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The cost of districting — when the best candidate wins no district at all

Two chapters of one club elect a single national delegate. Ana is adored in Northside and unknown in Southside; Beto is the mirror image; Cleo is everybody's solid second. Count chapter by chapter and Cleo wins neither — so a delegate picked from the chapter winners is Ana or Beto. Count all nine members as one electorate and Cleo wins outright, and she is also the highest-welfare candidate in the race and the Condorcet winner. Nothing about the ballots changed. Only the map did.

▶ Live on BetterVoting: vote · results ↗ (election 38b7fg, BV2274) — one election carrying all three counts as three races; it elects Ana, Beto, Cleo exactly as below, with no tie-break anywhere. In the two chapter races the other chapter's members leave the contest blank, which is how a districted election actually works: one ballot paper, and you vote only your own district's contest. A blank scores 0, so every total matches the 5- and 4-member tallies here exactly.

Level: 301 · deep dive The runnable companion to distributed voting — the measured price of counting by district, which is where the theorems live.

→ Its honest twin: Exercise 1 — two districts, one mayor, where districting costs nothing and the centralized count is the one that gives ground. Read the pair together or neither.

The whole story in one table

Nine members, three candidates, scores 0–5. Every winner below is engine-verified from the three cases/ files.

Electorate Ana Beto Cleo STAR winner
Northside (5) 23 0 19 Ana
Southside (4) 0 19 14 Beto
Both together (9) 23 19 33 Cleo

Cleo leads the combined field by ten points and wins neither half. That is not a paradox — it is arithmetic. Ana's 23 is concentrated in one chapter and Beto's 19 in the other, while Cleo's 33 is spread evenly across both. A district count reads concentration; a combined count reads total.

The three counts

Northside (5 members) — Ana wins. full report → cases/cases_tabulated/b38b7fg_districting_north_tabulated.txt

Abridged — scoring and runoff rounds only
Scoring Round
   Ana           -- 23 -- First place
   Cleo          -- 19 -- Second place
   Beto          --  0
Automatic Runoff Round
   Ana           -- 3 -- First place
   Cleo          -- 2

Southside (4 members) — Beto wins, by the mirror image of the same shape. full report → cases/cases_tabulated/b38b7fg_districting_south_tabulated.txt

Abridged — scoring and runoff rounds only
Scoring Round
   Beto          -- 19 -- First place
   Cleo          -- 14 -- Second place
   Ana           --  0
Automatic Runoff Round
   Beto          -- 3 -- First place
   Cleo          -- 1

Both chapters together (9 members) — Cleo wins. full report → cases/cases_tabulated/b38b7fg_districting_combined_tabulated.txt

Abridged — scoring and runoff rounds only
Scoring Round
   Cleo          -- 33 -- First place
   Ana           -- 23 -- Second place
   Beto          -- 19
Automatic Runoff Round
   Cleo          -- 6 -- First place
   Ana           -- 3

Cleo also beats both rivals head-to-head, so she is the Condorcet winner as well as the score leader — the two notions of "best" agree here, and the district map overrides both. Only Choose-One (Plurality) diverges from STAR on the combined ballots, electing Ana on first choices alone.

What it costs

Read the score totals as the members' values and the welfare arithmetic is immediate. Cleo at 33 is the utilitarian optimum; whichever chapter winner the over-rule picks, the club is worse off:

The delegate is… Chosen by Welfare (raw) Distortion Welfare (unit-sum) Distortion
Cleo counting all nine together 33 1.00 3.9167 1.00
Ana the district map (Northside's winner) 23 1.43 2.7639 1.42
Beto the district map (Southside's winner) 19 1.74 2.3194 1.69

(Both columns because distortion is defined on unit-sum-normalized values, not raw scores. The ordering is identical either way, so nothing here turns on the normalization.)

The theory says a k-district architecture multiplies the worst-case welfare loss by k, and that the loss survives even when every district counts perfect utilities. This is that mechanism at k = 2, small enough to check by hand: no ballot reform touches it, because every district here already ran a full 0–5 score count and still discarded the best candidate. What went wrong was the requirement that the winner be somebody's district winner.

Reading this fairly

A constructed example, and it shows — mirror-image chapters, three candidates, a candidate engineered to be everyone's second. It demonstrates the mechanism honestly; it says nothing about how often real district maps do this. Two guardrails, per the four-part test:

  • The published experiments cut the other way. On real-world data (the Jester dataset) the districting effect was far milder than the worst-case bound, because real electorates are homogeneous — when districts don't differ much, neither does the winner. The k is a worst case over adversarial partitions. A gerrymander is an adversarial partition; an ordinary map is not.
  • The library's other districting case shows the cost at zero. In Exercise 1 the districted answer is the welfare optimum and the centralized STAR count is the one that gives up 13%. Both cases are real; neither is the general rule. That is the whole point of pairing them.

And this measures welfare only. Districts exist for representation, local accountability, and federalism — none of which distortion scores. "Districting has distortion Θ(km)" is an argument about one axis, not a verdict on districts.

Run it yourself

.venv/bin/python STARVote_LH_tabulation_engine/starvote_larry_hastings.py method_comparisons/districting_cost/cases/b38b7fg_districting_combined.yaml
Case Ballots Winner Page Source
Northside chapter 5 Ana page yaml
Southside chapter 4 Beto page yaml
Both together 9 Cleo page yaml

See also