Elections: Single-Winner Methods

[Rob Lanphier]

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Here is the Election Methods List FAQ, version 1.  
-- 
Rob Lanphier
robla at eskimo.com
http://www.eskimo.com/~robla

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Properties, Standards & Criteria

for Evaluating & Comparing Multi-Alternative Single-Winner Social Choice
Methods.

And Comparison of Some Methods by Those Standards:

by Mike Ossipoff (dfb at cruzio.com)

OA Single-winner methods are for choosing 1 alternative from among several.
Typically, in partisan elections, the 1-Vote Plurality method is used.
People vote for 1 alternative (or candidate), and the one with most votes
wins. Other methods are proposed as improvements.

Various different single-winner methods, differing greatly from eachother,
are proposed. There's no consensus on standards for evaluating them. I'm
going to describe some standards for single- winner methods.

It's a question of what one wants from a single-winner method. What don't we
like about the single-winner method currently in use, the 1-Vote Plurality
method?

The "lesser-of-2-evils" problem. Voters know that if they vote for their
favorite, then they won't be able to vote for a more winnable compromise,
and won't be helping that compromise beat someone much worse. So they feel
compelled to vote for the "lesser-of-2-evils", the more winnable, but less
liked, compromise. They have to abandon their favorite. That's why we want
better single-winner methods.

So that's the problem. Now I'll list some proposed solutions, and then
discuss how well they solve the lesser-of-2-evils problem.

METHODS:

When discussing standards, & properties, I'm going to compare 5 methods by
those standards & properties:

1. Approval
2. Condorcet's method
3. Young's method
4. Copeland's method
5. Elimination (also known as "Preferential", "MPV", "the Alternative Vote",
"Hare's method")

Definitions of the methods:

1. Approval:

Differs from 1-Vote Pluality in that voters may vote for more than 1
alternative if they so choose, giving a whole vote to each alternative they
vote for.

Said to be a widely used alternative to 1-Vote Plurality.

The remaining 4 methods are rank-balloting methods. Voters rank some
alternatives in order of preference: 1st choice, 2nd choice, 3rd choice, ...
etc.

2. Condorcet's method:

A beats B if more voters have ranked A over B than vice versa. If 1
alternative beats each one of the others, then it wins.

If no 1 alternative beats each one of the others then the winner is the
alternative over which fewest voters have ranked any 1 particular
alternative that beats it. In other words:

For each alternative, determine which alternative that beats it is ranked
over it by the most voters. The number of voters ranking that alterntive
over it is the measure of how beaten it is. The winner is the alternative
least beaten by that measure.

Condorcet was the French 18th century founder of voting theory. With regard
to the lesser-evils problem, Condorcet's method has important properties not
possessed by the other methods. They'll be described after the methods are
defined.

I should say now that I claim that Condorcet's method is by far the best of
these methods, and the one that I recommend.

3. Young's method:

As with Condorcet's method, A beats B if more voters have ranked A over B
than vice-versa.

If 1 alternative beats each one of the others, then it wins.

If no alternative beats each one of the others, then the winner is the
alternative with the smallest "overall margin" against it.

The "overall margin" against an alternative is the sum of the margins by
which various alternatives beat it. The margin by which A beats B is the
number of voters ranking A over B minus the number of voters ranking B over
A.

4. Copeland's method:

As before, A beats B if more voters rank A over B than vice-versa.

The winner is the alternative for which the number of alternatives it beats,
minus the number of alternatives that beat it, is the greatest.

Copeland's method often returns a tie, and must have a tie-breaker method
specified with it.

5. Elimination: Repeatedly eliminate from the rankings the alternative
occupying highest position in fewest rankings. Very popular. Not very good.

*****

PROPERTIES & STANDARDS:

Lesser-of-2-evils guarantees:

Approval guarantees that if a full majority of all the voters have voted A
over B, that automatically defeats B, without anyone having to vote A over
any alternative which they prefer to A.

Condorcet's method guarantees that if a full majority of all the voters vote
A over B, then there's always a way that they can vote that will ensure the
defeat of B, without anyone voting A equal to or over any alternative that
they prefer to A.

In fact, under all reasonable & plausible conditions, with Condorcet's
method, if a full majority of all the voters vote A over B, that will
automatically defeat B, without anyone voting A equal to or over any
alternative that they prefer to A.

What do I mean by "reasonable & plausible conditions"?

There is a "Condorcet winner, and order-reversal doesn't occur on a scale
sufficient to change the election result.

A "Condorcet winner" is an alternative that would beat each one of the
others if everyone sincerely ranked all the alternatives-- in other words,
if everyone sincerely voted all their preferences.

"Order-reversal" is the practice of voting a less-liked alternative over a
more-liked one. Highly implausible voting behavior, very unlikely to happen
on a large scale.

