Election methods and criteria

View colorful table of methods and criteria compliances

My intention is to focus on criteria which seem important from a practical standpoint (so I would say, not consistency or reverse symmetry), and present methods that satisfy interesting combinations of these criteria. My hope is that this page will provide a selection of information quite different from other pages on this subject.

Thank you for visiting.

Kevin Venzke, stepjak@yahoo.fr.

(Last updated Dec 29, ‘06. But there have not been major changes since about May ‘05.)

Contents

Random Ballot. First Preference Plurality. Instant Runoff. Approval. Condorcet methods.

Descending Solid Coalitions. Improved Condorcet Approval. Minimum Opposition. Equal Majorities. The Vote For and Against.

Borda. Borda Elimination. Raynaud. Bucklin.

Plurality. Majority Favorite. Majority. Condorcet. Minimal Defense. Clone Independence. Monotonicity. Mono-add-top. Participation. Later-no-harm. Later-no-help. Favorite Betrayal. Sincere Favorite.

Proof that Plurality, Minimal Defense, and Later-no-harm are mutually incompatible.

When the result of an election is a tie, Random Ballot is often suggested to break it. A significant advantage that RB has over Random Candidate for this application is that the latter fails Clone Independence: A faction represented by more candidates will have an advantage. First Preference Plurality has this problem in the other direction: A faction will be penalized for being represented by multiple candidates.

There are some common criticisms of IRV: On my table of methods and criteria, IRV is the only method which fails Monotonicity. It is also quite difficult to count: The ballots need to be counted multiple times, and it's not possible to to find the IRV results on subsets of the ballots (for instance, in different cities) and somehow "merge" the results to get a final result.

A final major criticism is that IRV doesn't actually do a very good job of eliminating the spoiler effect. Consider this election:

49 voters: A
24 voters: B
27 voters: C>B

It is easy to imagine that the major competitors in this election are candidates A and B. But candidate C does seemingly well: He receives more first preferences than B. Unfortunately, this causes IRV to eliminate B. Then C is unable to beat A, so that A wins. But if C had not entered the race, or if C's supporters had (strategically) decided not to vote for C, then B would have won this election. It would be preferable for the C voters to not have incentive to vote so insincerely. (Candidate A being elected despite a majority preferring B to A in this scenario shows IRV's failure of the Minimal Defense criterion.)

A considerable advantage of IRV is that, even if voters have the option of truncating, they have no strategic incentive to do so. That is, by ranking additional candidates, one can't cause a higher preference to lose. This is called the Later-no-harm criterion. An attempt by Douglas Woodall to preserve this feature while improving on IRV's monotonicity is the Descending Solid Coalitions method.

In Approval, one votes for as many candidates as desired, and the candidate with the most votes is the winner. This method is no harder to count than Plurality, and it's very easy to explain.

Approval also has some attractive properties. It satisfies the Participation criterion, meaning that one can't get a preferable candidate elected by abstaining from voting as opposed to casting a (sincere) vote. (Plurality satisfies this as well.) Approval also satisfies the Sincere Favorite criterion, meaning that one can't get a preferable result by not voting one's favorite candidate in the highest position. Besides Random Ballot, no other method satisfies both of these criteria in the general case.

An obvious shortcoming of Approval is that the voter is very limited in expressing his preferences: He must divide all candidates into two groups. This results in a serious failure of Later-no-harm, since a set of voters could cause their preferred candidate to lose, by approving an additional candidate. Voters sensitive to this might consequently approve too few candidates, in which case Approval would be barely different from Plurality, and would suffer from spoiler candidates just as badly.

However, if voters are strategic, and have a reasonable guess as to which candidates are likely to win, this problem shouldn't occur too often: Voters preferring likely winners would not approve other candidates, and voters supporting unlikely winners would realize the need to additionally approve a more likely winner, as a compromise.

Condorcet methods satisfy the Condorcet criterion, which says that if there is a candidate who could beat any other candidate in a one-on-one election (i.e., if there is a candidate who "pairwise beats" every other candidate), this candidate must be elected. This candidate is called the Condorcet winner.

These methods tend to go a long way towards addressing the spoiler effect. Whereas IRV can inadvertently eliminate a major candidate due to a low number of first preferences, Condorcet methods look directly at the head-to-head vote totals.

