[EM] (with typos fixed) Self-contained organizational voting-system recommendations

[Michael Ossipoff]

I'd like to make some recommendations for voting in your organization.

The choice of voting systems (often referred to as "methods") depends
on how similar the alternatives are, how strongly their
merit-differences are felt. That determines whether you insist on
automatic majority rule enforcement, as opposed to just maximizing the
likedness of the outcome. The choice is also influenced by how
amicable the organization is. That determines how compromising the
voting system should be..

So let me suggest voting systems for various conditions:

1. Maximizing overall satisfaction is the important thing, more
important than automatic majority rule.   ...&/or maximum count-ease
is desired:

Use Approval or Score.

Approval:

Each voter can approve one or more alternatives–as many as s/he wants
to approve. To avoid vulnerability to people strategically taking
advantage of previous voting, the ballots shouldn’t be displayed until
they’re all voted.

A voter approves an alternative by writing its name, or by marking a
box next to its name, on a ballot.

Score:

Each voter gives to each alternative a rating from 0 to N, where N can
be any pre-specified number.

For instance, in 0-10 Score, each voter rates each alternative from 0
to 10. In 0-100 Score, each voter rates each alternative from 0 to
100. Of course Approval amounts to 0-1 Score.

Approval has the simplest and easiest handcount, but Score adds
flexibility, allowing built in fractional ratings , which are useful
when Approval’s yes/no would be difficult to decide, or when one
desires to strategically give somewhat less than full support to a
potential rival.
 .
 2. Automatic majority rule is desired

 …a) Conditions are amicable:
 ….Then, you want a method that elects the Condorcet winner (CW)
compromise. To not do so would be uncompromising and a bit inimical. .
I’ll define the CW:

X is socially preferred to Y if more voters prefer X to Y than prefer Y to X.

The CW is an alternative that is socially preferred to each one of the
other alternatives.

The Condorcet Criterion says:

If there's a CW, and if everyone votes sincerely, then that CW should win.

[end of Condoret's Criterion definition]

Definition of sincere voting:

A voter votes sincerely if s/he doesn’t vote an unfelt preference, or
fail to vote a felt preference that the voting system in use would
have allowed hir to vote in addition to the preferences that s/he
actually did vote.

To vote a felt preference is to vote X over Y when preferring X to Y.

To vote an unfelt preference is to vote X over Y when not preferring X to Y.

[end of sincerity definition]

Here are a few suggested voting systems for amicable conditions when
automatic majority rule is desired:.

If you want show-of-hands voting (or its Internet equivalent), &/or if
there are many alternatives, no counting software, and little time,
then do Sequential Pairwise Voting (SP):

Arrange the alternatives in a vertical list, in reverse order of their
order of proposal. …or hold a Vote-For-1 balloting, and order the
alternatives in reverse order of their vote-count scores.

Vote between the top 2 alternatives in the list. The winner then goes
against the next alternative in the list, in another 2-way vote. …etc.
Repeatedly, do a 2-way vote between the current winner and the next
alternative in the list. The winner is the last remaining unbeaten
alternative, the winner of the vote that includes the alternative at
the bottom of this list.

SP has a number of important properties:

SP meets the Condorcet Criterion:(defined above).

 SP meets that criterion.

Smith Criterion:

The Smith set is the smallest set of alternatives that all are
socially preferred to everything outside that set. If everyone votes
sincerely, then the winner should come from the Smith set.

[end of Smith Criterion definition]

SP meets the Smith Criterion, in addition to the Condorcet Criterion.

Mutual Majority Criterion:

A mutual majority (MM) is a set of voters, comprising a majority of
the voters, who all prefer a certain same set of alternatives to all
of the other alternatives. That set of alternatives is their
MM-preferred set.

If a MM vote sincerely, then the winner should come from their MM-preferred set.

[end of Mutual Majority Criterion definition]

Definition of sincere voting:

A voter votes sincerely if s/he doesn’t vote an unfelt preference, or
fail to vote a felt preference that the voting system in use would
have allowed hir to vote in addition to the preferences that s/he
actually did vote.

To vote a felt preference is to vote X over Y when preferring X to Y.

To vote an unfelt preference is to vote X over Y when not preferring X to Y.

[end of sincerity definition]

SP meets the Mutual Majority Criterion (MMC).

The chicken dilemma:

Say your faction prefer A, and another faction prefer B. Your 2
factions, combined are a majority, and both detest C, strongly
preferring A and B to C.

But if you support B, the B voters can take advantage of your
co-operativeness, by withholding support for A, and thereby winning at
your expense, even if A has more 1st choice support than B. They can
do so because you helped B. You don’t know if they’re going to
“defect” in that way, and that makes you hesitate to help B. But if
neither A not B voters help eachother’s alternative, the detested C
will win.

That’s the chicken dilemma.

For any method more complicated than Approval or Score, there's no
excuse to have the chicken dilemma.

