[EM] Condorcet/Range DSV

[Chris Benham]

Jameson,

This Condorcet-Range hybrid you suggest seems to me to inherit a couple of
the problems with Range Voting. 

It fails the Minimal Defense criterion.

49: A100,  B0,  C0
24: B100,  A0,  C0
27: C100,  B80, A0

More than half the voters vote A not above equal-bottom and below B, and yet
A wins.

Also I don't like the fact that the result can be affected just by varying the resolution
of  ratings ballots used, an arbitrary feature.

I think it would be better if the method derived approval from the ballots, approving all
candidates the voter rates above the voter's average rating of  the Smith set members.


"For strategies which don't change the content
of the Smith set, it does very well on other criteria, fulfilling
Participation, Consistency, and "Local IIA". "


The criteria you mention only apply (as a strict pass/fail test) to voting methods, not 
"strategies" (and  have nothing to do with strategy).

We know that Condorcet is incompatible with Participation  (and so I suppose also with
the similar Consistency).  I don't see how a method that fails Condorcet Loser can meet
Local IIA.

"And because it uses Range ballots as an input but encourages
more honest voting than Range,.."

That is more true of the "automated approval" version I suggested, and also it isn't
completely clear-cut because Range meets Favourite Betrayal which is incompatible
with Condorcet.

 
Chris Benham


Jameson Quinn wrote (25 June 2009) wrote:

________________________________
 
I believe that using Range ballots, renormalized on the Smith set as a
Condorcet tiebreaker, is a very good system by many criteria. I'm of course
not<http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-January/014469.html>the
first one to propose this method, but I'd like to justify and analyze
it further.

I call the system Condorcet/Range DSV because it can be conceived as a kind of
Declared Strategy Voting system, which rationally strategizes voters' ballots for them assuming that
they have correct but not-quite-complete information about all other voters.
Let me explain.

I have been looking into fully-rational DSV methods using Range ballots both
as input and as the underlying method in which strategies play out. It turns
out to be impossible, as far as I can tell, to get a stable, deterministic,
rational result from strategy when there is no Condorcet winner. (Assume
there's a stable result, A. Since A is not a cond. winner, there is some B
which beats A by a majority. If all B>A voters bullet vote for B then B is a
Condorcet winner, and so wins. Thus there exists an offensive strategy. This
proof is not fully general because it neglects defensive strategies, but in
practice trying to work out a coherent, stable DSV which includes defensive
strategies seems impossible to me.) Note that, on the other hand, there MUST
exist a stable probabilistic result, that is, a Nash equilibrium.

Let's take the case of a 3-candidate Smith set to start with. (This
simplifies things drastically and I've never seen a real-world example of a
larger set.) In the Nash equilibrium, all three candidates have a nonzero
probability of winning (or at least, are within one vote of having such a
probability). Voters are dissuaded from using offensive strategy by the real
probability that it would backfire and result in a worse candidate winning.
This Nash equilibrium is in some sense the "best" result, in that all voters
have equal power and no voter can strategically alter it. However, it is
both complicated-to-compute and unnecessarily probabilistic. Forest Simmons has
proposed an interesting
method<http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011028.html>for
artificially reducing the win probability of the less-likely
candidates,
but this method increases computational complexity without being able to
reach a single, fully stable result. (Simmons proposed simply selecting the
most-probable candidate, which is probably the best answer, but it does
invalidate the whole strategic motivation).
There's an easier way. Simply assume that any given voter has only
near-perfect information, not perfect information. That is, each voter knows
exactly which candidates are in the Smith set, but makes an ideosyncratic
(random) evaluation of the probability of each of those candidates winning.
That voter's ideal strategic ballot is an approval style ballot in which all
candidates above their expected value are rated at the top and all
candidates below at the bottom. However, averaging over the different
ballots they'd give for different subjective win probabilities, you get
something very much like a range ballot renormalized so that there is at
least one Smith set candidate at top and bottom. (It's not exactly that, the
math is more complex, especially when the Smith set is bigger than 3; but
it's a good enough approximation and much simpler than the exact answer).

Let's look at a few scenarios to see how this plays out.

First, the typical minimal Condorcet scenario. You have Va voters who say
A>B>C, and think on average that B is b% as good as A compared to C (that
is, if they rank them 70, 60, 20, then b=(60-20)/(70-20)=80%); Vb voters who
say B>C>A with C on average c% as good as B; and Vc voters who vote C>A>B
with A at a%. Without loss of generality, Va > Vb or Vc, but Va < Vb + Vc
(or A would be a Condorcet winner). The renormalized Range tiebreaking
scores are A=Va + a*Vc; B=Vb + b*Va; and C=Vc + c*Vb. What does that mean?
* If all of a, b, and c are 50%, then the candidate with the most
exceptionally strong win or weak loss, wins (that is, if the two strongest
wins are farther apart than the two weakest losses, then the strongest win,
otherwise the weakest loss).
* If one of a, b, and c is near 100% where the others are near 0%, then that
candidate wins. You could say that the XYZ voters' opinion of Y is acting as
the tiebreaker for Y.
* In general, for honest ballots, the winner is the candidate with the least
renormalized Bayesian regret. Assuming the effects of renormalization are
random, this will tend to be the candidate with the least Bayesian regret
overall.

