ScaledUserProp Mechanism: Fair Revenue Allocation

• ScaledUserProp is a revenue allocation mechanism that scales user engagement using a unique threshold to ensure fairness and resistance against manipulation.
• It employs a cap on individual contributions, balancing genuine activity rewards with strategic disincentives for excessive or fraudulent engagement.
• The mechanism guarantees fraud-, bribery-, and sybil-proof properties while achieving near-optimal welfare approximations in large market and crowdsourcing applications.

ScaledUserProp is a revenue allocation mechanism designed to disincentivize manipulation in subscription platforms and large market settings, while upholding fairness, budget-feasibility, and truthfulness. It operates by scaling each participant’s influence using a unique threshold that links total engagement or utility to the revenue pool, avoiding common vulnerabilities in prior mechanisms. This approach is applicable both to the division of subscription revenue among creators and budget-feasible selection in large-scale crowdsourcing, with rigorous guarantees on manipulation-resistance and approximation to optimal welfare (Ghosh et al., 6 Nov 2025, Anari et al., 2014).

1. Formal Definition and Mechanism

Subscription Platforms

Let $n$ users ($N = \{1, ..., n\}$) each pay a fixed fee for unlimited access to content, and $m$ artists ($C = \{1, ..., m\}$) receive shares based on engagement weights $w_{ij} \ge 0$ (the activity of user $i$ for artist $j$). Platform keeps share $1-\alpha$, so creators split $\alpha n$ in total. User $i$'s total engagement is $L_i = \sum_{j \in C} w_{ij}$.

ScaledUserProp mechanism:

• Find the unique $\gamma > 0$ such that

$\sum_{i=1}^n \min(\gamma L_i, 1) = \alpha n.$

• Each artist $j$ receives

$$\phi_I(j) = \sum_{i=1}^n \min(\gamma L_i, 1)\cdot \frac{w_{ij}}{L_i}.$$

Intuitively, each user contributes up to \$1 (their fee), modulated by$\gamma$ and capped at 1. This ensures no single user’s intensified engagement can disproportionately increase an artist’s payout.

Large Market Mechanism Design

For $n$ workers with private costs $c_i$ and public utilities $u_i$, and a hard budget $B$:

• Define an allocation curve $f(x) = \ln(e-x)$, with $f_r(x) = f(x/r)$ for scale $r > 0$.
• Allocate fraction $x_i = f_{r^*}(c_i/u_i)$ of each worker, where $r^*$ is chosen so total payment equals $B$:

$\sum_{i=1}^n u_i Q_{r^*}(c_i/u_i) = B$

with

$Q_r(x) = x f_r(x) + \int_x^{\infty} f_r(y) dy.$

This scaling generalizes the proportional-share rule to achieve optimal approximation and budget-exhaustion.

2. Manipulation-Resistance Axioms and Proofs

The mechanism satisfies three core axioms (Ghosh et al., 6 Nov 2025):

• Fraud-proofness: Adding $k$ fake users cannot profit the adversary by more than \$k. This holds since each new user can only contribute at most 1 to the revenue pool via the capped usable fee.
• Bribery-proofness: Any coalition of $k$ users, by changing engagement arbitrarily, can increase targeted artists’ revenue by at most $k$.
• Sybil-proofness: No artist gains by splitting into multiple identities; when an artist $j$ is replaced by clones $j_1, j_2$ such that $w_{i, j_1} + w_{i, j_2} = w_{ij}$ for every user $i$, total payout is preserved.

Sketches for these proofs rely fundamentally on the cap $\min(\gamma L_i, 1) \in [0, 1]$, establishing that no manipulation can exceed the "usable fee" per user.

3. Fraud-Proof Structure and Mechanistic Rationale

Prior mechanisms exhibit deficiencies:

• GlobalProp: A bot with large $L_i$ can extract an unbounded fraction of the pool, resulting in arbitrarily high payout relative to cost.
• UserProp: While fully fraud-proof and bribery-proof, it does not reward intensification of engagement (no benefit for users streaming an artist repeatedly).
• ScaledUserProp: Capping the per-user fee at 1 dilutes heavy bot activity but still rewards increased engagement up to a threshold $\gamma$; after exceeding the cap, further engagement yields no extra payoff.

This suggests ScaledUserProp interpolates between rewarding genuine activity and preventing exploitative manipulation.

4. Algorithmic Procedures and Complexity

Subscription platform pseudocode:

1. Compute $L_i$ for all $i$.
2. Sort $L_i$ in ascending order.
3. For each $k = 0, \ldots, n$ (number of capped users), solve

$$k \cdot 1 + \gamma \sum_{i=k+1}^n L_{(i)} = \alpha n$$

to find feasible $\gamma$.

1. For each artist $j$, sum over users:

$$\phi(j) = \sum_{i=1}^n \min(\gamma L_i, 1)\frac{w_{ij}}{L_i}$$

Total complexity: $O(n\log n + nm)$.

Large market budget-feasible allocation:

1. Use $f(x) = \ln(e-x)$, scale allocations according to $f_{r^*}(c_i/u_i)$.
2. Determine $r^*$ such that total payment equals $B$.
3. For each worker, compute allocation and Myerson-style payment.

5. Comparative Analysis to Alternative Rules

The following table compares four revenue division mechanisms by key properties (Ghosh et al., 6 Nov 2025):

Rule	Manipulation Resistance	Engagement Monotonicity	Pigou–Dalton Consistency
GlobalProp	Fails fraud-/bribery-proofness	Yes	Yes
UserProp	Fraud-/bribery-/Sybil-proof	Yes	Fails
UserEQ	Fraud-/bribery-proof	Yes	Yes
ScaledUserProp	Fraud-/bribery-/Sybil-proof	Yes (partial cap)	Fails in corners

GlobalProp offers simplicity and fairness but is highly manipulable. UserProp is robust to manipulation but fails to reward engagement intensity. UserEQ is coarse and vulnerable to Sybil attacks. ScaledUserProp combines resistance with engagement monotonicity, partial reward for stream intensification, and empirical reduction in price-per-stream envy. It does not satisfy Pigou–Dalton in some cases.

6. Theoretical Guarantees and Performance Bounds

• Resistance theorems: ScaledUserProp satisfies all manipulation-resistance axioms.
• Engagement monotonicity: Increasing artist engagement always weakly increases payoff.
• Approximation: In large markets, the mechanism achieves the optimal $1-1/e \approx 0.63$ approximation to fractional knapsack welfare; no truthful mechanism can do better (Anari et al., 2014).
• Pigou–Dalton: Small transfers can break consistency, but price-per-stream envy is lower empirically than in other fraud-proof rules.
• Submodularity: Extensions obtain $\tfrac12-o(1)$ or $\tfrac13-o(1)$ approximation for submodular utility objectives in large markets.

7. Empirical Evaluation and Impact

Experiments on real-world ($583$K users, $439$K artists, $27$B streams) and synthetic ($10$K users, $1$K artists) datasets reveal:

• ScaledUserProp exhibits the smallest disparity between top and bottom pay-per-stream (PPS) artists, particularly for $\alpha<1$.
• UserEQ is least fair, treating casual and heavy users equally.
• For $\alpha=1$, UserProp and ScaledUserProp coincide; for $\alpha<1$, ScaledUserProp smoothly interpolates between GlobalProp (for moderate-activity users) and UserProp (for heavy-activity users).

A plausible implication is that ScaledUserProp provides a robust, empirically fair allocation for revenue division and worker selection in contexts susceptible to adversarial manipulation, with best-known theoretical guarantees among scalable mechanisms.