Dear Remi,
welcome on this list! When you dig a little into the archives of it, you
will find that it provides loads of thoughts on and examples of
probabilistic voting systems and strategic voting, certainly more than
you will find in any book. Some of those discussions eventually lead to
this
Forest wrote:
Now for the interesting part: if you use this strategy on your approval
ballot, the expected number of
candidates that you would approve is simply the sum of the probabilities of
your approving the individual
candiates, i.e. the total score of all the candidates on your
Dear Stephen,
you wrote:
democratic implies majority rule
This is by no means clear. You might want to consult this list's
archives to find some discussion of this.
My opinion is rather that democracy and majority rule are
incompatible since the latter assigns all power to 50%+epsilon in the
Hi again,
I wrote:
Unfortunately, I fear Short Ranked Pairs might not be monotonic. One
would habe to check. And I'm not sure your description of an algorithm
for Short Ranked Pairs is valid -- after all, I only defined it
abstractly by saying that one has to find the lexicographically
Hi Kristofer,
you wrote:
I think Short Ranked Pairs also passes all these. To my knowledge, Short
Ranked Pairs is like Ranked Pairs, except that you can only admit XY if
that will retain the property that every pair of affirmed candidates
have a beatpath of at most two steps between them.
Dear Rob,
you wrote:
here's a fundamental philosophical question: why is it better, even in a
two-candidate race, to elect the majority winner?
I think the question is ill-posed in at least two ways: First, you say
better but not better than what. Second, after you settled for an
Dear Abd ul-Rahman,
some short remarks to your claims:
Random ballot does nothing to encourage compromise!
Perfectly true. That's why nobody suggests it to be used for decisions.
We suggest to use it instead as a benchmark any reasonable method must
improve upon. More precisely, we want that
- 1 0
C 0 - 0
D 2 3 -
B D
B - 0
D 2 -
D
D -
Yours, Jobst
Am 31.10.2010 18:35, schrieb Jobst Heitzig:
Hi Markus,
on 29.04.2010 20:33 you asked:
is Jobst Heitzig's river method identical
to Blake Cretney's goldfish method
Hi Markus,
on 29.04.2010 20:33 you asked:
is Jobst Heitzig's river method identical
to Blake Cretney's goldfish method?
I'm sorry that I have not read any list posts for months, so this caught
my attention just now. I will check the differences! You probably refer
to the method from Blake's
Hi folks,
I just recalled that four years ago I constructed a sophisticated
example which is somewhat similar:
http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2005-May/015982.html
Happy New Year!
Jobst
Warren Smith schrieb:
This point-set also works:
A=(1,0) B=(0,4)
Folks,
you probably overlooked that I have already described a variant which
works *completely* without Random Ballot and will definitely elect one
of the top-3 range options (as determined from the 'strategic' ballot):
Method Range top-3 runoff (RT3R)
===
1.
Even simpler is this:
Method Top-3 approval sincere runoff (T3ASR)
==
1. Each voter separately supplies
a nomination approval ballot and a runoff range ballot.
2. From all nomination ballots, determine
the options A,B,C with the top-3 approval
Hello Forest,
Most of the credit should be yours; in fact, the proof and all of the
ingredients are yours. I hurried to post the message this morning, because I
was sure that you were going to beat me to it! I would certainly believe you
if
you said that you had already thought of the
Dear folks,
although Forest's posting comes along so matter-of-factly, let's make it
absolutely clear that it is an
ENORMOUS MILESTONE!
Why so?
He describes a very SIMPLE, EFFICIENT, and FAIR method which
REVEALS THE TRUE UTILITY VALUES
of all voters who are rational in the
Dear Raph,
you wrote:
Likewise, you might as well pick your favourite as favourite.
This is, unfortunately, not true: The labelled favourite influences the
expected ratings against which possible consensus options are compared
on each ballot, so you can have the incentive to exaggerate by
Dear folks,
there is another assumption in Arrow's theorem which people almost
always forget: Determinism. Methods which use some amount of chance can
easily meet all his other criteria, the most trivial example of this
being again Random Ballot (i.e. pick a ballot uniformly at random and
copy
Dear Matthew,
you wrote:
Suite of complicated systems that strive to reach Condorcet ideals.
1. No regular bloke would ever trust 'em because you can't explain
how they work in one or two sentences.
