[EM] Bayesian Regret analysis of Bucklin, Top-Two-Runoff, and other methods

[Abd ul-Rahman Lomax]

Various factors that affect real elections have been neglected in the 
simulations which have been done to compare performance of various 
voting systems. The analysis which has been done, so far, is quite 
valuable and represents the best data we have on voting system 
performance, but the neglect of real voting patterns and factors has, 
I suspect, produced warped comparisons of systems.

The technique of simulating underlying absolute preferences has too 
quickly moved into an assumption that preferences can be normalized 
and that all members of the simulated population will actually vote. 
In fact, real voter behavior can be predicted to vary with preference strength.

As an example, if I'm correct, analysis of Bucklin made the 
assumption that all voters would rank all candidates, which is 
actually preposterous
Further, with Top Two Runoff, a assumption has been made that all of 
the original voters will then vote in a runoff, so the simulation, of 
course, simulates a Contingent Vote that accomplishes the same thing 
with a single ballot, unless, of course, voters truncate, and 
truncation hasn't been simulated, to my knowledge.

In fact, voters with low preference will not turn out to vote in a 
special election runoff. If the system is implemented with the 
primary as a special election (as in Cary, NC) and the runoff in the 
general election, then we'll see the effect of low preference 
strength on turnout in the primary instead.

It is a common assumption that low turnout in an election is a Bad 
Thing. However, I've seen little analysis that does anything more 
than make partisan assumptions; allegedly, low turnout favors 
Republican candidates. If so, then the source of the problem would be 
large numbers of voters who might otherwise favor a Democrat, but who 
have, in fact, low absolute preference strength, and Baysian regret 
analysis of the whole population would likely reveal that the 
Republican would be the social utility winner.

Turnout differential shifts any voting system toward social utility 
optimization and away from majority, plurality, or Condorcet 
criteria. Overlooking this has caused voting systems theorists to 
overlook the power of runoff systems, which have been assumed to be 
mere technical methods.

In fact, runoffs have not only this value, but the values noted by 
Robert's Rules of Order for repeated ballot in general. (Robert's 
Rules does not support "runoffs," i.e., elections with 
rigidly-determined candidate eliminations, it requires repeated 
elections, in toto, if there is a failure to find a majority of all 
those voting in the election, accepting the winner.)

Those values are: the ability of the voters to make decisions, 
including compromises, based on the results of the first ballot, and, 
as well, to make more informed decisions based on better knowledge of 
the candidates. We know that real runoff elections, nonpartisan, do 
produce "comeback" elections in about one-third of the cases. This 
phenomenon does not happen with Instant Runoff Voting, so the 
instantaneous preference profiles of voters in the primary are not 
simply replicated in the runoff, or, at least, that is the most 
likely explanation. There is also truncation as an explanation.

Some study of Bucklin has been based on an assumption that 
Later-No-Harm is important to voters. It's clearly important to 
*some* voters, specifically the most partisan. A partisan Democrat is 
not likely to add a second-rank vote for Bush in a Bush v. Gore 
election. But an independent voter *might*. And more to the point, if 
the voter buys the Tweedle-dum and Tweedle-dee argument of a 
candidate like Nader, they might well not cast any second-rank vote 
at all, even if they have some preference for one candidate over another.

We know, however, that in nonpartisan elections, the addition of 
additional ranked candidates was reasonably common in actual Bucklin 
elections. However, there is a basic voting phenomenon which has been 
inadequately considered, even though it was first noted, to my 
knowledge, by Charles Dodgson (Lewis Carroll) in about 1883. Many or 
most voters only have a good idea of their first preference, and, in 
an STV system, may be likely to truncate below that, and bullet 
voting, or at least some level of truncation, makes sense as a 
sincere vote for such a voter.

Out of some idea that "majority" is important, but not understanding 
the *purpose* of seeking a majority, some places and some theorists 
have advocated mandatory full ranking, which is combined, in 
Australia, with mandatory voting. It's illegal not to vote there, and 
if a ballot is cast (except in places which have Optional 
Preferential Voting), and does not fully rank the candidates, the 
ballot is "informal" or discarded.

