Technical Evaluation of Election Methods

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.

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.

Monotonicity Criterion (MC)

Statement of Criterion

Complying Methods

Commentary

A simple example will prove that IRV is non-monotonic. Consider, for example, the following vote count with three candidates {A,B,C}:

8:	A,C
5:	B,A
4:	C,B

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:

6:	A,C
5:	B,A
4:	C,B
2:	C,A

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

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.)

Condorcet Criterion (CC)

Definitions

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

Complying Methods

Commentary

Non-ranking methods such as Plurality and Approval could not possibly comply with the Condorcet Criterion. But IRV allows voters 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}:

8:	A,B
7:	C,B
5:	B

Generalized Condorcet Criterion (GCC)

Definitions

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

Complying Methods

Commentary

Strategy-Free Criterion (SFC)

Definitions

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

Complying Methods

Commentary

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.

Generalized Strategy-Free Criterion (GSFC)

Definitions

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

Complying Methods

Commentary

Strong Defensive Strategy Criterion (SDSC)

Statement of Criterion

Complying Methods

Commentary

Weak Defensive Strategy Criterion (WDSC)

Statement of Criterion

Complying Methods

Commentary

Favorite Betrayal Criterion (FBC)

Statement of Criterion

Complying Methods

Commentary

Summability Criterion (SC)

Statement of Criterion

Complying Methods

Commentary

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 map into 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, which is 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.

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.