Math Detours · Grade 8 · Connections · 9

Winning with fewer votes

Draw the districts and you choose the winner. Packing, cracking, and a number that measures it: the efficiency gap.

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Tonight's idea

A town of 50 voters elects a council of five, one seat per district of ten. Twenty voters back Purple, thirty back Orange: 60% for Orange. Who controls the council? Whoever draws the map decides.

Map 1: Purple wins 3–2 6–4, 6–4, 6–4, 1–9, 1–9 Map 2: Orange wins 5–0 4–6 in every district
Fifty voters, 20 Purple and 30 Orange. Same voters, same votes, two maps. Thick lines are district borders.

Map 1 puts nine Orange voters into each of two districts, landslides, and splits the rest so Purple wins the other three 6–4. Purple runs the council with 40% of the vote. Map 2 spreads Purple's voters evenly, four per district, and Orange wins every seat.

Cramming your opponent's voters into a few districts they win hugely is packing. Spreading them thin so they lose narrowly everywhere is cracking. Together they are gerrymandering, named in 1812 after Governor Gerry of Massachusetts and a district shaped like a salamander.

Both tricks create wasted votes. A vote for a losing candidate is wasted. So is every winning vote above half the district: the surplus. The efficiency gap compares how many each side wasted:

\[\text{gap} = \frac{\text{Orange's wasted votes} - \text{Purple's wasted votes}}{\text{all votes}}.\] On a fair map both sides waste about the same number, so the gap is near zero.

Watch one: Map 1

Count district by district. In a 6–4 district Purple wins. Half of 10 is 5, so Purple wastes its 1 surplus vote and Orange wastes all 4. Three such districts: Purple 3, Orange 12. In a 1–9 district Orange wastes its surplus \(9 - 5 = 4\) and Purple wastes its lone vote. Two such districts: Purple 2, Orange 8.

Totals: Purple wastes 5 votes, Orange wastes 20. The gap is \((20 - 5)/50 = 0.30\), 30% in Purple's favour. The two researchers who proposed this measure in 2015 suggested that a gap of a few percent is ordinary and one above about 8% deserves a second look. Thirty percent is not subtle.

Your turn

  1. Do the same count for Map 2. How many votes does each side waste, and what is the gap? Which side does it favour?
  2. A state has four districts of 100 voters. Party A wins three of them 55–45 and loses the fourth 20–80. (a) What share of all votes does A have? (b) What share of the seats? (c) Count each party's wasted votes and find the gap.
  3. Back in the 50-voter town, a seat needs 6 of 10 votes. (a) Can Purple win 4 of the 5 seats? (b) What is the smallest number of Purple voters that could still win 3 seats? (c) Now 1,000 voters in five districts of 200. What is the smallest share of the vote that can win 3 seats?
  4. In the town, a map gives Purple exactly 2 seats: 6–4, 6–4, 3–7, 3–7, 2–8. (a) Find the gap. (b) Another map gives Purple 1 seat: 6–4, 4–6, 4–6, 3–7, 3–7. Find the gap. (c) Is a gap of exactly zero possible for this town? Explain.

Stretch: the shortcut

When all districts are the same size, the gap has a shortcut: \[\text{gap} = (S - \tfrac12) - 2(V - \tfrac12),\] where \(S\) is a party's share of the seats and \(V\) its share of the votes. Check it on Map 1 for Purple and on Map 2 for Orange. Then: a party with 60% of the vote wins 70% of the seats. Biased?

Answers

  1. Every district is 4–6. Purple loses each, wasting all 4 votes: \(5 \times 4 = 20\). Orange wins each with 1 vote to spare: \(5 \times 1 = 5\). Gap \((5 - 20)/50 = -0.30\): 30% the other way, in Orange's favour.
  2. (a) \(185/400 = 46.25\%\). (b) 3 of 4, 75%. (c) A wastes \(5 + 5 + 5 + 20 = 35\); B wastes \(45 + 45 + 45 + 30 = 165\). Gap \((165 - 35)/400 = 32.5\%\) in A's favour. B is packed into one district and cracked across three.
  3. (a) No: four seats need at least \(4 \times 6 = 24\) Purple voters, and there are 20. (b) \(3 \times 6 = 18\) voters, 36% of the town, wins a majority of seats. (c) Three districts at 101–99 is 303 votes: 30.3%.
  4. (a) Purple wastes \(1 + 1 + 3 + 3 + 2 = 10\), Orange wastes \(4 + 4 + 2 + 2 + 3 = 15\). Gap \(5/50 = 10\%\) for Purple. (b) Purple wastes \(1 + 4 + 4 + 3 + 3 = 15\), Orange \(4 + 1 + 1 + 2 + 2 = 10\). Gap 10% for Orange. (c) No. If Purple wins \(k\) districts it wastes \(20 - 5k\) votes (its 20 votes minus the 5 that count in each win), and Orange wastes \(5 + 5k\). The gap is \((10k - 15)/50\), never zero for a whole number \(k\). The fairest maps give Purple 1 or 2 seats.

Stretch. Map 1, Purple: \((0.6 - 0.5) - 2(0.4 - 0.5) = 0.1 + 0.2 = 0.3\). Map 2, Orange: \((1 - 0.5) - 2(0.6 - 0.5) = 0.5 - 0.2 = 0.3\). Both match. With \(V = 0.6\) and \(S = 0.7\): \(0.2 - 0.2 = 0\), so not biased by this measure. The formula says a fair map gives the larger party a bonus: each extra 1% of the vote should bring about 2% of the seats.

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