Balance local results with statewide representation.
statewide representation.
District elections represent local voters and assemble a statewide delegation. What should the rule do when those goals conflict?
Model workspace
North Carolina, 2018
Statewide weight
Model result
0 Democratic seats
·
0 Republican seats
Model controls ready
Choose an election and see how the delegation changes
Select a state and year, move the weight, then compare allocation rules.
Historical district vote-share profile
Author-supplied normalized two-party district shares, shown on a schematic map.
Statewide tradeoff
Current prediction
Goal: Proportionality
Move left for local results and right for statewide proportionality.
0 switches
What the model predicts
The same votes and model settings determine the map, seat total, and diagnostics.
Model result
Model prediction
Model-predicted seat allocation
Move the statewide-weight slider to see when the predicted delegation changes.
Loading scenario
Model prediction
0 D / 0 R
Optimal model allocation shown
Move the statewide weight to update the model prediction.
Democratic allocation
Republican allocation
Reverses local plurality
Democratic 0%
Republican 0%
Model prediction at this weight
Selected
0 D / 0 R
w = 0.500 · 50% local · 50% statewide
North Carolina · 2018
Author-supplied election data
Dataset notes
Updated 2026-08-04
Model output · not an official election result
Compare allocations
Switch the allocation rule while holding the same district votes fixed.
The map shows the model allocation.
Current allocation · diagnostics
Definitions
Election diagnostics
Compare another measure on the map
Minimize an allocation diagnostic or open the mean–median vote profile while keeping district votes fixed.
Map: model prediction
Three allocations
Compare seat allocations
District-plurality winners, the model optimum at the selected weight, and the closest statewide-target allocation.
Seat-change ticks update with the model.
03
Why the model chooses this allocation
Compare the possible seat totals and see when the optimum changes.
Seat-change ticks update with the model.
Model loss by possible Democratic seat count
The selected prediction minimizes total misrepresentation.
District
Statewide
Weights at which the optimum changes
Prediction detail
Model assignments by district
Seat-change ticks update with the model.
District
D vote
Observed
Model
Status
District loss
04
See the calculation
The formulas below show how the model combines district and statewide loss.
Formal model
Method summary
Objective function and calculation for the selected allocation
The analysis starts with the outcome above. This section shows the exact notation, component values, and robustness weights behind it.
Paper objective
Selected allocation
Objective components at the current weight
- District loss Dist
- 0.000
- Statewide loss State
- 0.000
- Minimum objective value
- 0.000
Stability across weights
Method notes
Weight ranges and switch points
- Paper support ρ/N
- 0.0%
- Source support
- 0.0%
- First switch from FPTP
- w = 0.000
- Proportionality seat switch
- w = 0.000
Show the calculation line by line
At an exact objective tie, the map shows the smallest Democratic seat count. Every tied seat total remains a valid model optimum.
Alternative specifications
Choose the aggregation rule, statewide target, statewide penalty, and district loss
Party A is Democratic; all quantities use seat units. Each listed option satisfies the model checks. Custom formulas are checked before they are applied.
Model updated
The map, optimum, losses, and switching regimes have been recalculated.
Current specification
Current specification
Figures from the paper
Four figures
See the model's geometry
Each figure uses the selected election and specification. Use a horizontal weight control, or choose a point in a figure, to update the allocation above.
Figure 01
Pareto frontier and feasible allocations
Every allocation is shown through 18 seats; larger delegations use a fixed 32,768-allocation sample. Red circles retain the exact lowest district loss at each seat total; green marks are Pareto-efficient.
- Allocations
- 0
- Pareto seat totals
- 0
- Shown on map
- 0 D / 0 R
Interactive plot of district loss against statewide loss. It shows every feasible allocation through 18 seats and a fixed deterministic sample for larger delegations, with exact top-seat and Pareto-efficient allocations emphasized.
Scroll horizontally to inspect the full chart.
All allocations or fixed sample
Lowest district loss at each seat total
Pareto frontier
Shown on map
Figure 02
The optimal threshold rule
The vote-share cutoff is solved at every weight. For nonlinear choices, the adjacent-allocation indifference point is computed directly and capped only when an observed district boundary is needed to implement the optimum.
Exact linear cutoff
Profile-focused axes · click the plot to set w.
Seat-change ticks update with the model.
Interactive line chart of the model's optimal district vote-share threshold as a function of statewide weight.
Scroll horizontally to inspect the full chart.
Figure 03
Optimal seat total as statewide weight changes
The step path records every model-optimal delegation and every switching weight under the active specification.
Exact optimal path
0 switches
Seat-change ticks update with the model.
Interactive step chart of the model-optimal Democratic seat total as a function of statewide weight, including switching weights and target references.
Scroll horizontally to inspect the full chart.
Figure 04
Iso-misrepresentation curves select the optimum
An iso-misrepresentation curve joins district- and statewide-loss combinations with the same total M at the current weight. The lowest contour that reaches the Pareto frontier identifies an optimum.
Active objective contours
Move the shared weight or choose a point to inspect its allocation.
Seat-change ticks update with the model.
- Current optimum
- 0 D / 0 R
- Objective M
- 0
- Comparison
- None
Interactive plot with district loss on the horizontal axis and statewide loss on the vertical axis. Each iso-misrepresentation contour joins loss pairs with the same total M at the selected weight. Moving toward the lower left cannot increase M. Pareto-efficient top-seat allocations form the frontier; an optimum lies on the lowest attained contour, and muted dominated points are no better on either loss and strictly worse on at least one. Changing the statewide weight redraws the contours and may change the optimal seat total.
Scroll horizontally to inspect the full chart.
Equal total misrepresentation
Pareto frontier
Dominated top-seat allocation
Current optimum
How to read it. Moving toward the lower left cannot increase M. Changing the weight reshapes the contours and can change the optimal seat total. Muted points are dominated: another top-seat allocation is no worse on either loss and better on at least one.
About the explorer
Download the paper
Data, definitions, and methodology
Election data
The bundled dataset contains 270 state-year election profiles. Each profile uses author-supplied normalized two-party district vote shares, statewide vote support, observed FPTP seats, the proportional target, switch weights, and flags for proxied races or unusually large third-party vote shares. District shapes are schematic and are not geographic boundaries.
Diagnostic definitions
- Efficiency gap
- Difference in wasted-vote parity for the displayed allocation.
- Partisan bias
- Seat advantage at equal statewide support under the selected rule.
- Seat–vote gap
- Displayed Democratic seat share minus equal-district Democratic vote share.
- Declination
- Asymmetry in the distribution of district wins and losses.
- Mean–median gap
- Statewide Democratic support minus the median district share.
- First switch
- First statewide weight at which the optimum departs from FPTP.
Model summary
The model chooses the feasible delegation that minimizes a weighted combination of district-level loss and statewide representation loss. At w = 0, only local district outcomes matter; at w = 1, only the statewide target matters. Open Formal model in the explorer to inspect the active formula, calculation trace, exact switch points, and alternative specifications.
This is a research and methodological explorer, not an official election result or a legal determination. Custom profiles remain in the visitor’s browser and are not sent to a server.