Technical Evaluation of Election Methods
An election method is a voting procedure and a set of mathematical
rules for determining the winner(s). The best election method gives the
electorate, to the maximum extent possible, the leaders they sincerely
prefer, and it minimizes their need to vote strategically (e.g., for the
"lesser of two evils"). The choice of an election method should not be
based on subjective notions, nor should it be designed to advance any
particular social, political or ideological agenda (other than fair
elections, of course). It should be based, rather, on a set of strictly
objective technical criteria. The criteria we choose are listed below,
followed by a compliance table. Of those criteria, we consider
monotonicity mandatory, and we consider compliance with Condorcet and
other criteria highly desirable.
The election methods considered here are far from a complete list of
all methods that have ever been seriously proposed, but they are the
methods we consider most important. We consider Condorcet the best
choice, because it is the only method that complies with both the
monotonicity and Condorcet criteria, as well as several others. We
consider Approval a good second choice, with the advantage of extreme
simplicity. Unlike Condorcet, Approval Voting has a realistic chance of
being adopted in the near term. Cardinal Ratings are strategically
equivalent to Approval but more difficult to implement, hence they are
not worth pursuing.
Plurality is important only because it is the current election
method. We consider Instant Runoff Voting (IRV) the worst choice, but it
is important because it is currently popular among electoral reform
organizations, unfortunately. IRV does have one point in its favor: it
requires the same voting equipment and voting procedures (ranking
candidates) as Condorcet voting, so it could possibly be a step
toward Condorcet voting. Borda is not a serious contender but is
included here because it is relatively well known.
Note that the Condorcet method has several possible variations for
resolving cyclical ambiguities, but Table 1 applies to the SSD (Schwartz
Sequential Dropping) method, which is explained elsewhere at this website.
Statement of Criterion
With the relative order or rating of the other candidates
unchanged, voting a candidate higher should never cause the candidate to
lose, nor should voting a candidate lower ever cause the candidate to
win.
Complying Methods
All the methods listed in the compliance table above are monotonic
except Instant Runoff Voting (IRV).
Commentary
In the ordinal methods (Condorcet, Borda, and IRV), a candidate is
"voted higher" by being ranked higher. In Approval Voting, a candidate
is "voted higher" by being "approved" rather than "disapproved."
In a conventional plurality system, a candidate can be "voted higher"
only by being voted for at all rather than not voted for.
Monotonicity is perhaps the most basic criterion for election methods.
Common sense tells us that good election methods should be
monotonic. Methods that fail to comply are erratic.
A simple example will prove that IRV is non-monotonic. Consider, for
example, the following vote count with three candidates {A,B,C}:
In this example, eight voters ranked the candidates (A,C), five
ranked them (B,A), and four ranked them (C,B). Candidate C was ranked
first by the fewest voters and is eliminated. Since all the voters who
ranked C first also ranked B second, B now has nine top-choice votes and
wins.
Suppose, however, that two of the voters who had ranked A first
reverse their first two preferences so their votes change from (A,C) to
(C,A). Now the vote count is:
Candidate B is now ranked first by the fewest voters and is
eliminated. Since the five voters who ranked B first also ranked A
second, A now has eleven top-choice votes and wins. Hence, the two
voters who demoted A from first to second choice caused A to
win. That is, they caused A to win by ranking A lower, without changing
the relative ordering of the other candidates. IRV therefore fails
monotonicity.
For an even more bizarre example, consider the following vote count
with four candidates {A,B,C,D}:
| 7: | A,B,C |
| 6: | B,A,C |
| 5: | C,B,A |
| 3: | D,C,B |
Applying the rules of IRV, candidate A wins. But suppose the three
voters who voted (D,C,B) now promote A from last choice all the way up
to first choice, without changing the relative order of the other
candidates. Now B wins instead of A. So by promoting A from
last to first choice, those voters caused A to lose instead of
win. An election method that allows such nonsensical anomalies is
erratic and should be rejected.