Condorcet methods aren't without disadvantages: They necessarily fail Participation, Later-no-harm, and the Sincere Favorite criterion (my version of the Favorite Betrayal criterion).

Condorcet methods also tend to be vulnerable to burial strategy: Ranking a candidate X insincerely beneath another candidate Y in attempt to make X lose without making Y win. A common scenario (relevant to the specific methods I consider here) is that some voters predict that candidate Y will receive so many votes against him in some pairwise contest that Y can't possibly be elected. In this case, ranking Y above X becomes safe: It may defeat X, and can't possibly elect Y. Condorcet methods which satisfy Minimal Defense allow this strategy to be thwarted when X's supporters simply don't rank the candidate whom the burying voters are trying to elect.

There are many Condorcet methods, due to the fact that the Condorcet criterion doesn't specify how to elect a candidate when there is no Condorcet winner. The "best" of these methods satisfy Monotonicity, Clone Independence, Majority, Minimal Defense, the Plurality criterion, and the Smith criterion (which says that the winner must come from the smallest set of candidates, such that each candidate in this set pairwise beats every candidate outside the set).

Examples of methods not satisfying all of these criteria include Borda Elimination, Raynaud, Minmax, and methods that measure defeat strength as the margin of defeat.

In this document I will focus on the Schulze method, but this is an arbitrary choice. I could also choose Tideman's Ranked Pairs method (but not the version using margins of defeat) or Jobst Heitzig's River method. All three methods satisfy the same properties. In the three-candidate case they elect the same candidates.

Using this definition, Schulze satisfies the Plurality criterion and Minimal Defense criterion. It is also possible to define the strength of a beatpath based on the margin of defeat: It would be the smallest value of v[i,j]-v[j,i]. But then these two criteria are not satisfied, so I don't recommend the use of margins. (In the table, this variation is labeled "Schulze (Margins).")

Another way to find the Schulze winner involves repeatedly finding the Schwartz set, eliminating non-members of it, and (if more than one candidate remains) replacing the weakest pairwise victory with a tie. The process stops once either only one candidate remains, or there are no more victories that haven't been replaced with ties.

T. Nicolaus Tideman's Ranked Pairs method: Every pairwise victory is sorted according to its strength. For instance, if v[a,b]=40 and v[b,a]=35, then the defeat A>B has a strength of 40 votes. From strongest to weakest, each defeat is taken in turn, and locked if locking it would not create a cycle of defeats (for instance, A>B>C>A) with defeats already locked. If a defeat can't be locked, it's ignored. Note that if the defeat A>B is locked, then A also obtains a locked win over every candidate over whom B has a locked win, even if A did not defeat these candidates. Once all defeats have been considered, typically there is only one candidate who does not have a locked defeat against him, and this candidate wins.

Jobst Heitzig's River method is identical to Tideman's method, except that one additionally ignores (without locking) defeats against candidates against whom some defeat has already been locked. This is possibly the easiest of the three to work out by hand.

DSC is a method devised by Douglas Woodall, partly in attempt to improve upon Instant Runoff. It is similar to his Descending Acquiescing Coalitions, or DAC, which he introduced in D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting Matters Issue 6 (1996), pp. 9–12. On page 10 he writes, "My aim is to find a system that satisfies majority and as many of the monotonicity properties as possible." He says that DAC "is the first election rule that I am really happy with," and that it is the only rule that he would be "prepared to recommend as preferable" to IRV.

I consider DSC here rather than DAC because DSC satisfies Later-no-harm and is easier to understand. DAC satisfies Later-no-help instead, and also two other of Woodall's monotonicity criteria which I don't consider here.

Of the criteria I am considering (see the table), DSC satisfies the same criteria that IRV does, and also satisfies Monotonicity and Participation. In fact, DSC and DAC are the most complicated methods I'm aware of which satisfy Participation.

The DSC procedure is as follows: The voter submits a ranked ballot. Each possible set of candidates is given a score equal to the number of voters solidly committed to those candidates. A voter is solidly committed to a set of candidates if he ranks every candidate in the set strictly higher than every candidate not in the set.

Then each set is taken in turn, from those with the highest scores to those with the least. When considering a set, each candidate in the set is determined to "finish above" every candidate outside the set, unless it was earlier determined that the other candidate would finish above this candidate. By the time all sets have been considered, it should be the case that exactly one candidate finishes above every other candidate, and this candidate is elected.

