Range voting (also called "Score voting"), Majority Judgment, and "Burr dilemma" (also called "chicken dilemma"), and Bayesian Regret
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Warren D Smith
May 25, 2013, 11:58:53 AM5/25/13
to electio...@googlegroups.com, Jameson Quinn, Jack Nagel, Steven J. Brams, balinski, laraki
Hello.
This idea arose from a post by Jameson Quinn (JQ), now with many
comments and a small amount of editing added by Warren D. Smith (WDS).
The point is that the Balinski-Laraki "Majority Judgment" voting method (which
is like highest-average-score-wins score voting, but instead using
median score and a tie-breaking procedure; an alternate tie-breaking
method called GMJ had been invented by Quinn which is probably a bit
better and is described here:
http://rangevoting.org/MedianVrange.html#GMJ )
is more-immune to Nagel's "Burr dilemma." This will be explained
below, after some muddling.
JQ:
My contention is that in the real world, Majority Judgment (MJ) will actually
handle situations [like the below] just as well, and closely-related situations
significantly better, than Score.
Let's look at some scenarios. First, a scenario where Score does better,
which I claim is unrealistic:
#voters....their vote
51: X100, Y75
49: X0, Y75
(Note: I'm using 100/75/50/25/0 for A/B/C/D/F in MJ)
In this situation, Score would correctly choose Y and MJ would incorrectly
choose X.
WDS COMMENT: and this also is an example where simple-majority-vote
gets it wrong.
However, score voting might also get it wrong if the voters exaggerated, as
I suspect a lot would often do especially in a 2-candidate race. In a
3- or more-candidate race, the voters might feel the urge to describe
X,Y relative to the others and hence not fully exaggerate X and Y to
100/0, but with only 2 candidates there is nothing holding them back
except for the desire to be "honest". Quite a lot might indeed be
honest, or at least partially so, for example in experimental
score-voting test elections it has often been the case that a large
percentage of voters did not use both the max-score and min-score on
their ballot, so they were thus intentionally self-weakening their
ballots. Therefore I consider it plausible that even in 2-candidate
elections it could often happen that score voting outperforms simple
majority.
JQ:
But this would never happen.
WDS: I dispute "never," see above.
JQ continues: Why? Because in the real world,
polarizing candidates tend to have polarized supporters.
WDS: What is a "polarizing candidate"? And
second, in the above scenario, how do we tell from those votes which
candidate (if any) is "polarizing"? Answer: since all opinions about
X are 100/0, I guess he is "polarizing" while since all scores for Y
are 75, he is "not." OK. But in less-simple scenarios I'm not sure
what it would mean.
JQ:
Here's a more realistic version:
#voters....their vote
40: X100, Y0 (honest 50), Z0
11: X100, Y75, Z0
49: Z100, Y75, X0
X is a polarizing candidate. Generally, this means a radical; either on the
right (like either of the Le Pens in France, Duke in Louisiana, or Solyom
of Fidesz in Hungary, Rios Montt in Guatemala, or even You-Know-Who in
1930s Germany) or on the left (like Allende in Chile 1970). X's supporters
are a bare majority, but their support is strong; only a few of them are
willing to vote for any other candidates. Y, the honest utility winner,
loses in both Score and MJ.
So far, that's one scenario (which I call unrealistic) where Score is
better, and one (which I call simplified but essentially realistic in its
outline) where both Score and MJ fail. So why do I think that MJ is better
in cases like this? Because if you take that last scenario, and slightly
modify just 2 of the X voters, then MJ gets it right but Score fails badly:
#voters.....their vote
40: X100, Y0 (honest 50), Z0
9: X100, Y75, Z0
2: X75, Y100, Z0
49: Z100, Y75 , X0
X and Y both have a median of 75 (or grade "B"), but both GMJ and MJ
will give the
win to Y because there are fewer 0s for Y.
WDS: meanwhile plain average score says X wins. Also plurality. Also IRV.
(Condorcet says Y wins.)
WDS: So far, I don't see what this has to do with "polarizing"
(whatever that is). It seems to me what is going on here is the usual
"1-sided exaggeration" scenario where X's supporters downgrade Y and
upgrade X both to the max, but assuming there is no counter-reaction.
(I am unaware of any evidence such 1-sided exaggeration has ever
really happened in the real world in a race between 2 major rivals,
but we do not know much on that question.)