These are hardly contrived theoretical examples without practical
relevance. IRV has serious problems both in theory and in
practice. In practice, voters would soon realize, or be advised, that
they cannot safely vote sincerely, and the political system would likely
remain bogged down in a two-party duopoly just as it is today. And that
is the optimistic scenario. If a third party somehow manages to become a
strong contender, it could throw the entire political system into chaos,
just as it could in our current plurality system. (See The Problem with IRV.)
Definitions
A sincere vote is one with no falsified preferences or preferences
left unspecified when the election method allows them to be specified
(in addition to the preferences already specified).
One candidate is preferred over another candidate if, in a one-on-one
competition, more voters prefer the first candidate than prefer the
other candidate.
If one candidate is preferred over each of the other candidates, that
candidate is the Ideal Democratic Winner (IDW).
Statement of Criterion
If all votes are sincere, the Ideal Democratic Winner should win
if one exists.
Complying Methods
The Condorcet method complies with the Condorcet Criterion, but none
of the other methods in the compliance table above comply.
Commentary
The Condorcet criterion is one of the most basic criteria for
election methods. When an Ideal Democratic Winner exists, common sense
tells us that ideally he or she should win. However, the only method
listed in Table 1 that complies with the Condorcet criterion is the
Condorcet method itself, which is designed specifically to comply with
the criterion named after it.
Non-ranking methods such as Plurality and Approval could not possibly
comply with the Condorcet Criterion because they do not allow each voter
to fully specify their preferences. But IRV allows each voter to rank
the candidates, yet it still does not comply. A simple example will
prove that IRV fails to comply with the Condorcet Criterion.
Consider, for example, the following vote count with three candidates
{A,B,C}:
In this case, B is preferred to A by 12 votes to 8, and B is
preferred to C by 13 to 7, hence B is preferred to both A and C. So
according to common sense and the Condorcet criteria, B should win. But
under IRV, B does not win. According to the rules of IRV, B is
ranked first by the fewest voters and is eliminated. Again, an election
method that allows such nonsensical anomalies should be rejected. (See
The Problem with IRV.)
Definitions
A sincere vote is one with no falsified preferences or preferences
left unspecified when the election method allows them to be specified
(in addition to the preferences already specified).
One candidate is preferred over another candidate if, in a one-on-one
competition, more voters prefer the first candidate than prefer the
other candidate.
The Smith set is the smallest set of candidates such that every
member of the set is preferred to every candidate not in the set. If the
Smith set consists of only one candidate, that candidate is the Ideal
Democratic Winner (IDW).
Statement of Criterion
If all votes are sincere, the winner should be a member of the
Smith set.
Complying Methods
The Condorcet method complies with the Generalized Condorcet
Criterion, but none of the other methods in the compliance table above
comply.
Commentary
GCC generalizes the Condorcet Criterion (CC) to the case in which no
Ideal Democratic Winner (IDW) exists, thereby covering all possible
cases. If no IDW exists, then a cyclical ambiguity exists among the
members of the Smith set, and that ambiguity must be resolved in such a
way that the winner comes from that set. The commentary for CC above
applies here also.
Definitions
A sincere vote is one with no falsified preferences or preferences
left unspecified when the election method allows them to be specified
(in addition to the preferences already specified).
One candidate is preferred over another candidate if, in a one-on-one
competition, more voters prefer the first candidate than prefer the
other candidate.
If one candidate is preferred over each of the other candidates, that
candidate is the Ideal Democratic Winner (IDW).
Statement of Criterion
If an Ideal Democratic Winner (IDW) exists, and if a majority
prefers the IDW to another candidate, then the other candidate should
not win if that majority votes sincerely and no other voter falsifies
any preferences.
Complying Methods
The Condorcet method complies with the Strategy-Free Criterion, but
none of the other methods in the compliance table above comply.
Commentary
The reader may be wondering how the IDW, if one exists, could
possibly not be preferred by a majority of voters over any
other candidate. The key is that some voters may have no preference
between a given pair of candidates. Out of 100 voters, for example, 45
could prefer the IDW over another particular candidate, and 40 could
prefer the opposite, with the other 15 having no preference between the
two. In that case, it is not true that a majority of voters prefer the
IDW over the other candidate, and SFC does not apply.
