volatility and Sensitivity: In insurance marketplaces, small market share entities face unbounded potential transfers due to the $1/s_i$ scaling in expectations and variance (Li et al., 2017).
• Manipulability: In music streaming, pro-rata is not fraud-proof or bribery-proof; adding even one high-contribution fake user or bribing a user can confer unbounded profit to colluding artists. Detection of profit-maximizing suspicious artist groups is NP-hard (Ghosh et al., 6 Nov 2025, Yu, 14 Jan 2026).
• Sybil Resistance: Pro-rata is immune to Sybil attacks; splitting or merging identities does not alter total allocation per engagement (Ghosh et al., 6 Nov 2025).
• Price of Anarchy: In concave pro-rata games, Nash equilibrium can be highly inefficient (price of anarchy $\Omega(n)$), since proportional incentive mechanisms align poorly under pooling with diminishing returns (Johnson et al., 2023).

Mitigation strategies include switching to or interpolating towards user-centric rules, applying weights, or capping maximum allowable individual allocations (Yu, 14 Jan 2026, Ghosh et al., 6 Nov 2025).

5. Comparative Evaluation and Alternatives

Pro-rata is compared with several other prominent rules:

Rule Family	Defining Property	Core/Participation
Pro-rata (Global)	Proportional to total claims	Can lie outside core
User-centric (UC)	Per-user proportional allocation	Always in core
Shapley Value	Average marginal contribution	Core; fairness
Weighted/Hybrid	Tuning via per-user or per-claim weights	Tunable
Uniform	Equal share to all agents	Most egalitarian

Pro-rata is often operationally simplest and compatible with existing data structures, but may lead to cross-subsidization from low-frequency to high-frequency participants, exacerbate inequality in revenue distributions, and induce perverse strategic responses. User-centric and Shapley-based rules align more closely with individual contributions but are computationally and logistically more demanding (Bergantiños et al., 2023, Bergantiños et al., 2024).

6. Computational and Practical Implementation

Implementation of pro-rata allocations is computationally efficient: summing agent-level contributions and computing a normalization via one pass over the data ($O(n), O(nm)$ for matrix settings). The simplicity facilitates parallelization and auditability in web-scale environments (Bergantiños et al., 2024, Zhang, 2023). Practical variants adapt to sector-specific metrics: e.g., Ticketed Point Mileage in air transport, play/event counts in streaming, standalone task capacity in edge-cloud, or classifier-derived engagement scores in AI data markets.

Robust operation in adverse environments (e.g., susceptibility to fraud or fake activity) often requires moving to hybrid rules, introducing thresholds (as in Spotify's modernized royalty system), or adding audit/regulatory layers.

7. Limitations and Recommendations

Despite its foundational role, the pro-rata rule fails several participation, lower-bound, and fraud/fairness properties in cooperative, adversarial, and non-cooperative environments. The absence of global lower bounds and resistance to targeted manipulation means pro-rata can systematically disadvantage niche or low-frequency agents (Bergantiños et al., 2023, Ghosh et al., 6 Nov 2025, Yu, 14 Jan 2026). In regulated environments, it exposes small participants to catastrophic swings (insurance), while in competitive or adversarial digital platforms it requires supplementation with capping/weighting mechanisms or a shift to more robust user-centric formulas (Li et al., 2017, Ghosh et al., 6 Nov 2025, Yu, 14 Jan 2026, Bergantiños et al., 2023).

In conclusion, the pro-rata revenue sharing rule is a mathematically rigorous, axiomatically determined allocation mechanism with broad sectoral applications. While it guarantees efficiency, symmetry, and transparency, its vulnerability to volatility, manipulation, and sub-core outcomes necessitates contextual analysis and, in many cases, the adoption of hybrid or user-tuned rules.