[Election-Methods] Implausible DH3 argument at Range Voting website

Steve Eppley SEppley at alumni.caltech.edu
Tue Jan 1 13:15:30 PST 2008

Several months ago, a critic of Condorcetian voting methods referred me 
to a webpage at the Range Voting website 
<http://rangevoting.org/DH3.html>. Here's an excerpt:
* * * * *

          What is DH3?

    It is simply this. Suppose there are 3 main rival candidates A, B, &
    C, who all have some good virtues. This happens a lot. (In fact,
    whenever it doesn't happen, the situation is uninteresting – only 2
    real contenders – and we might as well just be using the plurality
    <http://rangevoting.org/Plurality.html> voting system.) Let us
    suppose support is roughly equally divided among those three, say
    31%, 32%, and 37%, although the precise numbers do not matter much.
    Suppose also there are one or more additional "dark horse"
    candidates whom nobody takes seriously as contenders because they
    stink. For simplicity assume there is only one dark horse D, but
    what we are going to say also works (indeed works even more
    powerfully) with more than one.

    Now, what happens? The A-supporters say to themselves: "We are in
    trouble. Polls suggest A is going to lose if we just vote A>B>C>D as
    is our honest opinion. But if we exaggeratedly vote A>D>B>C
    downgrading A's main rivals as far as we can, then maybe A will have
    a chance." The B-supporters say "those rotten A-supporters for sure
    are going to exaggerate and effectively get /twice/ the A-versus-B
    discriminating power as if they were honest. We cannot sit still and
    just take that. We have to fight back by also exaggerating:
    B>D>C>A." And similarly the C-supporters say "we will not just sit
    back and be robbed of our deserved victory by those dishonest
    exaggerating scum. We will also exaggerate: C>D>A>B." (And by the
    way, they are completely right. C would definitely lose to A or B if
    they just sat there.)

        Pictorial example
        <http://rangevoting.org/rangeVborda.html#DH3ex> under Borda;
        careful examination
        <http://rangevoting.org/WinningVotes.html#DH3> in several kinds
        of Condorcet systems. 

    Incidentally, some purists may quibble: why did the A-fan voters
    decide to exaggerate? Well the C-fans felt forced to do so because,
    given that the A- and B-fans already chose to exaggerate, the
    C-voters knew that C could not win without exaggeration. But all
    three kinds of voters do not know what the others are going to do
    and how many of them are going to do it (nor even how many of them
    there /are/), and hence have to /guess/, and their guess is "most of
    those rotters are probably going to exaggerate"! So based on this
    guess, they feel they too must exaggerate to get any chance of
    victory. (And that feeling is always accurate in the sense that, if
    some appropriate fraction of the opposing voters exaggerated, then
    [1] our candidate would be sure to lose to a rival, but [2] by such
    exaggeration we could regain the victory.)

    The result of the new exaggerated votes is: D, the /worst/ candidate
    in the eyes of /all/, wins the election. Guaranteed. As we said,
    this happens with the Borda system and also with every Condorcet

* * * * *

That story of what would go through the voters' minds may be plausible 
for the Borda method. Given a close election using Borda, the outcome 
can be manipulated by a small number of voters downranking the rival 
candidate(s). I think the story about Borda would be more plausible if 
simplified so that each strategizing voter believes only a small number 
of voters are strategizing, since if they expect many voters will raise 
the turkey candidate, they would understand the risk that the strategy 
will backfire by electing the turkey, and would be reluctant to take 
that risk. (Gary Cox may have coined the term "turkey raising" in his 
discussion of Borda in his 1997 book "Making Votes Count.")

The DH3 story is implausible for voting methods that elect within the 
top cycle. Each voter considering raising the turkey candidate over 
rival candidates will know that, unlike with Borda, the outcome won't 
improve if only a small number of voters strategically raise the turkey. 
The outcome can change only if so many voters raise the turkey that the 
turkey reaches the top cycle. Strategically minded voters would 
understand that the strategic vote could only help if it also creates a 
large risk of electing the turkey. I consider it implausible that a 
massive number of voters would take that risk, since they consider the 
turkey to be much worse than the sincere top 3 candidates.

* * *

Some other writing at the Range Voting website may mislead readers into 
believing there is a real-world example of DH3 in a public election 
(involving the Borda method).

    DH3-like phenomena were also immediately observed
    <http://rangevoting.org/rangeVborda.html#kiribati> in the only
    government in the world (Kiribati
    <http://rangevoting.org/FunnyElections.html#kiri91>) that tried
    Borda voting.

Here's an excerpt from another webpage two links away:

    Kiribati was the only country in the world to adopt the Borda Count
    system. It failed immediately as a consequence of massive "strategic
    voting" ...

Only a careful reader who digs into a long, linked academic paper will 
discover that the Kiribati election was not a public election. The 
voters in Kiribati were the members of the national parliament, an elite 
strategically experienced group. So, I ask the authors of the Range 
Voting website to make it clear that the voters were not the general 
public, everywhere in the website that the Kiribati example is mentioned.

* * *

I have another anecdote about turkey-raising with Borda. (Salvador 
Barbera told this story during his course on strategy-proofness at 
Caltech several years ago, and he said it was a true story.) Once upon a 
time at a major university in Europe, the economics department was 
hiring a new colleague. There were 4 applicants: One was a world-class 
macroeconomist, one was a world-class microeconomist. The other two were 
mediocre, one clearly better than the other. When it came time to select 
one by voting using Borda, about half the department preferred the 
macroeconomist, with the microeconomist as their second favorite. The 
rest of the department preferred the microeconomist, with the 
macroeconomist as their second favorite. They each had a pretty good 
understanding of each other's sincere preferences, and thus expected the 
election would be close. You can guess what happened: When they voted, 
each voter raised the two mediocre candidates over his/her second 
favorite, hoping to manipulate the outcome if the Borda count was close. 
As a result, one of the mediocre candidates was voted everyone's second 
choice and had the largest Borda count.

Economists seem to like Borda despite that problem. (Barbera too said 
Borda was his favorite method, but admitted he had not studied voting 
methods much.) I suspect this is because economists deal mainly with 
scenarios in which the set of alternatives is fixed; most social choice 
literature assumes a fixed set of alternatives. Given a set of 
alternatives fixed by nature and assuming the number of strategizing 
voters is insignificant, Borda would be pretty good; the winner would be 
a reasonable compromise. In public elections, the assumption about 
sincere voting may be reasonable, but the assumption of a fixed set of 
alternatives is clearly not, and it would be too easy to manipulate 
Borda by nominating inferior clone alternatives.

--Steve Eppley

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