In order to understand SFC, one must also understand that there are
two types of insincere votes: false preferences and truncated
preferences. Voters truncate by terminating their rank list
before their true preferences are fully specified (note that the last
choice is always implied, so leaving it out is not considered
truncation). Voters falsify their preferences, on the other
hand, by reversing the order of their true preferences or by specifying
a preference they don't really have. Suppose, for example, that a
voter's true preferences are (A,B,C,D). The vote (A) or (A,B) would be a
truncated vote, and the vote (B,A,C) or (A,C,B) would be a falsified
vote.
SFC requires that the majority of voters who prefer the IDW to
another particular candidate vote sincerely (neither falsify nor
truncate their preferences), and it also requires that no other voter
falsifies preferences. SFC therefore implies that the minority that does
not prefer the IDW to the other candidate cannot cause the other
candidate to win by truncating their preferences. (In theory, that
minority could cause the other candidate to win by falsifying their
preferences, but that would be a very risky offensive strategy
that is more likely to backfire than to succeed.) The significance of
the SFC guarantee is that the majority has no need for defensive
strategy, hence the name Strategy-Free Criterion.
The Condorcet election method was shown to comply with both the
Condorcet and Generalized Condorcet Criteria (CC and GCC) above.
Although compliance with CC and GCC are important, those criteria apply
only in the theoretically ideal case in which all votes are sincere. The
Strategy-Free criterion goes further and shows that, under certain
reasonable conditions, a majority of voters have no incentive to vote
insincerely. The fact that the Condorcet also complies with SFC
therefore enhances the significance of CC and GCC considerably.
Definitions
A sincere vote is one with no falsified preferences or preferences
left unspecified when the election method allows them to be specified
(in addition to the preferences already specified).
One candidate is preferred over another candidate if, in a one-on-one
competition, more voters prefer the first candidate than prefer the
other candidate.
The Smith set is the smallest set of candidates such that every
member of the set is preferred to every candidate not in the set. If the
Smith set consists of only one candidate, that candidate is the Ideal
Democratic Winner (IDW).
Statement of Criterion
If a majority prefers a member of the Smith set to another
candidate who is not in the Smith set, then the other candidate should
not win if that majority votes sincerely and no other voter falsifies
any preferences.
Complying Methods
The Condorcet method complies with the Generalized Strategy-Free
Criterion, but none of the other methods in the compliance table above
comply.
Commentary
GSFC generalizes the Strategy-Free Criterion (SFC) to the case in
which no Ideal Democratic winner (IDW) exists, thereby covering all
possible cases. If no IDW exists, then a cyclical ambiguity exists among
the members of the Smith set and must be resolved. The commentary for
SFC above applies here also.
Statement of Criterion
If a majority prefers one particular candidate to another, then
they should have a way of voting that will ensure that the other cannot
win, without any member of that majority reversing a preference for one
candidate over another or falsely voting two candidates equal.
Complying Methods
The Condorcet method complies with the Strong Defensive Strategy
Criterion, but none of the other methods in the compliance table above
comply.
Commentary
Compliance with SDSC means that a majority never needs any more than
truncation strategy to defeat a particular candidate, even when
countering offensive order reversal by that candidate's voters.
Offensive order reversal is the only strategy that can create the need
for defensive strategy in a Condorcet voting system.
Statement of Criterion
If a majority prefers one particular candidate to another, then
they should have a way of voting that will ensure that the other cannot
win, without any member of that majority reversing a preference for one
candidate over another.
Complying Methods
The Condorcet and Approval methods comply with the Weak Defensive
Strategy Criterion, but none of the other methods in the compliance
table above comply.
Commentary
WDSC is identical to the Strong Defensive Strategy Criterion (SDSC),
except that the phrase "or falsely voting two candidates equal" is
removed from the end. That difference allows the Approval method to
comply.
Statement of Criterion
By voting another candidate over his favorite, a voter should
never get a result that he considers preferable to every result he could
get without doing so.
Complying Methods
The Approval method complies with the Favorite Betrayal Criterion,
but none of the other methods in the compliance table above comply.
