[EM] Voting Criteria 101, Four Criteria

Jameson Quinn jameson.quinn at gmail.com
Mon Jun 17 09:38:17 PDT 2013

2013/6/17 Kristofer Munsterhjelm <km_elmet at lavabit.com>

> 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.

Right. Kristofer's response here is better than mine was.

>  *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".

Kristofer gives the ranked version of Mutual Majority. The rated version
is: "If a group of candidates is listed at or above a certain rating, and
those not in the group below that rating, on a majority of ballots, then a
candidate in that group should win". This criterion, in at least one of its
versions, is a prerequisite for IoC. I prefer the rated version, but those
like Kristofer who are working within the Arrovian paradigm prefer the
ranked one.

(Note that the mere fact that the rated paradigm is newer than the Arrovian
one does not necessarily make it better. Saari's ranked-symmetry paradigm
is newer than Arrow's, and also in my opinion worse. So in this debate
between people like me and people like Kristofer, there is no short cut to
evaluating each side's arguments on their merits. I of course think I'm
right, but Kristofer is a very smart guy, and you would be unwise to ignore
his side.)

>> *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].

I agree 100% with what Kristofer has said here.

>  *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.

This is the fundamental bone of contention between ranked thinkers like
Kristofer and rated ones like me. I say that Arrow's theorem says you can't
have those things and a ranked system; and I'd far rather choose a rated
system than give up IIA. The downside of choosing a rated system is that
the problem of strategic ambiguity gets worse; for a given set of
preferences, there are many possible "honest" votes, which complicates the
analysis. Most rated voting advocates would say that this problem is
manageable if you assume voters have some knowable underlying cardinal
utilities for each candidate; and that, though that assumption is not
perfectly realistic, it is close enough for meaningful results.

> 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.)

Here I agree with Kristofer.

>  *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<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,

Wait... I think you're right. I thought I had a counterexample in my
earlier response, but now I realize there was a problem with it. Do you
know of a proof of this statement? Because right now it seems right to me,
but I can't obviously see how I would prove it.

> 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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