Double-Score Voting Mechanism

• Double-Score Voting Mechanism is a framework that uses two layers of quantitative evaluation, merging raw voter intensity with quality or credibility adjustments.
• The approach applies a monotonic transformation to raw scores and employs reweighting functions to balance intensity with aggregated support, enhancing proportionality and resistance to manipulation.
• This mechanism is applied in contexts like participatory budgeting, expert evaluations, and secure electronic voting, addressing computational challenges and ensuring fairness even with sparse inputs.

A Double-Score Voting Mechanism is a generalized voting framework that incorporates two distinct layers of quantitative evaluation for each alternative in a selection process. The approach aggregates both the raw intensity of support and an additional score layer—often reflecting quality, credibility, or strategic corrections—by combining these components in a mathematically principled way. Double-score mechanisms have been formulated to address issues in diverse domains, including proportional representation, secure computation, sparse input robustness, and trust-sensitive expert evaluations.

1. Formal Construction and Mathematical Models

The Double-Score Voting Mechanism extends conventional score voting by processing ballots in two separate but interdependent stages. Typically, each voter assigns raw scores $t \in [0,1]$ to each candidate or alternative. A monotonic transformation $\varphi(t)$ (such as $\varphi(t)=t^2$) is used to emphasize high intensity, yielding refined scores.

The general form of voter representation in multi-winner elections (Djukanović, 2017) is expressed as:

$$\omega_\mathcal{L} = \sum_{x \in \mathcal{L}} \frac{\varphi(x)^2}{|\varphi(x)|}$$

where $x$ is the vector of scores assigned to a candidate, $\mathcal{L}$ is the set of elected candidates, $\varphi(x)$ is applied component-wise, and $|\varphi(x)|$ is its aggregated norm.

The reweighting function, which modulates the influence of additional scores according to past representation, is generalized to:

$$r(y, \omega)_k = \frac{\varphi(y_k)}{2\omega_k|\varphi(y)| + \varphi(y_k)}$$

This framework allows for fairer, more proportional allocation by balancing intensity (via $\varphi$) and breadth of support (via aggregation and reweighting).

In the context of secure electronic voting, double-score mechanisms are realized by secret sharing of two independent score vectors per voter, with final aggregation and winner selection performed via secure multiparty computation (Dery et al., 2019):

$$\mathrm{FinalScore}_m = \lambda \, w^{(1)}(m) + (1-\lambda) w^{(2)}(m)$$

A plausible implication is that any number or function of score vectors can be accommodated, further increasing flexibility.

2. Algorithmic Families and Aggregation Schemes

Two main algorithmic families emerge from divisor-based generalizations (Djukanović, 2017):

• Phragmén–Sainte-Laguë Generalizations: These modify classic divisors (such as $2s+1$) to operate directly on continuous score ballots, coupled with reweighting functions that accentuate higher scores.
• Difference-Optimized Algorithms: At each step, candidates are ordered by the maximized norm $|r(x, \omega_{\mathcal{L}})|$; difference quotients are applied to select candidates that yield maximal “improvement” in representation.

For sparse, high-dimensional environments, Mehestan (Allouah et al., 2022) incorporates collaborative normalization and Lipschitz-resilient aggregation primitives: Quadratically Regularized Median (QrMed$_L$) and Lipschitz-Robustified Mean (LrMean$_L$). These ensure bounded per-voter influence and are applicable to double-score settings where each score is normalized and aggregated independently or jointly.

In participatory budgeting, strategyproof double-score mechanisms rest on matrix-based score functions $M$ that must satisfy the Constrained Change Property (CCP) to prevent strategic manipulation (Cohen et al., 2022). Designers may project any score function onto the nearest strategyproof set via quadratic optimization subject to CCP-derived constraints.

3. Domain-Specific Instantiations

Crowdfunding and Expert Evaluation

In CertiFund’s implementation (Hosni et al., 27 Sep 2025), double-score voting manifests as dual-layered expert evaluations:

• Each expert $e$ distributes 100 percentage points across multiple recommendation levels (e.g., HNR, NR, R, HR), expressed as $(w_e^\ell)$.
• Each vote is weighted by the expert’s credibility score $\lambda_e$.
• Aggregation is

$S_\ell = \sum_{e \in E} \lambda_e \cdot w_e^\ell$

with final recommendation $\ell^* = \arg\max_\ell S_\ell$.