MJ reacts better to 1-sided exaggeration than average-based score
voting, at least if there is the right amount of it. (If over 50% of
voters do 1-sided exaggeration, then MJ is completely overcome while
average still retains a large amount of reality. But with,
say, 20% doing it, average can get shifted while medians stay fixed or
move less.)
In the preceding scenario, there also was 1-sided exaggeration, but
in that case MJ and plain score both fell for it.
JQ:
In essence, this is a chicken dilemma. Y and Z have split their votes,
causing Score to elect X; but MJ still elects Y. Note that this can happen
even with all-honest voters.
WDS: Aha!! The light dawns.
So I think JQ really is making the valid point that MJ is less
vulnerable to this kind of "Burr dilemma" (as Nagel had called it
http://rangevoting.org/BurrDilemma.html ) than plain score voting.
In an X vs Y vs Z race, where X & Y are the main rivals and Z is
similar to Y, some
Y-supporters strategically downgrade Z and some Z-supporters downgrade
Y, as a result X wins with plain score voting. That's the Burr
dilemma.
Well, can X also win due to same dilemma with MJ? Less likely.
Example: X gets
51% mins and 49% maxes, mean is near scale midpoint but median=min. Meanwhile
Y and Z both with full exaggeration and no dilemma would win with 51% maxes, but
with the dilemma, ballots like "Y=100, Z=70, X=0" appear by voters who want to
indicate a slight superiority of Y>Z. Score actually seems better
than approval voting
in this respect since score voters can only partly downgrade Z without
feeling the need for full downgrade, thus "more safely" playing the
game of "chicken."
But nevertheless score voting in our example is busted by this
dilemma, even with only partial downgrading going on. Meanwhile MJ is
unhurt concluding median(Y)=median(Z)=70 win versus median(X)=0.
So I think this is a valid point in favor of MJ, but it is not about
"polarizing," it is about "the Burr dilemma" with "partially
strategic" voters.
JQ:
I find the above scenarios essentially realistic. I've also played with
less-simplified versions of them, and I find that the MJ advantage is
pretty robust. Yes, it is possible for MJ to fall prey to the chicken
dilemma; but in general, score begins to fail chicken even when all of the
majority voters are being moderately cooperative, while MJ doesn't fail
until at least some voters are being strongly uncooperative.
I find MJ's behavior better here, because if some voters are strongly
uncooperative, it is not the job of the voting system to read their minds
and imagine that deep down, they really the other near-clone to be better
than the non-clone. Another way of saying that is: Score frequently elects
the (honest and voted) Condorcet Loser in chicken dilemma situations, while
MJ does not elect that candidate unless they are at least the voted
Condorcet winner.
This, to me, is an important advantage for MJ.
WDS:
It seems to me this issue can arise in any X vs Y vs Z situation where
Y,Z are similar and X is different. Those are common.
Now what about "Bayesian regret"? Aha, there is a nasty issue...:
1. The whole problem is not detected in (since it does not arise in)
computer simulations with 100% naive-wholy-strategic voters (who
would always vote in approval-style: all scores max and min, no
intermediate scores).
2. It also is not detected if the simulation involves 100% honest voters.
3. It also is not detected if the simulation involves ANY MIXTURE of
some wholy-honest
voters and the rest wholy-naive-strategic voters.
1-3 are, in fact, what my previous computer simulations were doing.
4. BUT if you put "partially strategic and partially honest" voters into
the simulation (who will do things like dishonestly downgrade Y from
100 to 70 on 0-100 scale) THEN the problem will be detected by the
simulation, and will cause
MJ and GMJ to improve (measured by Bayesian Regret) relative to
average-based score voting.
SUMMARY:
After some confusion, an important valid argument has arisen in favor
of Majority-Judgment-style median+tiebreak score voting, versus
average-based-score voting:
MJ handles the "Burr dilemma" better.
Previous Bayesian regret measurements by my computer simulations comparing
different voting systems, had not detected this effect because the
simulations involved mixtures of wholy-honest and wholy-strategic
voters.
Such a mix is NOT equivalent to allowing partly-strategic-partly-honest voters
into the electorate. (I had not previously simulated those because I
was not sure how to do so -- how do such voters behave?) If the
simulator is modified to incorporate them, its conclusions might
change.
This is exciting. What we need to know now, is how this kind of
voter behaves so that we can simulate her. It should be easy once
that is answered, to modify my public-source simulator program IEVS.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
This idea arose from a post by Jameson Quinn (JQ), now with many
comments and a small amount of editing added by Warren D. Smith (WDS).
