Is it more effective to vote for a woman? 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 &lt;=4, he might be demoted, but if I vote for the woman and she enters position &lt;=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?)
Consider the following cases when you are the marginal voter (i.e. you decide the election):


Two men $m_1$ and $m_2$ have $&gt;n$ votes, two other men $m_3$ and $m_4$ have $n$ votes, and one woman $w_1$ has $&lt;n$ 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 $&gt;n+1$ votes, one man $m_x$ has $n$ votes and two women $w_1$ and $w_2$ have $k&lt;n$ 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 $k&lt;n$ 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.