The claim: All RV versions, including Approval, when voting is strategic, guarantee that, with a few plausible assumptions RV will maximize the number of voters for whom the utility of the winner is greater than their pre-election expectation for the election. I.e, with (approximately) strategic voting, RV maximizes the number of voters who are pleasantly surprised by the outcome.
Another claim (proven symmetrically): With (approximately) strategic voting, RV minimizes the number of voters who are unpleasantly surprised by the outcome.
In "approval voting," assume approximately that each strategic voter votes for candidates with utility above the amount they expect for the winner. (This seems a strategically fairly reasonable thing to do – since why bother trying to influence improbable events far from your expectation. And indeed it can be proven to be the strategically optimum way to vote, in a certain probabilistic model.) Then the claim follows trivially: the most-approved candidate wins, i.e. the candidate approved by the most voters, i.e. the candidate whom the most voters regard as above their expected utility value for the winner. Q.E.D.
Well – what "plausible assumptions" exactly?
The plausible assumption that strategic voters will (usually) vote for the candidates who seem better than their estimate of expected quality of the winner.
And why should they do that? The question is whether or not to vote for candidate i, looked at in isolation.
Say candidate i is better than your expectation for the election. It's obvious that, for your expectation for the election to be worse than i's utility, it must be that your conditional expectation in the event that someone other than i wins must be less than i's utility, to bring the overall expectation down.
Now let's assume approximately that when you vote for i, raising i's win probability, you lower everyone else's win probability by a single uniform factor.
So, in case i doesn't win, at least your vote for him doesn't worsen your expectation when someone other than him wins. In fact, it doesn't change it at all.
And so, since i is better than your expectation if someone else wins, you should vote for i. Q.E.D.
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