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

[Abd ul-Rahman Lomax]

At 03:58 PM 6/17/2013, Chris Benham wrote:
>Benjamin,
>The criterion ("criteria" is the plural) you suggest is not new. It 
>is called Mono-add-Top, and comes from Douglas Woodall.
>
>It is met by IRV and MinMax(Margins) but is failed by Bucklin. In my 
>opinion IRV is the best of the methods that meet it.
>
>26: A>Y>X
>25: B>Y>X
>17: C>D>X
>17: E>F>X
>17: G>H>X
>
>The majority threshold is 51 and X wins in the third round. But if 
>we add anywhere between 3 and 100
>X>Y ballots then Y wins in the second round.

Some error there. Total votes are 102. Majority is 52 votes. I just 
want it to be noticed how *crazy* this scenario is. Voting systems 
criteria can be like that. A totally insane situation, won't happen 
in a real election in a billion years, can be asserted to cause 
criterion failure. There is no *evaluation*, no consideration of harm 
or effect on social utility, just a raw definition of a criterion and 
an example showing failure. Or, sometimes, a proof that failure is 
not possible (with *any* scenario).

The deeper analysis is much more difficult. What are the conditions 
that allow mono-add-top failure? In the example above, X wins without 
the additional votes because X is the *unanimous* third choice of all 
the voters, while being the first or second choice of none. That's, 
for starters, preposterously unlikely. However, a somewhat more 
realistic version could be constructed.

The ballots that then shift the win to Y cross a minimal majority 
threshold for Y in the second round (With the 3 votes additional, 53 
votes is majority, so this would need to be two votes, not three).

I have generally suggested that in studying criteria performance that 
overall social utility be considered. Because Bucklin, especially 
Bucklin-ER -- which this election could be -- uses a Range ballot, 
that's what makes strategic voting sense -- it is possible to 
estimate social utility performance. In a hybrid system, which is 
what I've been coming to highly recommend as a more sophisticated 
reform, social utility and the Condorcet criterion can be tested, and 
problems, conflicts, can easily be handled with a runoff, and if the 
runoff *is the general election*, then the primary method is merely a 
nomination device. If the primary method never *eliminates* a 
Condorcet winner, then the overall method can fairly be considered 
Condorcet-compliant, with the final application of the criterion 
being *in the general election*. I.e., the Condorcet winner will win, 
*unless* the voters vary and that preference is not maintained. That 
pairwise majority will know the situation from the primary. How will they vote?

Now, X is the *social utility winner* if this is Bucklin. The voting 
pattern does not reflect -- at all -- real voting behavior with 
Bucklin, which we know. Many or even most voters will bullet vote. 
The frontrunners are A and B. A and B voters are unlikely to add 
lower preferences in second rank, in fact, they many not add them at 
all. And all of this reveals problems with the majority criterion and 
the multiple majority criterion.

I have pointed out that Bucklin uses a Range ballot to control voting 
in a series of approval elections. That is, the optimal Bucklin 
ballot will show utilities for candidates, as to those within the 
approved set, those where the voter is at all willing to support the 
candidate, to approve the candidate, and thus cause the election. 
Here, a deeper preference is revealed in the third rank. What does 
this do to a social utility estimate:

     4 3 2
26: A>Y>X
25: B>Y>X
17: C>D>X
17: E>F>X
17: G>H>X
---
102 voters, max score 408

Totals, as percentage of maximum possible (i.e, 4 points per voter 
per candidate)

A: 25.4%
B: 24.5%
C,E,G: 16.7%
Y: 37.5%
D,F,H: 12.5%
X: 50%

X is *obviously* the social utility winner. However, that's a wimpy 
decision, 50% of maximum range. Voting third rank in Bucklin is *bare 
minimum approval,* which is why I interpret it as 50% range.

Now, we add

2: X>Y

The two votes, when the second rank is amalgamated, lead to 53 votes 
for Y, a bare majority, Y wins, in the second round, with only 2 
votes for X at that level. That's because Bucklin collapses to 
approval voting as higher ranks fail to find a majority. A and B are 
still two votes short of a majority. However, the full ballots show a 
different story.

In this case, the two additional votes caused a bare majority to 
appear in the second round, thus concealing the *full approval* for X 
that only comes up in the third round.

Notice that the A,B voters also approve Y, all of them. None of them 
approve each other. These votes, as far as I can tell, make no sense, 
they are preposterously unreal. While I can easily create preference 
profiles that match the votes -- they would be these preferences 
translated into Bucklin votes, which are then Range 4 utilities -- 
the voters are *uncorrelated* with each other, and are behaving as if 
purely and completely isolated. It's as if they don't follow the same 
media, and are, as it were, from different planets. Yet they are 
organized into clear factions, and that high level of organization is 
what allows this criterion failure to appear.

Bucklin amalgamation is not perfect for social utility. Rather, it's 
following a somewhat different principle, the seeking of majority 
approval. I agree that the process here is interesting, and the 
hybrid methods I suggest would detect the SU maximizer. They would 
*not* award the win based on that; rather, the presence of such a 
beats-all winner (X is pairwise approved over all other candidates, 
plus X is the SU winner from the Bucklin ballots interpreted as Range 
ballots), or even simply one that beats the otherwise-winner, would 
trigger a runoff.

Who would the runoff candidates be? First of all, this ballot is the 
truncated Range 4 ballot of Bucklin. I'd suggest a complete Range 
ballot, not truncated. However, dealing with just the example before 
us, I'd suggest:

X, the SU maximizer and Condorcet winner.
Y, the Bucklin winner otherwise.

A and B voters have all settled on Y as second choice, and the SU 
figures show that. That is a weak preference expression, so we can 
assume that the A and B voters will be satisfied with the runoff as a 
choice. Nodoby is likely to be seriously dissatisfied if they voted sincerely.