Majority failure is an essential feature of single-ballot systems, 
all of them, it will occur with some considerable frequency, unless 
the "majority" is in some way coerced, or the options are limited to 
two, which also frustrates democratic purposes.

Hence repeated ballot is ideal, and only deprecated for reasons of 
expense and efficiency. The question then becomes, once we realize 
this, *how much damage is done in the name of efficiency?*

If the damage is trivial, it's not really a problem. But if the 
damage is major, and it can be, then to avoid runoff elections in the 
name of saving money and "trouble," is penny-wise and pound-foolish.

Rather, the question would become how to *avoid* runoffs when 
sufficient data can be collected from voters to make the runoff 
redundant and unnecessary. And there has been far too little study of 
this problem. Robert's Rules suggests preferential voting as a way to 
reduce the need for repeated ballot, and certainly considers it an 
improvement over accepting a plurality result, but apparently does 
not realize that the sequential elimination method that they describe 
is singularly inefficient at the goal of actually finding a majority 
of votes, but in no way are they deluded into thinking that the "last 
round majority" of IRV is a real majority, it obviously is not, and 
Robert's Rules requires that the election be repeated if a real 
majority of votes is not found.

Bucklin is quite a bit more efficient, because it counts all the 
votes. It's been argued that counting all the votes, as Bucklin will 
have done in any situation that is at or is approaching majority 
failure, will cause voters concerned about Later-No-Harm to 
truncation, and that's true, but only to a degree. It depends on the 
preference strength of the voters for their favorite over all 
alternatives. Thus the additional preferences that voters express in 
Bucklin are, in fact, sincere additional approvals, assuming 
reasonably educated voters. When Bucklin finds a majority, it is a 
true majority of voters accepting that result.

Further, selective truncation based on preference strength, quite 
likely, shifts real Bucklin results toward Range results, since low 
preference strength votes are suppressed.

 From these arguments, I suspect that in real social utility 
performance, Bucklin with a majority required, used in a primary, 
with plurality Bucklin reserved for runoffs (where it can be two-rank 
Bucklin), is very close to ideal, but this does, as well, depend on 
details of Bucklin which have sometimes been missed.

As an example of an important detail, some real and notable Bucklin 
implementations allowed multiple voting in the third rank. Thus the 
method was a closer implementation of Approval voting than has been 
realized. One really could vote this kind of Bucklin as an 
antiplurality, "anybody but Joe" method. An obvious improvement on 
the old Bucklin, then, would be to allow equal ranking in all ranks. 
It's not terribly important, overall, that equal ranking be allowed 
in the first two ranks, but it would allow better and more accurate 
expression by voters, as well as providing some safety net for the 
unusual situations where strategic voting in Bucklin could suggest 
preference reversal. Instead of reversal, what can be done is to vote 
equal rank, which is less harmful and more sincere. ("Equal" means 
"to be equally supported under the voter's understanding of election 
conditions." It does not mean that there is no preference, but that 
the preference is small compared to other issues, most particularly a 
strong dislike of some frontrunner. I actually think this condition 
would be so rare that I'm not stressed about the idea that equal 
ranking might continue to be disallowed in the first two ranks, but 
it's important, if Bucklin is to handle large candidate sets, that it 
be allowed in third rank.)

Bucklin with equal ranking thus becomes quite a close approximation 
of Range voting, and could even be implemented using a Range ballot, 
as long as approval cutoff is specified. In combination with runoff 
requirements when there is failure to obtain majority approval, most 
realistic election pathologies can be avoided. Bucklin is alleged to 
violate the Condorcet Criterion, but, in fact, it only does so, I 
consider likely, under conditions where the Condorcet Criterion fails 
to find an optimal winner but merely pretends to, based on an 
assumption that all preferences are equal.