Constrained Hybrid Borda (CHB) Rule

• CHB Rule is a preference aggregation mechanism that extends the classical Borda count by incorporating explicit popularity and consensus filters.
• It employs tunable parameters α, β, and λ to balance Borda and Plurality scoring, ensuring compliance with social choice axioms and computational efficiency.
• The mechanism guarantees almost-sure convergence in decentralized settings and scales linearly, as validated by empirical phase transitions and lock-in behavior.

The Constrained Hybrid Borda (CHB) Rule is a preference aggregation mechanism that extends and constrains the classical Borda count by incorporating explicit filters for popularity and consensus, and linearly blending the normalized Borda and Plurality scores. As introduced in the context of the Snowveil framework for Decentralised Preference Discovery (DPD), CHB is engineered to select outcomes that reflect both broad-based approval and core intensity of support, while satisfying essential social choice axioms and ensuring convergence in decentralized, asynchronous environments (Kotsialou, 20 Dec 2025).

1. Formal Structure and Definition

Let $P = \{p_1, \dots, p_m\}$ be a set of $m$ candidates. A sampled preference profile of size $k$ is $\Pi_k = (R_1, \dots, R_k)$, where each $R_i$ is a strict total order over $P$.

For each candidate $p_j$:

• First-place count:

$t_j = |\{ i : R_i(1) = p_j \}|$

• Raw Borda score: $$B(p_j) = \sum_{i=1}^k [m - \mathrm{pos}_{R_i}(p_j)]$$, where $\mathrm{pos}_{R_i}(p_j)$ is the rank of $p_j$ in $R_i$.

Three rule parameters govern CHB:

• $\alpha \in [0,1]$: popularity filter (minimum fraction of first-place support)
• $\beta \in [0,1]$: consensus filter (minimum fraction of maximal Borda score)
• $\lambda \in [0,1]$: hybrid weight (tradeoff between Borda and plurality)

Define:

• Normalized Borda:

$\bar B(p_j) = \frac{B(p_j)}{k(m - 1)}$

• Normalized plurality:

$\bar t_j = \frac{t_j}{k}$

Let $p^B \in \arg\max_{p_j} B(p_j)$ denote the Borda winner, ties lexicographically.

The eligible set:

$$C_{\text{eligible}} = \{\,p_j \mid t_j \ge \lceil \alpha k \rceil,\; B(p_j) \ge \beta \max_\ell B(p_\ell) \, \}$$

Hybrid score:

$$H(p_j) = (1 - \lambda) \bar B(p_j) + \lambda \bar t_j$$

CHB selection (single winner):

1. If $t_{p^B} \ge \lceil\alpha k\rceil$, return $p^B$.
2. Else if $C_{\text{eligible}} \neq \emptyset$, return $$p^* = \arg\max_{p_j \in C_{\text{eligible}}} H(p_j)$$ (lex tie-break).
3. Otherwise, return $p^B$.

2. Explicit Score Functions

Function	LaTeX Definition	Interpretation
Borda	$$B(p_j) = \sum_{i=1}^k (m - \mathrm{pos}_{R_i}(p_j))$$	Total positional score
Normalized Borda	$\bar B(p_j) = \frac{B(p_j)}{k(m-1)}$	$[0,1]$ scaled Borda
First-place count	$t_j = |\{i: R_i(1) = p_j\}|$	Plurality/first-rank count
Normalized plurality	$\bar t_j = t_j/k$	$[0,1]$ fraction of first-rank votes
Hybrid score	$H(p_j) = (1-\lambda)\bar B(p_j) + \lambda\bar t_j$	Linear blend for eligible candidates
Eligibility	$$C_{\text{eligible}} = \{p_j\mid t_j\ge \lceil \alpha k\rceil,\; B(p_j)\ge\beta\max_{\ell}B(p_\ell)\}$$	Constraints on viability

This structure enforces both intensity and breadth: candidates must meet both minimum first-place support and consensus Borda conditions to be considered for hybrid scoring.

3. Constraint Mechanisms and Hybridization

The CHB mechanism introduces two explicit constraint parameters that modify the Borda component:

• Popularity filter $\alpha$: A candidate must achieve at least an $\alpha$ fraction of first-place votes to win directly by Borda count, or to be eligible for hybrid scoring. This ensures that Borda winners with negligible core support (compromise candidates) can be filtered out.
• Consensus filter $\beta$: A candidate's Borda score must reach a $\beta$ fraction of the highest Borda score, excluding fringe candidates even if they have substantial first-place support but poor overall rankings.
• Hybrid fallback and blending ($\lambda$): If no candidate meets both constraints, the algorithm returns the (possibly unpopular) Borda leader, ensuring determinacy. For candidates passing the filters, the hybrid score $H$ provides tunable balance between social efficiency (Borda) and core intensity (plurality).

This architecture allows for policy control over the tradeoff between broad acceptability and intensity of support, offering tunable selection “philosophies” depending on community requirements.