SP doesn’t have the chicken dilemma. If the B voters defect against A
before it’s time to vote on B, then you (as an A-preferrer) will
notice that, and you can penalize them by refusing to support B when
it’s time to vote on B. If you’ve already helped B, before it’s time
to vote on A, then the B voters will  have no reason to defect against
A, because B must have already been eliminated by losing its 2-way
vote. …unless B's 1st vote, after you've helped B, is between B and A,
in which case it isn’t defection when the B voters sincerely vote B
over A.

In short, SP has no chicken dilemma.

All of the methods that I recommend, other than Approval and Score,
meet the Mutual Majority Criterion and have no chicken dilemma. SP is
such a method. There are a few more elaborately and laboriously
counted methods that have those desirable properties. If there are
more than just a few alternatives, then you might want to do a
computerized count, for these rank-balloting methods described below:

Benham’s method (also called “Benham” or “Condorcet IRV”):

Do IRV until there there is an un-eliminated alternative that beats
each of the other un-eliminated alternatives. Elect hir.

[end of Benham definition]

Definition of IRV (Instant Runoff):

Repeatedly, cross off or delete from the rankings the alternative that
tops the fewest rankings.

Of course (in plain IRV) the winner is the candidate who remains
un-eliminated, when all but one have been eliminated.

[end of IRV definition]

Actually, SP is as good as Benham, and a lot easier to count. But
Benham is deluxe, in the sense that it isn’t necessary for voters to
observe and penalize defection, because Benham automatically penalizes
defection. In Benham, it’s simply a matter of sincere ranking. A
mutual majority have no reason to do other than rank sincerely. In
fact no one has need to do other than rank sincerely.

Woodall:

Do IRV till only one member of the initial Smith set remains
un-eliminated. Elect hir.

[end of Woodall definition]

For the purpose of this definition, the Smith set is defined in terms
of actual votes:

X beats Y if more ballots rank X over Y than rank Y over X.

The Smith set is the smallest set of alternatives such that all the
set’s alternatives beat everything outside the set.

Though both Benham and Woodall always choose from the Smith set,
Woodall is more particular about which Smith set member it chooses.
For that reason, Woodall achieves slightly better social utility than
does Benham.

Schwartz Woodall:

Schwartz Woodall is like Woodall, except that it uses the Schwartz set
instead of the Smith set. Those 2 sets are identical if there are no
pairwise ties. But when there aren’t many voters, there can be
pairwise ties. Then, the Schwartz set is a bit more exclusive than the
Smith set. You can get into the Smith set by tying one of its members
and beating all the non-Smith-set alternatives. That won’t get you
into the Schwartz set. So the Schwartz set is more deluxe, for
small-electorate voting.

Schwartz Woodall:

Do IRV tilll only one member of the initial Schwarzt set remains
un-eliminated. Elect hir.

[end of Schwartz Woodall definition]

Let me define the Schwartz set. It has 2 equivalent definitions. Both
definitions define the same set:

Cycle definition of the Schwartz set:

The Schwartz set is the set of alternatives that don’t have a pairwise
defeat that isn’t in a cycle.

A pairwise defeat is what Y has, if X beats Y.

A cycle is a cyclical sequence of defeats, such as X beats Y beats Z beats X.

Unbeaten set definition of Schwartz set:

1. An unbeaten set is a set of alternatives none of which are beaten
by anything outside the set.

2. An innermost unbeaten set is an unbeaten set that doesn’t contain a
smaller unbeaten set.

3. The Schwartz set is the set of alternatives that are in innermost
unbeaten sets.

[end of Schwartz set definitions]

I recommend Schwarz Woodall as the deluxe voting system for amicable
conditions. Sequential Pairwise is really just as good, though not as
deluxe. Sequential Pairwise is, of course, much easier to count.

Benham and Woodall are discussed in a a journal article, by James
Green-Armytage, at:

http://econ.ucsb.edu/~armytage/hybrids.pdf

…b) When conditions are not amicable:

The a) methods (for amicable conditions) are ok under inimical
conditions too. But the methods described below might be preferred:

Plain IRV, instead of Schwartz Woodall. Plain IRV (defined above),
like all of the methods recommended here, other than Approval and
Score, meets the Mutual Majority Criterion and doesn’t have the
chicken dilemma. IRV doesn’t always elect the CW compromise (defined
above), making IRV less compromising, and maybe a bit inimical. But
IRV is simpler to count than Schwartz Woodall, or Benham. And, in
inimical conditions, the CW compromise might even not be desired. It
might be felt that only mutual majorities should be honored, and that
there’s no need to compromise with voters who aren’t in a mutual
majority.

b) for show-of-hands voting, or when there are many alternatives, no
counting software, and little time:

Exhaustive Balloting, also called “Elimination Voting”:

Do a Vote-For-1 vote among all the alternatives. Eliminate the
alternative that gets fewest votes. Repeat till only one alternative
remains.

Of course this could be done by show-of-hands, or its Internet
equivalent. It’s even easier to count than IRV.

Those are my recommendations. All of them except for Approval and
Score meet the Mutual Majority Criterion, and have no chicken dilemma.

Michael Ossipoff