[EM] Re: The Allure of IRV

[Richard Moore]

Dave Ketchum wrote:
>> So, let's say you have an election in which two sets of ballots are taken
>> simultaneously: one ranked set, and one approval set. The ranked ballots
>> are counted to find the CW, and the approval ballots are counted to find
>> the AW. If CW and AW are the same, then neither method has bested the
>> other. If CW and AW are different, then...
>>
>> 1) there is a majority N1 of voters who prefer CW to AW. Some of these
>> will have approved of CW and disapproved of AW; call this number N1A.
>>
>> 2) there is a minority N2 of voters who prefer AW to CW. Call the number
>> of voters within this group that approved of AW and disapproved of CW
>> N2A.
> 
> 
> 
> DWK:  This even sails over my head.  If I read it right, it says CW 
> should win a runoff between AW and CW, suggesting that looking for CW is 
> preferable to looking for AW.  So, what are we arguing over?

No argument as to who would win the runoff. But you ask for more expressive
power. You only responded to the first half of this argument, and ignored
the part that has to do with expressive power.

>> For simplicity I'll assume nobody voted any tied preferences between
>> AW and CW.
>>
>> Since AW is the approval winner, we know that N2A > N1A. We also know
>> that N1 > N2, meaning that (N2A/N2) > (N1A/N1). That is, the voters
>> who prefer CW to AW are less likely to express that preference on an
>> approval ballot, than the voters who prefer AW to CW are to express
>> that preference on an approval ballot. In other words, we have a good
>> reason to believe that the median strength of the CW>AW preferences
>> is less than the median strength of the AW>CW preferences.
>>
>> I don't claim that the Approval winner is always more strongly preferred
>> than the Condorcet winner when the two differ, but I think I've shown
>> that such will be the case more often than not. Hence, I am more willing
>> to trust Approval than Condorcet.

See, Condorcet did not provide as much expressive power, if we think of
that expressive power as a property of the whole electorate rather than
something granted to individual voters. Only relative preferences of
individual voters were expressed. But when it came to the Approval ballots,
the voters were forced to choose, and in choosing they revealed something
about the strengths of their preferences. The ones who felt most strongly
about their preference between the two candidates, and hence showed it in
their Approval ballots, were those who preferred the AW to the CW. Most
of those who preferred the CW chose to hide that preference on their
Approval ballot, so it would be hard to convince me that they hold as
strong a preference between these two candidates as the AW supporters
do.

I suppose this argument is meaningful only if you believe that there is
more to expression than the ordinal rankings by individual voters. When
the subject of low-utility Condorcet winners came up here several months
ago, a number of examples were posted that yielded such CWs (marginally
favored over each other candidate by a slim majority but detested by
everyone else), but not everyone on the list agreed that such CWs were a
bad thing, as I recall.

>> "...as far as practical" -- indicating that there are practical limits
>> to allowing individual expression. I think Approval does at least as
>> well as Condorcet at aggregating individual choices. If I have a weak
>> preference for A>B, and a strong preference for B>C, then voting a
>> ranked ballot of A>B>C adds more noise than voting an Approval ballot
>> of AB.
> 
> 
> 
> BOTH ballots showed least preference for C.  The Approval ballot did not 
> let me indicate a preference for A>B, no matter how strong it might be, 
> without giving up my right to show C as less preferred than either.

Again from the individual point of view I might feel I had more power
of expression by ranking all my choices. But -- from a systems point of
view -- I'm putting my preferences through a non-linear mapping when I
do so, and the result is distortion. Once distorted, a signal cannot be
recovered accurately. By comparison, with Approval I am simply quantizing
my vote, and the result is noise, but it is "white" noise that gets
averaged out when many votes are aggregated. An analogy I used once, from
the world of electrical engineering, is that of a delta-sigma converter
(or one-bit D/A converter), such as those often used in digital audio to
avoid the non-linearities of multi-bit converters.

I don't dislike Condorcet; in fact, I like it, and it has some nice
properties. There may well be applications where it is better than
Approval. But if the goal is to measure the aggregate preferences of a
large population of voters, I would have to say Approval is the best
tool for the job.

  -- Richard

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