# [EM] Deconstructing the Majority Criterion

Benjamin Grant benn at 4efix.com
Mon Jun 17 08:26:34 PDT 2013

Thanks for your reply, let’s see what I can grasp on this pass, shall we? ;)

From: Jameson Quinn [mailto:jameson.quinn at gmail.com]
Sent: Sunday, June 16, 2013 10:36 PM
Subject: Re: [EM] Voting Criteria 101, Four Criteria

2013/6/16 Benjamin Grant <benn at 4efix.com <mailto:benn at 4efix.com> >

Re: Majority Criteria:

To be honest, I am worried that some (or all) of your history lesson
regarding Arrow might not have landed as well as it should in my brain.

Sorry. Sometimes I tend to try to say things too succinctly, and end up
leaving my meaning a bit locked up in jargon or terminology. If you have any
specific questions about the "history lesson" I'd be happy to expand.

freely assign a score of 0 to the maximum amount to each candidate (say
100), the candidate with the greatest aggregate score wins) let me see how
this might fail.  Let’s say out of 1000 people 550 give candidate A scores
of “100”. Then let’s say that 700 people give candidate B scores of “80”
each. Let’s also say that everyone else falls short of either of those
totals.  A gets 55,000 total, B gets 56,000.  B wins.

Right.

On the one hand, one could say in one sense this violates Majority, but in
another sense one could perhaps with even more justification claim that B
actually has the larger majority.  Or maybe to put another way, Majority
criteria only applies to voters when the system is one person, 1 vote –
others perhaps Majority criteria applies to *votes*, not voters.

In other words, maybe Majority criteria should be worded thusly: If one
candidate is preferred by an absolute majority of *votes*, then that
candidate must win.

That would be stretching the criterion to the point of meaninglessness. The
majority criterion speaks of voters, and Range doesn't pass, but Bucklin
systems do.

The more controversial case for this criterion is approval. Some try to
define the criterion so that an internal preference which doesn't fit on the
ballot is enough to constitute a "majority"; others prefer to define it so
that a "majority" only means anything in terms of the ballots themselves. I
tend to side with the latter as a matter of definition, but I certainly
understand that as a practical matter approval's passing of the majority
criterion leaves much to be desired.

So my takeaway I think is that vis-à-vis voting systems, there are 3 kinds
of voting systems with regard to the Majority criterion: systems that fulfil
the criterion, systems that fail it, and systems in which “majority” makes
no sense.

I would say that First Past the Post would be an example of the 1st – it is
easy to see that FPTP fulfils Majority, as if over 50% of the votes cast are
for A, then A wins, always. An (admittedly lame) example of the 2nd – a
system that fails the Majority Criterion, is the following: Of all
candidates on the ballot, the one that gets the least votes, wins.   Call
this LPTP (Last Past the Post)

However, let’s look at Score Voting again – which I *think* can work like
this: each voter gives each candidate a score from 0 to 9 on their ballot,
with empty spots being treated as 0. Then add up all the scores for each
candidate, the one with the highest total score wins.

Now let’s look at the following election being run that way:

45 votes give Candidate A a score of 9, Candidate B a score of 6, Candidate
C a score of 0, and Candidate D a score of 3

20 votes give A:0, B:6, C:9, and D:3

20 votes give A:0 B:6 C:3 D:9

15 votes give A:6 B:9 C:3 D:0

The totals are A:495 B:645 C:285 D:375 – so B wins.

My thought is that perhaps in the context of this vote, the concept of
majority as applied in the first two example (FPTP and LPTP) doesn’t work
here.  I also think that changing the definition of “majority” so that it is
intelligible here will make it less understandable in the context of
FPTP/LPTP.

Maybe what I am wondering is, is the context of some of these voting system
so different that *some* concepts – like “majority” do not make sense in all
contexts, and that trying to alter the definition to make it fit better in
one context makes it fit worse in others?

