I live in a town with a council-weak Mayor form of government. We have four council members, the fifth vote being supplied by the Mayor. The Mayor is up for election every two years, and also every two years, we vote for two of the four council seats.
These are undistricted council seats. That means there's a single non-partisan election with N candidates for two seats on it every two years. This year N is 8. So the ballot is, vote for up to two of these eight candidates (the Mayor runs separately).
If I want any ONE particular candidate to win, then, it's actually in my best interest to vote for ONLY one candidate. If I vote for a second candidate, then if that candidate finishes second, and my masin candidate finishes third by one vote, then I've effectively nullified my vote by voting for and against my one candidate,
Let me try to do this mathematically. Let's say there are three candidates for council and I get two votes. There are 100 voters in the town. If 100 vote for Candidate A and B, 100 vote for Candidate B and C, and 100 vote for Candidates A and C, then it will be a three-way tie at 200 votes each. If I, on the other hand, vote for A and nobody else, then it's 200 for A, 200 for B, and 199 for C. I essentially get two votes for A by not voting for C.
If, however, I want to affect the majority on the council -- assuming the vote is split 1-3 with the Mayor in the minority -- I need to assure two candidates I agree with are elected. In order to do that, I need to vote for two candidates to maximize my chances of having my will carried out, and hope that they finish 1-2, instead of accidentally helping to elect a single candidate who disagrees with me in the 2nd slot who finished one vote ahead of the second candidate I agree with.
Voting for somebody I disagree with is never a viable strategy, unless I do the following. I find a supporter of Candidate A, whom I do not agree with. That supporter also says they support Candidate B. I am in favor of Candidates X and Y, whom the supporter of A and B disagrees with. We agreed that Candidate M is our third choice. I agree with support of A and B that we'll both vote for Candidate M, and thus we're left with only one vote for one of our own candidates. That in turn creates two votes for M and only one each for, say, A and X, which creates a greater possibility M will be elected. In turn, it decreases the chances of B being elected, and if B is more popular than one of X or Y, it increases the chances that one of X or Y will be elected relative to the state before the deal. Of course, this kind of deal is considered illegal in many places, for all I know it's illegal here, and it's non-enforceable since you never know how the other guy will actually vote.
There are some communities where they're experimenting with a full preference ballot -- vote for first choice, second, etc. all the way down the ballot. Each first preference would count the same as each other first preference, each second the same as every other second. It seems to me that would avoid the problem I've got -- one candidate I want elected, but real power only possible if two of my candidates are elected.
Now let's take a look at this from a different perspective. Let's say the council is tied 2-2. The best way I have to affect policies is to vote for Mayor, since the Mayor is the swing vote. So I should probably run for Mayor instead of council, because the aggregate possibilities of having my votes discounted that are present in the City Council election are absent in the Mayoral race. Consequently this makes the Mayor, even though s/he has no other powers, significantly more powerful politically.
In the meantime, this seems like a dumb way to elect representatives, because I don't get to choose between two alternatives. It's essentially diluting my vote for this office by half (2 of 8, if you wish, instead of 1 of 2). I'm sure somebody has written some political science paper about this somewhere, but I'd be curious to hear if others have this same kind of voter's dilemma.
I think for this election I'm just going to vote for my one guy -- he finished third last time -- because whoever I vote for in the second slot may accidentally (from my perspective) win.