At 10:12 PM 5/18/2010, [email protected] wrote:
Suppose that you were voting a range ballot and all you knew was
that candidate X was very likely to
win or be tied for first place. Then you should give max support to
each candidate that you prefer over X,
and zero support to the candidates that you like less than X. But
what about X ? How much support
should you give to alternative X ?
If you knew which candidate Y was most likely to be tied with X,
then you would give X full support if you
liked X better than Y, and no support if you liked Y better than
X. But we're assuming that this
information is unavailable.
"Full support" should be translated to "high support." The optimal
ballot uses the probability that an election pair is the relevant
one, i.e., the one in which your vote, if it counts, will make a
difference (covert a tie to a win for your preferred one in this
paid) to determine the expected utility-maximizing vote.
The condition set up here is a very unusual one, it should be
noticed. Win or tied is "very likely"? "Tied" happens with decreasing
frequency as the number of voters increases. So, again, to make this
meaningful, I'll read that as "close enough such that my vote may
make a difference." The "win or be tied" is way too vague.
How this voter could know this without knowing the identity of the
other candidate(s) likely to be tied is beyond me. I'm concluding
that, since any one of the other candidates could be this candidate,
and it is, ab initio, equally likely, the probability of each of
these candidates being the one to be in the running is the 1/2 * 1/N,
where N is the number of such other candidates.
So all we know is that X is very likely to either win or be tied for
first place in the range count. Obviously
if X is your favorite, you should give X top rating, and if X is
your most despised option, you should rate X
at minrange.
Yes.
Suppose that X was halfway in between your favorite and worst,
i.e. you would be
indifferent to having X or a coin flip between Favorite and
Worst. Then it seems natural that you would
give X a rating half way between the max and min range values. This
line of reasoning leads one to
conclude that you should just give X and anyone you like the same as
X your sincere rating.
Given the conditions, yes. But "sincere rating" is undefined!
Midrange can be defined, though, except that the probabilities must
be considered. The "coin flip" should be a choice between the
candidate being rated and the expected outcome of the election. When
one is indifferent to that, this is midrange. The "expected outcome"
of the election is not the midpoint between the favorite and worst,
that is what leads to the idea that Range fails IIA.
You only have one candidate for whom you know an expectation, you
have stated an inclusive outcome of two possibilities: win or tie. A
tie is a 1/2 possibility of a win (if no runoff). But "tie" is a red
herring. There is some probability of "fail," but it is very low.
Since it is not conceivable that there is a certain probability of
failing to win by one vote, but the probability of exactly tieing is
much more, I must consider that a tie is also very unlikely, unless
there are only a handful of voters.
What this reduces to is that it is very unlikely that X will lose,
period. So you might as well vote sincere utilities, and it's not
worth a lot of effort. The basic question is whether or not the win
of X is acceptable, and whether or not you want to struggle as much
as possible against that. If X is, indeed, here, your expected
outcome of the election, so it's pretty simple: unless X is your
favorite (lucky guy!) you should vote X as midrange, and then give
max vote to every other candidate you prefer to X and min vote to
every other candidate you dislike more than X.
There is one other possibility: full rating for every candidate you
prefer to X, zero rating to all others. Or bullet voting for your favorite.
I like Bucklin-ER, fed by a Range ballot. Given a good range ballot
and procedure, the system votes for you in a series of simulated
approval elections, each one with declining approval cutoff. If it
were, say, Range 10, the system would start by looking at the vote
for each candidate at a rating of 10. If a majority, done. Then it
would add in the votes at a rating of 9, etc., until a majority is
found. If no majority is found at or above a rating considered
approval cutoff (I'm recommending midrange, i.e., 5, in this case),
then a runoff is held. The identity of the candidates in the runoff
could be decided with better than simple top two, though with
Bucklin, top two might be pretty good. I've suggesting doing
Condorcet analysis on the full Range ballot; note that this method
would allow 11 candidates to be fully ranked. If there is a Condorcet
winner, that candidate should be in the runoff. If there is a Range
winner (sum of ratings as fractional votes), that candidate should be
in the runoff. And certainly the most approved candidate should be in
the runoff. It's likely that all three of these are the same
candidate, and the question of whether or not to even hold a runoff
is then open, for me. But if they differ, all of them should be in
the runoff. Thus it is possible that a runoff has three candidates,
and I'd use exactly the same ballot for it. But it would be
deterministic. All the voters would know, pretty well, what the
likely situation is.....
