[EM] Truncation dilemma

[Jameson Quinn]

The basic truncation dilemma is a familiar one on this list. There are two
near-clones A and B who split a majority of the vote - say, with 35% and 25%
honest-first-choice support - and one distinct candidate C who has a
plurality of first-choice support - say, 40%. Since C voters don't care much
which of A or B is elected, they are willing to truncate, even if A and B
are the frontrunners. So in many voting systems, including both
Approval-like and Condorcet-like systems, A and B voters are faced with a
collective prisoners-dillemma like situation; they can cooperate and perhaps
help elect their second preference, or truncate, giving the most possible
relative support to their favorite but risking the worst-case winner.

Note that I am speaking of a true truncation dilemma, where A and B voters
honestly see little a priori utility difference between the two (though the
strategic need to truncate may lead to demonization and/or bad feelings over
the course of the campaign). If they actually had strong preferences,
truncation might simply be more honest than strategic. So, in all that
follows, I'm assuming that A is the honest Condorcet, Range, and social
utility winner, by clear margins.

This is a true dilemma; it's no more an artifact of the voting system than
an honest Condorcet cycle. And I believe that it would be much more common
than such a cycle, to the point where the majority of apparent Condorcet
cycles would actually be caused by truncation/burial strategy.

So, what are the possible responses, from the point of view of voting system
design? There really are only a few.

1. Embrace the dilemma. Either A and B voters manage to cooperate, or the
system elects C; deal with it. This approach is perhaps best exemplified by
Approval.

1a. Probabilistic dilemma. If truncation causes a cycle, then there is some
probabilistic tiebreaker which always includes some chance of C winning.
This can act as a goad to A and B voters to cooperate. I suspect that some
system like this might be the theoretical optimum response if voters were
pure rational agents; however, real people tend not to like probabilistic
election systems.

2. Obfuscation; that is, hope that the voters don't really notice. I'd say
that the best example of this is Bucklin. One hopes that the voters are
satisfied by expressing a strong preference for their favorite, and they
don't notice the strategic dilemma in adding lower preferences. This is not
a vain hope. Between strategically naive voters and principled honest
voters, there may well be enough to ensure A is elected. However, it's still
obfuscation.

3. Elimination. IRV is the preeminent example of this response. If B is to
be eliminated first, then there's no strategic reason for B voters to
truncate. However, this can lead to other problems with the voting system -
IRV's nonmonotonicity and center squeeze are directly related to this issue.
Also, if it isn't clear which of A or B is the frontrunner, elimination
might not help, because the best strategy is to loudly pronounce that your
faction will truncate, and perhaps too many people will carry through with
the threat.

3a. Quasi-elimination. I believe that winning-votes Condorcet methods, like
Schulze, are an attempt to ensure that A wins even in the face of B's
truncation. However, this only works if C voters truncate rather than
splitting evenly between CBA and CAB. Other stronger quasi-elimination
systems that I know of have IRV-like problems.

4. Runoff. Viewed in an outcome-oriented game-theoretic vacuum, this is just
the same as elimination, and it suffers the same problems. However, if
voters have some negative utility for the runoff itself, then a system can
use the threat of one to motivate honest voting in the first round. Since
the scenario assumes that there is a clear winner with no cycles under
honest voting, that may be enough.

I think that's it. Does anyone have any other possible responses?

To me it's clear that option 4 is the best. Like options 1 and 1a, it's
using the threat of something voters don't want to motivate honest voting.
However, a runoff is a less extreme threat than N years of bad leadership,
and so much more palatable if it actually comes to pass.

This has clear implications for system design. If the main purpose of a
runoff possibility is to motivate honest voting and thus never actually have
a runoff, it's important to be as decisive as possible in the first round.
In general, I think that at a minimum, if there's a first-round Condorcet
winner evident from the ballots, there should be no need for a second round.

This analysis suggests that, in response to this dilemma, two-round
Condorcet systems deserve a closer look. I'd also suggest simpler systems
which make a good approximation of that: for instance, 2-approved-rank
Range, with a runoff if the winner's approval score doesn't beat all other
Range scores.

I think that two-round Condorcet systems have been neglected because the
Condorcet matrix offers a seductive plethora of tiebreaking possibilities.

Jameson Quinn
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