On 06/16/2013 06:55 PM, Benjamin Grant wrote:
With your kind indulgence, I would like some assistance in understanding
and hopefully mastering the various voting criteria, so that I can more
intelligently and accurately understanding the strengths and weaknesses
of different voting systems.
So, if it’s alright, I would like to explain what I understand about
some of these voting criteria, a few at a time, perhaps, and perhaps the
group would be willing to “check my math” as it were and see if I
actually understand these, one by one?
No problem :-)
*Name*: *_Plurality_*
*Description*: If A gets more “first preference” ballots than B, A must
not lose to B.
Be careful not to mistake Plurality, the criterion, from Plurality the
method. Plurality, the criterion, says: "If there are two candidates X
and Y so that X has more first place votes than Y has any place votes,
then Y shouldn't win".
The Plurality criterion is only relevant when the voters may truncate
their ballots. In it, there's an assumption that listed candidates are
ranked higher than non-listed ones - a sort of Approval assumption, if
you will.
To show a concrete example: say a voter votes A first, B second, and
leaves C off the ballot. Furthermore say nobody actually ranks C. Then C
shouldn't win, because A has more first-place votes than C has any-place
votes.
*Name: _Majority_*
*Description*: If one candidate is preferred by an absolute majority of
voters, then that candidate must win.
That's right. More specifically, if a candidate has a majority of the
first place votes, he should win. There's also a setwise version (mutual
majority) where the criterion goes "if a group of candidates is listed
ahead of candidates not in that group, on a majority of the ballots,
then a candidate in that group should win".
*Thoughts*: I might be missing something here, but this seems like a
no-brainer. If over 50% of the voters want someone, they should get him,
any other approach would seem to create minority rule? I guess a
challenge to this criteria might be the following: using Range Voting, A
gets a 90 range vote from 60 out of 100 voters, while B gets an 80 from
80 out of 100 voters. A’s net is 5400, but B’s net is 6400, so B would
win (everyone else got less). Does this fail the Majority Criterion,
because A got a higher vote from over half, or does it fulfill Majority
because B’s net was greater than A’s net??
There are usually two arguments against the Majority criterion from
those that like cardinal methods.
First, there's the "pizza example": say three people are deciding on
what piza to get. Two of them prefer pepperoni to everything else, but
the last person absolutely can't have pepperoni. Then, the argument
goes, it would be unreasonable and unflexible to pick the pepperoni
pizza just because a majority wanted it.
Second, there's the redistribution argument. Consider a public election
where a candidate wants to confiscate everything a certain minority owns
and then distribute the loot to the majority. If the electorate is
simple enough, a majority might vote for that candidate, but the choice
would not be a good one.
Briefly: the argument against Majority is "tyranny of majority". But
ranked methods can't know whether any given election is a
tyranny-of-majority one, and between erring in favor of the majority and
in favor of a minority (which might not be a good minority at all), the
former's better. Condorcet's jury theorem is one way of formalizing that.
Rated methods could distinguish between tyranny-of-majority cases, were
all the voters honest, but being subject to Gibbard and Satterthwaite
just like ranked methods, they too can be gamed. There's usually a way
for a majority to force a win if they absolutely want to, too[1].
*Name: _Participation_*
*Description*: If a ballot is added which prefers A to B, the addition
of the ballot must not change the winner from A to B
*Thoughts*: This seems to make sense. If we do not require this, then
we permit voting systems where trying to vote sincerely harms your
interests. Also, any voting system that would fail Participation would
be I think fragile and react in not always predictable ways – like IRV.
SO this seems to me to be a solid requirement, that I can’t imagine a
system that failed this Criterion to have some other benefit so
wonderful to make failing Participation worth overlooking – I cannot
imagine it.
Welcome to the unintuitive world of voting methods :-) Arrow's theorem
says you can't have unanimity (if everybody agrees that A>B, B does not
win), IIA (as you mention below) and non-dictatorship. Since one can't
give up the latter two and have anything like a good ranked voting
method, that means every method must fail IIA.
The trade-off with Participation is similar. It is impossible, for
instance, to have a method that passes both Participation and Condorcet,
so one has to choose which is more important. Similarly, it's impossible
to have a method that passes Later-no-harm, later-no-help, mutual
majority and monotonicity. (IRV passes them all except monotonicity; DAC
and DSC pass them all except one of the Later-no criteria; and Plurality
pass them all except mutual majority.)
*Name: _Independence of Irrelevant Alternatives (IIA)_*
*Description*: Adding a new candidate B to an election that previously A
would have won must not cause anyone apart from A or B to win. That is,
If A would have won before B was added to the ballot, C must not win now.
*Thoughts*: This also seems fairly non-controversial. This I think is
the repudiation of the spoiler effect – that just because Nader enters
the race shouldn’t disadvantage the candidate that would have won before
that happened. This would seem (to me) to also be a good Criterion to
hold to in order to encourage more than just two Candidates/Parties
always dominating the scene. I wonder what the downside would be to
strongly embracing this criteria?
From the ranked-ballot side of things, one usually says "okay, so IIA
is impossible, but how far can we get?". This leads to things like local
IIA (removing the winner or loser of an election shouldn't change the
output ranking for the other candidates), independence of clones (which
I'll get to later), and independence of Smith-dominated alternatives (if
X is not in the Smith set, removing X shouldn't make the winner change).
There's also a heuristic argument that IIA is too strong. It goes that
the introduction of additional candidates may tell you that things
aren't the same before and after the introduction of the same
candidates. See
https://en.wikipedia.org/wiki/Independence_of_irrelevant_alternatives#Criticism_of_IIA
for more information.
Also note that IIA and majority is incompatible. The same link shows why.
*Question*: It seems to me that another criterion I have heard of –
Independence of Clones(IoC) – is a subset of IIA, that if a system
satisfies IIA, it would have to satisfy the Independence of Clones
criterion as well – is that correct? If not, what system what satisfy
IoC but **not** satisfy IIA?
Methods that pass IIA also pass IoC, yes, but not all methods that pass
IoC pass IIA. Schulze and Tideman are simple examples of rules that are
cloneproof (pass IoC) yet, being deterministic ranked ballot methods
reducing to majority when there are only two candidates, must fail IIA
itself.
*Question*: it seems like the two above criteria – Participation and IIA
– would be related. Is it possible to fail one and not the other? Or
does either wind up mandate the other – for example, a system with IIA
must also fulfill Participation, or vice versa?
Trying to come up with counterexamples usually is a simple task, because
one can design an obviously outrageous system. As long as the system
provides a counterexample, it doesn't matter how unsuitable it otherwise is.
So for Participation and IIA, consider a method that works like Range as
long as there are fewer than 100 voters, but reverses the order of the
winners if there are more than 100 voters - i.e. the Range loser becomes
the new winner.
This method passes IIA since both Range and Anti-Range (as it were) does
so. Yet it obviously fails Participation. Say you're voter number 100,
and you prefer the Range winner. Then submitting your ballot will make
the Range loser win instead, so you're better off not doing so.
So let me stop there for now – I know there are other Criteria, but let
me pause so you guys can tell me what I am getting right and what I am
getting wrong.
Thanks.
-Benn Grant
[1] I'm kind of seeing a strategy-stealing argument here, which if
right, would mean a majority could force a win in any anonymous rated
system that fails Majority. But I could be wrong and I don't want to
clutter the text proper with it.
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