The point is that the Balinski-Laraki "Majority Judgment" voting method (which
is like highest-average-score-wins score voting, but instead using
median score and a tie-breaking procedure; an alternate tie-breaking
method called GMJ had been invented by Quinn which is probably a bit
better and is described here:
http://rangevoting.org/MedianVrange.html#GMJ )
is more-immune to Nagel's "Burr dilemma." This will be explained
below, after some muddling.
JQ:
My contention is that in the real world, Majority Judgment (MJ) will actually
handle situations [like the below] just as well, and closely-related situations
significantly better, than Score.
Let's look at some scenarios. First, a scenario where Score does better,
which I claim is unrealistic:
#voters....their vote
51: X100, Y75
49: X0, Y75
(Note: I'm using 100/75/50/25/0 for A/B/C/D/F in MJ)
In this situation, Score would correctly choose Y and MJ would incorrectly
choose X.
WDS COMMENT: and this also is an example where simple-majority-vote
gets it wrong.
However, score voting might also get it wrong if the voters exaggerated, as
I suspect a lot would often do especially in a 2-candidate race. In a
3- or more-candidate race, the voters might feel the urge to describe
X,Y relative to the others and hence not fully exaggerate X and Y to
100/0, but with only 2 candidates there is nothing holding them back
except for the desire to be "honest". Quite a lot might indeed be
honest, or at least partially so, for example in experimental
score-voting test elections it has often been the case that a large
percentage of voters did not use both the max-score and min-score on
their ballot, so they were thus intentionally self-weakening their
ballots. Therefore I consider it plausible that even in 2-candidate
elections it could often happen that score voting outperforms simple
majority.
JQ:
But this would never happen.
WDS: I dispute "never," see above.
JQ continues: Why? Because in the real world,
polarizing candidates tend to have polarized supporters.
WDS: What is a "polarizing candidate"? And
second, in the above scenario, how do we tell from those votes which
candidate (if any) is "polarizing"? Answer: since all opinions about
X are 100/0, I guess he is "polarizing" while since all scores for Y
are 75, he is "not." OK. But in less-simple scenarios I'm not sure
what it would mean.
JQ:
Here's a more realistic version:
#voters....their vote
40: X100, Y0 (honest 50), Z0
11: X100, Y75, Z0
49: Z100, Y75, X0
X is a polarizing candidate. Generally, this means a radical; either on the
right (like either of the Le Pens in France, Duke in Louisiana, or Solyom
of Fidesz in Hungary, Rios Montt in Guatemala, or even You-Know-Who in
1930s Germany) or on the left (like Allende in Chile 1970). X's supporters
are a bare majority, but their support is strong; only a few of them are
willing to vote for any other candidates. Y, the honest utility winner,
loses in both Score and MJ.
So far, that's one scenario (which I call unrealistic) where Score is
better, and one (which I call simplified but essentially realistic in its
outline) where both Score and MJ fail. So why do I think that MJ is better
in cases like this? Because if you take that last scenario, and slightly
modify just 2 of the X voters, then MJ gets it right but Score fails badly:
#voters.....their vote
40: X100, Y0 (honest 50), Z0
9: X100, Y75, Z0
2: X75, Y100, Z0
49: Z100, Y75 , X0
X and Y both have a median of 75 (or grade "B"), but both GMJ and MJ
will give the
win to Y because there are fewer 0s for Y.
WDS: meanwhile plain average score says X wins. Also plurality. Also IRV.
(Condorcet says Y wins.)
WDS: So far, I don't see what this has to do with "polarizing"
(whatever that is). It seems to me what is going on here is the usual
"1-sided exaggeration" scenario where X's supporters downgrade Y and
upgrade X both to the max, but assuming there is no counter-reaction.
(I am unaware of any evidence such 1-sided exaggeration has ever
really happened in the real world in a race between 2 major rivals,
but we do not know much on that question.)
MJ reacts better to 1-sided exaggeration than average-based score
voting, at least if there is the right amount of it. (If over 50% of
voters do 1-sided exaggeration, then MJ is completely overcome while
average still retains a large amount of reality. But with,
say, 20% doing it, average can get shifted while medians stay fixed or
move less.)
In the preceding scenario, there also was 1-sided exaggeration, but
in that case MJ and plain score both fell for it.
JQ:
In essence, this is a chicken dilemma. Y and Z have split their votes,
causing Score to elect X; but MJ still elects Y. Note that this can happen
even with all-honest voters.