Condorcet's method has an additional property:

Even if everyone mistakenly believes a certain alternative to be the
necessary compromise, and everyone includes that alternative in their
ranking, and no one includes in their ranking any alternative that they like
less than it, that can't give the election away to that alternative if
there's a Condorcet winner which a majority of the voters have ranked over
it.

This is a re-wording of one of Condorcet's method's lesser-of-2-evils
guarantees. It's a very important property. It's quite possible for all the
voters to mis-estimate winnability, to underestimate the winnability of some
candidate, for instance.

I call this property "invulnerabilty to mis-estimate".

***** Condorcet's method, Young's method, & Copeland's method are members of
a class of methods called "Condorcet criterion methods. They meet
Condorcet's criterion, which requires the election of an alternative that
beats each one of the others.

The election of the Condorcet winner is important. Since it possesses a
majority over each one of the other alternatives, meaning it's preferred to
each one of the other alternatives by a majority of those voters having a
preference between the two, it's the rightful winner, and it can obviously
easily be made to beat any other alternative. It's important that members of
a majority preferring it to another alternative be able to make it beat that
other alternative without abandoning more liked alternatives. As you can
tell, this is a re-statement of the desirability of the lesser- of-2-evils
guarantees described earlier. In this section I'll approach the subject from
a different starting point.

The desirability of electing the Condorcet winner is perhaps the only
proposition approaching a consensus, on the subject of voting systems.

The Condorcet criterion methods, as I said, always elect an alternative that
beats each of the others. The Condorcet winner is the alternative that would
beat each of the others if everyone voted a sincere ranking of all the
alternatives.

In a Condorcet criterion method, there are 2 things that can defeat a
Condorcet winner: trunction & order-reversal.

Truncation is the voting of a short ranking that doesn't include all the
alternatives (it can result in the Condrocet winner's defeat if it doesn't
include the Condorcet winner). Order-reversal, as I said earlier, is the
ranking of a less-liked alternative over a more-liked one.

So then, since it's generally agreed that we should elect the Condorcet
winner, and there are only 2 things that can defeat a Condorcet winner in a
Condorcet criterion methods, it's of interest how to prevent those things
from defeating a Condorcet winner:

A method is "truncation-resistant" if truncation can never gain the election
of an alternative over which a full majority of all the voters have ranked
the Condorcet winner.

Condorcet's method is the only truncation-resistant method, of the 3
Condorcet criterion methods compared in this article.

A method is "order-reversal resistant" if:

If a full majority of all the voters have ranked the Condorcet winner over a
certain other alternative, then, even if order-reversal is used, there's
always a way that members of that majority can vote to ensure the defeat of
that other alternative, while still voting any preferences that they have
between alternatives that they prefer to that other alternative.

This is an awkward re-statement of one of Condorcet's methods's
lesser-of-2-evils guarantees. The definition of truncation-resistance is
based on Condorcet's method's other lesser-of-2-evils guarantees.

The purpose here is to put these properties in terms of protecting the
Condorcet winner against the only things that can defeat a Condorcet winner
in a Condorcet criterion method.

Condorcet's method is the only one of the 3 Condorcet criterion methods
comprared in this article that is order-reversal resistant.

A method is "Condorcet protective" if it is a Condorcet criterion method,
and is truncation-resistant, and is order-reversal resistant.

Condorcet's method is the only Condorcet protective method described in this
article. ***

Condorcet's method's lesser-of-2-evils guarantee, I should have added in
that section, is unmatched by any other method described in this article.
Condorcet's method & Approval are the only one of these 5 methods that have
such a guarantee.

Something much weaker can be said for Copeland's method & Young's method:
Under the conditions that I called "reasonable & plausible", Copeland &
Young guarantee that if a full majority of all the voters rank A over B,
there's a way that they can vote that will ensure the defeat of B, without
voting A over any alternative they prefer to A. (they might however have to
vote A equal to an alternative they prefer to A).

*****

Academic criteria.

The academics have written several criteria. I'll name a few that are most
often violated:

1. Condorcet criterion
2. Smith Criterion
3. Condorcet loser criterion
4. Majority criterion

There are a few issues of interst about these criteria. They're sometimes
defined according to voters' psychological preferences, and sometimes in
terms of actual votes. When defined in terms of psychological preference,
they actually aren't met by any methods.

I define them in terms of actual votes, as I did earlier when I defined the
Condorcet criterion as requirement that we elect any candidate that beats
each one of the others.