4. Axiomatic Characterization

The following axiomatic properties are satisfied:

Definition 1 (Determinism and Uniqueness): $F$ is deterministic and unique if for every profile $\Pi$, exactly one $p \in P$ satisfies $F(\Pi) = p$.

Definition 2 (Positive Responsiveness):

• (a) Monotonicity: If $F(\Pi) = p$, then for any profile $\Pi'$ attainable via a single-voter improvement for $p$, $F(\Pi') = p$.
• (b) Responsiveness: For each $p$, there exists a profile $\Pi$ with $F(\Pi) \neq p$ and a single-voter improvement $\Pi'$ for $p$ such that $F(\Pi') = p$.

Theorem 4.1: CHB is deterministic, unique, and computable. That is, it returns precisely one winner and does so in polynomial time.

Proposition 4.2: CHB satisfies positive responsiveness (both monotonicity and responsiveness).

Definition 4.3 (Fine-Grained Responsiveness, FGR): $F$ is FGR if a minimal improvement (raising Borda by exactly one without increasing first-place counts) can flip $p$ from loser to winner in some profile.

Theorem 4.4: CHB admits FGR. In “close-race” profiles, even minimal Borda improvements may alter the outcome.

These properties place CHB within a select class of aggregation rules that blend strict computability, social-choice sensitivity to profile changes, and a tunable spectrum between compromise and decisiveness.

5. Convergence and Process Analysis

Within the Snowveil paradigm, voters iteratively update and “lock in” to candidates based on local CHB invocations. The system state is $S_t = (s_1(t), \dots, s_n(t))$, with each voter either locked on unique $p_j$ or unlocked $(\perp)$. Define $N_j(t) = |\{v_i : s_i(t) = p_j \}|$ and potential $\Phi(S_t) = \sum_{j=1}^m N_j(t)^2$.

Key process properties:

• Positive lock probability: From any non-terminal state, any unlocked voter will eventually lock on a candidate with strictly positive probability.
• Plurality amplification: If $N_a > N_b$, unlocked voters are more likely to lock on $p_a$ than $p_b$, given positive responsiveness.
• Submartingale potential: $\{ \Phi(S_t) \}$ forms a bounded, strict submartingale—every lock strictly increases the potential.

Theorem 5.3: Almost-sure single-winner convergence: The process almost surely reaches an absorbing state where some candidate attains a majority quorum in finite time, due to the strict submartingale and transience of fragmented states. Iterating this winner-selection $m-1$ times (removing the previous winner) constructs a full ranking in $O(mn)$ expected steps.

6. Algorithmic Realization

Pseudocode for computing $\mathrm{CHB}(\Pi_k)$:

Function CHB(Πₖ, P; α, β, λ): Input: Πₖ = { R₁,…,Rₖ }, P={p₁…pₘ}, thresholds α,β, weight λ Compute for each p_j: t_j ← |{i: R_i(1)=p_j}| B(p_j) ← ∑_{i=1}^k (m - pos_{R_i}(p_j)) Let p^B ← argmax_j B(p_j) (lexicographically break ties) If t_{p^B} ≥ ceil(α·k): return p^B Let B_max ← max_j B(p_j) Build C ← {p_j : t_j ≥ ceil(α·k) and B(p_j) ≥ β·B_max} If C ≠ ∅: For each p_j∈C compute H(p_j) = (1−λ)·(B(p_j)/(k(m−1))) + λ·(t_j/k) return argmax_{p_j∈C} H(p_j) (lex tie‐break) Else: return p^B

This procedure is implementable in polynomial time and is modular with respect to the profile and parameter inputs.

7. Empirical Phenomena and Observations

• Tunable “policy window” $(\alpha, \beta)$: Profiles with strong Borda and Plurality contenders exhibit phase transitions in winner selection as $(\alpha, \beta)$ vary. This confirms CHB’s parameter-driven adaptability to different philosophies regarding consensus versus core support.
• Cautious-voter paradox: Increasing the robustness parameter $\gamma$ (number of CHB rounds before voter “lock”) decreases total convergence time. Higher local decision confidence dampens oscillatory early lock-ins and accelerates system-wide consensus.
• Scalability: Both in Impartial-Culture and Polarised preference models, the number of CHB invocations required for majority convergence scales linearly in the number of voters $n$, affirming the $O(n)$ bound.
• $\lambda$-robustness: System convergence performance is insensitive to the hybrid weight $\lambda$, so the consensus-plurality tradeoff can be set purely as a normative governance parameter without algorithmic penalty.

In summary, the Constrained Hybrid Borda rule operationalizes a hybrid between Borda and Plurality voting, modulated by explicit viability constraints, and satisfies determinacy, responsiveness, fine-grained sensitivity, and scalability. Within decentralized, gossip-based protocols such as Snowveil, CHB guarantees almost-sure, finite-time convergence to stable consensus outcomes, with empirical validation indicating efficient and tunable performance across diverse electorate configurations (Kotsialou, 20 Dec 2025).