Voting strategy when you can vote for multiple candidates?

Question

What is the best voting strategy when you're allowed to vote for multiple candidates, and the candidates with the most votes win? For example, say there's a race where you may vote for up to three candidates. You like two of them, feel ambivalent about one, and dislike the others. Should you vote for the one you feel ambivalent about, or not? What factors should you consider?

This is a common scenario in US state and local elections.

Edit: As noted in the comments, I should probably define "best." Obviously, you want the candidate(s) you like to be elected. Ideally, the candidate(s) you are ambivalent about would finish ahead of the candidates you dislike. The dilemma is that voting for the neutral candidate(s) may aid them in defeating your preferred candidate(s).

Answer 1

It sounds like you're talking about approval voting, in which each voter votes either "yes" or "no" for each possible candidate.

Voting strategy depends on predicting who is likely to win. Regardless of who you predict to win, though, one principle always holds in an approval vote:

If you decide to vote for a particular candidate, you should also vote for every candidate you prefer to that candidate.

In order to ensure that your vote has some effect on the election, you should vote in a way that distinguishes between the candidates that are likely to win. Thus:

Out of the (two or more) candidates you predict as likely to win, vote for at least one of them, but not for all of them.

It's unlikely that you'll be able to predict the outcome of the vote well enough for any detailed tactics, but even with perfect knowledge, the above two rules are sufficient. Predict which candidates are likely to win (usually two possibilities), and vote in a way that distinguishes between those candidates, without violating the first rule.

In cases where you can't predict other voters' behavior well enough to narrow the election down to 2 candidates, one very good (possibly optimal?) strategy is to:

Vote for your favorite of the likely-to-win candidates, and do not vote for your least favorite of them. For all candidates between those two candidates in terms of your preference ordering, assign a rating from 0 to 100 based on just how much you like those candidates compared with the previous two. Using some random source, pick a random number from 1 to 100; vote for all candidates that you rated greater than or equal to your random number.

This effectively changes the vote into range voting.

Answer 2

This depends on the details, of course.

Is it more likely that voting for the ambivalent candidate will push out a liked candidate, or a disliked candidate? Sometimes you can determine this with polling data, but often these types of elections are local and little to no data is available. If you have no data, then perhaps you can assume that if there are more disliked candidates than liked candidates, you're more likely to push out a disliked candidate and you should go ahead and vote for the ambivalent one.

If you're electing multiple members of a government body, you can also take into account the composition of the body. How many seats do you need to tip the scales? If you need three, then you should vote for all three. If you just need one seat, and there are only 2 disliked candidates, then voting for the ambivalent candidate could result in the worst possible outcome of disliked-disliked-ambivalent, so you probably should only vote for the liked 2. If you need 2 seats, and there's only 1 disfavored candidate, then voting for the ambivalent candidate could result in the worst possible outcome of liked-disliked-ambivalent, so again you should not vote for the ambivalent candidate.

Answer 3

The answer to this would depend on how this particular voting system works, since there are multiple systems where you're able to cast a vote for more than one candidate, and they are completely different from each other. In a system called approval voting, you can approve multiple candidates, or in other words, you are given multiple votes, but you can't use more than one vote on a single candidate. In another system called cumulative voting, your vote is split up between different candidates if you vote for more than one. Voting for two candidates in cumulative voting is statistically no different than flipping a coin over which of two candidates to vote for in a traditional first-past-the-post election.

Tactical voting is the practice of voting for one candidate to help prevent an even worse candidate from winning, instead of voting for who you actually like the best. Once the first-past-the-post voting system degenerates in to a system with two leading candidates or parties, a large number of votes are tactical votes.

In an approval voting system, you would probably want to vote for the better half of the candidates and not vote for the other half (or vote against them if that's an option). If the most undesirable candidate is likely to win, then you should add in some tactical voting and vote for every other candidate besides that one.

Answer 4

What factors should you consider?

it would depend on your objective function, your (ex-anti) estimate of winning probability for each candidate, and the variance matrix for such estimates.

a few considerations:

1) if you are reasonably sure of your candidate's chance of winning, obviously vote for him;

2) if not, your strategy is likely to vote for all the ones other than the ones that you dislike.

3) if the ones you dislike are to win, vote for the ones that you dislike the least.

Answer 5

For longer more detailed answer, you could try mathematics. Voting theory is a branch of mathematics. @Brilliand answered this correctly.

(I looked for a good website on voting theory in the pure math sense, haven't found one, everyone has taken to explaining it politically).

The multiple voting system is a good system, people sometimes don't like it cause people can say "he got 3 votes and I got just 1", but everyone has the option to use 3 votes, so it's not unbalanced, it's just different. On average, multiple votes hurts polarizing candidates and helps candidates that the other party doesn't "hate", so it tends towards a more moderate outcome. That's the problem with any voting "fix" if applies is at the end of the day, one candidate is helped another is hurt and the one hurt is bound to cry foul.

Take the MVP system, writers vote for baseball players in a 10 to 1 scale, the best player getting 10 points, the 10th best, 1 point. And the points are added up and the player with the most points (but not necessarily the most first place votes) wins the MVP award. This system can be questioned when the player who finishes 2nd with 16 first place votes, behind the player who finishes first with 10, cries foul, but it's still a pretty good system. But if the 16 voters of the guy they voted #1 want him to win, they could vote for the guy who gets 10 first place votes much later, and game the totals, but it would be fairly obvious when a guy who's up for #1 gets a bunch of 9 and 10 votes that some foul play was afoot.

One person one vote is tough to fix, but candidates (not voters) can do it by running and effectively stealing votes from another candidate (Anderson taking votes from Carter, Ross Perot, (perhaps) taking votes from Bush the dad, and the infamous Ralph Nader taking votes from Al Gore). That's not voter theory, but it demonstrates the point that it's very hard to run a fair election. I'm getting off subject though - sorry.

To expand on Brilland's answer - you vote for the candidate you like best among those who have a chance, that's just common sense.

And if you have two votes left, and you want to support, the green party or the economy first party or the religion of your choice party, that probably won't win, but you want to show support - vote away.

In short, it makes sense to vote for the candidate that you like among those with a shot at wining, and it makes sense to also vote for the candidate (or two) you like the best, but there's also something to be said for just recognizing something you agree with (an environmentalist voting green party for example).

Likewise, if there's someone you sort of like and he's favored to win, and you like person #2 better, you can not vote for the person you sort of like in the hopes that your #2 will win the election, but if enough people do that, maybe some other person's #2 - a guy you hate will win, so there's an argument to weighing chance of wining against how much you like someone and not overthinking.

The difficulty with showing this (and maybe someone on the math board can give a true mathematical argument), but the difficulty is that the way to vote changes based on how much you like each person, and how good a chance each person has on wining. If you like A better than B, but B has a 30% chance of wining and A has a 20% chance of wining, the math gets very murky and there's no clear answer. You need to weigh whether you're wiling to vote for B (cause he's much better than C), or not vote for B cause A is much better than B. - for an individual vote, a probability chart could be drawn up where you weigh chances against desired outcome. . . . but that hinges on the poles and the projected outcomes being accurate (and Dewey didn't beat Truman).

For an individual with a specific election, a probability chart could still be drawn out and a proper strategy could be devised, but in general terms, there's no way to assign a precise strategy, though I think Brilland's answer is as close as anyone is going to get to a general answer to this. (though I don't like the use of the term random - as it's not random, but math only works with precise input. The question the way it's asked is in terms of general input, so there's no precise answer.

I know that people will ask for citations, but mathematical stations for answers to general questions are hard to comeby. I was a math major in college, I really did used to study this stuff.