Just to be explicit about the application of this to equal ranking.
At 10:28 AM 2/15/2007, Chris Benham wrote:
>Pasting from Mike's page:
>>
>>Some definitions useful in subsequent criteria definitions:
>>
>>A voter votes X over Y if he votes in a way such that if we count
>>only his ballot, with all the candidates but X & Y deleted from it,
>>X wins. [end of definition]
Equal ranking of X and Y is clearly not voting X over Y. If we modify
the ballot as stated and this is the only voter, it's a tie.
>>Voting a preference for X over Y means voting X over Y. If a voter
>>prefers X to Y, and votes X over Y, then he's voting a sincere
>>preference. If he prefers X to Y and votes Y over X, he's
>>falsifying a preference.
Equal voting is neither voting a preference nor falsifying a
preference. It is not expressing a preference.
>>A voter votes sincerely if he doesn't falsify a preference, and
>>doesn't fail to vote a sincere preference that the balloting rules
>>in use would have allowed him to vote in addition to the
>>preferences that he actually did vote.
I find the application unclear. What is undefined is what it means
for an election method to "allow" the expression of a preference.
Plurality allows the expression of a preference. Unfortunately, it
only allows the express of a preference for one candidate over all
others. Approval allows the expression of a preference for a set of
candidates over all others.
If the ballot allows complete ranking and allows equal ranking, then
any use of equal ranking where a preference actually exists, no
matter how small, would be considered a failure to "vote sincerely."
While one can define terms any way one likes, it would seem
inadvisable to define them in a way which flies in the face of ordinary usage.
But what if a ballot does not allow complete ranking, or does not
have enough rating levels to accomodate all candidates?
It appears that the interpretation being used is that these methods
don't satisfy SFC, but this would be because they don't satisfy the
criterion even without "falsification."
>>
>>SFC:
>>
>>
>>
>>
>>If no one falsifies a preference, and there's a CW, and a majority
>>of all the voters prefer the CW to candidate Y, and vote sincerely,
>>then Y shouldn't win.
>>
>>[end of definition]
Now, this has been about the definition of the criterion. Even if
equal ranking in the presence of a sincere preference is not
falsification, Approval, for example, fails SFC. Yet, I've argued, it
fails SFC because it does better. It is clearer with Range:
If no one falsifies a preference in Range of sufficiently high
resolution, and all preferences are expressed, that is, equal rating
is only used for absolute equality of rating, then Range still fails
SFC, and it is easy to construct scenarios where it does so by
choosing a winner who is clearly "better" for society and for the
members of society individually, than the Condorcet winner.
This is because of preference strength. If the CW is preferred, by a
majority to a candidate A by a majority with a very small preference,
such that, for practical purposes, these voters will be equally happy
with the election of the CW or A, and a minority of voters strongly
prefer A, such that they will be happy with A and seriously unhappy
with the CW, it is quite clear that A should win. A makes *everyone*
happy, the CW in this situation only makes a bare majority happy.
Thus, we conclude, the Condorcet Criterion *must* be violated in some
elections by an optimal method, and thus this theoretical optimum
method must fail the criterion and others similar to it, such as the
Majority Criterion and SFC.
Of course, we need a definition of "optimal." I've been suggesting
that it should be explicit. Too often, when we consider methods by
election criteria, we assume that a criterion is desirable, entirely
apart from whether or not it chooses the optimum winner. It's
*assumed*, very easily, that the majority choice is the optimum
winner -- and therefore it is desirable to satisfy the Majority
Criterion -- when this is certainly not clear enough to be reasonably
an axiom. Any person or business which makes decisions failing to
consider the strength of preferences will soon run into trouble
Perhaps I should be more explicit about this. In considering a
decision among many choices, I may consider the effect of each choice
on various aspects of my life. With each aspect, I may have a
preference among the choices. If we model the importance of an aspect
by a number of voters voting according to that, then systems which
only rank but do not consider preference strength can seriously fail
to make an optimum decision. The additional necessary element is to
incorporate preference strength.
Decision-making strategies often use this, quite explicitly. One will
give weight to various aspects of a decision, and for each aspect a
numerical rating can be used. Then, for each
