At 10:38 PM 5/31/2012, Michael Ossipoff wrote:
As I was saying in a recent previous post about this, Approval's count
results will tell Green-preferrers whether or not they need Dem to protect
against Repub.
And I gave a reason why that is: Preferrers of the middle of 3 parties have
no reason to approve either extreme. I told of a reason why that is.
Perhaps you did, but you are framing this as a general truth, when
your arguments, here at least, seem to be based on a very particular
assumption about the two major parties, not the general case at all,
nor do I think that most readers of this list will agree with you.
Further, improved voting systems, of the kind that are most-discussed
here, generally lead to increases in the number of candidates.
Now I'd like to tell of another:
On EM, it's been shown by at least three people, in at least two ways, that
the expectation-maximizing strategy of Approval is to approve the
above-expectation candidates.
I think that's so.
It's obvious why that's so: Your expectation is the sum, over all of the
candidates, of the product of a candidate's win-probability and hir utility.
It's obvious that when you increase the win-probability of a candidate who
is better than your expectation (you do that when you approve hir), that
will raise your expectation.
Seems sound.
Well, suppose you're a Democrat-preferrer (if there really are any). Say
it's Green, Dem, Repub. If it's certain that some particular candidate will
win, then your expectation is the utility of that candidate. Otherwise your
expectation is somewhere between the utility of the Green and the Dem, or
somewhere between the utility of the Repub and the Dem.
Say it's somewhere between the Green and the Dem. As I said above, your best
expectation-maximizing strategy is to approve (only) all of the
above-expectation candidates. By assumption, the Green is farther from you
than is the point representing the utility equal to your expectation. So you
don't approve the Green.
What would it take to make your expectation worse than the Green? No, even
if it were almost certain that the Repub would win, that wouldn't do it,
because Dem and Repub are so close that you couldn't squeeze an amoeba
between them.
This is in your view, Mike, not necessarily the view of the voters. I
find it bizarre and disappointing that you would base a general
argument about three-party elections on something so flimsy. I expect
more of you.
Someone who considers the two major parties to be Tweedle-Dum and
Tweedle-Dee, is almost by definition out on the left or right edge.
The major parties do tend, frequently, to nominate toward the center,
hoping to attract the real middle. Indeed, that's the force behind
Center Squeeze. The *parties* may have great differences, but the
candidates are pushed toward the center, squeezing out a true centrist.
The expected utility for you would have to be a candidate
farther away from you than the Green and the Repub. And that would be
impossible with just 3 candidates.
Let's see how this works without the assumption of almost-identity
for the voter of the two major parties. Let's assume a left-right
scale of 5 (left) to -5 (right). Say the Green is 5, the Democrat is
1, and the Republican is -5. Just picking numbers. Suppose a voter is
located at 2.5. The voter prefers the Democrat (regret 1.5), then the
Green (regret 2.5), and the regret for the Republican is 7.5.
Normalized to a scale of 0-6, this is 0 Dem, 1 Green, 6 Repub, and
inverting to show utilities, it ends up 0 Repub, 5 Green, 6 Dem.
(I'd have preferred to represent this with four parties, including a
rightist party, more to the right than the Republicans, but I'm
sticking with Mike's three-party situation.)
Suppose the expectation is that the election is close between the
Democrat and the Republican, and the Green is unlikely to win. The
expected utility is then 3, so the above-expectation strategy is to
approve both the Democrat and the Green.
Further, our voter is in the left wing of the Democratic party. The
voter may wish to encourage the Democrats to move to the left, and
while the voter, by definition, prefers the Democrat to the Green,
the loss of utility if the Green wins, because of so many voters
pushing in that direction by approving the Green, is small. But the
Republican winning is a disaster to this voter.
What we don't know, really, is how many supporters of a center party
will actually approve the closer of the left or right party
candidates. There are indications from Bucklin history that there
will be quite a bit of such crossing of party lines.
In reality, there are more than three parties in U.S. Presidential
elections, and voters vote as they vote for complex reasons.
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