As originally written, I think, the Majority criterion states that: “if one
candidate is preferred by a majority (more than 50%) of voters, then that
candidate must win”

Well, in the above Score Voting system context, the concept of preference as
an all or nothing trait makes no sense. You could has scores of A:9 B:6 C:3
D:1 and be said to in some sense express some amount of preference for each
of them. The only way in which the criterion would make sense is if we
mutated the criterion somewhat like this: “if one candidate is preferred (at
the highest score or ranking, where such exists) by a majority (more than
50%) of voters, then that candidate must win”

OK, let’s create a new Score Voting election, four candidates, 100 voters,
0-9 scores:

51 votes: A:9 B:7 C:4 D:0

29 votes: A:0 B:5 C:6 D:9

11 votes: A:1 B:7 C:4 D:0

9 votes: A:2 B:0 C:9 D:2

Results: A:488 B:555 C:515 D:255

Well, in this case over half of the group ranked A the highest, but ranked B
just a little lower. I guess we could say that in one sense A did not win
even though he got the highest vote from over half. And yet I don’t feel bad
about B winning – since the 51 percent likes B almost as much, and B had
much more support than A with both the 29% and the 11%.  It seems intuitive
to say that B had more support than A in that election.  And yet, it also
seems intuitive to me to say that the Last Past the Post (LPTP) election
about B winning the above election?

Maybe we ought to toss out the Majority criterion altogether, and replace it
with something more on point? (I know, who the heck am I to suggest such a
thing, I am not sure I am suggesting it seriously as much as following the
thought process to its conclusion.)

Maybe the underlying idea is behind the desire for the Majority Criterion is
two-fold:

1)      We don’t want to see losers win. We don’t want someone winning who
is obviously (in some sense) received less support than someone they beat.
For example, LPTP: in a single vote system, where each person votes for only
a single candidate on the ballot, and then all votes are added, and the one
with the least votes wins – we do not want to see that outcome, assuming
that the intent of the ballot is presented as voting for your favorite
candidate.

2)      We also don’t want to see a FPTP election where five people run for
office, one candidate gets 26 percent of the vote and wins, but 74% of the
voters got their preferences overruled by the 26% - minority rule. So
perhaps that’s another criteria dealing with something like “the greatest
good for the greatest number” or something?

The first one can probably be worded something like “A more preferred
candidate should not lose to a less preferred candidate”, and then if needed
people can argue over what it means to be more preferred or less preferred
in different  contexts. However, in the ones where it matters, ie, LPTP it
should be obvious.

As to the second – I think that is what the Majority criterion is trying to
encapsulate. At any rate, it is the only piece that I *personally* need from
a system. I am not sure how to word it it. Perhaps something like this: “If,
as quantified by the voting system, the preference that a candidate be
elected is less than the preference that they NOT be elected, then they must
not win.”

Or more strongly, perhaps, “If, as quantified by the voting system, a
candidate cannot muster more support for their election than against it,
they must not win.”

OK, so now we have two criterion:

Losers Can’t Win: A more preferred candidate must not lose to a less
preferred candidate.  (This one may be so trivially obvious that no one ever
bothered to name it or worry about it.)

No Minority Rule: If, as quantified by the voting system, a candidate cannot
muster more support for their election than against it, they must not win.

Now for all I know, either or both of the above already exist as a criterion
that I just haven’t bumped into yet – if so, please tell me what their
proper name is so I can use it. Here’s my main question though:

As understood, under what circumstances can a voting system FAIL the
Majority Criterion, but NOT fail either Losers Can’t Win or No Minority
Rule, as defined above?

Or to put another way, if we retain LCW and NMR, do we really need the
seemingly more vague Majority Criterion?

Thoughts? Anyone? Beuler?

Will respond to points on the other Criteria in a following post, this is I
am sure quite long enough on it’s own.

-Benn Grant

eFix Computer Consulting

<mailto:benn at 4efix.com> benn at 4efix.com

603.283.6601

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