There are two clear disadvantages to DSC. One is that it is relatively difficult to count (although unlike IRV, the ballots can be counted in multiple locations, with the results summed afterwards). The other disadvantage is that DSC fails the Sincere Favorite criterion worse than IRV does: If you vote a very weak candidate as your first preference under IRV, most likely the candidate will be quickly eliminated, and your vote will still be useful. Under DSC, having a weak candidate at the top of your ranking will prevent your vote from being counted towards any sets other than those containing this candidate, so that it's very likely that your vote will have no effect (just as under Plurality). (However, since DSC satisfies the Participation criterion, unlike IRV, it is guaranteed that at least your vote won't harm the result from your perspective.)

DSC is not as truncation-resistant as IRV. That is, under Instant Runoff, adding or removing preferences beneath some candidate on some ballots can never change whether this candidate wins or loses. Under DSC, adding preferences is never harmful, but removing preferences could be. This is a failure of Woodall's Later-no-help criterion. I don't consider this criterion to be as clearly important as Later-no-harm, since Later-no-help failures typically reward voters who provide complete rankings.

This method is an attempt to bring Sincere Favorite compliance to a Condorcet-like method, without failing Plurality or Minimal Defense as Minimum Opposition does. In order to maximize the number of criteria satisfied, I will present it as a ranked ballot method, although I suggest that it's more intuitive to use it on three-slot ballots.

The voter provides a ranking of the candidates. Those candidates for whom the voter explicitly ranks are considered "approved;" in order to maximize the number of criteria satisfied, suppose that the voter may explicitly rank all candidates, even his strict last preference.

A candidate X pairwise beats another candidate Y, if the number of voters ranking X over Y plus the number ranking X and Y tied in the top position is greater than or equal to the number of voters ranking Y above X. This is slightly different from the ordinary definition of a pairwise win.

Another version of the method, which seems more agreeable to some, considers not the number of voters ranking X and Y tied in the top position, but instead the number of voters ranking X and Y tied and approved. There is not a difference in merit.

If there is at least one candidate who pairwise beats every other candidate, then the one of these candidates approved on the most ballots wins. Otherwise, whichever candidate is approved on the most ballots wins.

This method comes very close to satisfying Condorcet, since it always elects a candidate who pairwise beats every other candidate. But it doesn't strictly satisfy it, due to the fact that "pairwise beat" is defined slightly differently, so that it is possible for two candidates to both beat each other pairwise.

This difference in definition is what permits the method to satisfy the Sincere Favorite criterion: Lowering a candidate from the top rank can't cause another top rank candidate to pairwise beat the lowered candidate, because "tied at the top" rankings are counted as votes for each side over the other. (The final step of the method is equivalent to Approval, which clearly satisfies Sincere Favorite.)

I suggest that this method could be used with three-slot ballots as a nice expansion of Approval. I suggest that the top two slots be considered "approved."

The method is essentially Condorcet//Approval; i.e., if there is a Condorcet winner, he is elected; otherwise the Approval winner is elected. (The special treatment of being "tied at the top" is just a patch to satisfy Sincere Favorite, and doesn't greatly change the method.)

I consider this method rather intuitive. The voter divides the candidates into three groups, hoping that a candidate in the first group will be the decisive winner. If there is no decisive winner, then no voter's first-slot candidates are as viable as hoped, and every voter reluctantly compresses the top two slots. (And note that voters unhappy with a decisive winner could not defeat him using this compression.) When this is done, there is inevitably a decisive winner (since under Approval voting there can't be cyclic defeats).

Assuming it is not possible to rank among disapproved candidates, this method isn't as vulnerable to the burial strategy that Condorcet methods tend to suffer from. Burying a candidate X insincerely beneath candidate Y can only be effective if this prevents X from being a decisive winner (one who pairwise beats everyone else). But then the burying voter is obligated to approve candidate Y, which is very risky and can't help in defeating X.

Minimum Opposition is a Condorcet-like method which is generally overlooked due to the fact that it doesn't strictly satisfy the Condorcet criterion.

Suppose that v[a,b] is the number of voters ranking A above B. Let opposition[a] signify the greatest value of v[x,a] where X is some other candidate. Then Minimum Opposition elects the candidate(s) will the least opposition. (I assume that ties will be broken by Random Ballot.)

