# DFI Special Research Request: Electoral Disproportionality Assessment

**Country to be assessed:** [INSERT COUNTRY NAME HERE]

## Introduction to the Democratic Fitness Indicator (DFI)

The Democratic Fitness Indicator (DFI) measures how effectively citizens can exercise democratic power at the national level. This assessment focuses on a single component of the **Elections** category (25% of total DFI): **Electoral disproportionality** — the degree to which the electoral system mathematically translates voter preferences into seats.

## What this assessment measures — and what it doesn't

**Electoral disproportionality** is a property of the electoral system itself: the size of the gap between how citizens voted and how the resulting parliament is composed. A First-Past-The-Post (FPTP) system can produce substantial disproportionality even with fairly-drawn district boundaries; a proportional representation (PR) system with high district magnitude produces minimal disproportionality regardless of boundary shape. The assessment is mathematical, not investigative.

**This assessment does NOT measure intentional gerrymandering** — the deliberate manipulation of district boundaries for partisan or discriminatory advantage. That separate question is handled by the *Intentional gerrymandering* special research request and is scored as a distinct row in the DFI framework. A country can score high on disproportionality without intentional manipulation (UK 2024 — Severe disproportionality, independent boundary commissions), or low on disproportionality with intentional manipulation in a sub-region (a national PR system with one gerrymandered party-list district), or both simultaneously (the U.S. House under partisan-drawn maps in FPTP states).

Researchers must complete both assessments — disproportionality here, intentional manipulation in the separate file — and report each result independently.

## The metric: Gallagher Index

The DFI uses the **Gallagher Index** (also known as the Least Squares Index, LSq) as the single measure of electoral disproportionality. The Gallagher Index captures the system-level distortion: the cumulative effect of all parties' vote-to-seat translation gaps, not just the most-distorted single party.

### Why Gallagher rather than max-asymmetry

A measure that reports only the largest single-party distortion would miss the cumulative effect of small parties being systematically under-represented across a multi-party system. Gallagher captures this. In fragmented multi-party systems (Latvia, Netherlands, Switzerland), Gallagher is more honest about the system's overall behavior. In two-party FPTP systems, Gallagher and max-asymmetry produce similar magnitudes.

### Calculating the Gallagher Index

For each chamber separately:

1. For each party (including independents and named minor parties), compute:
   - Vote share % = (party votes ÷ total valid votes) × 100
   - Seat share % = (party seats ÷ total seats in chamber) × 100
   - Squared difference = (vote share − seat share)²
2. Parties below 1% of the national vote may be aggregated into a single "Other" bucket and included in the calculation as one entry, **but only when the combined "Other" share is itself small (≤2% of valid votes)**. When the sub-1% tail collectively exceeds 2% — typical in fragmented multi-party systems where 20+ small parties compete (e.g., Costa Rica 2022 with 30+ sub-1% parties totaling 21% of votes) — list each party individually instead. Aggregating a large tail into a single bucket mathematically treats independently-competing parties as if they were one party, dramatically inflating the squared-difference contribution and distorting the metric. Each entry in the calculation must represent a single party for the formula to behave correctly.
3. Sum the squared differences across all parties (including the "Other" bucket if used).
4. Divide by 2.
5. Take the square root.

```
Gallagher Index (LSq) = √( Σ (Vᵢ − Sᵢ)² ÷ 2 )
```

where Vᵢ is party i's vote share percentage and Sᵢ is party i's seat share percentage.

### Example calculation — UK House of Commons 2024

```
Party        Vote %    Seat %    (V−S)²
Labour        33.7      63.2      870.25
Conservative  23.7      18.6       26.01
Reform UK     14.3       0.8      182.25
Liberal Dem   12.2      11.1        1.21
Green          6.7       0.6       37.21
SNP            2.5       1.4        1.21
Other          6.9       4.3        6.76

Sum of squared differences = 1124.90
Divided by 2 = 562.45
Square root = 23.7

Gallagher Index = 23.7 → Severe (>15%)
```

## Classification thresholds

The Gallagher Index value, computed per chamber, maps to a DFI bracket:

| Gallagher Index | Classification |
|---:|---|
| <7 | Minimal |
| 7 to 15 (inclusive both edges) | Limited |
| >15 | Severe |

So Gallagher 7.00 is Limited (not Minimal); Gallagher 15.00 is Limited (not Severe); Gallagher 15.01 is Severe.

**Per-chamber penalty schedule** (per Scoring Guidelines §3.1.7 Gerrymandering table — penalties for the disproportionality row inherit the historical gerrymandering schedule):

| Chamber affected | Limited | Severe |
|---|---:|---:|
| Lower house only | −50% | −100% |
| Upper house only | −25% | −50% |
| Both houses | −50% | −100% |
| Single chamber (unicameral) | −50% | −100% |

The lower-house / upper-house asymmetry preserves the DFI convention that upper-chamber distortions count for half — upper houses often serve territorial-balance functions and are not direct mass-democratic representations.

