A Survey of Basic Voting Methods

Given a field of candidates running in an election for the same position, a single winner method will select only one. Choosing a governor would be an example of this; if there can only be one governor of a state, then the governor must be decided by a single-winner method. A multiple winner method will select more than one candidate, for example an election that fills ten seats in a legislature, with one electorate choosing from a field of ninety candidates.

I.A. Non-ranked ballot systems

I.A.1. Plurality

Plurality is the most common single winner method used in the world today. The plurality method simply gives the victory to the candidate who receives the most votes (a plurality). This may sound intuitive, but unfortunately it has severe flaws. The problems stem from the difference between a plurality and a majority.

Example 1: One candidate from the far right receives 20% of the vote, and eight candidates from the left receive 7%, 9%, 10%, 10%, 10%, 10%, 11%, and 13% of the vote. The right winger will win a plurality election, despite the fact that 80% of the voters preferred a leftist candidate.

It is believed that on average there will only be two viable candidates for any given election under the plurality system. This is because rather than picking the candidate who is their sincere favorite, most voters are likely to instead vote for the one of the perceived front-runners whom they prefer, since this is the best chance they have of their vote making a positive difference. This tendency is known as Duverger's law[*], and is thought to be the primary cause of two-party systems where they exist.

Given the existence of two major party candidates who dominate an election field together, the entrance of a new candidate is most likely to split the vote of the major party candidate whom they have the most in common with, thus giving the other candidate an advantage, and going directly against the wills of would-be supporters of the emergent candidate. This 'spoiler effect' is an extremely strong deterrent against new parties and candidates entering a race where a close competition has already been established between two major parties. This is a fatal problem for the competitiveness of political races and the accountability of politicians. Standards are very low for political candidates because they only need to be preferred over a single other viable candidate, rather than over a large field of viable candidates. This dynamic also encourages negative campaigning, and severely limits the range of political discourse.

Criteria summary for plurality

Criteria passed: majority, monotonicity, participation, consistency, Pareto, later preferences

Criteria failed: mutual majority, Condorcet, Condorcet loser, Smith, independence of clones

Strategic vulnerability: Very strong and very damaging compromising-reversal incentive.

I.A.2. Two round runoff

The second most common single-winner method is the two round runoff. The rules for two round runoff vary slightly, but the most common procedure is this: An initial election is held, and if any candidate gets a majority of the votes, she is declared the winner. If not, then a second election is held between the two candidates who received the most votes in the first election. This assures that the resulting winner is preferred by a majority to at least the one other candidate who makes it to the second election.

In example 1, a second election would be held between the right wing candidate with 20% and the leftist candidate with 13% of the initial vote. Since an 80% majority of the voters preferred leftist candidates, the remaining leftist would be likely to win with ease.

Of course, by the same logic as above, it is not certain that the candidate with 13% is the best representative of the leftists. This is the basic limitation of the two round runoff’s effectiveness.

Criteria summary for two round runoff:

Criteria passed: majority, later preferences, Pareto, Condorcet loser

Criteria failed: mutual majority, Condorcet, Smith, participation, consistency, monotonicity

Strategic vulnerability: There is a somewhat strong and potentially quite damaging compromising-reversal incentive. There is some vulnerability to paradoxical strategies. There is no burying vulnerability.

I.A.3. Approval voting

In approval voting, you can vote only once for each candidate, but you may vote for as many candidates as you like. The winner is the candidate with the most votes. For example, if A, B, C, D, and E are running in an election, you can vote only for A, you can vote for A and B, or you can vote for A, B, and C. You could vote for all five of the candidates if you like, but doing so is essentially equivalent to not voting at all, since your vote affects all the candidates equally.

Approval voting is probably an unambiguous improvement over plurality. For one thing, where plurality often forces people to choose between voting for a candidate whom they perceive to be viable and a candidate whom they strongly agree with, approval allows people to do both. Thus, it arguably gives candidates a fair chance to prove themselves on election day even if they are not expected to be one of the top two contenders. Some also argue that approval would do fairly well at choosing strong compromise candidates. They make an interesting case, but I would like to see this happen in practice before I accept it as being true.

Approval does have significant limitations. Take example 1.1, a race between a Conservative party candidate, a Labor party candidate, and a Democratic party candidate. 45% of the voters prefer the Conservative candidate over both the Democratic candidate and the Labor party candidate, and will only approve him. 30% of the voters prefer the Labor candidate as their first choice, but strongly prefer the Democratic candidate over the Conservative candidate. 25% of the voters prefer the Democratic candidate as their first choice, but strongly prefer the Labor candidate over the Conservative candidate.