This method is defined, for instance, in Jonathan Levin, Barry Nalebuff, "An Introduction to Vote-Counting Schemes", Journal of Economic Perspectives, vol. 9, no. 1 (1995), pp. 3–26, as the "Simpson-Kramer min-max rule." (They assume all voters cast a complete ranking, in which case this method does satisfy Condorcet.)

Condorcet methods need to compare v[a,b] with v[b,a] to establish which of A and B defeated the other pairwise. In Minimum Opposition, these comparisons don't take place, which explains why Condorcet is not strictly satisfied. However, this simplicity also allows Minimum Opposition to satisfy two valuable criteria that are difficult to satisfy: Later-no-harm and Sincere Favorite.

Consider the scenario discussed under Instant Runoff:

49 A
24 B
27 C>B

Minimum Opposition returns a tie between B and C. If Random Ballot is used to pick a winner, then B is elected with 24/51sts or 47.06% probability, and C is elected with 27/51sts or 52.94% probability. This shows a considerable weakness of the method: There can be ties in seemingly unlikely scenarios. Another weakness apparent here is that C is elected with greater probability (52.94%) than A (0%), even though A has more first preferences than C has any preferences. This is a failure of the Plurality criterion.

Here is a scenario which illustrates Minimum Opposition's failure of Clone Independence, Majority, and Minimal Defense when there are more than three candidates:

20 A>B>C>D
20 B>C>A>D
20 C>A>B>D
13 D>A>B>C
13 D>B>C>A
13 D>C>A>B

Minimum Opposition elects D here. This is a failure of Clone Independence, since if A, B, and C were combined into a single candidate, that candidate would win. It's a failure of Majority, because more than half of the voters prefer all of A, B, and C to D, yet all three lose. Finally, this is a failure of Minimal Defense, because although more than half of the voters have ranked A (for example) above D, and haven't ranked D above anybody, D is still elected.

Minimum Opposition is vulnerable to burial strategy, like most Condorcet methods.

The CDTT is a set of candidates defined by Woodall to include every candidate A such that, for any other candidate B, if B has a majority-strength beatpath to A, then A also has a majority-strength beatpath back to B. (See Schulze for a definition of a beatpath.) Another definition (actually, the one Woodall chooses to use) of the CDTT is that it is the union of all minimal sets such that no candidate in each set has a majority-strength loss to any candidate outside this set. (Candidate A has a "majority-strength loss" to candidate B if v[b,a] is greater than 50% of the number of cast votes.)

Markus Schulze proposed this set earlier, in 1997. His wording was to take the Schwartz set resulting from replacing with pairwise ties, all pairwise wins with under a majority of the votes on the winning side.

In this document, I will use the term Equal Majorities to mean electing a candidate from the CDTT using Random Ballot. (This is intended to be in the same vein as the name of Steve Eppley's Maximize Affirmed Majorities method.) I view Equal Majorities as an attempt to correct Minimum Opposition's failure of Clone Independence, Minimal Defense, and Majority, while sacrificing as little Later-no-harm compliance as possible.

(Rather than Random Ballot, the CDTT could be paired with Plurality, IRV, DSC, or even Minimum Opposition, as these methods all satisfy Later-no-harm. One would first use the companion method to generate a ranking of the candidates, and then elect the highest-ranked candidate who is in the CDTT. However, only DSC can be used in this way without sacrificing either Monotonicity or Clone Independence.)

When increasing v[x,y] can never harm any candidate other than Y, the Later-no-harm criterion is satisfied. Under Equal Majorities, increasing v[x,y] usually cannot hurt any candidate other than Y. In the three-candidate case, this is guaranteed. In the four-candidate case, a few vulnerabilities crop up, which I will list here. In all of the following scenarios, "A>>B" means "A has a majority-strength win over B," and there are four candidates lettered A through D.

1. Suppose B>>D and D>>A. The CDTT is {b,c}. Now suppose some A>B votes are added so that A>>B. Now the CDTT is {a,b,c,d}, so that C's probability of election may be reduced.
2. Suppose B>>D, D>>C, and C>>B, so that the CDTT is {a,b,c,d}. Add in some A>B votes, so that A>>B. Now the CDTT is {a}, so that candidates C and D may be harmed.
3. Suppose B>>D, D>>C, C>>B, and C>>A, so that the CDTT is {b,c,d}. Add in some A>B votes, so that A>>B. Now the CDTT is {a,b,c,d}, so that candidates C and D may be harmed.