Note that this could elect a Condorcet loser. For instance, if you had
ballots A>B>D>>>C, B>C>D>>>A, and C>A>D>>>B, (that is, each ballot rates
candidates at 100, 99, 98, and 0) then D is the Condorcet loser but has
higher utility than any other candidate, and wins. But this could only
happen if there is a Condorcet tie for winner. In general, I find the
scenario pretty implausible, and the result still optimal for that scenario.
(Because I think such results are optimal, I advocate using the Smith set
and not the Schwarz set for renormalizing).

How does this method do on other criteria? It fulfills Condorcet (by
definition) and is monotonic. For strategies which don't change the content
of the Smith set, it does very well on other criteria, fulfilling
Participation, Consistency, and "Local IIA". However, as the content of the
Smith set changes, it can fail all of those latter criteria - but only by
moving *towards* the renormalized utility winner, who is arguably the
correct winner anyway. I believe that, because of its construction, it will
have relatively low Bayesian Regret among Condorcet systems.

How resistant is it to strategy? When the Smith set is unchanged, all useful
strategies are at worst asymptotically on the honest side of semi-honest -
that is, they only require ranking equal (or, for nearly all of the
strategic benefit, nearly equal) candidates who are not honestly equal.
Moreover, I think that the DSV construction of this system gives it
excellent resistance to real-world strategies. Unless you have more
information than the agent which strategizes your ballot, you simply vote
honestly and allow the system to strategize for you. Thus, you wouldn't be
motivated to use strategy unless you felt you knew not only the exact
possibilities and chirality of the Smith set, both before and after your
strategizing, but also the probable winner from that Smith set. Under
realistic polling information, I think that such scenarios will be rare; if
you know of a possible Condorcet tie, then you will not generally know much
about the likely winners of this tie.

(It may even be provable that if you assume voters exist in some kind of
continuous, unimodal distribution in ideology space, and motivate Condorcet
ties by having non-euclidean but continuous distance measures, then there
will always naturally exist enough "honest defensive votes" to make any
strategy backfire).

Note that in regard to resisting simple strategies, this method is a serious
improvement over either Range or Approval. Because it is a kind of DSV, it
"does the strategy for you". So a candidate will gain no significant
advantage if their voters are more strategic (that is, more dichotomous and
better at evaluating the expected winners of the election) than other
candidates' voters. It allows for naive votes of many kinds, including
potentially "don't know" votes for certain candidates, simple approval-style
votes, simple ranking-style votes, and others, giving all approximately the
same power. And because it uses Range ballots as an input but encourages
more honest voting than Range, it enables the society to see (as an academic
question) who is the true Range winner, when that differs from the Condorcet
winner. I believe that, in successive elections, enough voters would
reevaluate such candidates to make one of them win in both senses.

Nonetheless, I will present one scenario where strategy might be employed.
Say the candidates are Nader, Gore, and Bush, and assume (contrafactually)
that this is simple a matter of 1-dimensional ideology and that all voters
agree that the candidates are, left to right,
nader..gore..........................bush (that is, Nader and Gore in this
scenario are considered much closer than Gore and Bush). Assume also that
there is some dearth of center-left voters who prefer Gore to Bush only
weakly. So the honest preferences are something like

               Nader  Gore  Bush
11%         100      99      0
40%         99        100    0
49%         0          1       100

If all of the 11% of Nader voters dishonestly "bury" Gore under Bush, then
Nader wins with 50.6% (that is, 11% + (99%*40%)). However, if even 2 of the
40% Gore voters honestly vote Gore, Bush, Nader - or EVEN if those 2% simply
give Nader 1 point instead of 99 points - then the strategy backfires and
elects Bush. Given the honest utilities assumed, the Nader voters would not
use this strategy if they thought this was even 1% likely. In the real
election, of course, there was in fact a non-negligible minority of honest
Gore>Bush>Nader voters. (Gore voters were probably likelier to support Bush
second than Nader voters, and I've seen polls that say 1/4 of Nader voters
supported Bush second (!); 1/4 of Gore voters would be around 12%). So this
strategy would never have worked in real life, and in fact it requires an
artificial gap in the voter distribution right near the Gore>Bush>Nader zone
(extending on both sides of that zone, because for instance Gore>>Nader>Bush
votes hose the strategy too).
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