Maximizing geography of losing candidates

[muon2]

Mathematically, there is an equivalent, simpler procedure. Take the margin of victory for the loser in those counties won by the loser, call it L. Count the number of counties won by the winner, call it C. L+C is the winning margin for the loser if all of the winner's counties just flipped to the loser by one vote. To solve the problem select the smallest number of counties that voted for the winner (n < C), such that the total margin of victory w in those counties is greater than L+C-n.

Since C and n are generally going to be small compared to the difference between w and L they can be usually be neglected. Then the problem simplifies to finding the smallest set of counties that gives a value for w that is larger than L.

For example, for IA Gov, Hatch only won Johnson county and the margin was 10,568 (Politico). Branstad won the other 98 counties. The only county Branstad won by more than that was Scott, with a margin of 15,073 (Woodbury was just short with 10,245). So flipping all other counties except Scott to Hatch would give Branstad a margin of 15,073 - (10,568 + 98 - 1) = 4,408.

[muon2]

Wow that California map is insane

It could be even more insane. Kashkari's winning margin in the states he won was 215,555 (again Politico numbers). Los Angeles county provided a margin for Brown of 405,230 so it alone could balance Kashkari's win plus all other flipped counties, not unlike the way Cook balances the rest of IL.

But it gets better, Alameda had a 226,726 vote margin for Brown which is larger than Kashkari's in his winning counties. So, you can take politicallefty's CA map and flip San Francisco, too and it's still a Brown win - with only Alameda!

[muon2]

Mathematically, there is an equivalent, simpler procedure. Take the margin of victory for the loser in those counties won by the loser, call it L. Count the number of counties won by the winner, call it C. L+C is the winning margin for the loser if all of the winner's counties just flipped to the loser by one vote. To solve the problem select the smallest number of counties that voted for the winner (n < C), such that the total margin of victory w in those counties is greater than L+C-n.

Since C and n are generally going to be small compared to the difference between w and L they can be usually be neglected. Then the problem simplifies to finding the smallest set of counties that gives a value for w that is larger than L.

For example, for IA Gov, Hatch only won Johnson county and the margin was 10,568 (Politico). Branstad won the other 98 counties. The only county Branstad won by more than that was Scott, with a margin of 15,073 (Woodbury was just short with 10,245). So flipping all other counties except Scott to Hatch would give Branstad a margin of 15,073 - (10,568 + 98 - 1) = 4,408.

Very interesting! Now I'll just have to read it over about seven more times before I get it, since I'm so bad with equations.

It's really not very complicated if you don't think about the totals but only the margin between the candidates. The loser has a fixed total margin in the counties won by the loser. Each county flipped adds one to the margin for the loser. To keep the winner in front requires one or more counties not flip where the winner's total margin exceeds the total held by the loser.

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Is there another combination of Ramsey + something smaller that also holds for Dayton? Hennepin supplied about 60K more votes than needed.

[muon2]

Is there another combination of Ramsey + something smaller that also holds for Dayton? Hennepin supplied about 60K more votes than needed.

Maybe I'm not following, but the general idea was to flip each county based on the margin of victory for the winning candidate, starting with the counties with the smallest margins of victory and ascending from there. The secondary goal is to see how many counties that would result in the losing candidate winning, while still ultimately losing, but keeping the counties that flip in line with that first goal was the idea.

However, I just realized that I messed up Minnesota by flipping Cook County and St Louis County, both of which had larger margins of victory for Dayton than Hennepin. Unfortunately, after subtracting these from Johnson's total and then adding Hennepin in, Johnson would be ahead by 20,000 or so, so this is the actual map based on the criteria I outlined instead:

Mark Dayton 989,100 49.43%
Jeff Johnson 904,543 45.21%

Perhaps the confusion was mine. Is your OP about percentage margin or vote margin? When margin is used without an adjective it usually means vote margin. Your revised MN map suggests that you mean percentage margin.

[muon2]

Perhaps the confusion was mine. Is your OP about percentage margin or vote margin? When margin is used without an adjective it usually means vote margin. Your revised MN map suggests that you mean percentage margin.

Yes, sorry: I meant percentage margin. I figured that would be more realistic for such a scenario (relatively speaking).

I also went with your statement in the OP of the overall problem.

Analyze a county-by-county election result for a 2014 gubernatorial or senatorial contest, and without changing the overall result, maximize the number of counties the losing candidate won without pushing them over the top.

I took your statement that followed the quote to be your method, rather than the problem itself. As clarified, the problem becomes a brute force math exercise of moving through the sequence of counties from narrowest to widest percentage. My math solves the problem in the quote above, but not by a percentage margin basis, nor by your sequence. It creates the opportunity to explore individual county results with especially large vote margins.