Final Generic Vote-Share Calibration Experiment — Methodology and Production Recommendation

1. Executive Summary

This document presents the final calibration experiment comparing generic election-day vote-share models for Swedish Riksdag elections.

We evaluate three models across both marginal (8-party CRPS) and joint multivariate (Energy Score) criteria:

1. base: OpinionState v1.1 + Dynamics v2 = symmetric_all_history (no election-day residual layer).
2. pp_centered_noise (canonical ID for pp_noise_only): Historical percentage-point residuals centered around historical mean bias ($\mathbf{r}_e - \bar{\mathbf{r}}$) with bounded simplex transfer.
3. pp_symmetric_noise: Raw historical percentage-point residuals multiplied by an independent random sign draw ($S \cdot \mathbf{r}_e$ with $S \in {-1, +1}$) with bounded simplex transfer.

Final Production Recommendation

Canonical Production Model: pp_centered_noise

• Rationale:
  1. Consistent CRPS Improvement in Both Target Elections: Improves 8-party CRPS in both 2018 (0.9867 vs 1.0623 base) and 2022 (0.6173 vs 0.6364 base). In contrast, pp_symmetric_noise degrades 2022 CRPS to 0.6435 due to excessive dispersion.
  2. Short-Horizon Superiority: At the critical final horizons ($h=7$ and $h=14$ days), pp_centered_noise delivers the lowest CRPS (0.4038 at $h=7\text{d}$ and 0.5329 at $h=14\text{d}$), avoiding the interval over-widening of sign-symmetric noise.
  3. Joint Multivariate Energy Score: Achieves the best overall Energy Score (2.8668 vs 2.8730 for symmetric and 2.9200 for base).
  4. Compact, Sharp Calibration: Reaches 76.8% overall 90-coverage (and 90.7% in 2022) with compact 3.51 pp intervals and virtually zero transfer attenuation (mean $\lambda = 0.9992$).
  5. Selection Rule Compliance: Because pp_symmetric_noise alternates between elections and over-disperses at short horizons, pp_centered_noise is selected by the predefined decision rule.

2. Mathematical Formulation

Let $x \in \Delta^9$ be the paired base composition from state_plus_dynamics ($\sum x_p = 100\%$, $x_p \ge \epsilon = 0.01\%$), and let $\mathbf{r}e \in \mathbb{R}^9$ be a historical 14-day percentage-point residual vector ($\sum \mathbf{r}{e,p} = 0$).

Bounded Simplex-Safe Transfer

For any transfer vector $v$ with $\sum v_p = 0$: \(\lambda = \begin{cases} 1.0 & \text{if } \{p : v_p < 0\} = \emptyset \\ \min\left(1.0,\; \min_{v_p < 0} \frac{x_p - \epsilon}{-v_p}\right) & \text{otherwise} \end{cases}\) \(x' = x + \lambda v\)

Model Definitions

• base: $x’ = x$
• pp_centered_noise: $v^{(i)} = \mathbf{r}{k^{(i)}} - \bar{\mathbf{r}}$ where $k^{(i)} \sim \text{Uniform}(1..K)$ and $\bar{\mathbf{r}} = \frac{1}{K}\sum \mathbf{r}{e_k}$.
• pp_symmetric_noise: $v^{(i)} = S^{(i)} \cdot \mathbf{r}_{k^{(i)}}$ where $k^{(i)} \sim \text{Uniform}(1..K)$ and $S^{(i)} \in {-1, +1}$ with $P(S=1)=0.5$.

3. Multivariate Energy Score

For a 9-category probabilistic composition forecast $F$ and actual certified outcome $y \in \mathbb{R}^9$:

Where $X, X’ \sim F$ are independent sample draws.

• For continuous Monte Carlo ($N=5\,000$ to $20\,000$), $ES$ is computed exactly on 5,000 deterministic sample vectors without stochastic subsampling.
• For finite discrete distributions ($M$ support points), $ES$ is computed directly from the finite support.
• Invariant Verified: For a single deterministic point forecast $X = x$, $ES(F, y) \equiv |x - y|_2$.

4. Standalone Exact Forward Evaluation from 14-Day Consensus (2010–2022)

Evaluated directly on $x = \text{PollConsensus}_E$ using exact discrete finite support ($K$ points for centered, $2K$ points for symmetric):

5. Full Pipeline Hindcasts (State + Dynamics, $N=5\,000$, Seed 12345)

6. High-Sample ($N=20\,000$) Multi-Seed Stability Audit

Evaluated across three independent fixed seeds (12345, 24680, 98765):

7. Horizon Breakdown (8-Party CRPS across Horizons)

8. Substantive & Methodological Interpretations

1. Broad Historical Error Representation:
   • The election-day residual layer captures broad historical survey-to-result discrepancies (differential turnout, late decision-making, house effects, and polling noise).
   • It should not be interpreted as a mechanistic model of individual strategic voting.
2. Small Historical Training Sample:
   • Training pools are limited to $K=2..5$ general elections. While non-parametric resampling provides an empirically grounded dispersion layer, future downstream layers (such as strategic threshold behavior or parliamentary mandate simulations) should be built with explicit structural domain logic.

9. Final Decision & Stop Condition

Decision: pp_centered_noise is frozen as the canonical production vote-share model.

Generic vote-share calibration work is complete. All downstream election simulations, seat allocations, and coalition models will build directly upon this calibrated foundation.