Well, here's a very simple Condorcet system which can easily be
explained in two
Dear Robert,
you wrote:
Round Robin tournament, Ranked Ballot: The contestant who wins in a
single match is the candidate who is preferred over the other in more
ballots. The candidate who is elected to office is the contestant who
loses to no one in the round robin tournament.
that's
Hello Kristofer,
you wrote:
However, my point was that Range goes further: a minority that acts
in a certain way can get what it wants, too; all that's required is that
the majority does not vote Approval style (either max or min) and that
the minority does, and that the minority is not too
Dear Kristofer,
both Approval Voting and Range Voting *are* majoritarian: A majority can always
get their will and suppress the minority by simply bullet-voting.
So, a more interesting version of your question could be: Which *democratic*
method (that does not allow any sub-group to suppress
Dear Peter,
I claimed that SEC...
make sure option C is elected in the following situation:
a% having true utilities A(100) C(alpha) B(0),
b% having true utilities B(100) C(beta) A(0).
with a+b=100 and a*alpha + b*beta max(a,b)*100.
(The latter condition means C has the
Dear folks,
earlier this year Forest and I submitted an article to Social Choice and
Welfare (http://www.fair-chair.de/some_chance_for_consensus.pdf)
describing a very simple democratic method to achieve consensus:
Simple Efficient Consensus (SEC):
=
1. Each
Dear Abd ul-Rahman,
you wrote:
Well, I find it hard to believe how wrong-headed this is.
Well, thank you very much.
In a real society that is
large enough, the consensus urn will never choose a winner unless there
is a true consensus process already in operation, people will not
naturally
Dear Michael,
very interesting, I don't think I saw anything like this before.
When trying do evaluate a new method, I always try to check very simple
criteria first, like neutrality and anonymity (obviously fulfilled
here), Pareto efficiency, monotonicity, etc. Concerning the latter two,
I was
Dear Raph,
Schulze and ranked pairs are the only methods that meet clone
independence and the condorcet rule.
Nope. River, too, of course, meets all three criteria...
Does ranked pairs fail the Smith criterion?
I would change B to If there is a group of candidates all preferred
over
Hi friends,
Raph wrote:
On Sat, Feb 14, 2009 at 7:29 PM, fsimm...@pcc.edu wrote:
Note well that in the case of a tie among lotteries the tie is broken by a
voter, not the voter's ballot, and the tie breaking voter only decides among
the
tied lotteries, not directly picking the winner.
It
Hi Raph,
this was suggested before, but I can't remember by whom. The problem is,
as you have noted, that the method gives everyone an incentive to reduce
her approvals for non-favourites as long as this doesn't change the
approval order.
For example, consider our example in which the true
Hi Raph,
The odds of it actually working are pretty low. For it to work, all
voters must be aware that C is a valid compromise.
Sure, that's the flipside of it being so ultimately simple. The easiest way to
safeguard against a small number of non-cooperative voters would be to require
only,
Jobst Heitzig schrieb:
Hello folks,
I know I have to write another concise exposition to the recent
non-deterministic methods I promote, in particular FAWRB and D2MAC.
Let me do this from another angle than before: from the angly of
reaching consensus. We will see how chance processes can
random ballot from the favourites urn to the
possible consensus result (C) and therefore vote (e.g.) for A in their
consensus ballot.
Juho
--- On Sun, 1/2/09, Jobst Heitzig heitzi...@web.de wrote:
From: Jobst Heitzig heitzi...@web.de
Subject: [EM] Some chance for consensus revisited
Hi Adb ul-Rahman,
still, Asset Voting is majoritarian and therefore not democratic. The
reason why we have been studying methods with chance components (that
is, non-deterministic methods) is that we wanted to find a democratic
method, i.e. one that does not give any subset of the voters (no
Does anybody know why the wiki on www.electorama.com is still not
working properly?
Election-Methods mailing list - see http://electorama.com/em for list info
poorest firstasset
voting with predefined lists.
2008/11/16 Jobst Heitzig [EMAIL PROTECTED] mailto:[EMAIL PROTECTED]
Hi Kristofer,
That's just the multiwinner adaptation of IRV.
I don't think so! The point is that the *candidates* provide the
ranking from which the vote
the idea?
