[EM] Voting Criteria 101, Four Criteria

[Kristofer Munsterhjelm]

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.