This information can be summarized in handy ways, as below. The notation “30%: Labor > Democrat > Conservative” means that 30% of the voters would prefer the Democrat to the Labor candidate, and the Labor candidate to the conservative candidate. The notation “45%: Conservative > Labor = Democrat” means that 45% of the voters most prefer the Conservative candidate, and consider the remaining candidates to be roughly equal to each other in desirability or undesirability.

45%: Conservative

30%: Labor > Democrat > Conservative

25%: Democrat > Labor > Conservative

Approval voting in itself offers no mechanism to resolve this kind of situation. If the Labor and Democratic voters unite in approving a candidate, then that candidate will beat the Conservative, something that both Labor and Democratic voters want. But which will be the winner, Labor or the Democrats?

If all of the Labor and Democratic voters approve both the Labor and Democratic candidates, then there will be a tie. If all of them except one approves both, and the remaining one only approves the Democrat, then the Democrat will win. If one voter only approves the Democrat, and two only approve Labor, then Labor will win. Thus both Democratic and Labor voters have an incentive to approve only one, but as the numbers grow who follow this incentive, the chances of electing the Conservative increase.

The result is essentially a game of chicken between the Labor and Democratic voters, where approving both candidates is analogous to swerving, approving only their favorite candidate is equivalent to staying on course, and the car crash is the election of the Conservative. It seems that this might cause instability in some cases.

Put another way, the choice between the Labor and Democratic candidates is a relatively haphazard one. Most of the Labor and Democrat voters need to approve both of them in order to beat the Conservative, and most of the Conservative voters won't approve either Labor or Democrat (or else the Conservative won't have a chance), so the choice between L and D is made by a few outliers, or a few people who make a lucky gamble.

The problem is that approval voting only offers voters two levels of support (1 vote, or 0 votes), while the Labor and Democratic voters both have three distinct tiers of preference. What is desirable, then, is a way to give voters a way to express as many levels of support as there are candidates, that is to say, a ranked ballot.

One of the attractive features of approval voting is does not require different voting equipment from plurality voting. In addition, it is very easy to explain. Hence, the cost of switching to approval voting from plurality is very low compared to many other systems which utilize ranked ballots and have more complex rules. However, there may be many situations where this added cost is more than compensated for by the more sophisticated communication from voters to government that is made possible by some ranked ballot methods.

In general, I feel that it is difficult to accurately predict how approval voting will play out in practice, since we cannot map preferences directly onto votes, and we rely on our own understanding of rather subtle dynamics of voter psychology, voter interaction, voter strategy. It will be very interesting for approval to be tried on larger scales, to see how it handles more contentious election scenarios. Perhaps approval voting will have a powerfully positive impact on democracy; it's hard to know at this point. However, it does seem clear that it has substantial advantages to plurality with few disadvantages, so in my opinion it is probably worth supporting as an alternative to plurality.

Criteria summary for approval voting

Criteria passed: monotonicity, participation, consistency

Criteria failed: Pareto, majority, mutual majority, Condorcet, Condorcet loser, Smith, independence of clones, later preferences

Strategic vulnerability: "Sincere vote" difficult to define for approval. Compression of preferences forced by the ballot, thus causing similar problems to methods with strong compromising-compression and burying-compression problems.

Note: As it fails the mutual majority criterion (and even the more basic majority criterion), I don't think that approval should be known as a majority rule method. However, there may be some situations where approval voting is preferable to majority rule, especially in electorates that are not very contentious, i.e. where the members are more inclined to seek consensus.

I.B. Ranked ballot methods

Ranked ballot methods allow voters to list the candidates in order of preference, that is first choice, second choice, and so on. Many but not all ranked methods allow voters to give an equal ranking to more than one candidate, for example I can list A as my first choice, B as my second choice, C and D as tied for third, E at fourth, and so on. The standard version of IRV does not allow equal rankings, but all Condorcet versions below do.

In some places, such as Australia, voters are required to rank all of the candidates in order for their ballot to be counted. However, this is not at all necessary from the standpoint of the methods themselves; any ranked system can work from ballots that do not rank everyone, or are ‘truncated’ after a certain point. Generally, ranked ballot methods consider all candidates not ranked on a ballot to be tied for last place.

I.A.1. Borda

Borda is a point count system, where a first choice vote is worth a fixed number of points, a second choice vote is worth a fixed number of points, and so on. The winner is the candidate with the most total points. One common formula for Borda is to make the last place on a ballot be worth zero points, the second to last place worth one point, and so on until the first place, which is worth one less than the number of candidates running. Variations exist which can effect the result of a given election, but the principle remains the same.