Forest
- Original Message -
From: Jobst Heitzig
Date: Thursday, November 6, 2008 3:37 pm
Subject: Re: [EM] Some chance for consensus (was: Buying Votes)
To: [EMAIL PROTECTED]
Cc: [EMAIL PROTECTED], election-methods@lists.electorama.com,
Raph Frank , Kristofer Munsterhjelm
of the ratings, where Q is the rating
on the drawn ballot...
Any thoughts?
Jobst
Jobst Heitzig schrieb:
Hi folks,
I think I know what the problem is with the idea of somehow
automatically match pairs or larger groups of voters who will all
benefit from a probability transfer: It cannot be monotonic
Dear Greg,
you wrote:
Nondeterminism is a delightful way of skirting the
Gibbard-Satterthwaite theorem. All parties can be coaxed into exposing
their true opinions by resorting or the threat of resorting to chance.
Actually, if I remember correctly, that theorem just said that Random Ballot
Dear Raph,
you wrote:
I was thinking of a 'stable marriage problem' like solution.
Good idea! If it works, the main difficulty will be to make the whole process
monotonic, I guess...
Yours, Jobst
Each voter rates all the candidates.
Each voter will assign his winning probability to
Dear Kristofer,
you wrote:
With more candidates, a minority might find that it needs to approve of
a compromise with just slightly better expected value than random
ballot, if the majority says that it's not going to pick a compromise
closer to the minority than that just-slightly-better
Dear Forest,
good to hear from you again!
You said:
Not quite as important, but still valuable, is achieving partial cooperation
when that is the best that can be
done:
25 A1AA2
25 A2AA1
25 B
25 C
Here there isn't much hope for consensus, but it would be nice if the first
two
one
contracting group of voters were allowed.)
What do you think?
Jobst
-Ursprüngliche Nachricht-
Von: Raph Frank [EMAIL PROTECTED]
Gesendet: 31.10.08 15:35:30
An: Jobst Heitzig [EMAIL PROTECTED]
CC: [EMAIL PROTECTED], election-methods@lists.electorama.com, [EMAIL
PROTECTED
Dear Greg,
you wrote:
I wish I knew how FAWRB worked...
I will give a new step-by-step exposition of FAWRB in a new thread
during this day.
Most people say the majority criterion is a good one. I, for one,
doubt its importance. I was merely saying that a counter-criterion
inconsistent
Dear Greg,
you wrote:
I'm not speaking about majoritarianism in this case, although you are
correct that it alleviates many of the problems. What I meant was
there is the potential for vote buying under any voting method where
voting is verifiable and non-unanimity can pass a policy.
OK, I
the link at the website above to MAM vs
PathWinner.
Regards,
Steve
--
Jobst Heitzig wrote:
Hi folks,
you're right, the option ending up second in the ranking constructed
in RP is also the one that wins when you exclude
Dear Steve,
you wrote:
Jobst, is there any justification for the criterion you mentioned below
that's satisfied by Beatpath but not by MAM or River?
I personally don't think so. In the beginning, I liked the criterion
however for its mathematical beauty, and since it's a natural
Dear Diego,
you wrote:
The risk of minority will remains. How does FAWRB perform in binary issues?
What you mean by risk of minority? That a minority favourite may win?
Well, that is just the *feature* of FAWRB: It gives each part of the
electorate full control over an equal share of the
Dear Greg,
you wrote:
Group membership is difficult to define. With ranked ballots it's
simple, but in the majority criterion debate, I argue that a score of
60% represents 60% of a first preference, not the preference between 59%
and 61%.
Sorry, I don't get your meaning here.
However, it
Dear Raph,
you answered to me:
a) FAWRB is not a random but a very specific and quite sophisticated
method. It only uses a certain amount of chance, just as many things in
our life do. Chance should not be mixed up with arbitrariness. Used in a
rational way, FAWRB will usually elect good
Dear Greg,
I will focus on the question of majoritarianism in this message.
First my working definition of majoritarian method: A method is majoritarian
if for every option X and every group G consisting of more than half of the
voters, there is a way of voting for G which makes sure X wins
Dear Raph,
you replied to me:
That leads me to the main problem with Range (as with any other majoritarian
method): It is simply not democratic. It cannot be because every
majoritarian method gives 100% of the power to less than 100% of the people
(the demos in greek).
