Snowveil Framework for Decentralised Preference Discovery

• Snowveil Framework is a decentralised system that aggregates individual preferences via local sampling and thresholding without exposing the full global profile.
• It employs a modular, gossip-based protocol combined with the Constrained Hybrid Borda rule to balance consensus and plurality in single and multi-winner decisions.
• The framework guarantees almost-sure convergence with linear communication complexity and robust performance under both impartial and polarised voter regimes.

Snowveil is a framework for Decentralised Preference Discovery (DPD), designed to aggregate subjective preferences across large electorates under constraints typical of distributed systems: censorship resistance, partial information, and asynchronous communication. Snowveil departs from traditional centralised approaches in computational social choice by employing a modular, iterative, gossip-based protocol that achieves consensus without requiring any participant to see the global preference profile. Its architecture enables both single-winner and multi-winner decisions, supporting a generalized class of aggregation rules with formal guarantees on convergence and scalability (Kotsialou, 20 Dec 2025).

1. Foundations of Decentralised Preference Discovery

A DPD instance is rigorously formalized as a tuple $$(V, \mathcal{P}, \{R_i\}_{i \in V}, \mathcal{N}, \mathcal{C})$$:

• $V = \{v_1, ..., v_n\}$: Voters.
• $\mathcal{P} = \{p_1, ..., p_m\}$: Projects (candidates).
• Each voter $v_i$ possesses a private strict ranking $R_i \in S_m$ over $\mathcal{P}$.
• $\mathcal{N}$: An implicit, fully connected peer-to-peer network, satisfying:
  • Censorship resistance: no central authority; any two honest agents can eventually communicate.
  • Partial information: no agent observes the full profile $\Pi = (R_1, ..., R_n)$, accessing only random peer samples.
  • Asynchronous communication: arbitrary but finite message delays; absence of global synchronization.
• $\mathcal{C}$: A consensus rule mapping up to $k$ ballots to a unique winner.

The protocol’s objective is to progress all voters from an Unlocked state ($\bot$) into a Locked state on candidates through local sampling and update steps, until some candidate achieves a global quorum ($Q > 1/2$). By iterating this mechanism and sequentially removing winners, the framework yields a complete global ranking (Kotsialou, 20 Dec 2025).

2. The Snowveil Protocol: Sampling and State Evolution

Snowveil’s central mechanism is an iterative, gossip-based sampling protocol. At each discrete step, a single Unlocked voter $v$ is activated, draws a uniform random sample of $k$ peers, applies the aggregation rule $F$, and updates its state according to an internal thresholding routine:

VoterUpdate Algorithm (Summary)

Input: Unlocked voter v, candidate set P, peers V, parameters k, γ, τ_max, τ_min C[p] ← 0 for all p∈P for r in 1…γ: S_r ← Sample(V\{v}, k) p_r ← F(S_r, P) # e.g. CHB(S_r,P) C[p_r] ← C[p_r]+1 if C[p_r] ≥ τ_max: return LOCK(p_r) C_max = max_{p∈P} C[p] if C_max < τ_min: return NO-LOCK else: W = { p∈P : C[p]=C_max } if |W|=1: return LOCK(the only p∈W) else: return LOCK(TieBreakByRecency(W))

Activation continues until a candidate reaches the required number of locks ($\lceil Qn \rceil$), designating it as the winner for the round. The formal structure supports uniform random sampling (without replacement), with analytic approximations by i.i.d. draws justifiable when $k \ll n$. This structure is largely agnostic to the choice of aggregation rule, provided it meets axiomatic requirements detailed below (Kotsialou, 20 Dec 2025).

3. Aggregation via Constrained Hybrid Borda (CHB)

Snowveil introduces the Constrained Hybrid Borda (CHB) rule for modular, consensus-oriented aggregation. For a sample of $k$ ballots:

• Borda score:

$$B(p) = \sum_{R \in \Pi_{\text{sample}}} \bigl(m - \text{position}_R(p)\bigr)$$

• First-place count:

$$t(p) = |\{ R \in \Pi_{\text{sample}} : p \text{ is ranked 1st in } R \}|$$

Parameters: $\alpha, \beta, \lambda \in [0,1]$.

