Voting for candidates you "approve" is of course the most obvious way to vote in approval, but it is neither the most likely (in my opinion) or the most effective.
Rather than vote for candidates you like more than average (if that is what approving means) the more reasonable strategy is to vote for the candidates you like more than your expected return from the election. The only time that means "vote for better than average" is when you know nothing about the polls, which is a ridiculous assumption in a real election.
While this strategy (vote for the candidates you like more than your expected return from the election) can seem quite complicated, and in fact is in some situations, it's very easy in other, common situations..
John B. Hodges wrote:
The essence of his argument is in the math, the theorems he claims to prove and his interpretation of what they mean. Isolated examples and "nightmare scenarios" don't really prove anything, because no voting system is perfect. But I'll pass on one of the two examples he gives, as an "approval voting nightmare". You have a town of 10,000 people, choosing a Mayor. 9,999 people regard A as excellent, B as mediocre but passably competent, and C as a disaster. One voter (possibly C himself) regards C as excellent, B as passably competent, and A as a disaster. Everyone follows the recommended strategy for Approval voting of voting for all candidates that offer "above average" utility for the three candidates; so everyone votes for their top two. Tally is C gets one vote, A gets
9,999 votes, and B gets 10,000 and wins.
In this scenario, C is essentially irrelevant, and realistically everyone knows this. So the expected return on the election (for the 9,999 people) is something between excellent and mediocre. So they will approve only the candidate they like more than that, which is A. A therefore wins the vote 9,999 to 1 to 1.
More generally, if there is a clear front-runner and #2 candidate, then the expected return on the election is just on the #2 candidate side of the front-runner. So, if my utility for frontrunner is 60, and my utility for secondplace is 80, then my expected return on the election is probably in the low 60s. On the other hand, if my utility for frontrunner is 80, and my utility for secondplace is 60, then my expected return on the election is probably in the high 70s
From this I derive what I usually give as the optimal approval strategy, which is, "approve everyone you like more than the front-runner, and approve the front-runner if you like that candidate more than the second-most popular candidate." This is similar, but not identical to the commonly prescribed "vote for your favorite of the two front-runners, and for anyone you like more". The only difference is whether you sometimes vote for candidates you like less than one but more than the other.
So, in summary, Saari's nightmare scenario does not seem at all realistic to me. It requires a profoundly ignorant electorate.
-Adam
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