**Important — "Both houses" is not additive.** The "Both houses" row is the worst-case-lower-house equivalent, not the sum of lower + upper. A bicameral country with both chambers in the Limited bracket gets −50% (not −75%); both Severe gets −100% (not −150%). The framework's logic is that the lower house dominates the parliamentary-scope effect; upper-house issues do not compound on top when the lower house is already affected. The Limited −25% / Severe −50% values for the upper house exist for the upper-only case, where there's no lower-house effect to dominate.

**Mixed-severity cases (asymmetric Lower × Upper combinations).** The original framework enumerates only the four symmetric cases (Lower-only, Upper-only, Both, Single). For asymmetric pairs, apply the **max-per-chamber-penalty-value convention** — take the larger of the two chamber-specific penalty values (not the higher classification). This is the only convention that extends the framework without introducing new methodology assumptions and matches the four enumerated cases.

| Lower house | Upper house | Penalty | Reasoning |
|---|---|---:|---|
| Minimal | Limited | −25% | max(0, −25) = −25 |
| Minimal | Severe | −50% | max(0, −50) = −50 |
| Limited | Minimal | −50% | max(−50, 0) = −50 |
| Limited | Limited | −50% | "Both houses — Limited" |
| Limited | Severe | −50% | max(−50, −50) = −50 — *not* Severe; Upper Severe is only −50% to begin with |
| Severe | Minimal | −100% | max(−100, 0) = −100 |
| Severe | Limited | −100% | max(−100, −25) = −100 |
| Severe | Severe | −100% | "Both houses — Severe" |

In all asymmetric cases, document the per-chamber Gallagher values in the YAML comment so the asymmetry remains visible to readers. The framework option chosen should be the symmetric case whose penalty value matches the max-per-chamber result (e.g., Limited + Severe → use "Lower house — Limited" if lower-house dominated, or the closest matching option; the YAML comment carries the asymmetric reality).

## When the metric is not applicable

Some chambers are structurally non-applicable for Gallagher Index assessment because they don't translate citizen votes into seats in the conventional sense:

### Indirectly elected or appointed chambers

Examples: UK House of Lords (appointed/hereditary), Canadian Senate (PM-appointed), Netherlands Eerste Kamer (elected by Provincial States — not directly elected by citizens).

**These chambers don't have a citizen-vote-to-seat translation function.** Mark Gallagher as N/A. The chamber's accountability deficit is captured by other DFI rows (unelected upper house veto distortions in §1.6).

### Fixed territorial-unit chambers

Examples: U.S. Senate (2 per state regardless of population), Swiss Council of States (2 per canton). The seat allocation isn't a function of electoral district boundaries — it's constitutional. Gerrymandering of state/canton boundaries is institutionally implausible; Gallagher doesn't apply in the conventional sense.

**Mark as Not applicable; the framework's "Fixed territorial units" applicability cascade handles these chambers separately.**

### Note on pure-PR-with-nationwide-constituency

Some chambers (e.g., Netherlands Tweede Kamer — single nationwide list-PR constituency) will mechanically produce Gallagher values near zero — vote share equals seat share by design, subject to small rounding from D'Hondt or Sainte-Laguë. **Compute Gallagher anyway** rather than marking N/A. The mechanic produces a low value, which is the correct answer; the metric does apply, it just lands at Minimal. This keeps the corpus consistent — every applicable chamber has a number.

## Step-by-step research protocol

### Step 1 — Identify the chambers to assess

Determine the country's legislative structure:
- **Unicameral** → one chamber to assess
- **Bicameral** → two chambers, assessed independently

For each chamber, classify whether Gallagher applies:
- **Directly elected with single-member or mixed districts** → Gallagher applies
- **Pure PR with nationwide constituency** → Skip (effectively Minimal by design; document this)
- **Indirectly elected, appointed, or fixed-territorial-unit** → Skip (mark N/A)

### Step 2 — Collect election data

For each chamber where Gallagher applies, collect from the most recent national election:

**National-level data** (sufficient for Gallagher calculation):
- Total valid votes cast
- Votes received by each party that won ≥2% of national vote
- Seats won by each party
- Total seats in chamber

**Sources, in priority order:**
1. Official electoral commission website (typically has both votes and seat allocations)
2. Official PDF reports with annexes
3. International IDEA Electoral System Design dataset; Inter-Parliamentary Union Parline database
4. Academic compilations (Lijphart's *Patterns of Democracy*, Gallagher's own electoral systems dataset)

District-level data is **not required** for Gallagher Index calculation — only national-level vote and seat totals. (District-level data is required for the separate *Intentional gerrymandering* assessment.)

### Step 3 — Compute the Gallagher Index

Using the formula above, compute for each applicable chamber. Show the calculation in your output (party-by-party squared differences, sum, divide by 2, square root) so the reader can verify.

Round to one decimal place. The DFI brackets are coarse enough that fine precision doesn't change the classification.