Definitions:

• $p$ is $\alpha$-popular if $t(p) \ge \lceil \alpha k \rceil$
• $p$ is $\beta$-consensual if $B(p) \ge \beta \max_q B(q)$

Procedure:

1. If $p^B = \arg\max_p B(p)$ is $\alpha$-popular, declare $p^B$ the winner.
2. Otherwise, for all $p$ meeting $\alpha$-popularity and $\beta$-consensus thresholds:

$$H(p) = (1-\lambda) \frac{B(p)}{k(m-1)} + \lambda \frac{t(p)}{k}$$

Return the $p$ maximizing $H(p)$.

1. Default to $p^B$ if no eligible candidate.

CHB’s tunable $\lambda$ parameter offers an explicit spectrum between Borda (consensus) and Plurality (plurality support): $\lambda = 0$ yields pure Borda, $\lambda = 1$ recovers Plurality among $\alpha$-popular candidates. The rule is proven deterministic, unique, computable in $O(k m)$, and satisfies Positive Responsiveness, ensuring both monotonicity and single-ballot-tipping responsiveness (Kotsialou, 20 Dec 2025).

4. Convergence and Markov Process Analysis

The protocol evolution constitutes a finite Markov chain over global states $$S_t = (s_1(t), ..., s_n(t)) \in (\mathcal{P} \cup \{\bot\})^n$$. A potential function,

$$\Phi(S_t) = \sum_{j=1}^m [N_j(t)]^2,\quad N_j(t) = |\{i : s_i(t) = p_j\}|,$$

is employed to demonstrate convergence.

Key results:

• In any non-terminal state, a randomly chosen Unlocked voter locks with strictly positive probability.
• The likelihood that an Unlocked voter locks onto the plurality-leading candidate dominates, directly resulting from the Positive Responsiveness property of $F$.
• $\Phi(S_t)$ forms a strictly bounded submartingale, increasing in expectation with each voter activation.

The main theorem asserts almost-sure convergence in finite time: for any aggregation rule meeting Determinism, Positive Responsiveness, and Computability, Snowveil terminates at a state with a unique winner attaining the quorum threshold. Repeated application constructs a full permutation, with as many rounds as candidates ($m$) (Kotsialou, 20 Dec 2025).

5. Computational Complexity and Scalability

Assuming constant sample size $k$, winner margin $\delta_{\min}$, and thresholds $\gamma, \tau_{\max}, \tau_{\min}$, the framework achieves:

• Constant lower bound $c_2>0$ on the probability that an UpdateVoter call results in LOCK, independent of the number of voters.
• Expected convergence time per winner is $O(n)$ UpdateVoter activations, with per-activation communication complexity $O(k)$.
• The protocol thus requires total $O(kn) = O(n)$ messages to decide a winner, given constant $k$ (Kotsialou, 20 Dec 2025).

6. Empirical Validation and Parameter Regimes

Simulations utilize discrete-event modeling, focusing on LOCK dynamics and first-place abstraction for ballots. Two synthetic preference regimes are benchmarked:

• Impartial Culture (IC): fully random voter rankings.
• Polarised: electorate partitioned into two nearly antipodal preference blocs.

Metrics of interest include:

• Convergence time (total UpdateVoter activations to first quorum).
• Decision accuracy (frequency of matching canonical CHB winner on the entire profile).

Empirical results establish:

• Linear scaling of convergence time with $n$ in both IC and Polarised models.
• Significant diminishing returns for sample sizes $k \gtrsim 15$.
• Increased robustness rounds ($\gamma$) facilitate more rapid consensus, a phenomenon labeled the “cautious voter paradox.”
• Near-unanimity quorum thresholds ($Q \to 1$) induce superlinear costs.
• $(\alpha, \beta)$ parameter maps elucidate a distinctive plurality–consensus performance boundary and its associated trade-offs.
• The influence of $\lambda$ on convergence time is negligible across regimes.

Comprehensive appendices provide full theoretical proofs, parameter sensitivity analyses, and extended candidate count simulations (Kotsialou, 20 Dec 2025).