I wrote a simulation to compare Equal Majorities (selecting a CDTT winner using not Random Ballot, but Minimum Opposition with ties broken by Plurality) with Schulze in four-candidate, five-faction scenarios. I found that under these conditions, Equal Majorities' Later-no-harm failure rate was under 7% of Schulze's. I suggest that this is extremely good performance.

Equal Majorities retains what is probably Minimum Opposition's greatest defect: The failure of the Plurality criterion. Additionally, it is vulnerable to burial strategy, like most Condorcet methods.

This method is not currently advocated by anyone, but is of interest in my opinion because it's very easy to count (no more complicated than Plurality or Approval) and offers an interesting property, which could be called the Majority Last Preference criterion: The strict last preference of a majority must not be elected. This, combined with Majority Favorite, implies satisfaction of Majority in the three-candidate case. But other than these and Monotonicity, VFA satisfies none of the criteria used here.

Under VFA, the voter votes "for" one candidate, and votes "against" one candidate. If a candidate receives more than half of the "against" votes, this candidate is not allowed to win. Subject to this, the winner is the candidate with the most "for" votes, just as under Plurality.

Consider again the scenario discussed under Instant Runoff. In VFA the votes might end up as follows:

49 for A, against B
24 for B, against A
27 for C, against A

Candidate A is disqualified, and C is elected. In contrast, Plurality would surely elect A. Approval might succeed in electing B or C if those voters cooperate. But VFA manages to elect B or C without those voters having any need to cooperate, outside of agreeing that candidate A is their least favorite.

This scenario shows also that VFA fails the Plurality criterion, for the same reason Minimum Opposition and Equal Majorities do: A has more first preferences than C has any preferences (the B voters' unstated preference for C, assuming it exists at all, does not count), yet C is elected.

VFA fails Clone Independence for obvious reasons: Dividing up one candidate's "against" votes could be beneficial, while dividing up one's "for" votes would be harmful. These seem to cancel each other out: Suppose that there are two major parties, and each considers how many candidates they should enter in the race. If the party believes they will beat the other party, then they have no incentive to run two candidates, because a single candidate won't be in danger of being disqualified, whereas running two candidates could split the "for" votes so that both lose. If the party believes they will lose to the other party, then running two candidates could split the "against" votes so that neither is disqualified. But this will also split the "for" votes, so that they would only be able to win if the other party also runs two candidates, which they won't do if they believe they can win.

A nice property of VFA is that if there are two major candidates, one or the other of which is expected to receive more than half of the "against" votes, and there is a third candidate in the race, then voters preferring this third candidate to the two major candidates have no incentive to not vote "for" the third candidate. This is because, if a candidate is disqualified, he would still be disqualified no matter how many voters move their "for" vote to him.

In terms of criteria, Borda satisfies Monotonicity and Participation, which implies Mono-add-top. It fails Majority Favorite, Majority, Condorcet, Minimal Defense, Later-no-harm, and Clone Independence.

If the above treatment of equal rankings is used, or if equal ranking is not permitted and my treatment of this case is used, Borda fails the Plurality criterion and Sincere Favorite. However, if you allow equal ranking and score to each candidate one point for every candidate beneath them or tied with them, and don't award any points to truncated candidates, both criteria are satisfied. (Example: If the ballot reads A=B>C=D(>E=F) where E and F are truncated, award 5, 5, 3, 3, 0, and 0 points to each candidate (going down the ranking).)

Borda seems unusable for public elections for three major reasons:
1. It will not necessarily elect a majority's first preference.
2. It favors factions which are represented by more candidates. Surely the winner of an election should not depend on which side has more candidates.
3. It has great burial incentive. That is, if there are two major candidates, everyone's optimal strategy is to rank their preferred of the two at the top, and the other candidate at the bottom. But when all voters do this, the winner will never be one of the major candidates (unless of course there are only two candidates).

Methods satisfying the same properties and more include Approval and DSC.

Under Borda Elimination, one repeatedly eliminates the current Borda loser, with scores based on non-eliminated candidates. This method has been suggested to correct some of Borda's shortcomings. It does fix some: Borda Elimination satisfies Majority Favorite, Majority, and Condorcet. But in exchange, it fails Participation, Mono-add-top, and Monotonicity.

Since Borda Elimination doesn't have any desirable properties lacked by, for instance, Schulze, I can't see a reason to use it.