They do have an
even adds a
delegable proxy component to it: http://62.75.149.22/groucho_fawrb_dp.php
Yours, Jobst
Terry Bouricius schrieb:
What does FAWRB stand for?
Terry Bouricius
- Original Message -
From: Jobst Heitzig [EMAIL PROTECTED]
To: Greg Nisbet [EMAIL PROTECTED]; Raph Frank
[EMAIL
Dear Raph,
you wrote:
The thing is that in such a case, it isn't really a single 'demos'.
It is two groups voting as one.
Do you mean to say democracy is only for societies which are
sufficiently homogeneous?
That doesn't help because then the majority on issue A will still
overrule the
Dear Diego,
But randomness of FAWRB can cause institutional conflicts, especially if
the minority faction leader was the winner.
My focus has always been to decide issues, not to elect people.
My suggestion if your
scenario exists is:
1. Perform simultaneously an approval election and a
Dear Kristofer,
you wrote:
Since Condorcet is a majoritarian method (as it needs to be in order to
be a good single-winner method) ...
A good single-winner method *must not* be majoritarian but must elect C
in the situation of
55% voters having A 100 C 80 B 0 and
45%
Dear Kevin,
your wrote:
The problem is that if you do not guarantee the majority that they will
get their favorite if they vote sincerely, then they will stop telling
you who their compromise choices are.
No. In D2MAC there is no such guarantee (since it is not majoritarian) and this
fact is
Hi, you wrote:
encourages people to vote honestly
What makes you believe this?
Yours, Jobst
Election-Methods mailing list - see http://electorama.com/em for list info
Hi folks,
you're right, the option ending up second in the ranking constructed in
RP is also the one that wins when you exclude the first winner. Even
more so, the whole new ranking is just the old one with the top removed.
There is, by the way, a related major difference between RP and the
Dear Greg,
I appreciate your trying to sum up the recent discussion. Some remarks:
You wrote:
Range Voting:
There are two types of arguments against this system:
There is another one - in my opinion the most serious problem of Range
Voting: It fails to achieve exactly that which it claims
Dear Greg,
Reasons why Range is better and always will be.
I would like to end the truce.
That won't work I guess. Using the term better alone is a major flaw of many
discussions here. Obviously, it all depends on what goals a method is expected
to achieve.
I'll be generous to the
at once instead of making it a surprise as each stage shows up.
Alternately, or additionally, there could be a side-bar flow chart
that lists all the stages, probably vertically. Bonus points if the
stages light up as the process moves through them.
On Aug 25, 2008, at 4:10 PM, Jobst Heitzig
Hello all,
although I did not follow all of the discussion so far, the following
question strikes me:
Why the hell do you care about proportional representation of minorities
when the representative body itself does not decide with a method that
ensures a proportional distribution of power?
Dear Forest,
This is indeed astonishing from a slightly different point of view: if I rate the winner well below the benchmark lottery, my action tends to cause other voters'
accounts to lower, but does not cause mine to increase, yet the total is constant. Magic!
Not so astonishing after
Dear folks,
I must admit the last versions of RRVC (Representative Range Voting with
Compensation) all had a flaw which I saw only yesterday night. Although
they did achieve efficiency and strategy-freeness, they did not achieve
my other goal: that voters who like the winner more than the
A first typo: It must read C(i) instead of A(i) under Input...
--
Dear folks,
I must admit the last versions of RRVC (Representative Range Voting with
Compensation) all had a flaw which I saw only yesterday night. Although
they did achieve efficiency and strategy-freeness, they did not
will lead to the standard
deviation having an order of at most O(1) instead of O(sqrt(D)) or
even O(D).
Perhaps someone can analyse this averaging in more detail and come up
with am estimation of that standard deviation?
Yours, Jobst
Jobst Heitzig schrieb:
Another small remark:
With N
not shrink with growing N.)
Jobst
Jobst Heitzig schrieb:
Dear folks,
this night I had two additional ideas for RRVC, so here's two new
versions of it.
In the first version, the fee F is determined from the benchmark ballots
so that the expected price a deciding voter has to pay from her
Dear Warren and list members,
you wrote:
The main differences, however, are these:
- Voting money is transferred on a regular basis, not only in the very rare
case of swing voters, thus making the strategic incentive much stronger.
--this is good. However, in one of your revised schemes you
Dear Warren,
you wrote:
But I do not fully understand it yet and I think you need to
develop+clarify+optimize it further... plus I'd like you to unconfuse me!
