Constrained Hybrid Borda (CHB)

• Constrained Hybrid Borda (CHB) is a modular aggregation mechanism combining Borda scoring with plurality support for decentralized preference discovery.
• It employs parameterized filters (α and β) and a tunable hybrid parameter (λ) to balance broad consensus with strong first-place backing.
• CHB guarantees a unique, deterministic winner and computational efficiency while satisfying key social choice axioms within the Snowveil framework.

The Constrained Hybrid Borda (CHB) rule is a modular aggregation mechanism engineered to balance broad consensus (Borda count) with strong plurality support in collective choice settings. CHB operates over samples of strict rankings and implements parametrized filters and hybrid scoring to selectively elevate candidates that combine sufficient first-place backing with general acceptability. It is a central component within the Snowveil framework for decentralized preference discovery, ensuring uniqueness, deterministic outcomes, and compliance with core social choice axioms (Kotsialou, 20 Dec 2025).

1. Formal Definition and Mechanism

Let $P = \{p_1, ..., p_m\}$ denote the set of $m$ alternatives and $\Pi = \{R_1, ..., R_k\}$ a multiset of $k$ strict rankings from sampled voters. For candidate $p_j$:

• Borda score:

$B(p_j) = \sum_{i=1}^k \left(m - 1 - \ell\right)$

where $\ell$ is such that $p_j$ occupies rank $\ell+1$ in $R_i$.

• Plurality count:

$t_j = |\{i : p_j \text{ top-ranked in } R_i\}|$

Three parameters ($\alpha, \beta, \lambda \in [0, 1]$) control the aggregation:

Filter/Weight	Description
Popularity $\alpha$	Enforces $t_j \geq \lceil \alpha k \rceil$ for eligibility
Consensus $\beta$	Requires $B(p_j) \geq \beta \cdot \max_\ell B(p_\ell)$
Hybrid $\lambda$	Governs convex tradeoff: $(1-\lambda)$ Borda + $\lambda$ plurality

Define normalized hybrid score:

$$H(p_j) = (1-\lambda)\cdot\frac{B(p_j)}{k(m-1)} + \lambda\cdot\frac{t_j}{k}$$

CHB winner selection proceeds:

1. Borda filter: $p_B =$ Borda winner. If $t_{p_B} \geq \lceil\alpha k\rceil$, return $p_B$.
2. Hybrid stage: Let $$E = \{p_j: t_j \geq \lceil \alpha k\rceil, B(p_j) \geq \beta B_{\max}\}$$. If $E \neq \emptyset$, return $\arg\max_{p\in E} H(p)$.
3. Fallback: Otherwise, return $p_B$.

Exact comparisons and lexicographic tie-breaking yield unique outputs.

2. Design Intuition and Parameter Roles

CHB is constructed to mitigate the known drawbacks of pure Borda and pure plurality. Borda alone may select compromise candidates with negligible active support, while plurality alone can elevate polarizing choices disliked by most participants. The $\alpha$-filter screens out fringe candidates lacking baseline first-place support. The $\beta$-filter excludes candidates with insufficient aggregate approval, even if they have plurality blocs. The $\lambda$ parameter modulates the tradeoff between consensus and majority support, offering a tunable convex blend:

• $\lambda=0$ gives pure Borda within $\alpha$-eligible candidates.
• $\lambda=1$ yields pure plurality among $\alpha$-popular, $\beta$-acceptable candidates.
• Intermediate $\lambda$ specifies the hybridization.

This design ensures CHB avoids “extreme” candidates, promoting winners with both a core of high support and broad acceptability.

3. Social Choice Axiomatic Properties

A rule $F$ is Snowveil-compatible if it is deterministic, unique, positively responsive (May-style), and computable in finite time. CHB is proven to satisfy:

• Determinism & Uniqueness: Always a unique winner in finite time.
• Monotonicity: Up-ranking a winning candidate in any ballot cannot cause its defeat.
• Responsiveness: There exist profiles where a single rank improvement for $p$ switches the winner to $p$.
• Fine-Grained Responsiveness: An adjacent swap (one Borda point) is sufficient to flip outcomes in tailored profiles.

These properties ensure predictable, manipulability-adverse, and locally-sensitive operation.

4. Integration in Snowveil and Convergence Guarantees

In the Snowveil protocol for Decentralised Preference Discovery (DPD), CHB is called by the VoterUpdate procedure. Unlocked voters iteratively sample $k$ peer states, aggregating using CHB to determine potential candidates to lock onto. Locked voters exclusively advertise their locked choice, introducing positive feedback.

Global state $S_t$ is tracked by locked ballot counts $N_j(t)$ per candidate. The potential function:

$\Phi(S_t) = \sum_{j=1}^m [N_j(t)]^2$

quantifies concentration. Key convergence lemmas for all positively responsive rules (thus CHB):

• Strictly Positive Decision Probability: Any non-terminal state allows a voter to lock in one update.
• Plurality Amplification: If $N_a>N_b$, probability to lock on $p_a$ is strictly higher.

$\Phi(S_t)$ is a strict submartingale; the process almost surely converges in finite rounds to a stable, single-winner state, by the Martingale Convergence Theorem. Iteration across $m-1$ rounds constructs a full ranking.

5. CHB Pseudocode and Evaluation Workflow

The CHB procedure for sample $\Pi$, parameters $\alpha$, $\beta$, $\lambda$:

function CHB(Π, α, β, λ): # Step 1: Compute Borda and plurality counts for each p in P: B[p] ← sumᵢ (m−1−rank_of(p) in Rᵢ) t[p] ← |{i : top(Rᵢ)=p}| B_max ← max_{p} B[p] # Step 2: α‐popular Borda winner? p_B ← arg maxₗ B[ℓ] # break ties lex if t[p_B] ≥ ceil(α k): return p_B # Step 3: Hybrid stage E ← {p : t[p] ≥ ceil(α k) and B[p] ≥ β·B_max} if E≠∅: for p in E: H[p] = (1−λ)*(B[p]/(k*(m−1))) + λ*(t[p]/k) return arg max_{p∈E} H[p] # tie break lex # Step 4: default fallback return p_B

6. Computational Complexity and Empirical Behavior

• Borda score computation: $O(km)$
• Plurality count computation: $O(k)$
• Hybrid scoring/filtering: $O(m)$

Overall evaluation: $O(km)$ per sample.

Within Snowveil, the expected number of UpdateVoter calls to reach a winner is $O(n)$ due to constant probability bounds per lock event (demonstrated via Chernoff bounds).

Empirical simulation, under both Impartial Culture and strongly polarized electorates (up to $10^4$ voters), confirms:

Parameter	Observed Effect
$k$	Diminishing returns after $k \approx 15$
$\gamma$	Increasing rounds (higher $\gamma$) accelerates global convergence
$Q$	Cost grows super-linearly as quorum $Q \to 1$
$\lambda$	Runtime insensitive to $\lambda$, policy can tune freely
$\alpha, \beta$	“Policy window” controls winner transition, slowest convergence at regime boundary

Decision accuracy (canonical CHB winner selection in Snowveil) remains near $100\%$ for $k \geq 10$.

7. Context and Extensions

CHB provides a computationally lightweight, tunable mechanism for decentralized systems requiring stable consensus from subjective and diverse preferences. Its analytic transparency via deterministic operation, responsiveness, and fine-grained adjustment establishes a formal toolkit for preference discovery in settings constrained by censorship resistance, partial information, and asynchrony. The generic framework is applicable beyond CHB, wherever the aggregation rule meets Snowveil-compatibility (deterministic, unique, positively responsive, and finite-time computable) (Kotsialou, 20 Dec 2025).