But if they're defined that way, in terms of actual votes, in terms of what
beats what, then, strictly speaking, they're met by Approval & 1-Vote
Plurality, two methods not usually considered to meet these criteria. For
instance, Approval or 1-Vote Plurality can be considered a system where one
can rank alternatives either 1st or last. In Plurality one can only rank 1
alterntive in 1st place. So then, if in Approval or Plurality, a certain
alternative beats each of the others, by being "ranked" over it by more
voters than vice-versa, that means it will have more votes than each of the
other alterntives, and will win.

If it's felt desirable to define the Condorcet criterion, and the other 4
criteria in that list, so that Approval & Plurality don't meet them, then it
might be best to add a requirement, to those criteria, that the method allow
as many rank positions as the voter wants to use.

***

Copeland's method meets all 4 of those criteria. Condorcet's method meets
the Condorcet criterion & the majority criterion. (any method that meets the
Condorcet criterion meets the majority criterion).

To me, those 4 academic criteria, except for the Condorcet criterion, seem
less important than the properties & standards described earlier, which deal
with the lesser-of-2-evils problem. That is, the academic criteria other
than the Condorcet criterion don't address the reason why we want better
single-winner methods. So, though Condorcet's method, in its plain form, as
I've defined it, doesn't meet all the academic criteria, I claim that the
properties that it has are much more important than the properties that it
doesn't have. I'll later show how any method, including Condorcet's method
can be made to meet all the academic criteria.

Definitions of those 4 academic criteria:

1. I've already defined the Condorcet criterion.

2. The Smith criterion says that if there's a set of alternatives such that
every alternative in the set beats every alternative outside the set, then
the winner should be chosen from that set.

3. The Condorcet loser criterion says that if an alternative is beaten by
every other alternative, then it shouldn't be the winner.

Note that when Condorcet's method fails that criterion, it's when _every_
alternative is beaten, and every other alternative is more beaten by some
other alternative than is the winner chosen by Condorcet's method. Also, as
I said, I claim that the important properties that I've described for
Condorcet's method are more important than these more cosmetic properties
that it doesn't have.

4. The majority criterion says that any alternative that is the favorite of
a majority should win.

Copeland's method meets all 4 of these criteria. Condorcet's method , in its
plain form as I've defined it, meets the Condorcet criterion & the majority
criterion, but not the Smith criterion or the Condorcet loser criterion.
That's also true of Young's method. Elimination, of these 4 criteria, meets
only the majority criterion.

***

But any method can be made to meet all 4 of those criteria by limiting its
choice to a _subset_ of the alternatives, called the "Smith set":

The "Smith set" is the smallest set of alternatives such that every
alternative in the set beats every alternative outside the set.

I should add that Condorcet's method retains all the properties that I've
described for it, whether or not its choice is limited to the Smith set.

Obviously it would be desirable to define Condorcet's method so that its
choice is limited to the Smith set, to avoid unnecessary criticism. But when
it is defined that way, people tend to say the definition is too
complicated. Condorcet's method is well accepted when the Smith set isn't
included in its definition. So I'd suggest that Condorcet's method be
proposed in its plain form, without any mention of the Smith set. Then, at
some later time, the provision to limit its choice to the Smith set could be
added, as a separate initiative or proposal, after the adoption of
Condorcet's method, or at least after its plain definition is well-known.

If anyone brings up the objection that Condorcet's method doesn't meet
Smith's criterion or (especially) the Condorcet loser criterion, that would
be a good opening for adding to the proposal the provision to limit the
choice to the Smith set.

*****

A note about Elimination: It's popularity makes it necessary to mention some
of its problems:

Elimination is the only rank-balloting method that can fail to elect a
Condorcet criterion even if everyone ranks it 1st or 2nd.

Elimination possesses no lesser-of-2-evils guarantees. Even if a full
majority vote A over B, in Elimination, sometimes the only way they can
ensure the defeat of B is to vote A by itself in 1st place. (Elimination
doesn't allow voting more than 1 alternative in 1st place, or in any other
rank position). A, I emphasize, may not be those voters 1st choice, but they
must sometimes rank it alone in 1st place to defeat a more disliked
alternative, B.

*****

A few notes on other methods:

Runoff-Pairwise shares Condorcet's method's important properties, relating
to getting rid of the lesser-of-2-evils problem & protecting the Condorcet
winner.

Runoff-Pairwise:

If no 1 alternative beats each one of the others, then simply hold a 2nd
balloting between all the alternatives (or all the alternatives in the Smith
set), using 1-Vote Plurality or (preferably) Approval.

The results of the 1st balloting, showing what beats what, will make it
quite obvious to everyone how far, if at all, they need to compromise in the
2nd balloting.

Only a few close relatives of Condorcet's method share the important
properties that it & Runoff-Pairwise have.

***

The commonly used municipal "Runoff" system possesses all the disadvantages
& problems of Elimination.

Mike Ossipoff --

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