I'll try...
Of course, this is far from being a new idea so far, and it is not yet
the whole idea since it has an obvious problem:
Another small remark:
With N voters total and B benchmark voters, the size D of the deciding
group should probably be O(sqrt(N-B)).
This is because the amount transferred to an individual deciding voter's
account is roughly proportional to D times a typical individual rating
difference,
Dear folks,
some time ago we discussed shortly whether it was possible to design a
strategy-free ratings-based method, that is, a method where voters give
ratings and never have any incentive to misrepresent their true ratings.
If I remember right, the methods that were discussed then were only
Hi again.
There is still another slight improvement which might be useful in
practice: Instead of using the function 1/(5-4x), use the function
(1 + 3x + 3x^7 + x^8) / 8.
This is only slightly smaller than 1/(5-4x) and has the same value of 1
and slope of 4 for x=1. Therefore, it still
Dear Forest,
well - thanks.
Anyway, there is still room for improvement.
Our last version was this: Let x be the highest approval rate (=approval
score divided by total number of voters). Draw a ballot at random. With
probability 1/(5-4x), the option with the highest approval score amoung
to make the probability of using Random Ballot
depend
on some degree of cooperation.
Perhaps we can salvage it.
Forest
- Original Message -
From: Jobst Heitzig
Date: Sunday, June 1, 2008 2:57 pm
Subject: Re: [english 90%] Re: Challenge Problem
To: [EMAIL PROTECTED]
Cc: election-methods
of cooperation.
Perhaps we can salvage it.
Forest
- Original Message -
From: Jobst Heitzig
Date: Sunday, June 1, 2008 2:57 pm
Subject: Re: [english 90%] Re: Challenge Problem
To: [EMAIL PROTECTED]
Cc: election-methods@lists.electorama.com
Dear Forest,
glad you find time to delve
Dear Forest,
glad you find time to delve somewhat deeper into these questions. Looks
like a good idea to make the probability of using Random Ballot depend
on some degree of cooperation.
Two notes as of now:
I guess you meant
RABMAC*doc^M + RB*(1-doc^M)
instead of
RB*doc^M +
to manipulation by insincere ballots.
- Original Message -
From: Jobst Heitzig
Date: Saturday, May 24, 2008 10:04 am
Subject: Re: [english 94%] Re: D(n)MAC
To: [EMAIL PROTECTED]
Cc: election-methods@lists.electorama.com
Dear Forest,
your analysis was right from the beginning while mine
, 0, else y^n)
pC=(x+y)^n - when(X+YP, 0, else x^n) - when(X+YQ, y^n, else 0).
Miraculously, these probabilities add up to 1 !
The two faction expectations are
EA = pA + R*pC, and
EB = pB + R*pC
From there, it's all downhill.
My Best,
Forest
- Original Message -
From: Jobst
Dear Juho,
you wrote:
Yes, but as I see it the reasons are different. In a typical non-
deterministic method like random ballot I think it is the intention
to give all candidates with some support also some probability of
becoming elected.
Not at all! At least in those non-deterministic
Dear Juho,
you wrote:
One observation on clone independence and electing a centrist
candidate using rankings only and when one of the extremists has
majority.
...
It is thus impossible for the algorithm in this case and
with this information (rankings only) to satisfy both requirements
and
Dear Raphfrk,
you wrote
There needs to be some system for providing an incentive for people
to give their honest ratings.? A random system with trading seems
like a reasonable solution.
I am glad that I am no longer alone with this opinion...
If a majority has a 100% chance of getting their
Dear Raphfrk,
it did not think through all you wrote yet, but one point troubles me:
Also, it is majority compliant. If a majority support a candidate first
choice (i.e. first choice and nominate him), then he cannot lose.
If that is true, your method cannot be a solution to the given
Dear Raphfrk,
I also see no obvious way how the Anti-STV approach might become
clone-proof when voters (or factions) can add options.
So, the method AMP (and variants thereof) still seems to be the only
solution yet...
I wonder if anyone comes up with a different approach. In particular,
) that will be eliminated.
Only C remains and is the winner.