This suggests that double-score voting enables far more granular and reliable project assessment than traditional binary expert validation.

Secure Election Systems

Secure double-score voting collects multiple score vectors per voter, secret-shares each vector independently, and combines them under privacy-preserving MPC circuits (Dery et al., 2019). Aggregation rules may blend scores linearly or hierarchically without disclosing ballots or intermediate results, supporting perfect secrecy.

Robust Sparse Voting

Mehestan’s architecture (Allouah et al., 2022) uses separate normalization and scaling for each score dimension, integrating collaborative scaling, translation, and joint robust aggregation. This approach is vital when voters score only a fraction of alternatives, and scores arrive at different or non-commensurate scales. Double-score mechanisms benefit by compensating for input sparsity and adversarial attacks.

4. Fairness, Proportionality, and Robustness

Double-score mechanisms provide several theoretical advantages:

• Proportional Representation: The two scoring layers allow schemes to account for both “raw popularity” and representation across the electorate, addressing over-representation by small but intense supporter groups (Djukanović, 2017).
• Lipschitz Resilience: Robust aggregation primitives (QrMed$_L$, LrMean$_L$) guarantee that no individual voter can sway any alternative’s score substantially, a property essential for manipulation resistance (Allouah et al., 2022).
• Strategyproofness: By ensuring CCP holds for the underlying score function, double-score systems provide resilience against incentive-driven deceit (Cohen et al., 2022).
• Mitigation of Information Asymmetry and Bias: Expert-based double-score systems explicitly weight expertise, improving the trustworthiness and informativeness of aggregate signals (Hosni et al., 27 Sep 2025).

A plausible implication is that the mechanism’s transparency and robustness could increase truthful voting and reduce manipulation in both electoral and expert-driven contexts.

5. Implementation and Practical Considerations

Implementing double-score mechanisms poses several challenges:

• Computational Complexity: Calculating reweighted norms, conducting secure MPC, or optimizing projections onto the strategyproof cone may be computationally intensive, especially at scale (Djukanović, 2017, Dery et al., 2019, Cohen et al., 2022).
• Parameter Selection: Choosing the transformation function $\varphi(t)$ and setting aggregation parameters (e.g., Lipschitz constants, expert weights $\lambda_e$) impacts both fairness and system behavior. Calibration is necessary to avoid dominance of one score layer (Djukanović, 2017).
• Transparency and Usability: More complex mechanisms may be less transparent to voters and administrators, requiring user interfaces supporting granular scoring and sophisticated backend aggregation (Hosni et al., 27 Sep 2025).
• Robustness to Sparse and Heterogeneous Inputs: For very sparse or heterogeneous scoring patterns, collaborative normalization and robust aggregation via Mehestan primitives become critical (Allouah et al., 2022).
• Security and Privacy: In secure electronic voting, ensuring authenticated channels, honest-majority among talliers, and endpoint security are fundamental (Dery et al., 2019).

6. Comparative Analysis and Limitations

Double-score voting mechanisms surpass traditional binary voting, simple score aggregation, and approval methods in expressiveness and robustness. Key comparative advantages include:

Mechanism	Score Granularity	Manipulation Resistance
Binary Approval	Low	Low to moderate
Single Score Voting	Moderate	Sensitive to outliers
Double-Score Voting	High	High (with suitable design)

Potential limitations stem from increased complexity, parameter tuning, determining credibility weights accurately, and computational demands for robust aggregation or secure operations (Djukanović, 2017, Dery et al., 2019, Hosni et al., 27 Sep 2025). Ensuring fairness across both scoring dimensions requires ongoing assessment and may introduce subjective components if expert credibility scores are not transparently assigned.

7. Future Directions

Double-score voting frameworks can be further extended to:

• Multi-dimensional and hierarchical voting: Beyond two score layers, systems may incorporate additional measures of quality, impact, or risk.
• Algorithmic transparency: Developing user-facing explanations and visualizations for double-score evaluations.
• Machine-learned credibility estimation: Credibility weights $\lambda_e$ may be inferred from large-scale evaluation outcomes or reputation systems.
• Integration with complex strategic environments: Extending CCP-based projections to multidimensional or time-evolving voting contexts (Cohen et al., 2022).

As applications expand in fields such as participatory budgeting, content moderation, expert validation, and crowdfunding, the empirical and theoretical foundations of double-score mechanisms will likely be refined to optimize fairness, efficiency, and trust.