Commentary
Election methods that meet this criterion provide no incentive for
voters to betray their favorite candidate by voting another candidate
over him. FBC is the only criteria that favors Approval over
Condorcet. In fact, it is the only criteria that favors any of
the methods listed in Table 1 over Condorcet. Although Condorcet
technically fails to comply with FBC, the probability is very small that
a voter can cause a preferable result by not voting for his or her
favorite in a Condorcet system.
Statement of Criterion
Each vote should map onto a summable array, where the summation
operation is associative and commutative, and the winner should be
determined from the array sum for all votes cast.
Complying Methods
All of the methods in the compliance table above comply with the
summability criterion except Instant Runoff Voting (IRV).
Commentary
The summability criterion is the only criteria discussed on this
webpage that addresses implementation logistics. Election methods that
comply with the summability criterion are substantially easier to
implement with integrity than those that do not. All the election
methods listed in Table 1 comply except Instant Runoff Voting (IRV).
In plurality voting, each vote is equivalent to a one-dimensional
array with a 1 in the element for the selected candidate, and a 0 for
each of the other candidates. The sum of the arrays for all the votes
cast is simply a list of vote counts for each candidate.
Approval voting is the same as plurality voting except that more than
one candidate can get a 1 in the array for each vote. Each of the
selected or "approved" candidates gets a 1, and the others get a 0.
In Condorcet voting, each vote is equivalent to a two-dimensional
array referred to as a pairwise matrix. If candidate A is ranked above
candidate B, then the element in the A row and B column gets a 1, while
the element in the B row and A column gets a 0. The pairwise matrices
for all the votes are summed, and the winner is determined from the
resulting pairwise matrix sum.
IRV does not comply with the summability criterion. In the IRV
system, a count can be maintained of identical votes, but votes do not
correspond to a summable array. The total possible number of unique
votes grows factorially with the number of candidates. The larger the
number of candidates, the more error-prone and less practical it becomes
to maintain counts of each possible unique vote. It becomes impractical
with more than about six candidates.
Suppose, for example, that the number of candidates is ten. In our
current plurality system, the votes at any level (precinct, county,
state, or national) can be compressed into a list of ten numbers. The
same is true for an Approval system. For a Condorcet system, a 10x10
matrix is needed. In an IRV system, however, the number of possible
unique votes is over ten factorial -- a huge number.
Under IRV, therefore, every individual vote (rank list) must be
available at a central location to determine the winner. In a major
public election, that could be millions or even tens of millions of
votes. The votes cannot be compressed by summing as in other election
methods because votes may need to be transferred according to which
candidates are eliminated in each round.
IRV therefore requires far more data transfer and storage than the
other methods. Modern networking and computer technology can handle it,
but that is beside the point. The biggest challenge in using computers
for public elections will always be security and integrity. If many
thousands of times more data needs to be transferred and stored,
verification becomes more difficult and the potential for fraudulent
tampering becomes substantially greater.
To illustrate this point, consider the verification of a vote tally
for a national office. In our current plurality system, each precinct
verifies its vote count. The counts for each precinct in a county can
then be added to determine the county totals, and anyone with a
calculator or computer can verify that the totals are correct. The same
process is then repeated at the state level and the national level.
The point is that once the votes are verified at the lowest
(precinct) level, the numbers are available to anyone for independent
verification, and election officials could never get away with "fudging"
the numbers. At the lowest level, ballot problems such as "hanging
chads" could be a problem, but adding the vote counts will certainly not
be a problem. And this applies not only to conventional plurality
elections, it applies also to Condorcet, Approval, and even Borda -- but
not IRV
In an IRV election, the voting data cannot be "compressed" by adding
the vote totals together at each level, so verification of the tally
results becomes nearly impossible. The final result depends on all the
votes, but even if the individual votes are all counted correctly,
nobody can verify that the total pool of votes has not been tampered
with at some level of the tallying process. And with IRV's erratic
properties, someone could lower the rankings of a candidate to
make him win or raise the rankings of a candidate to
make him lose. It's a prescription for disaster and voter
cynicism.
If IRV were superior otherwise, then its failure to comply with the
summability criterion might be excusable. But IRV has been shown to fail
with respect to every one of the criteria listed, including such basic
criteria as monotonicity. To accept the additional security risks that
IRV poses would therefore be the epitome of folly.
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