- I used only rankings = also worse than 52 point compromise
candidates would be elected
- I didn't use any lotteries = C will be elected with certainty
Juho
On May 2, 2008, at 22:29 , Jobst Heitzig wrote:
Dear Juho,
I'm not sure what
On Apr 28, 2008, at 20:58 , Jobst Heitzig wrote:
Hello folks,
over the last months I have again and again tried to find a
solution to
a seemingly simple problem:
The Goal
-
Find a group decision method which will elect C with near certainty in
the following situation
Hello folks,
over the last months I have again and again tried to find a solution to
a seemingly simple problem:
The Goal
-
Find a group decision method which will elect C with near certainty in
the following situation:
- There are three options A,B,C
- There are 51 voters who prefer A
Dear Clay,
you wrote:
the point is that we know it
exists. a very simple economic concept called revealed preference
demonstrates this. it works like this.
say you prefer apples to oranges to bananas. i give you a guarantee
of having to eat an orange, or a 50/50 chance of having
Dear Juho!
You wrote:
I could imagine a voting system that might address this issue for larger
groups, but it isn't Range.
One could have elections that take into account e.g. proportionality in time
(that could be called one kind of reciprocity) (favour a republican after a
democrat,
trading.
My Best,
Forest
Jobst Heitzig wrote:
Dear Forest,
Perhaps candidates should be required to publish their range ballots
before the election, and their trading of assets should be required
to be rational relative to these announced ratings?
I had
Dear Forest,
Perhaps candidates should be required to publish their range ballots
before the election, and their trading of assets should be required
to be rational relative to these announced ratings?
I had this idea, too. But upon closer inspection, it is not quite easy
to define
Dear Abd ul-Rahman,
I dislike, by the way, describing voters as selfish if they vote in
their own interest. That's the default, they *should* vote in their
own interest.
That is probably a language problem again. I thought selfish was a
synonym for acting in my own interest only, is it
Dear Adb ul-Rahman,
I dislike, by the way, describing voters as selfish if they vote in
their own interest. That's the default, they *should* vote in their
own interest.
That is probably a language problem again. I thought selfish was a synonym
for acting in my own interest only, is it
Dear Abd ul-Rahman,
I am most concerned about majority *consent.* Jobst is ignoring the
fact that I'm suggesting majority *consent* for decisions;
What exactly is majority consent? In my understanding consent means
*all* voters share some opinion...
what do you call it when a minority
Dear Abd ul-Rahman,
the ratings that Jobst fed us as a distraction.
You're doing it again -- please stop it.
That was not an insult. It made the challenge more interesting. I'm
sorry that you thought it critical.
I don't think it was insulting. You just repeatedly attribute opinions
or
Dear Abd ul-Rahman,
No. It's an understanding of what utilities mean.
If you think so...
If A does not win,
the supporters of A lose something. They are in the majority. If each
of them grabs a B supporter and wrestles with him, or her, I suppose,
the excess A supporters can then
Dear Steve,
Although Jobst may not have intended this assumption, I will continue to
make the assumption that the B minority's preference intensity for the
compromise C over A is much greater than the A majority's preference
intensity for A over C.
Sorry, I had just not read carefully
Dear Abd ul-Rahman,
Okay, here is my solution. The B voters gain some very substantial
advantage for the election of C over the favorite of the A voters,
who have only a substantially smaller preference for A over C.
So the B voters offer something of value to the A voters to
compensate
Dear Forest,
The main thing I overlooked was vote trading.
So there are two main devices for solving the challenge: vote trading
and randomness.
There is a third one! One of the oldest voting methods that have been studied
can also solve it at least in part. I wonder who will first see
Dear Kevin,
Hi,
It seems to me there might be a use for something like the method that
was proposed awhile ago that had to do with offering voters incentives
to give sincere ratings. For example, the majority would give the
sincere score to their compromise in exchange for their vote
.
It is paradoxical that randomness, usually associated with uncertainty,
is the key to making C the certain winner.
Look up D2MAC in the archives for a more quantitative analysis.
I hope that this doesn't prematurely take the wind out of the challenge.
Forest
From: Jobst Heitzig
A common situation: 2 factions 1 good compromise.
The goal: Make sure the compromise wins.
The problem: One of the 2 factions has a majority.
A concrete example: true ratings are
55 voters: A 100, C 80, B 0
45 voters: B 100, C 80, A 0
THE CHALLENGE: FIND A METHOD THAT WILL ELECT THE
99 matches
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