Geographic Projection v1 (IPF / Raking)

1. Executive Summary

GeographicProjection v1 translates simulated 9-category national vote totals (M, L, C, KD, S, V, MP, SD, REST) into plausible constituency-level vote matrices across all 29 Swedish Riksdag constituencies.

The algorithm uses deterministic Iterative Proportional Fitting (IPF / Biproportional Raking) anchored to a historical baseline election matrix ($B_{c,p}$).

In historical forward evaluations ($2014 \to 2018$ and $2018 \to 2022$), deterministic IPF achieves:

• Constituency party-share MAE: 0.65% – 0.67%
• National share reproduction error: $< 10^{-6}$ (exact conservation)
• Total Riksdag Seat Error: 0 seats (exact certified seat reproduction for all parties in both Oracle and Production modes).

These are retrospective historical forward-evaluation results for the stated 2014→2018 and 2018→2022 fixtures. They do not certify every cell-level tie-break outcome in arbitrary forecast scenarios; the simulator’s separate freeze audit reports national-seat agreement and cell-level diagnostics explicitly.

2. Mathematical Methodology (IPF / Raking)

2.1 Problem Formulation

Let:

• $B_{c,p} \ge 0$: Baseline constituency $\times$ party vote count (from previous election, e.g. 2014 for 2018, 2018 for 2022).
• $R_c > 0$: Target total valid votes in constituency $c \in {1 \dots 29}$.
• $C_p \ge 0$: Target total national votes for party $p \in {1 \dots 9}$.
• Conservation invariant: $\sum_{c=1}^{29} R_c = \sum_{p=1}^9 C_p = T_{\text{total}}$.

The biproportional scaling problem seeks multipliers $a_c$ and $b_p$ such that: \(X_{c,p} = a_c B_{c,p} b_p\) subject to: \(\sum_{p=1}^9 X_{c,p} = R_c \quad \forall c \in \{1 \dots 29\}\) \(\sum_{c=1}^{29} X_{c,p} = C_p \quad \forall p \in \{1 \dots 9\}\)

2.2 Numerical Algorithm

Initialize $X^{(0)} = B$. Iterate for $k = 0, 1, 2, \dots$:

1. Row Step (Constituency Scaling): \(X_{c,p}^{(k+1/2)} = X_{c,p}^{(k)} \times \frac{R_c}{\sum_q X_{c,q}^{(k)}}\)
2. Column Step (National Party Scaling): \(X_{c,p}^{(k+1)} = X_{c,p}^{(k+1/2)} \times \frac{C_p}{\sum_d X_{d,p}^{(k+1/2)}}\)
3. Convergence Criterion: \(\max\left(\max_c \left|\sum_p X_{c,p}^{(k+1)} - R_c\right|, \max_p \left|\sum_c X_{c,p}^{(k+1)} - C_p\right|\right) < 10^{-8}\)

Convergence is deterministic and typically achieved in 9 to 10 iterations.

3. REST Category Handling

REST is not a single political party. It represents the aggregate of all minor non-parliamentary parties (e.g. Medborgerlig Samling, Alternativ för Sverige, Piratpartiet, local lists).

1. In Geographic Projection: REST is treated as a modeled category in IPF, receiving proportional geographic allocation based on historical minor-party strength.
2. In Mandate Allocation: REST is automatically mapped to OTHER_INELIGIBLE:
   • Its votes contribute to valid-vote denominators in constituencies and nationally.
   • It is strictly ineligible for the 4% national threshold and 12% constituency threshold.
   • It never receives seats.

4. Constituency Vote Total Modes

4.1 Oracle Mode

Uses the actual certified valid vote total $R_c$ for each constituency in the target election. This mode isolates the pure geographic party-distribution error from turnout volume error.

4.2 Production Mode

Predicts constituency valid votes $R_c^{\text{target}}$ prior to the election using the target electorate and prior turnout rate: \(R_c^{\text{target}} = \text{Eligible}_c^{\text{target}} \times \frac{\text{ValidVotes}_c^{\text{previous}}}{\text{Eligible}_c^{\text{previous}}}\)

5. Historical Forward Evaluations

5.1 Evaluation: 2014 Geography $\to$ 2018 Election

• National Votes Target: Actual certified 2018 national party votes ($N=6,476,725$).
• Baseline: 2014 certified constituency $\times$ party matrix.

Party-Level Constituency Share MAEs (2014 $\to$ 2018):

• M: 1.15%
• S: 0.90%
• C: 0.98%
• SD: 0.83%
• KD: 0.73%
• V: 0.54%
• REST: 0.34%
• L: 0.24%
• MP: 0.21%

5.2 Evaluation: 2018 Geography $\to$ 2022 Election

• National Votes Target: Actual certified 2022 national party votes ($N=6,477,970$).
• Baseline: 2018 certified constituency $\times$ party matrix.

Party-Level Constituency Share MAEs (2018 $\to$ 2022):

• S: 1.33%
• SD: 1.07%
• M: 0.95%
• V: 0.67%
• C: 0.63%
• KD: 0.58%
• MP: 0.38%
• L: 0.23%
• REST: 0.22%

6. Primary Practical Findings

Question: If the future national vote result were known exactly, how much seat error is introduced solely by the geographic projection?

Finding: Zero seat error. In both the 2018 and 2022 elections, deterministic IPF raking on prior-election geographic baselines introduces 0 total seat error when evaluated against certified Riksdag outcomes.

Why Deterministic Raking is Exceptionally Accurate in Sweden

1. Constitutional Proportionality: 39 adjustment seats (utjämningsmandat) absorb minor constituency-level swing differences and restore strict nationwide proportionality among qualifying parties.
2. Persistence of Regional Relative Strength: Relative regional party bastions (e.g. S in Norrland, M/L in Stockholm, KD in Jönköping, SD in Skåne) shift gradually over 4-year cycles.
3. No Overhang Distortion: Because no party reached an overhang threshold in 2018 or 2022, national party seat totals were dictated entirely by national proportional entitlement.

Conclusion on Geographic Residual Uncertainty

Deterministic geography via IPF is fully adequate for the baseline Riksdag simulator. Additional geographic stochastic noise is unnecessary and would introduce artificial variance without improving national seat calibration.