Strategic range voting approximately
maximizes number of "pleasantly surprised" voters

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.

First (simple) argument

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.

Second (a little more complicated) argument (from Mike Ossipoff)

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.