This appendix specifies the full methodology for comparing electoral systems geometrically. The protocol is designed to be applied to any electorate with available multi-dimensional preference data.
The comparison requires voter positions on the political preference manifold. The recommended instrument elicits positions on each of the six dimensions using a battery of survey items.
Each dimension score is the mean of its constituent items (after reverse-coding where indicated), standardized to mean 0 and standard deviation 1 across the sample. The resulting six-dimensional vector is the voter’s position on the political preference manifold $\mathcal{P}$.
The $6 \times 6$ covariance matrix $\Sigma$ is estimated directly from the sample:
$$\hat{\Sigma}_{ij} = \frac{1}{n-1} \sum_{k=1}^n (x_k^{(i)} - \bar{x}^{(i)})(x_k^{(j)} - \bar{x}^{(j)})$$
Compute the eigenvalues $\lambda_1 \geq \cdots \geq \lambda_6$ of $\hat{\Sigma}$. The effective dimensionality is the number of eigenvalues needed to explain 85% of the total variance:
$$d_{\text{eff}} = \min\left\{k : \frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^6 \lambda_i} \geq 0.85\right\}$$
The Mahalanobis distance serves as the manifold distance estimate:
$$d_\mathcal{P}(v_1, v_2) = \sqrt{(v_1 - v_2)^T \hat{\Sigma}^{-1} (v_1 - v_2)}$$
For each electoral system $E$ to be compared:
Simulate the election. Using the voter positions and candidate positions, simulate the election under system $E$. For plurality, count first-choice votes. For RCV, perform iterative elimination. For approval voting, use a threshold rule (voters approve all candidates within $\tau$ manifold distance units). For Condorcet, compute all pairwise comparisons.
Compute information preservation. Using the formulas from Chapter 5 and Chapter 13:
Condorcet: $R = \min(d, \binom{k}{2})/d \cdot \text{(ordinal correction)}$
Compute the Political Bond Index. $BI(S, E) = \mathbb{E}[d_\mathcal{P}(v, r(v)) - d_\mathcal{P}(v, r^*(v))]$, where $r(v)$ is the voter’s representative under system $E$ and $r^*(v)$ is the manifold-optimal representative.
Compute dimension-specific BI. For each dimension $i$: $BI_i(S, E) = \mathbb{E}[|v^{(i)} - r(v)^{(i)}| - |v^{(i)} - r^*(v)^{(i)}|]$.
The electoral system comparison produces:
| Metric | Definition | Interpretation |
|---|---|---|
| $R$ | Information preservation ratio | Fraction of manifold information surviving the contraction |
| $BI$ | Political Bond Index | Average excess manifold distance between voter and representative |
| $BI_i$ | Dimension-specific BI | Which dimensions the system serves and which it ignores |
| $GI$ | Gerrymandering index | Manifold coherence of districts (for district-based systems) |
| Condorcet efficiency | Probability of electing the Condorcet winner | How often the system selects the manifold-optimal candidate |
| Representativeness | $1 - BI / d_{\max}$ | Normalized representation quality |
For reliable manifold estimation on a 6D space, a sample of at least $n = 500$ voters is recommended (approximately 80 per dimension). For precise metric estimation (including off-diagonal covariance terms), $n = 2{,}000$ or more is needed.
The Political Bond Index is a sample mean, so its standard error is:
$$SE(BI) = \frac{s_{BI}}{\sqrt{n}}$$
where $s_{BI}$ is the sample standard deviation of the individual Bond Index contributions $d_\mathcal{P}(v, r(v)) - d_\mathcal{P}(v, r^*(v))$. A 95% confidence interval is $BI \pm 1.96 \cdot SE(BI)$.
For gerrymandering detection, generate $N = 10{,}000$ random compact, contiguous, population-equal district maps using Markov chain Monte Carlo on the space of valid districtings. Compute $GI(D_i)$ for each random map. The actual map’s $GI$ percentile rank in this distribution is the gerrymandering score. A score above the 95th percentile is evidence of manifold gerrymandering.
To apply the comparison protocol to a specific electorate:
The protocol is designed for reproducibility: given the same input data (voter positions and candidate positions), the comparison metrics are deterministic (for deterministic systems like plurality and RCV) or converge to stable values (for stochastic systems like ensemble gerrymandering analysis).