T.N.Tideman [in email]:
More precisely, you could approve candidates you like slightly less than the preferred frontrunner. For example, if you like the Dem and dislike the Repub, and you like the Green slightly less than the Dem but a lot more than the Repub, then it wouldn't hurt to approve the Green also, and you might want to do so just to encourage the Greens to keep trying.
... If I wanted to devise an evaluation of Approval that I regarded as informative, it would need to involve a statistical process that reflected this understanding of how approval works in practice. But even if that evaluation turned out well, it would make me uncomfortable to have a voting procedure in which getting the most out of one's vote required one to begin by figuring out who the two leading candidates were.
You have to estimate how others will vote with almost any voting method, if you want to vote strategically. Let's compare several voting methods with two situations: two party dominated situations (like Florida 2000), and multiway elections with several strong candidates and no reliable polling data ("non partisan" municipal elections where the incumbent is not running for re-election can be like that; or the TX governor race in 2006).
Plurality: It's usually obvious when a race is two-party dominated. If you are satisfied with voting for one of the frontrunners, then it's easy to vote. If you want to vote for a third party, then you need to estimate whether your favorite candidate would act as a spoiler.
Approval: Easy! Vote for your favorite frontrunner and anyone you like better.
IRV (and Condorcet, I think): Easy! Just vote honestly.
Suppose there are three candidates, Good, Middling, and Bad, who are about equally likely to win. (Note that this is fairly common in nonpartisan municipal elections.)
Plurality: It's a crap shot. If Bad wins and Middling comes in second, you may kick yourself if you voted for Good.
Approval: Vote for Good. And if you like Middling better than the midpoint between Good and Bad, vote for Middling also.
IRV: Voting by ranking is seductive. It seems as if you can just rank the candidates honestly, and the voting method will make some reasonable and fair choice of winner. But, as we know, IRV can fail to choose the Condorcet winner, and it can reward strategic voting, among other nasties. When there are three candidates of roughly equal strength, you need to estimate whether voting Good > Middlling > Bad could cause Bad to win, when strategically voting Middling > Good > Bad might cause Middling to win instead.
Condorcet: If there are some "dark horse" candidates in the race, Worse and Worst, voters will be tempted to rank their own personal Good candidate 1st, then strategically rank the other two expected frontrunners, Middling and Bad, behind the dark horse candidates. In other words, vote Good > Worse > Worst > Middling > Bad. This voting behavior, if widely practiced, can cause Worse to win. Really nasty result!
Range: I think this is a situation where it makes some sense to vote an intermediate value for Middling. Suppose there is just one other voter besides yourself, and that voter will vote max for Bad and 0 for Good, but you don't know what he will vote for Middling, so assume a uniform distribution between 0 and max (and say max=1, i.e. 0-1 Range with real values allowed). Say your utility for Good is 1 and for Bad is 0, and for Middling is v. Then Good and Bad will both get average scores of 0.5. The probability that Middling's average score is greater than 0.5 is equal to the score you give to Middling. (So if you give Middling a score of 1, Middling's average score has probability=1 of being greater than 0.5, and Middling would win.)
If Middling loses, the expected utility of the election is .5. If Middling wins, the utility of the election is the utility of Middling, which is v (for you). If you vote 1 for Middling, the utility of the election is therefore v. If you vote 0 for Middling, the utility of the election is 0.5.
Suppose you vote v for Middling. Your expected utility of the election would be v² + 0.5(1-v). When v=0, 0.5 or 1, the expected utility of the election from voting v is equal to the expected utility from voting approval-style. The expected utility differs the most when v=0.25 or .75. When v=.75, for example, voting Middling=.75 gets you an expected utility of 11/16, which is 1/16 less than you would expect from voting Middling=1. When v=0.25, voting Middling=0.25 gets you an expected utility of 7/16, which is 1/16 less than you would expect from voting Middling=0.
I usually vote honestly under Plurality voting, figuring it's worth more to support my favorite party than to try to help the lesser evil win. Therefore, I would have no qualms about voting v for Middling, even if my expectation is lower by up to 1/16. One way of looking at it is, I am "negotiating" with my opponent voters. If our average opinion of Middling is better than the average of Good and Bad, then Middling will win.
Some people complain that 0-10 (integer) Range Voting is too fine-grained. They think some people would be paralyzed with indecision about whether to give someone a 6 or a 7. But Approval Voting is too coarse for me. If v=0.5, I would have a hard time deciding whether to approve Middling or not. I would find it less stressful to give Middling some intermediate value than to have to choose all-or-nothing for Middling.
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