Ideal voting methods (was Re: No Subject)

[Steve Eppley]

Hi Tom et al,

Tom Round posted (widely):
>Date:          Wed, 18 Sep 1996
>Subject:       No Subject
-snip-
>The "ideal voting situation" would have the following features.
>2    Every voter has, say, 1,000 "points" which may be
>     allocated among candidates, without restriction, in any
>     way the voter thinks will maximise his/her preference
>     satisfaction.

It's debatable whether distributing points is ideal, but if the
number of prop rep seats S is more than a few, the minutiae of the
rules aren't very critical: so what if one of the runners-up is
slightly more "qualified" than one of the marginal winners?  The
effect on society will be lost among more important forces.  But 
if S = 1 they're much more critical.

For reasons of simplicity for the voters, letting them rank the
choices in preference order (and treating choices left unranked as
if the voter had ranked them last) may be better than asking them to
allocate points.  Even a child can sort best to worst, but many
adults are math-challenged and certainly strategy-challenged.

You might also want to relax some of your axioms: the legislative
weight of each seat doesn't have to be equal, and the number of seats
could be affected by the voting results.

>3    The voting proceeds in one or more rounds. On each round,
>     the voters take turns, in the same (initially random)
>     order, to allocate their points.

What's the point of this thought experiment with multiple rounds?
How does it help decide what one-balloting method should be used?

>     A voter is deemed to
>     prefer seeing her 1st, 19th and 20th preferences elected
>     (out of 20 candidates) more than seeing her 2nd, 3rd and
>     4th preferences get up, if it comes down to a choice
>     between the two.

Some voters won't see that much difference between their 1st, 2nd,
3rd & 4th choices; they would presumably prefer electing 2-3-4
instead of 1-19-20.  (Also, there will be quite a few voters who will 
not want to express preferences on 20 candidates, even though 20 may 
seem like a small number to you.)

>Okay. Given these criteria of the model, what sort of system would
>emerge? This is the point at which opinions may legitimately
>differ. 

I guess my opinions are illegitimate?  :-)

>     For instance, what would happen in the ideal system if a
>standard "cycle" exists (ie A beats B, B beats C, C beats A
>pairwise)? On the first count, whichever has most primaries is
>leading. But then this lead isn't "stable", it won't last the next
>999 rounds.

Another reason to use one balloting of preference-ordering and a good
tally algorithm, instead of rounds and point-summing.

>I should add at this point that since 1992 I have become (after
>much soul-searching) a supporter of Condorcet pairwise systems - as
>mandatory for referendums, as highly encouraged for single-position
>elections, and as desirable even for multi-seat elections.

My opinion is similar: yay for Condorcet!  I don't think you and I
define Condorcet's method the same way, though.  Your definition
doesn't cover the cases where no candidate manages to win all
his/her pairings.  (See below for more.)  

I'd go further regarding single-winner offices: make it mandatory.  

For multiseat elections there are also reasonably good non-pairwise
methods, and I don't think the health of the democracy is affected
much by achieving a few degrees more representational accuracy. 
It's affected far more by whether the balloting is party-based or
candidate-based, and other factors (access to the intrusive media,
for instance).  A "kitchen sink" method might let each voter's
ranked ballot intermix parties, independent slates, and individual
candidates (and maybe even automatically append to the end of each ballot
the published rankings of the parties, slates, & candidates who were
explicitly ranked, in order to avoid wasting ballots which foolishly
only ranked a few losers).

It's also possible to adapt Condorcet's single-winner method to
multiseat elections to achieve proportional representation in a
non-eliminative manner.  I won't go into detail here, but here's 
the basic algorithm:

  for each seat to be filled
     elect one of the remaining candidates with Condorcet's method;
     delete the ballots which elected this candidate;
  endfor

>     In a way, pairwise comes close to squaring the circle in
>producing "PR for a single seat" - not literally, but in ensuring
>that every party over a certain (medium-low) primary vote threshold
>has a chance to win a seat

