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I have a problem in which I have two parties. Party 1 is Purple. Party 2 is Yellow. There are a total of 135 votes. Purple has 75 of them while yellow has 60. There are 9 districts total. Each district has to have 15 votes in each of them, making votes in each district even. I'm supposed to set up the districts to maximize the number of Yellow-majority districts with efficiency gap 7%. I can't figure out how to arrange it to be this way. Any help?

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    Since 7% is the cutoff efficiency gap to be considered gerrymandering, it seems like your teacher is trying to make you work out how gerrymandered something can be without being recognized as gerrymandered by the efficiency gap system. In any case, we don't do homework for you. Try some stuff and see what you can get. I don't think he has onecorrect solution in mind. Oct 4, 2023 at 23:57
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    I’m voting to close this question because we're not doing your homework for you.
    – SQB
    Oct 5, 2023 at 9:57

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Hint: since this an equal districts arrangement, the efficiency gap has a simplified formula that only depends on the proportions of votes and the proportion of seats won by one party EG=(PS-1/2)-2(PV-1/2). Working the 60/135 fraction (as the proportions of votes) in that formula and letting PS range from 0 to 1 (in increments of 1/9) will give you range of values for EG. The minimum EG (as an absolute value) is 5.55% and attained for a PS of 3/9 or 4/9 (i.e. 3 or 4 seats won by the minority party). The other EG values turn out to be over 7%. Since we're asked to maximize PS (for the minority party), you want a PS of 4/9, of course, i.e. 4 seats. As for the actual packing and cracking, that's left as an exercise for the reader.

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    There is a slightly subtle issue that if you work out the vote by districts using an integer threshold (of 8) you get an [min] efficiency gap of 5.185%. That's because the simplified formula doesn't account for odd-size rounding effects; to get the EG predicted by the simplified formula, you have to use a threshold of 7.5 votes per district. Oct 5, 2023 at 3:35
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    Alternatively, the prediction of the simplified formula can be corrected by subtracting 0.5/135, i.e. half a vote from the total pool. Oct 5, 2023 at 3:48

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