Is it more effective to vote for a woman?

Question

A certain political party wants to encourage women to participate in their primary elections, so they decide, that the 4th position is reserved for a woman. That is, if there is no woman in the top 4 positions, then the woman with the largest number of votes will be promoted to the 4th position, and the candidates at positions 4 and below (5, 6, 7...) will be demoted one position (of course, if there is initially a woman in one of the top 4 positions, then no promotion/demotion will take place).

There are two candidates that I support equally, one is a man and the other is a woman. Is it true that, if I vote for the woman, my vote is more effective?

In a more extreme case, where the 1st position is reserved for a woman, it's clear that my vote is most effective when I give it to the woman, because this is my only chance of sending my favorite candidate to the 1st position; voting for the man, in this case, will never bring my favorite candidate to the 1st position.

Intuitively, it seems to be the same with the 4th position reserved, because, if I vote for the man and he enters position <=4, he might be demoted, but if I vote for the woman and she enters position <=4, she might be promoted, so my single vote may be worth a lot.

However, I am looking for a formal proof that this is the case (or maybe a disproof?)

Answer 1

Consider the following cases when you are the marginal voter (i.e. you decide the election):

• Two men $m_1$ and $m_2$ have $>n$ votes, two other men $m_3$ and $m_4$ have $n$ votes, and one woman $w_1$ has $<n$ votes. If you support $m_3$ and $w_1$ equally and support $m_4$ less, then you should vote for $m_3$, such that both $m_3$ and $w_1$ will be in the top four.
• Three men have $>n+1$ votes, one man $m_x$ has $n$ votes and two women $w_1$ and $w_2$ have $k<n$ votes each. If you support $m_3$ and $w_1$ equally, you should vote for $w_1$ regardless of $x$ because $m_x$ will definitely lose, and $w_1$ will only win with certainty with your vote.
• Men $m_3$ and $m_4$ have $n$ votes, two women $w_1$ and $w_2$ have $k<n$ votes. What happens in the case of a tie is unclear, and (given one vote) it will end in a tie either way.

Thus, it seems as though there is no advantage to voting for either a woman or a man over the other.

Answer 2

Here is my attempt to solve this - comments are welcome:

• Mark our favorite candidates with M and W.
• Our vote is obviously effective only when the candidate I vote for is in tie, either before or after the vote. Assume that in case of tie the order is decided at random; in this case, our vote increases the chance that the candidate will be promoted by 50%.
• If the tie is in positions 1, 2 or 3, then there is no difference between voting for M or for W - in any case, our vote will promote the candidate in 1 position (with 50% probability).
• Also, if another woman got more votes than the candidates in the tie, then there is no difference between M and W, because only the first woman is promoted.
• If the tie is in positions 4, 5, 6, ... and there is no woman that got more votes, then there may be a difference:
  • If the tie is with another man, then it's useless to vote for W, because she will win the 4th position anyway. Voting for M will promote him 1 position (with 50% probability).
  • If the tie is with another woman, then it's useless to vote for M, because the woman will win the 4th position anyway. Voting for W will promote her 1 or more positions (with 50% probability), directly to the 4th position.

If the probability of these latter two cases is equal, then voting for W is more effective, because it may promote her more than 1 position.

However, without further information about how other people are going to vote, the only information we can use is the relative number of women and men candidates:

If the number of men candidates is larger, the first case (tie with a man) is more probable, however, there are fewer women candidates, so in the second case (tie with a woman), the distance between W and the 4th position may be larger, so the promotion will be more effective.

Intuitively, it seems that the two effects cancel, so, the "expected number of promoted positions" is equal whether we vote for M and for W. Again, I am not sure. What do you think?