Specifically, Raynaud satisfies Majority, Condorcet, and Clone Independence. It fails Plurality, Minimal Defense, Monotonicity, and Mono-add-top. As with all Condorcet methods, Raynaud fails Participation, Later-no-harm, and Sincere Favorite.

As with Borda Elimination, I can't see a reason to use Raynaud rather than e.g. Schulze, which satisfies the same properties and more.

Bucklin can be interpreted as an Approval variant designed to satisfy Majority. It also satisfies Minimal Defense. But it loses compliance with Clone Independence (that is, if Approval is considered to satisfy it), Mono-add-top, Participation, and Sincere Favorite. Strategy is also less straight-forward: Since there's a race to achieve a majority, voters have incentive to order their rankings based on how viable each candidate seems. For these reasons, I don't consider Bucklin to be preferable to Approval. However, some feel that satisfying Majority is a sufficient reason to prefer Bucklin.

In this section, I will describe and comment on the criteria I have used in the compliance table. I have taken some liberties in how I have worded some of the criteria.

These criteria tend to be written with the assumption that an election method can accept any preference order, with full, partial, and equal rankings all acceptable. But not all methods do accept these. So I offer some suggestions on interpreting these criteria:

Full rankings not permitted. This is the case in Plurality and some proposals of Instant Runoff; Approval has this problem in a different way. I propose to interpret that the voter always may intend to submit a complete ranking, and that limited methods may just ignore those rankings. As an example, consider Plurality and the Majority criterion: Even if voters somehow specified complete rankings, Plurality wouldn't consider them, so the criterion is failed.

Partial rankings not permitted. Common implementations of IRV and Borda are examples of this. The problem is that some criteria (in particular Minimal Defense and Later-no-harm) assume that voters are able to "not vote" for any number of candidates. I propose to interpret that the voter always may intend to not vote for any number of candidates, and if the method does not support this, then the voter ranks these candidates completely arbitrarily (beneath the other candidates).

Equal rankings not permitted. I suggest to treat this just as above. As a hypothetical example, suppose a criterion says that If more than half of the voters rank three candidates in equal first, then the winner must be one of these candidates. Suppose the method does not allow equal ranking. Then I suggest that the only way this method could satisfy this criterion is if a majority of the voters ranking the three candidates in arbitrary order in the top positions is sufficient to elect one of the three.

To summarize, when criteria refer to how voters vote, this refers to how they intend to vote, in the above senses. In this way, ballot restrictions don't provide methods an easy way to pass criteria.

The Plurality criterion says that C must be elected with no greater probability than A. If C is elected, it could be viewed as an obvious mistake, as there is no way to adjust the ballots voting for C so that there are as many C first preferences as A first preferences. But Minimum Opposition and Equal Majorities both elect C with greater probability than A. (See VFA for a similar scenario.)

A possible excuse for these methods to fail Plurality is that it isn't possible for a method to satisfy all three of Plurality, Later-no-harm, and Minimal Defense. (A proof of this.) Methods satisfying Plurality have to give up one of the other two. Schulze does without Later-no-harm, and First Preference Plurality and Instant Runoff do without Minimal Defense.

A few methods sacrifice Plurality compliance for no benefit. For instance, Schulze when margins are used as the measure of defeat strength, Raynaud, and Borda Elimination do not satisfy any of the above three criteria.

(A footnote: Any two of these three criteria are compatible in the three-candidate case. With more than three candidates, it seems likely that Later-no-harm and Minimal Defense are incompatible.)

It's difficult to imagine voters accepting the results of a ranked ballot method which doesn't strictly comply with this criterion. The only commonly discussed methods which fail this criterion are Approval, which can't detect first preferences, and Borda.

Many ranked methods satisfy Majority, including IRV, DSC, Equal Majorities, and Schulze. Methods which don't accept ranked ballots can't satisfy it. Minimum Opposition fails Majority when there are more than three candidates, in cases where the majority prefers at least three candidates, all locked in a majority-strength cycle.

Majority is somewhat limited in protecting majority rule, since it only protects a majority who rank each of their common candidates uninterrupted in the top positions: That is, they must make up a solid coalition. A voter who is part of the majority, except that his preferred candidate is unusual, and ranked above the majority's candidates, is not counted towards the size of the majority. If enough voters give unusual candidates high rankings, then the Majority criterion may not see any majority to protect.

