gerrymandering to the max
Another Fiddler puzzle on gerrymandering an electoral map when all the votes are already known. There are seven districts with identical population sizes and a party needs at least half of the votes to win a district.
What is the smallest total fraction of population needed to vote the gerrymanderer’s party to ensure winning a majority of the seven districts?
The minimal number of districts to secure is four, where at least half of the votes need be for this party, meaning that the extreme solution where no such votes are seen in the three other districts works out for only two out of seven voters are behind the gerrymanderer’s party!
The second puzzle asks for gerrymandering a map by choosing a set of probabilities \(p_1,\ldots,p_7\) of winning the corresponding districts, under the constraint that the average of these probabilities is \(\frac12\). Meaning that the overall population is balanced between both parties. What is the strategy maximising the chances of winning at least four of the seven districts? at least five? And what are these probabilities?
Setting two \(p_i\)’s to zero and all other five \(p_i\)’s to \(7/10\) leads to a dismal \((7/10)^5\) probability of getting five districts, worse than having \(\frac12\) for all districts. While setting three \(p_i\)’s to zero and all other four \(p_i\)’s to \(7/8\) leads to a \((7/8)^4\) that beats \(\frac12\). (The Fiddler calls the first option dummymandering since it produces the consequence opposite to the intended one! In the real, US, world, dummymandering happened for the midterms in most places the Republican majority redrew the maps, as in Texas, Ohio, NC, Florida, &tc, since the redistricting did not integrate the shift in voters’ preferences) I checked on Claude that this was the optimal allocation, with it backing the optimality on a result of Hoeffding (Annals of Mathematical Statistics, 1956) I had never heard of.
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