[EM] "good method" ? , was "IRV ballot pile count (proof of closed form)"

[Abd ul-Rahman Lomax]

At 08:22 PM 2/12/2010, Chris Benham wrote:
>Rob LeGrand wrote (11 Feb 2010):
>><snip>
>>
>>35:A
>>32:B>C
>>33:C,
>>
>>by which I mean
>>
>>35:A>B=C
>>32:B>C>A
>>33:C>A=B.
>>
>>In this example, C is the Condorcet winner even though C does not have a
>>majority over B.  I can see how this example could be seen as an
>>embarrassment to the Condorcet criterion, in that a good method might not
>>choose C as the winner.
>
>Well I can't. Electing A would be a violation of the Minmal Defense criterion,
>and electing B would violate Woodall's Plurality criterion and 
>Condorcet Loser.
>
>What "good method" do you have in mind that might not elect C?
>
>And what's good about it?

Standard deliberative process, used for elections by every 
deliberative body I'm aware of, including those in Australia....

What they would normally do is allow only vote for one, which with 
the profiles shown, would result in

35:A
32:B
33:C

which is majority failure. And then they would continue balloting, 
same rules, until one candidate has a majority.

For efficiency, they could use a ranked method. Bucklin, in fact, is 
the most efficient, but IRV might seem to work. Let's use IRV on this:

35:A
32:B>C
33:C

B is eliminated, leaving C the winner, with a majority, 65:35. It is 
obvious that this method allows truncation. So if the B voters were 
really approving C by voting for C, then the result of C is 
legitimate. But if they were not, then it is not legitimate. In the 
use of such a method for election under parliamentary procedure, it 
would be proper for the election to be ratified by vote as a yes or 
no on the question, "Shall C be elected." This might merely be a vote 
on the question of accepting the report of the clerk or other 
election official.

The answer to Chris's question is repeated ballot until a majority is 
found. With such a process, and if a preferential ballot is used, 
voters may have an incentive to truncate, because the later-no-harm 
property of IRV, as an example, fails if a true majority is required. 
We can see in this example that the B voters, by ranking a second 
choice for C, have elected C, whereas if they did not, balloting 
would have been repeated, and B might have won. Indeed, there is very 
little difference between B and C in first preference, nor between 
any of these candidates in that respect. There might even have been a 
Condorcet cycle:

35:A>B
32:B>C
33:C>A

What's the best result? To my mind, it's impossible to tell from the 
preferences without knowing preference strengths. It would seem, 
however, from the voting pattern that *probably* we'd have something like:

35:A>>B>A
32:B>C>>A
33:C>>A>B

In a range approximation, using Range 3:

35: A, 3 / B, 1 / C, 0
32: B, 3 / C, 2 / A, 0
33: C, 3 / A, 1 / B, 0

Range totals:

A: 138
B: 131
C: 163

C is the utility optimizer, by a decent margin. But this is based on 
a speculation about preference strengths. This election is really so 
close, as far as what can be seen from preferential ballot, that the 
electorate does not appear to me to be necessarily ready to make a 
decision, unless we can truly take the B>C votes as approval of both 
candidates. C's a good choice, though, from the votes, and it might 
be enough to submit the matter to ratification under the rules. 
That's one reason I like asset. In this case, though, Asset would not 
have made a difference, because the B voters explicitly provided a 
second choice, thus resulting in a true majority of votes for C.

That they violated LNH doesn't matter, does it?

LNH is incompatible with a majority requirement. I think that should 
be explicitly understood and acknowledged. (Coercing votes, as in 
some Australian elections where Optional Preferential Voting is not 
used, does not generate true majority results, it creates a majority 
of the votes, all right, as a mathematical certainty, but by tossing 
any votes that result from voters refusing to vote for a candidate 
when, perhaps, they detest the candidate. All that has to happen is 
that they detest more than one....)