With Bucklin, you still have the question of whether or not to
approve X; if not your favorite, X is at your approval cutoff
(factoring for probabilities). Toss a coin, follow a hunch, whatever.
And then approve of all candidates better than X and don't approve of
all candidates worse than X. You can still rank approved candidates,
and with a relatively high-resolution ballot, (Range 10 is probably
quite adequate), simply ranking them is an obvious solution; spread
them equally across the range and it's likely to be reasonable, but
if you have trouble deciding how to rank two candidates, then rank
them equally.
Bucklin still leaves the necessity of deciding approval for a
candidate, but, really, it's only necessary to decide it for the
worst approved candidate; all the rest fall into line. Most any
decent voting strategy involves making some overall classification of
candidates into classes of approximately equal utility, and that can
start with a rank order except where ranking is difficult, in which
case the pair of candidates can be treated as one, ranked and rated
the same. It is not necessary, with Bucklin, to worry about
electability, more than to approve at least one of two or more
frontrunners; and that decision then sets the midrange approval cutoff.
Bucklin is a more sophisticated Approval method that can use Range
data, and I've been claiming that a 3-rank Bucklin ballot, if ER is
allowed, is a Range 4 ballot with ratings 0 and 1 combined, and it's
an obvious step to allow the extra disapproved rating, and then use
it for Range and Condorcet analysis if a runoff is necessary. And
then another obvious step to increase the Range resolution of the ballot.
In fact, a Range 4 ballot could be interpreted as Range 7 by using
the possible "rating overvote" combinations in an obvious and
easy-to-understand way. Overvoting in ratings creates an
interpretation problem; original Bucklin counted a vote for the same
candidate in, say, second and third rank as being a vote in the
higher rank. But it would be equally possible and sensible to
interpret it the other way: third rank. So ... count it as rank 2.5.
Let the voters know this, that if they vote for a candidate in more
than one rank, it will be treated as the average of the ranks voted.
And rank 2.5 is its own Bucklin round. It comes after the second
round and before the third.
Bucklin makes the strategic voting decision easier, in the scenario
given; the only difficult decision, at all, is whether or not to
approve of X. I handle that as an absolute. Do I approve of X? Will I
be pleased to hear that X has been elected? Or will I be displeased?
There is my decision, and if it's hard to anticipate, toss a coin,
that's really where I'm at! If the method uses a range ballot as I've
stated, maybe rate X just below the approval cutoff, so I'd be more
likely to have an opportunity to reconsider in a runoff.
I haven't proven it, by any means, but my strong sense is that
Bucklin, fed by a Range ballot, with a runoff and good rules if there
is no majority, will encourage the casting of sincere Range ballots.
It's possible that straight deterministic Range/Bucklin would do the same.
It's Approval voting with a bot deciding when to toss your vote for a
candidate in, the bot being guided by your ratings of the candidates.
In sum, if your sincere ratings for X, Y, and Z were all equal to
r, then you should rate these alternatives
at level r, all of the alternatives you like better than them with
the top rating, and all of the alternatives
you like less with the bottom rating.
Suppose that when all voters use this strategy it results in X
getting the highest range vote. Then we
could say that X was a stable range winner.
But sometimes it will be the case that no matter which candidate X
this strategy is used on, some other
candidate Y will end up getting the highest range total, i.e. there
is no stable range winner.
It seems to me that we should then seek the alternative that comes
nearest to being a stable range
winner.
How could we measure how close candidate X was to being the stable
range winner?
Perhaps we could take the difference in the voted range totals of Y
and X as the measure of distance
from stability, and elect the candidate X that minimizes this difference.
We can automate this strategy by converting each range ballot into a
pairwise matrix as follows:
The (i,j) element of the matrix is the maxrange value if the ballot
prefers alternative i over alternative j. It
is the minrange value if the ballot prefers alternative j over
alternative i. Otherwise, it is just the common
rating for alternatives i and j.
All of these matrices are summed to a matrix M.
The winner is the alternative j that has the smallest value of max{
M(i,j)-M(j,j) | given i not equal to j}.
It seems to me this winner would have an excellent claim as the
nearest to stable range winner.
Comments?
Forest
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