Election Result Layer v2 (Bounded Percentage-Point Transfers) — Methodology and Report

1. Executive Summary

This study implements and evaluates Election Result Layer v2, testing whether historical poll-to-election residuals improve election forecasts when applied via bounded percentage-point transfers on the simplex, rather than multiplicative shifts in Centered Log-Ratio (CLR) space.

Core Findings

1. Percentage-Point vs CLR Space:
   • Unlike CLR residuals (which produced catastrophic distortions due to log-ratio multiplier leverage on historically unpolled parties), percentage-point transfers remain strictly well-behaved and preserve the simplex geometry.
2. pp_noise_only Consistently Outperforms base:
   • 8-Party Mean CRPS: Improves overall from 0.8494 (base) to 0.8001 (pp_noise_only).
   • Election-by-Election: Improves in both target general elections:
     • 2018: 0.9845 vs 1.0623 (base)
     • 2022: 0.6158 vs 0.6364 (base)
   • Horizon-by-Horizon: Improves CRPS across all six forecast horizons ($h \in {112, 84, 56, 28, 14, 7}$).
   • Calibration: 90% coverage improves from 54.6% to 76.9% (reaching 90.7% in 2022) with only modest interval widening ($+1.22$ pp).
   • Minimal Attenuation: Mean $\lambda = 0.9991$; over 99.7% of samples required zero or negligible attenuation ($\lambda \ge 0.90$).
3. Standalone Forward Evaluation (2010–2022):
   • Evaluated directly from 14-day pre-election polling consensus, pp_noise_only reduced 8-party CRPS in all 4 historical elections (2010, 2014, 2018, 2022), improving overall CRPS from 0.9537 to 0.7195.
4. Static Bias (pp_bias_only) Adds No Value:
   • Applying a static historical mean bias vector degrades full hindcasts (8-party CRPS worsens to 0.8886) because party-specific errors vary across political eras.

2. Mathematical Formulation & Transfer Scaling

Let $x \in \Delta^9$ be the base composition from state_plus_dynamics ($\sum x_p = 100\%$, $x_p \ge \epsilon = 0.01\%$), and let $r \in \mathbb{R}^9$ be a joint residual vector satisfying $\sum_{p=1}^9 r_p = 0$.

Bounded Simplex-Safe Transfer

The maximum feasible uniform transfer scale $\lambda \in [0, 1]$ is:

The transferred composition is: \(x' = x + \lambda r\)

Theoretical Guarantees

1. Exact Sum to 100%: $\sum x’_p = \sum x_p + \lambda \sum r_p = 100.0\%$.
2. Strict Positivity: $x’_p \ge \epsilon > 0$ for all parties $p$.
3. Directional Preservation: If $\lambda > 0$, the relative transfer proportions $\frac{x’_p - x_p}{x’_q - x_q} = \frac{r_p}{r_q}$ are strictly preserved.

3. Four Model Variants

1. base: $x’_i = x_i$ (unmodified state_plus_dynamics).
2. pp_bias_only: $x’i = x_i + \lambda^{(i)} \bar{\mathbf{r}}$ where $\bar{\mathbf{r}} = \frac{1}{K}\sum{k=1}^K \mathbf{r}_{e_k}$.
3. pp_noise_only: $x’i = x_i + \lambda^{(i)} (\mathbf{r}{k^{(i)}} - \bar{\mathbf{r}})$ with $k^{(i)} \sim \text{Uniform}(1..K)$.
4. pp_bias_plus_noise: $x’i = x_i + \lambda^{(i)} \mathbf{r}{k^{(i)}}$ using the exact same sampled index $k^{(i)}$ as pp_noise_only.

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

Evaluated directly on $x = \text{PollConsensus}_E$ using exact discrete distributions:

5. Full Pipeline Hindcasts with OpinionState + Dynamics (2018 & 2022)

Evaluated across 6 horizons ($112, 84, 56, 28, 14, 7$ days) $\times$ 9 categories $\times$ 5000 paired Monte Carlo samples ($N=432$ rows):

Overall Comparison (2018 + 2022)

By Target Election

By Horizon (8-Party CRPS)

6. Attenuation Diagnostics ($\lambda$)

• pp_noise_only:
  • mean_lambda: 0.9991
  • p05_lambda: 1.0000
  • fraction_lambda_lt_0_99: 0.30%
  • fraction_lambda_lt_0_90: 0.30%
  • fraction_lambda_lt_0_75: 0.10%
• Conclusion: Percentage-point transfers virtually never violate simplex boundaries on real Swedish compositions, confirming that $\lambda$-scaling serves as a reliable guardrail without artificially compressing transfer vectors.

7. Substantive & Methodological Interpretations

1. Why pp_noise_only Works:
   • pp_noise_only injects realistic, empirical joint election-day co-movement without assuming static bias directions.
   • It captures the historical scale of late-breaking / election-day error while preserving cross-party correlations (e.g. Left/Green intra-bloc consolidation and Alliansen co-movements).
2. Double-Counting Caveat:
   • A 14-day poll-to-election residual includes genuine late campaign movement as well as polling error.
   • At long horizons ($h=112$ days), Dynamics already accounts for most opinion drift. However, as $h \to 7$ days, Dynamics uncertainty contracts toward zero, leaving state_plus_dynamics under-dispersed. pp_noise_only fills this gap naturally, reducing $h=7\text{d}$ CRPS from 0.4469 to 0.4032.
3. Small Sample Size Warning:
   • Training pools contain only $K=2..5$ elections. The empirical improvements are clear and consistent, but must be treated as a parsimonious empirical dispersion layer rather than an exhaustive asymptotic distribution.

8. Final Recommendation for Production Simulator

Recommended Production Vote-Share Layer: pp_noise_only

• Architecture: \(\text{Sample } \theta_t^{(i)} \sim \text{OpinionState v1.1}\) \(\mathbf{z}^{(i)} = \text{clr\_to\_composition}(\text{CLR}(\theta_t^{(i)}) + S^{(i)}\mathbf{\Delta}_h^{(i)}) \quad \text{via Dynamics v2}\) \(x_E^{(i)} = \mathbf{z}^{(i)} + \lambda^{(i)}(\mathbf{r}_{k^{(i)}} - \bar{\mathbf{r}})\)
• Why: It improves 8-party CRPS in both 2018 and 2022, outperforms across every forecast horizon, elevates 90% coverage to ~77–91%, operates with virtually zero attenuation ($\lambda \approx 1.0$), and avoids all pathological distortions of CLR log-ratio models.