I'm not sure you'd want to call this PR even in a figurative sense.
Condorcet's method succeeds at electing the voters' centrist choice,
without the muss or fuss of organizing a compromise coalition to
support the centrist, by what could be described as an ad hoc
coalition of voters...  By eliminating the "lesser of evils" dilemma
(which holds the two "big tent" parties together, both by discouraging
potential candidates from competing because they'd throw the election
to the greater evil and discouraging voters from "wasting" votes on
minor candidates because that would throw the election to the greater
evil), it would bust up the two-party system and enable multiparty
democracy.  It should also help voter turnout considerably, since
voters would be free to rank their true and inspiring favorites
first, ahead of the centrist, without wasting their votes.  What you
get is multiparty viability (along with an unshackling of the media
and the political discourse), but centrist winners instead of prop
rep.  Having said that, though, I'll understand if you prefer to 
continue using the term prop rep for this single-winner multiparty 
centrism.  

(As an important side note, have you considered the effects of 
using Condorcet's method for single-winner-takes-district
legislative elections instead of prop rep?  It's not quite as
democratic as prop rep, in my opinion, but on the other hand it's 
a lot easier to get there from here in the U.S., since the
Constitution is silent about voting methods and leaves voting 
methods to each state, but federal law or the Constitution must be
changed to enable prop rep.  It would bust up the two-party system
and thereby make it easier to achieve PR afterward by making the
incumbents in Congress more interested in sharing power proportionally.  
I think there are even ways that states could negate the two-party
effect of the Presidential Electoral College by cleverly electing the
delegates who will give all their votes to the viable candidate with
the highest Condorcet score, with viability being defined by whether
adding these delegates to the delegates already won in the less-
enlightened states is enough to win a majority of the College.)

In the long run, nearly all single-winner offices could be
eliminated.  (I don't favor the parliamentary system, though, in
which the largest party in the legislature gets first crack at
choosing the executives.  For one thing, that still maintains a
significant two-party dynamic since some voters will vote for a
"lesser of evils" large party to prevent a worse party from choosing
the government--Israel provides evidence for this, since in the
recent election in which the Prime Minister was for the first time
elected directly by the people the two big parties lost a lot of
seats in the Knesset.  Better to let the people elect the executive
using Condorcet's method, and not allow the executive to be toppled
easily by a vote of no confidence.)  A single person is needed only 
to deal with emergencies in which a group would be too slow to
respond. So the government could consist of a series of "concentric"
circles: the small inner circles would have the advantage of speed,
but the larger outer circles would have the advantage of authority. 
Possibly 4 circles would work best: one person to handle emergencies,
a small council, a large legislature, and the citizens as a whole.
But to be flexible, any large circle ought to be able to create and 
terminate smaller circles.

>     As to how to apply the pairwise criterion ... Obviously a
>candidate who wins every comparison against every other candidate
>is Condorcet-preferred and should be elected. What if there is no
>such winner? Several options:
>
>(1)  The candidate who wins the most comparisons is elected.
-snip-
>(2)  An alternative: Elimination proceeds in something similar
>     to the traditional low-man-out manner except that, before
>     eliminating any candidate, the two lowest candidates are
>     compared pairwise.
-snip-

There are a lot more pairwise cyclic-tie-breakers than these two. In
my opinion, the best single-winner "cyclic tie" breaker is based on
what the Marquis de Condorcet hinted at: pick the one whose largest
pairing-loss is the smallest (where the size of a pairing-loss is
defined as the number of voters who ranked the pairing-winner ahead
of the pairing-loser).  There are some academic criteria not
satisfied by this method, and if you want to appease the academics
you could first apply the Smith filter to eliminate from contention
the candidates who were pair-beaten by the smallest possible set of
candidates who pair-beat everyone not in that set.  (Personally,
I don't think the Smith filter is that important, but I wouldn't 
want to provide ammunition to the anti-reform crowd when there's a
campaign fight going on over it.  The Smith filter has another use,
since Ken Arrow's impossibility theorem is undermined by defining
as "relevant" all candidates in the Smith set.)

The 1st method you cited is known as Copeland, and it fails some
criteria I think are important: majority rule, independence from
twins, and no "lesser of evils" (LOE) dilemma.  The 2nd method is a
variation of a well-known method with many names (MPV, Hare's
method, Instant Runoff, etc.) and it also fails majority rule and 
LOE.  I want to point out that methods which fail LOE don't
do as good a job at establishing true multiparty democracy, for the 
reasons cited above.

