[EM] Participation Criteria and Bucklin - perhaps they *can* work together after all?

[Abd ul-Rahman Lomax]

At 03:41 PM 6/17/2013, Jameson Quinn wrote:
>2013/6/17 Abd ul-Rahman Lomax 
><<mailto:abd at lomaxdesign.com>abd at lomaxdesign.com>
>At 01:23 PM 6/17/2013, Jameson Quinn wrote:
>
>2013/6/17 Benjamin Grant <<mailto:benn at 4efix.com>benn at 4efix.com>
>
>Is *this* an example of Bucklin failing Participation?
>
>5: A>B>C
>
>4: B>C>A
>
>A wins
>
>Right
>
>But add these in:
>
>2: C>A>B
>
>  B wins.
>
>
>Yes, with your "tiebreaker".
>
>
>This is not participation failure. Adding ballots ranking C highest 
>did not cause C to lose.
>
>
>Abd, you're wrong. Adding B>A ballots caused A to lose; that is a 
>participation failure.

I did err in my analysis. However, I would urge anyone tempted to 
write "you are wrong" to be very careful. It's a big red flag that 
one is, oneself, making some mistake. Would we not expect the 
addition of B>A ballots to have the possibility of causing A to lose?

No, the added ballots were A>B ballots, at a lower ranking. The 
failure comes from causing majority failure, thus pulling in deeper votes.

I had somehow failed to notice the B votes from the 5 voters. A wins 
in the first round without the added votes, with a simple majority. 
This is the  sequence, which Jameson did not explore specifically, 
merely stating his result.

>>5: A>B>C
>>
>>4: B>C>A
>>
>>A wins
>>
>>Right
>>
>>But add these in:
>>
>>2: C>A>B
>>
>>  B wins.


A wins in the 9-voter case, by a simple majority. However, the 
11-voter case has a new majority requirement, 6 votes instead of 5.

We have the situation here that a majority favoring A votes second 
rank for B. In Range equivalents, often proposed as examples of "bad 
Range behavior," we see the same kind of phenomenon asserted.

Under straight Bucklin, if there are two frontrunners, A and B, we 
would expect *very few* additional votes for B, because the election 
is very likely to reduce, then, to A vs B. So this is like examples 
of alleged majority criterion failure based on the majority 
suppressing its preference by voting for another candidate, who gets 
a greater majority once they do that.

First round: (majority is 6)
A:5, B:4, C:2

Second round:
A: 7, B:9, C:6

*All three candidates have a majority?* Who is the ideal winner of 
this election? The A voters elected to vote A>B rather than A>.>B or 
just A. The only reason B can win is because they set it up. By 
second-ranking B, which indicates weak preference, they gave the 
election to B. The C voters merely opened that box.

The back-up Bucklin system under discussion would still award the 
victory to A, because there was more than one majority, so the result 
would back up to the first rank and A would still win. Is that a 
desired result, or otherwise?

What do we see here WRT utilities? First of all, I don't know what is 
really meant by the .>.>X preferences? Are there more than three 
candidates in this election? Are those actually approvals? If so, 
every voter ultimately approved every candidate. Jameson and another 
seemed to assume Bucklin votes from a preference order, which is naive.

That's why I suggested that these might have been written the way 
below, if they were *not* approvals, i.e, if the third rank shown was 
the *worst* rating. Yeah, in analyzing ranked voting systems, this is 
common practice, to give the complete rankings. But Bucklin is *not* 
a simple ranked system, it uses an Range ballot, in the traditional 
form, with the range only covering the approved categories. You 
cannot translate sincere preferences to Bucklin votes, especially 
Bucklin-ER. There are *families* of sincere votes.

So Bucklin votes, perhaps:

5: A>B
4: B>C

If this is 3-rank Bucklin (standard), then the voters also had X>.>Y 
possibilities, or bullet votes. If they second-rank a candidate, that 
indicates *weak preference*. Bucklin analysis here only looks at the 
first rank, because a majority is found there.

Range analysis gives me this:

9-voter election
5: A,4; B,3
4: B,4; C,3

So full range analysis: A,20; B,31; C,12.

So the A win is actually weak, mere majority criterion compliant, an 
example of the failure of the majority criterion! This election is 
then *vulnerable* to more voter participation, and that is what happens.

5: A>B
4: B>C
2: C>A

This vote expresses, by default, A>B. However, the *primary 
expression* is a vote for C. The A vote is a subsidiary preference. 
Are these weak or strong preferences (Bucklin allows four levels of 
preference strength, i.e., a Range increment of 4, 3, 2, or 1. The 
voters here all elected to express weak preference with the top two 
and strong preference with the third candidate.)

5: A,4; B,3
4: B,4; C,3
2: C,4; A,3

A,26; B, 31; C, 20

B is *still the utility optimizer.*

Yes, there is technical participation failure. 2 votes that did 
express, as a lower preference, A>B, did cause A to lose to B.

Will the C voters be upset? On the face, yes. They got C, the worst 
candidate, because they voted. However, what they really did was to 
trigger a deeper consideration, that revealed that B was the more 
widely preferred candidate. This would have been a case where voting 
A=C would have made much more sense.

*Multiple majorities* were not common with Bucklin. It's a problem 
where there is an abrupt introduction of many additional approvals. 
In the system above, we get a *triple majority.* That is a sign of 
weak preferences, as expressed. It could be naive voters, note that 
the *majority of voters in the 9-candidate election* first or 
second-rank the true utility winner. Bucklin simply doesn't consider 
that, because of the prior majority. I've argued that, in a two-round 
Bucklin system, it should.

I have pointed out that multiple majorities indicate that the 
majority *may not have made a choice,* and I've suggested that these 
go to a runoff. The backup method, where a multiple majority backs up 
to the previous rank and goes for the plurality winner there (among 
the majority-approved set), seems to solve the problem. The C winners 
will be happy with that, A will win.

*However,* social utility took a whack! Thus, I would suggest, a 
runoff election between A and B would resolve the issue, clearly. 
Contrary to what may voting systems analysts have expected, A will 
not necessarily win. The C voters, in particular, may have less 
motivation to vote. The A voters may decide that they prefer to find 
closer to unanimity, they have weak preference, and that's what real 
people do in small-group social dynamics. The majority has a 
*choice,*  now narrowed and uncomplicated by the presence of C.

The C voters may be unhappy, but the B voters are pleased, and there 
are more of them. As to maximimum displeasure with the result, if A 
wins, we have 4 seriously unhappy, and if B wins, we have 2. This 
assumes sincere votes, under the Bucklin system, which *only allows approvals.*

In that system, if the A voters had voted A>.>B, say, or A only, A 
could win both ways. Those voters elected to vote as they did. (Of 
course, this was a made-up scenario, but in making it up, were the 
options open to the A voters considered? As well, what about the C 
voters? If the system allows them to vote A=C, why didn't they vote 
that way? They'd have gotten A, no problem.)

Once we realize the possibility of two-round elections, with the 
primary being exactly that, a primary election to choose general 
election candidates, these issues resolve. The general election will 
have wider participation. It can and should use an advanced voting 
system. Certainly it should *allow* multiple approvals. But Bucklin 
still allows favorite expression.

I *do* want to allow voters to vote sincere preferences, like the C 
voters. The problem only arises because of multiple majorities, and 
the presence of a true utility optimizer, apparently, with the C 
voters causing majority failure in the first rank and thus a deeper look.