For this reason, I see Minimal Defense as a better guarantee of majority rule. Instead of requiring a majority to rank common candidates uninterrupted in the highest positions, it requires that they not rank any of the candidates they wish to defeat.

Condorcet-compliant methods, such as Schulze, directly consider the number of voters favoring one proposition over another, and as a result tend to reduce most voters' need to use strategy to get the best outcome from the election.

On the negative side, there isn't a graceful way to choose a winner when there is no Condorcet winner. Many methods have been proposed, but the criterion is so demanding that no resolution method (including Random Ballot) is able to avoid failing certain criteria, such as Participation, Later-no-harm, and Sincere Favorite.

Consider this scenario:

49 A
24 B
27 C>B

There is no Condorcet winner. Methods which satisfy Minimal Defense (according to which A may not be elected) and Plurality (according to which C may not be elected), such as Schulze, must elect B. But if the B voters add a preference:

49 A
24 B>C
27 C>B

Now there is a Condorcet winner, and it's C, so that C is elected. The 24 B voters caused B to lose by ranking an additional preference, which is a typical failure of Later-no-harm in Condorcet methods.

One way to defeat this failing is to not explicitly regard whether one candidate pairwise beats another. For example, Minimum Opposition counts the number of votes for each side, but doesn't consider who beats whom. Equal Majorities doesn't count the votes for one candidate over another unless it's a majority, in which case necessarily the other candidate must lose pairwise. In both of these methods, candidate C (in the scenario above) has so few votes over B (just 27) that essentially this pairwise win is ignored.

Steve Eppley has defined and discussed Minimal Defense here and here. Satisfaction of this criterion implies compliance with Mike Ossipoff's strong defensive strategy criterion, although the reverse is not necessarily true. That criterion can be found here.

Note that the ballot must accept all preference orders; in particular, the voter must be able to rank multiple candidates above no one (usually by truncation), and to strictly rank any number of candidates. If the word "strictly" were dropped, then Approval would satisfy, as could other methods using a "limited slot" ballot. (Approval satisfies Mike Ossipoff's weak defensive strategy criterion for this reason.) In my opinion, the word "strictly" should be dropped, since Approval can already be made to satisfy Minimal Defense just by allowing the voter to number his approved candidates, without analyzing the ballot any differently.

Minimal Defense deals with the issue of what a majority need to do to get their opinion counted. Specifically, if they are united in preferring candidate A to candidate B, all they have to do is not rank B. They need not do anything special regarding A. For instance, on these ballots:

49 B
13 C>A
13 D>A
13 E>A
12 F>A

The A>B voters (i.e., the voters preferring A to B) are a majority, and do not rank B at all, so that Minimal Defense guarantees that B won't win. However, candidate B is the winner in e.g. Plurality, Instant Runoff, and Descending Solid Coalitions. Under these methods, if the A>B majority want to prevent B from being elected, they need to insincerely raise candidate A in their rankings.

A more general way to view this problem is by noting that this election is primarily a contest between A and B. In that light, it would be very undesirable for the election method to elect the wrong one of these two. Minimal Defense ensures that the method can't be "confused" by the introduction of weaker candidates preferred to the major candidates.

As for the methods which satisfy Minimal Defense, Schulze elects A in the above scenario (as would any Condorcet method), and Equal Majorities elects either C, D, E, or F. (This is because all candidates besides B are in the CDTT; when Random Ballot is used to break the tie, only these four candidates have any first preferences. Minimum Opposition gives the same result, incidentally, although it doesn't satisfy Minimal Defense when there are more than three candidates.)

For example, under Plurality, a party's probability of winning is greatly reduced when the party runs multiple candidates, since this divides up the number of first preferences each candidate can receive. On the other hand, under Borda or Random Candidate, a party's chances are improved by running multiple candidates.

It's very intuitive to suppose that when the winner of an election receives higher rankings on some ballots, this should never cause him to lose. The only method on my table which fails Monotonicity is Instant Runoff, which possesses some strong properties which make Monotonicity impossible to satisfy. Specifically, IRV would have to sacrifice either Majority (as Plurality does), or some of its truncation resistance (as DSC does), for it to be mathematically possible to satisfy Monotonicity. (Douglas Woodall proves this in "Monotonicity of single-seat preferential election rules", Discrete Applied Mathematics 77 (1997), pp. 81–98.)