WDS: Aha!! The light dawns.
So I think JQ really is making the valid point that MJ is less
vulnerable to this kind of "Burr dilemma" (as Nagel had called it
http://rangevoting.org/BurrDilemma.html ) than plain score voting.
In an X vs Y vs Z race, where X & Y are the main rivals and Z is
similar to Y, some
Y-supporters strategically downgrade Z and some Z-supporters downgrade
Y, as a result X wins with plain score voting. That's the Burr
dilemma.
Well, can X also win due to same dilemma with MJ? Less likely.
Example: X gets
51% mins and 49% maxes, mean is near scale midpoint but median=min. Meanwhile
Y and Z both with full exaggeration and no dilemma would win with 51% maxes, but
with the dilemma, ballots like "Y=100, Z=70, X=0" appear by voters who want to
indicate a slight superiority of Y>Z. Score actually seems better
than approval voting
in this respect since score voters can only partly downgrade Z without
feeling the need for full downgrade, thus "more safely" playing the
game of "chicken."
But nevertheless score voting in our example is busted by this
dilemma, even with only partial downgrading going on. Meanwhile MJ is
unhurt concluding median(Y)=median(Z)=70 win versus median(X)=0.
So I think this is a valid point in favor of MJ, but it is not about
"polarizing," it is about "the Burr dilemma" with "partially
strategic" voters.
JQ:
I find the above scenarios essentially realistic. I've also played with
less-simplified versions of them, and I find that the MJ advantage is
pretty robust. Yes, it is possible for MJ to fall prey to the chicken
dilemma; but in general, score begins to fail chicken even when all of the
majority voters are being moderately cooperative, while MJ doesn't fail
until at least some voters are being strongly uncooperative.
I find MJ's behavior better here, because if some voters are strongly
uncooperative, it is not the job of the voting system to read their minds
and imagine that deep down, they really the other near-clone to be better
than the non-clone. Another way of saying that is: Score frequently elects
the (honest and voted) Condorcet Loser in chicken dilemma situations, while
MJ does not elect that candidate unless they are at least the voted
Condorcet winner.
This, to me, is an important advantage for MJ.
WDS:
It seems to me this issue can arise in any X vs Y vs Z situation where
Y,Z are similar and X is different. Those are common.
Now what about "Bayesian regret"? Aha, there is a nasty issue...:
1. The whole problem is not detected in (since it does not arise in)
computer simulations with 100% naive-wholy-strategic voters (who
would always vote in approval-style: all scores max and min, no
intermediate scores).
2. It also is not detected if the simulation involves 100% honest voters.
3. It also is not detected if the simulation involves ANY MIXTURE of
some wholy-honest
voters and the rest wholy-naive-strategic voters.
1-3 are, in fact, what my previous computer simulations were doing.
4. BUT if you put "partially strategic and partially honest" voters into
the simulation (who will do things like dishonestly downgrade Y from
100 to 70 on 0-100 scale) THEN the problem will be detected by the
simulation, and will cause
MJ and GMJ to improve (measured by Bayesian Regret) relative to
average-based score voting.
SUMMARY:
After some confusion, an important valid argument has arisen in favor
of Majority-Judgment-style median+tiebreak score voting, versus
average-based-score voting:
MJ handles the "Burr dilemma" better.
Previous Bayesian regret measurements by my computer simulations comparing
different voting systems, had not detected this effect because the
simulations involved mixtures of wholy-honest and wholy-strategic
voters.
Such a mix is NOT equivalent to allowing partly-strategic-partly-honest voters
into the electorate. (I had not previously simulated those because I
was not sure how to do so -- how do such voters behave?) If the
simulator is modified to incorporate them, its conclusions might
change.
This is exciting. What we need to know now, is how this kind of
voter behaves so that we can simulate her. It should be easy once
that is answered, to modify my public-source simulator program IEVS.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
Jameson Quinn
May 25, 2013, 12:53:12 PM5/25/13
to Warren D Smith, electio...@googlegroups.com, Jack Nagel, Steven J. Brams, balinski, laraki
To Warren: Thanks for your response.
To the scholars who received this forwarded message from Warren: I mostly agree with what Warren says, but it may be a bit hard to follow in this choppy form; if you're interested, let me know, and I'll write up my own views on this more cleanly.
In general: Warren's comments are helping me clarify my ideas on this, and I hope that my research will shed yet more light. I should get back to programming that research, rather than commenting here.
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