Hamilton
D · Group D and E · Group E lose a seat as the pool grows. The next allocation is recomputed from the same weights.
This is a house-size paradox. Changing the pool changes both whole quotas and remainder rankings.
Earlier equal priorities may change award order without changing the final totals. Inspect the trace below.
Compare the allocations
4 groups receive different totals across the rules
This compares the same weights, number of places and tie priorities. Agreement here does not mean the rules always agree.
- Group A [A]: Hamilton 7 · D’Hondt / Jefferson 8 · Sainte-Laguë 8. Range 7–8 places.
- Group B [B]: Hamilton 7 · D’Hondt / Jefferson 8 · Sainte-Laguë 8. Range 7–8 places.
- Group D [D]: Hamilton 3 · D’Hondt / Jefferson 2 · Sainte-Laguë 2. Range 2–3 places.
- Group E [E]: Hamilton 3 · D’Hondt / Jefferson 2 · Sainte-Laguë 2. Range 2–3 places.
Use the brief to discuss the differences, then record your group’s chosen rule and rationale separately. It is not a decision made on your behalf.
| Group | Weight share | Exact quota | Hamilton | D’Hondt / Jefferson | Sainte-Laguë | Selected rule vs quota |
|---|---|---|---|---|---|---|
| A · Group A | 29.412% | 37500 / 5100≈ 7.353 · bounds 7–8 | 728% of seats | 832% of seats | 832% of seats | -0.353 seatsWithin lower/upper quota |
| B · Group B | 29.412% | 37500 / 5100≈ 7.353 · bounds 7–8 | 728% of seats | 832% of seats | 832% of seats | -0.353 seatsWithin lower/upper quota |
| C · Group C | 17.647% | 22500 / 5100≈ 4.412 · bounds 4–5 | 416% of seats | 416% of seats | 416% of seats | -0.412 seatsWithin lower/upper quota |
| D · Group D | 9.804% | 12500 / 5100≈ 2.451 · bounds 2–3 | 312% of seats | 28% of seats | 28% of seats | +0.549 seatsWithin lower/upper quota |
| E · Group E | 9.804% | 12500 / 5100≈ 2.451 · bounds 2–3 | 312% of seats | 28% of seats | 28% of seats | +0.549 seatsWithin lower/upper quota |
| F · Group F | 3.922% | 5000 / 5100≈ 0.98 · bounds 0–1 | 14% of seats | 14% of seats | 14% of seats | +0.02 seatsWithin lower/upper quota |
Where does adding a seat cause a loss?
Checks every adjacent total from 1 → 2 through 199 → 200 using Hamilton. Selecting a transition applies its starting total; it does not apply unfinished group edits.
Whole quotas, then remainder seats
quota = seats × weight / total weight
Allocate each whole quota, then give the remaining seats to the largest remainders. Recompute the entire allocation when the seat total changes.
Inspect whole quotas and all 6 remainder ranks
Whole-quota seats: A = 7 · B = 7 · C = 4 · D = 2 · E = 2 · F = 0. “Extra” means one of the remaining seats, not a sequential allocation across changing house sizes.
| Rank | Group | Exact remainder | Extra seat | Equal remainder groups |
|---|---|---|---|---|
| 1 | F | 5000 / 5100 | Yes | None |
| 2 | D | 2300 / 5100 | Yes | D, E |
| 3 | E | 2300 / 5100 | Yes | D, E |
| 4 | C | 2100 / 5100 | No | None |
| 5 | A | 1800 / 5100 | No | A, B |
| 6 | B | 1800 / 5100 | No | A, B |
What the numbers do not decide
Proportionality is one principle, not a complete definition of fairness. Group weights do not express need, rights or the quality of representation.
Lower and upper quotas are the ideal share rounded down and up. They coincide when the quota is already a whole number. Hamilton stays between these bounds; divisor rules can exceed them. A rule’s behavior is a tradeoff to examine, not a verdict.
Educational proportional-allocation model; not a full election system or a universal fairness score. No thresholds, guaranteed seats, district boundaries, eligibility rules or real-world tie procedures. Inputs stay in this browser unless you download them.
Primary sources
- US Census Bureau — Hamilton and Jefferson methods
- European Parliament — Understanding the D’Hondt method (2019)
- New Zealand Electoral Commission — Sainte-Laguë formula
Reviewed 8 September 2026. The lab uses the core formulas, not each jurisdiction’s complete election rules. The fixed tie convention is ours.