Fiddler Nation has an upcoming election between its two major political parties: the Suns and the Moons. You are drawing a new electoral map, and you have been tasked with gerrymandering on behalf of your party, the Moons. Fortunately, you know exactly how every citizen will vote in the election.
Fiddler Nation has seven districts, each with an equal population. For simplicity, assume that the Moons win any district in which they receive at least half of the vote.
What is the smallest total fraction of Fiddler Nation’s population needed to vote Moon such that it’s possible for the Moons to win a majority of the seven districts?
Let's let $p_i \in [0,1]$ be the proportion of voters in district $i$ that vote for the Moons. We see that the Moons will win district $i$ if and only if $p_i \geq 0.5.$ So without loss of generality, we assume that $p_1 \leq p_2 \leq \dots \leq p_6 \leq p_7,$ so if the Moons are only going to win 4 districts, then they would win districts 4 through 7, that is we would need $p_i \geq 0.5, i = 4, \dots, 7.$ Since each district has the same population, say $1/7$ of the total, we have that the total number of votes for the Moons would be $\frac{1}{7} \sum_{i=1}^7 p_i.$ For the classic problem, we want \begin{align*} \min \,\, & \frac{1}{7} \sum_{i=1}^7 p_i \\ \text{suct that} \,\, & 0 \leq p_1 \leq p_2 \leq \dots \leq p_6 \leq p_7 \leq 1 \\ & p_i \geq \frac{1}{2}, \, i = 4, 5, 6, 7\end{align*} We see that the minimizer is $$p^*_1 = p_2^* = p_3^* = 0, p^*_4 = p^*_5 = p^*_6 = p_7^* = 1/2,$$ so the smallest total proportion of the entire voting population to vote for the Moons that could result in a Moon victory is $$p^* = \frac{1}{7} \left( 3 \cdot 0 + 4 \cdot \frac{1}{2} \right) = \frac{2}{7}$$ of the population, which is indeed a pitifully low portion of the populace.
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