Here's an example which illustrates the majority rule and LOE
failures of the two methods you listed:

Suppose 3 candidates (Left, Center, and Right) are competing for 1 
seat.  Suppose 100 voters vote as follows:
   46:  L > C=R
   20:  C > L=R
   34:  R > C > L
  ---
  100
Neither wing is a majority, and everyone has ranked the Center
candidate as either first or second choice.  
  Tallying the pairings:
  R loses to L (34 < 46, with 20 abstentions)
  L loses to C (46 < 54)
  C loses to R (20 < 34, with 46 abstentions)
The first method you listed produces a 3-way tie, since each
candidate has one pair-win, so you'd better identify an additional
tie-break algorithm to deal with this common occurence.  The second
method you listed eliminates C, then R, electing L, even though
there's a majority of 54 who want to defeat L.  (The fallacy of the 
elimination method is that is erroneously defines "candidate 
weakness" by relying on voters' highest preferences, ignoring their
nearly-as-high preferences.)  

The method I defined above, which we in election-methods-list call
Condorcet's Method, elects C here since the size of C's largest
(only, in this small example) pair-loss (34) is smaller than the
size of the other candidates' largest pair-losses (46 & 54).  

Because those two methods elect L--I'm making an assumption about
your unstated choice of Copeland-tie-breaker--the voters who prefer
R more than C have the LOE dilemma.  The only way they can defeat
their greater evil L is to vote the lesser evil C at least as high 
as their true favorite R.  This doesn't contribute to multiparty 
democracy...

I won't elaborate here about Copeland's violation of the
Independence from Twins criterion, since it takes a 5-candidate
example to illustrate how it can rip off a poor party that has the
best candidate, when richer parties can afford to field more 
candidates.

>Perhaps as well as the threshold suggested above, the pairwise rule
>should also "kick in" only if there is one seat to fill, or once
>only one seat out of several remains vacant and eliminations begin
>to take place. (It would not apply if, say, three candidates are
>all elected on either primary or surplus votes before anyone needs
>to be excluded from the count.)

I agree with that last sentence, but there are additional cases "in
between" the ones covered by those two sentences (since eliminations
may begin in STV when more than one seat remains to be filled), so
maybe Condorcet's method could "kick in" when eliminations begin.
Perhaps I'm failing to understand what you're saying about a 10% 
threshold?

>     Maybe the rule in such cases could be "eliminate the lowest,
>unless (a) the lowest is over the 10% threshold, and also (b) in a
>pairwise comparison against the lowest, the second-lowest polls
>less than 1/ [vacant seats remaining plus 1] of the remaining
>votes". Ie, when two seats remain, the second-lowest only needs
>one-third to escape elimination.

I don't know why you're making it so complex, since it's important
that the voters comprehend how the system will work.  To speed up a
tally which will be done by computers anyway?  STV already has a
quota needed to win a seat; if you're including a second threshold I
think it's not important. 

Or perhaps you're talking about electing a large legislature
at-large where thousands of candidates would be competing and the
pairwise tally time would be horrendous.  That's true, but this
system would place too much of a burden on the voters anyway to be
viable: don't expect them to rank much more than about a dozen each.
 Which means either an at-large party-based system (where, ideally,
voters can intermix parties, slates, and candidates in their
rankings) or a multi-winner (around 11 to 19, perhaps) districted
system in which only a few dozen candidates would be expected to
compete in each district.  If it's a multiwinner districted system
and you want to take into account the voter turnout, reduce the
importance of district population differences, and create an
additional incentive for the people to vote, the number of seats 
in each district could be set after the voting to be proportional 
to the district's turnout.  (Watch out for ballot box stuffing, 
though.  :-)

* *

Tom's message, to which I'm replying below, was cc:ed to quite a few
addresses, and I saw it in election-methods-list.  If you want me to
see your replies to this, please include election-methods-list in
your To: or Cc: line.

I was unaware of the existence of the c-p-r and voting-methods lists.
I'm not sure what the scope of those lists are, but perhaps we
should think about consolidating some of these lists.  And
personally, I'll think about subscribing to them.

--Steve Eppley,
  list administrator for <elections-reform at igc.apc.org>