[EM] Bucklin strategy (was: Fwd: [ESF #1118] Another bullet vote argument)

[Jameson Quinn]

I was going to write a message to this list about approval strategy. I ended
up writing it as a response to a message on the ESF list. The generally
relevant part of my post is excerpted below (if you're on the other list,
it's unedited, so dont bother reading it twice).

---------- Mensaje reenviado ----------
De: Jameson Quinn <jameson.quinn at gmail.com>
Fecha: 13 de mayo de 2010 23:43:25 UTC-5
Asunto: Re: [ESF #1118] Another bullet vote argument
Para: electionsciencefoundation at googlegroups.com

...Sure, Bucklin is no harder than approval. The correct strategic vote is
semi-honest, so you have at least a 1/(n-1) chance of getting it right
(where n is number of candidates), and if you know something about others'
preferences, you can do better than that.

But tell me, what is the correct rational strategy for Bucklin? I've been
thinking about this, and it's not a trivial problem. I have most of an
answer, and I doubt that it's what you expect.

First off, consider that for any cabal (perfect information, perfect
coordination) strategy in Bucklin, there's an equally good or better
approval-style cabal strategy. So with perfect information, you should (or
at least, might as well) give all candidates the top or bottom rank. As with
approval, if there is a clear first and second place by the condorcet
criterion, then the condorcet winner will win with their score against the
second place, which will be the only majority by any candidate.

In approval, if you don't have good information and thus aren't sure which
the first- and second-place frontrunners are, you can vote for any candidate
over your mean utility. This maximizes the utility of pairwise races in
which you vote for only one side. If everybody votes that way, candidate
scores will be distributed around 50%, with about half of them over.

So, with rational voters in approval, somewhere between 1 and half of the
candidates will attain a majority; so a random voter can expect to
contribute between 1/2 vote and 1/4 of the number of candidates to a
majority. Ideally, as a Bucklin voter you'd like to lower some votes to
lower rankings, so that exactly 1 of your first round votes counts towards
or against a majority; that's 1/2 of a vote for and 1/2 of a vote against
So, if everyone else is voting approval-style, and nobody has any info about
anybody else, you might lower all votes but 1 to a lower ranking.

But if you're doing that, then everyone else is. And if everyone is lowering
even one vote to a lower rank, then the average number of votes in top rank
is under 50%, and so, if votes are uncorrelated and there are many voters,
there is essentially no chance of any candidates attaining a first-round
majority. So, if you want to maximize your first-round impact, you should
vote all but one of your approved candidates in the first round; there's
essentially no risk that you'll be part of two majorities (and thus regret
voting for your less-preferred of those two). You still lower one vote,
because you can; it's an extra free chance for your opinion to have power.

Now you have exactly one candidate who you're voting below top-rank. The
whole point of lowering this candidate was that you held out a small hope
that a majority would be obtained using your other votes, without coming to
this one. Thus, it's almost certain that you want to lower this candidate as
low as possible, to the bottom approved rank.

Iff you use the Bucklin variant where multiple majorities obtained in round
n are distinguished, not by their round-n totals, but by their round-1
totals, there is some motivation to add extra votes at the lowest rank,
which you wouldn't have voted for in approval. I'm not sure, but I think
that in general you would want to use 1 such "extra" vote.

This, by the way, is why I favor 2-rank Bucklin over 3- or more-rank. There
is asymptotically NO rational strategic reason to use anything but the top,
second to last (lowest approved) and last (unapproved) rank. Sure, real
people might use them, and they can serve a purpose for expressiveness; but
if that comes at the cost of the possibility of regret and the resulting
loss in legitimacy of the system and its results, the price is too high
(since you can obtain expressiveness for free, by just adding a nonbinding
range-style poll to the ballot).

So: if my analysis is correct, Bucklin strategy is almost just Approval
strategy. You vote the better of the two frontrunners as the lowest approved
rank, and any candidates you prefer at the highest rank. If the first-round
votes are to be used to distinguish multiple majorities, you also vote the
next candidate down in the lowest approved rank, unless they're also a
frontrunner.

I do NOT think that this is how most people will vote. I think people are
much more likely to put a single candidate in the top rank, and then all
other candidates down to the top frontrunner in the lowest approved rank.
And the practical difference between this and the "correct" strategy is
minimal. If all voters do this, the result is just about the same in
standard Bucklin, or perhaps slightly better from a societal point of view
in top-rank-tiebreaker Bucklin.

I like Bucklin a lot. It is almost as easy to explain as Approval yet more
expressive, it is probably easier to explain than Condorcet, human nature
comes very close to optimal strategy, it is psychologically easier to vote
than Approval or Range and cognitively easier than Condorcet or Range,
there's no risk of dishonest and thus pathological strategy, and the
systematic advantage that strategic voters have over even the most ignorant
semi-honest voters is minimal (unlike Range). But these reasons have more to
do with human nature than with the abstract characteristics of the system.
As I think the foregoing strategic analysis shows, fully-rational Bucklin
strategy is not obvious, nor is it particularly easy relative to other
voting systems (it requires knowing the frontrunner), nor does it give
results significantly better than Approval.

...




JQ
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