Here is a scenario demonstrating IRV's failure:

Candidate A is elected. Now suppose two of the B>C>A voters decide that in fact A is the best candidate, so that these are the ballots:

This increase in support causes A to lose to C.

Methods satisfying Participation inherently satisfy Mono-add-top. I include it here because, of Woodall's monotonicity criteria, Mono-add-top is the only significant one which Instant Runoff doesn't fail, and it is one which is not necessarily satisfied by other methods, such as virtually all Condorcet methods.

It is easy to see why IRV satisfies it: At each stage, the candidate with the fewest traveling votes is eliminated. The winner is never eliminated. Adding ballots whose first preference is this winner can have no effect but to increase the number of votes which this candidate possesses at each stage.

It's clearly desirable for a method to satisfy Participation. Unfortunately, every complying method simply counts and compares points: Plurality, Borda, and Approval all do nothing but assign points to candidates, and elect the candidate with the most. The most complicated method satisfying Participation, Descending Solid Coalitions, assigns points to each ballot's preferred sets of candidates. This allows it to satisfy Majority.

There is some criticism that if a method failures Participation, it will be hard to convince people to vote. I don't think this is likely. If voters have enough information to determine that abstaining is a better strategy than voting sincerely, they probably also have enough information to devise a way of voting which is more effective than abstaining.

"Adding a preference" means, for instance, changing A>B to A>B>C. In Approval, it means voting for an additional candidate. Methods which don't allow additional preferences, such as Plurality, are considered to satisfy the criterion, since at least there is no way for a voter to ever complain that the criterion was violated.

Later-no-harm is important in order to persuade voters to provide as much information in their rankings as possible. If voters fear that by providing lower preferences, the method will use this information to elect a lower preference than it otherwise would have, then voters will have strong incentive to withhold some of their preferences.

Unfortunately, very few non-random methods satisfy Later-no-harm: Plurality, Instant Runoff, DSC, and Minimum Opposition.

This is the other side of Later-no-harm. Failing this criterion is not obviously harmful; to fail this but satisfy Later-no-harm would mean that a voter never has incentive to submit an incomplete ranking.

Failing Later-no-help is mainly a concern where it facilitates burial strategy, where voters have incentive to rank obscure candidates insincerely high, in attempt to make more viable opposing candidates look worse than the obscure candidates. This is often a concern of those who oppose Condorcet methods.

Whether burial strategy would be a common strategy under Condorcet remains to be seen. Arguably there are factors that would keep it from being a major problem. For one thing, it is simply not intuitive to explicitly rank worse candidates over better candidates, at least when truncation is allowed. Additionally, there is usually a risk that burial strategy will backfire and cause the election of a candidate that nobody likes.

Mike Ossipoff defines this criterion here. It essentially says that a group of voters with a common favorite candidate must have a way of voting such that their favorite is ranked below no one, and the elected candidate is not liked less than the best candidate who could be elected if this group doesn't rank their favorite below no one. In other words, voters should not have a reason to not give their favorite candidate the top position.

Rather than this criterion, I want to propose a modified version, below, which is easier to work with. To my knowledge, the same methods (of those considered here) pass and fail this version:

This criterion is difficult to satisfy. It is incompatible with the Condorcet criterion, and is failed by every non-random method that doesn't allow tied candidates in the top position, such as Plurality and Instant Runoff. (See the beginning of this section for an explanation of how to evaluate methods which don't allow equal ranking.)

It is most obviously satisfied by Random Ballot and Approval. It's also satisfied by Minimum Opposition, since lowering X can only hurt X's score, and help the scores of candidates outside of S. Finally, a new method I've contrived, Improved Condorcet Approval, satisfies this criterion.

Due to Minimal Defense, candidate A is not allowed to win. Due to Plurality, candidate C is not allowed to win. So it must be that candidate B is elected with 100% probability.

Now suppose that the B voters add a later preference:

48 A
26 B>C
26 C>B

Due to Later-no-harm, B must still win with 100% probability. But candidates B and C are indistinguishable, so that their probability of election must be identical.

The only way around this is for B or B's supporters to be privileged. But even this is not adequate. Suppose that the C voters then truncate:

48 A
26 B>C
26 C

Due to Later-no-harm, deleting a lower preference must not help C. But as before, due to Plurality and Minimal Defense, C is the only candidate allowed to win.