**Mixed electoral systems** (parallel or compensatory MMP): compute Gallagher on the chamber's final seat allocation in aggregate, treating the chamber as a single voting-to-seats translation. Do not separate SMD and PR portions for the calculation. The voter cast votes; the chamber has its composition; Gallagher measures the gap. Examples: Japan's House of Representatives (289 SMD + 176 PR) is computed as one chamber against the final 465-seat allocation; Germany's Bundestag (compensatory MMP) is computed against the final seat distribution after compensation.

**Satellite-party situations.** Where major parties field PR-only "satellite" lists separate from the parent party (e.g., South Korea 2020 onwards, where the parent contests SMDs and a satellite contests PR), the PR ballot does not cleanly express party preference: parent-party voters cast tactical PR ballots for the satellite, and the satellite's vote share appears against the parent's full (SMD+PR) seat allocation. In these cases, the standard PR-vote-share denominator inflates Gallagher dramatically without identifying any voter-to-seat translation harm — voters who chose the satellite *got* the parent winning, so voter intent and outcome aligned. The satellite arbitrage is a measurement problem in the metric, not a representational outcome problem.

For satellite-party cases, compute Gallagher using the **constituency (SMD) vote share** against final seat allocation, treating each parent-and-satellite as a single party. Independents and PR-only parties (which fielded no SMD candidates) get zero constituency-vote share against their actual final seat count — this correctly captures their structural disadvantage in the partial-compensation tier. Document the methodology choice and the constituency-vote source in the per-country file.

Worked example — South Korea 2024 (illustrative). PR-vote framing: Gallagher 26.6 (Severe) — but 99% of the squared-difference total comes from satellite-arbitrage and PR-only-party artifacts. Constituency-vote framing (DP-coalition ~50%, PPP ~45%, Rebuilding Korea 0%): Gallagher ~9 (Limited) — captures the genuine FPTP amplification on the 254 SMD tier without conflating satellite-tactical voting with vote-to-seat distortion.

### Step 4 — Apply classification and report

Look up the chamber's bracket from the table. Apply the per-chamber penalty.

## Output format

```
COUNTRY: [Name]
ELECTION: [Year and date]

CHAMBER 1: [Name, e.g., House of Commons]
ELECTORAL SYSTEM: [e.g., FPTP single-member districts]
GALLAGHER INDEX: [X.X]
CLASSIFICATION: [Minimal / Limited / Severe / Not applicable]
PENALTY: [None / -50% / -100% etc.]

[Repeat for second chamber if bicameral]

CALCULATION DETAIL:
[Per-party table: Vote %, Seat %, (V-S)²]
[Sum of squared differences]
[Divided by 2]
[Square root → Gallagher Index]

DATA SOURCES:
- [Source name with URL]
- [Source name with URL]

NOTES:
- [Any data caveats — minor parties consolidated, vote totals reconciled, etc.]
```

## Common pitfalls

- **Using only major parties.** Including only the top two or three parties under-counts the small-party distortions and produces an artificially low Gallagher value. Include every party with ≥2% national vote share.
- **Using turnout-weighted instead of valid-vote percentages.** Vote share should be computed against total valid votes, not total registered voters or total turnout including invalid ballots.
- **Mixing chambers.** Each chamber gets its own Gallagher calculation with its own party-vote-and-seat data. A bicameral country with different election dates may use different recent-election years for its two chambers — use each chamber's most recent election.
- **Interpreting Gallagher as evidence of intent.** A high Gallagher value indicates a disproportionate electoral system. It does not establish whether boundaries were drawn deliberately for advantage. The intentional gerrymandering question is handled by the separate special research file. Don't speculate about intent in this output.
- **Using bracket precision the threshold doesn't have.** The brackets use strict inequalities: <7 is Minimal, 7 to ≤15 is Limited, >15 is Severe. So Gallagher 7.00 is Limited (not Minimal); Gallagher 15.00 is Limited (not Severe). Values within ±0.5 of a boundary should be flagged in the output's notes so the YAML author can sanity-check.

## Reference values from the existing corpus

| Country | Chamber | Gallagher (most recent election) | Bracket |
|---|---|---:|---|
| UK | House of Commons (2024) | ~23.7 | Severe |
| Botswana | National Assembly (2024) | 23.05 (cited in research file) | Severe |
| USA | House of Representatives (2024) | low single digits — ~3–5 expected | Minimal–Limited boundary |
| Latvia | Saeima | typically low single digits (D'Hondt PR) | Minimal |
| Netherlands | Tweede Kamer | <2 (nationwide PR — D'Hondt rounding only) | Minimal |
| Switzerland | National Council | low single digits (cantonal list PR) | Minimal |
| Japan | House of Representatives | mixed system; SMD portion contributes to Gallagher | typically Limited |

These reference values are for sanity-checking only — re-compute against the country's most recent election data.

---

*This assessment measures mathematical disproportionality only. The separate Intentional gerrymandering assessment investigates whether currently-in-force boundaries were drawn through a process documented to have intentional partisan or racial-discriminatory advantage.*
