Can strategy be used to win a majority in an election which uses cumulative voting? This is a voting theory question. [closed]

Scenario: A town is having an election for its Board of Trustees. There are six seats on the Board. A bloc of voters wants to win a majority, or four seats.

The election uses cumulative voting, an election method that permits a voter to cast multiple votes. In this case, each voter may cast up to six votes for one or more candidates.

Question: If the bloc puts forward four candidates in the election, is there a strategy it can use to allocate the six votes per voter to four candidates to maximize the probability of winning four seats?

Solution 1:

Let $p$ be the proportion of the town's voter's that the bloc (which I'll call P) has.

Suppose that the bloc gives its supporters the following instructions:

• Cast one vote each for candidates A, B, C, and D.
• If you're a woman, cast extra votes for A and B. If you're a man, cast extra votes for C and D. (The choice of how to assign the two A/B and C/D sub-blocks is arbitrary, but should be as close to a 50/50 split as possible.

Then each of the bloc's four candidates gets an average of $\frac{3p}{2}$ of the total vote.

Now, suppose that a competing bloc Q with proportion $q$ of the electorate decides to nominate $n$ candidates, with a similar scheme to divide votes equally between candidates. Each of their candidates gets $\frac{6q}{n}$ of the vote.

For this competing block to have their candidates get more votes than the original bloc's $\frac{6q}{n} > \frac{3p}{2}$. Equivalently, this restricts them to nominating $n < \frac{4q}{p}$ candidates. But they must also nominate a minimum of 3 candidates, or the original bloc would still win 4 seats by default. So, for Q's strategic nomination to work, they must have $3 \le n < \frac{4q}{p}$, which requires $q > \frac{3}{4}p$. Assuming a two-party system where $p + q = 1$, this means $q > \frac{3}{7}$.

So bloc P has the ability to win all four seats only if they have at least $\frac{4}7$ (57.1%) of the voters. Otherwise, a competing faction can use the same split-your-votes-equally strategy to win at least 3 seats.