Not Voting Helps the "Enemy" I have often heard people say "If candidate X is nominated, I'm not going to vote for anyone; I'm just going to stay home on Election Day."  A common response is "If you don't vote for X then you're helping Y get elected."  The logic seems to be you have to vote for someone you don't like, otherwise you are helping the other party.  I bring this up because of the recent election possibilities in the US. but also because I have heard this in past elections.  What does logic or math say about this kind of statement?
Please note that this is not a "political" question.  I am purely interested in what logic or mathematics says about this question.  Logically, is this statement valid?  Is there a line of reasoning that can help one decide such a statement?  In other words, an answer should take the form "Given a pool of voters and a choice of two candidates, if this person does this...."  This would help bring some rational thinking to the topic.
 A: Let's say $a$ other people will vote for party/candidate $A$ and $b$ other people will vote for party/candidate $B$. Presumably their decision is independent of yours (unless this post motivates you to go and try to persuade them to change their mind :-).
If you stay home, candidate $A$ will receive a share $\frac a{a+b}$ of the votes and candidate $B$ will receive a share $\frac b{a+b}$.
If you vote for candidate $A$, candidate $A$ will receive a share $\frac{a+1}{a+b+1}$ of the vote and candidate $B$ will receive a share $\frac b{a+b+1}$.
If you vote for candidate $B$, candidate $A$ will receive a share $\frac a{a+b+1}$ of the vote and candidate $B$ will receive a share $\frac{b+1}{a+b+1}$.
Now lets compare these shares. We have
$$
\frac{a+1}{a+b+1}-\frac a{a+b}=\frac{(a+1)(a+b)-a(a+1+b)}{(a+b)(a+b+1)}=\frac b{(a+b)(a+b+1)}\gt0
$$
and
$$
\frac a{a+b+1}-\frac a{a+b}=\frac{a(a+b)-a(a+b+1)}{(a+b)(a+b+1)}=-\frac a{(a+b)(a+b+1)}\lt0\;.
$$
So candidate $A$ receives a greater share of the vote if you vote for here than if you stay at home, and she receives a lower share of the vote if you vote for $B$ than if you stay at home. Likewise (since the shares add to $1$), candidate $B$ receives a lower share of the vote if you vote for $A$ than if you stay at home, and he receives a greater share of the vote if you vote for $B$ than if you stay at home.
Thus you have three options, and we can strictly order them in the order of their effects on the vote shares. The vote share will be most favourable for $A$ if you vote for her, most favourable for $B$ if you vote for him, and in between if you stay home.
Thus, if your goal is to maximise the chances of one candidate winning and the other candidate losing (the two go hand in hand), you should vote for the candidate whom you want to win. Of course you may have other goals – to save the effort of going to vote, or to express protest at the available choices, or to destabilise the system, or to be cool and cynical. None of these can be captured in this calculation. But if the focus is on influencing the outcome of the election by your actions, and you have any preference at all for one candidate over the other, it makes no sense to stay home, since in that case the vote share of the preferred candidate will be lower than it would have been if you'd gone to vote, and that of the unpreferred candidate will be higher.
A: The thing about this is if you are not voting and you only for the party x stands for then if you don't vote you are not helping X to reduce Y's lead.
In voting the the number of votes is always limited, so when you vote for X you are reducing Y's 1 vote and increases X's lead.In order to overcome one vote of Y,X needs two votes and vice versa
A: A vote not for a person is not a vote for that person.  This statement is a tautology.
If we are to assume that one of two people will win, then a non-vote or a vote for a third person (not able to win) is the same.  However this is not the same as a vote for the person you dislike more--it's a non-vote--you do not contribute to the voting process.  You only contribute if you feel that you should have voted for one of the two candidates over the other.  In that sense you have subtracted a vote from the candidate you preferred--but it's still not an endorsement of the other candidate--and it's still less than an actual vote for that less desirable candidate.
In an actual election, it should be theoretically possible that we do not know who will win and thus any vote is a vote for that person and no other, regardless of likely outcomes.  Unfortunately, that's not the nature of voting.