Borda is a highly inelegant system that has little merit for use in public elections. For one thing, the strength of a given person's vote is highly variable as it affects the competitions between different candidates. For example, let's say there is a race with two strong frontrunners and some candidates with only a slim chance of winning. If I vote for my own long-odds favorite first, and my preferred compromise candidate second, the strength of my vote as it affects the race between the frontrunners is less than someone else in a similar situation who left their sincere favorite off the ballot. Hence, there are strong and frequent incentives for voters to rank someone other than their sincere favorite in first place. Also, it is extremely common for Borda to offer strong incentives for strategic truncation.

Further weirdnesses abound in the Borda system. For one thing, say that there are two elections with identical ballots cast. The only difference is that in the second election, an extra candidate runs, who is not ranked anywhere on the ballots of any of the voters. Borda is the only system discussed here weird enough that the results of the two elections can be different under these circumstances.

Borda is virtually alone among ranked ballot methods in failing the majority criterion, which states that if a candidate is voted over all other candidates by more than half of the voters, he or she should win.

Borda spectacularly fails the independence of clones criterion, which will be defined later. The basic upshot of this, though, is that it can be a huge advantage for a given constituency to be represented by a large number of candidates in an election, rather than a single candidate or only a few. That is, a constituency can crowd out other constituencies by flooding the election field with similar candidates.

Also, Borda should not be used in multiple-winner elections when proportional representation (which will also be defined later) is appropriate, because it does not produce fully proportional results.

Click here for a more detailed critique of the Borda count.

Criteria summary for the Borda count

Criteria passed: monotonicity, participation, consistency, later preferences, Pareto, Condorcet loser

Criteria failed: majority, mutual majority, Condorcet, Smith, independence of clones

Strategic vulnerability: Strategic vulnerability is a major issue for Borda. Very strong compromising and burying incentives. Also, teaming incentives significantly greater than any other method considered here.

I.A.2. Instant runoff voting / the alternative vote / Hare

This system, called the alternative vote internationally, or sometimes the Hare[*] method or simply 'preference voting', uses a ranked ballot to simulate a process much like a multiple-round runoff election, hence the American name for it, instant runoff voting.

Each ballot is initially assigned to candidates who are listed as the first choice on that ballot. If any candidate already has a majority of the votes at this point, then they automatically win the election. If no one has a majority yet, then the candidate with the fewest top choice votes is eliminated, and the votes cast for them are transferred to the next choice on each ballot. This process continues until one candidate achieves a majority, or until only one candidate remains.

Example 2: The candidates running are “Far Right,” “Right,” “Left,” and “Far Left.”

Again, the notation “5%: A>B>C” means that 5% of the voters indicate A as their first choice, B as their second choice, and C as their third choice. If there are more candidates than A B and C, then they are considered to all be tied for last place on this ballot. (The notation is being used the same way as in example 1.1, except that in that case we were dealing with “internal” voter preferences which couldn’t be expressed on an approval ballot, and now we are dealing with voter preferences as expressed on actual ranked ballots.)

5%: Far Right > Right > Left > Far Left.

40%: Right > Far Right > Left > Far Left

36%: Left > Far Left > Right > Far Right

19%: Far Left > Left > Far Right > Right

The IRV count would go like this:

Far Left Left Right Far Right

19% 36% 40% 5%

round one: nobody has a majority, so Far Right is eliminated, transferring 5% to Right.

+5% 5%

19% 36% 45%

round two: Far Left now has the fewest votes, so she is eliminated, transferring 19% to Left.

~~19%~~ +19%

55% 45%

round three: Left now has a clear majority and wins.

IRV is a step in many of the right directions, as it allows for as many levels of preference as there are candidates, and as a person’s vote retains its full original value after it is transferred. However, it is not perfect, as I will explain in the next section.

Criteria summary for IRV

Criteria passed: majority, mutual majority, later preferences, Pareto, Condorcet loser, independence of clones

Criteria failed: Condorcet, Smith, participation, consistency, monotonicity

Strategic vulnerability: If equal ranking is not allowed, a somewhat strong and potentially quite damaging compromising-reversal incentive. If it is allowed, a relatively less-damaging compromising-compression incentive, along with a reduced compromising-reversal incentive. Because it fails monotonicity, IRV is one of the few methods vulnerable to paradoxical strategies. However, paradoxical strategies in IRV probably tend to be difficult and risky, and the vulnerability is probably not severe.

I.A.3. Condorcet methods

Methods based on the Condorcet[*] principle first use the ranked ballots to ask whether there is any one candidate who would win in a head to head election against every other candidate individually.

To do this, it breaks down the election into a series of pairwise comparisons between every candidate and every other candidate. In a pairwise contest between candidate A and candidate B, a ballot counts as one vote for candidate A if he is ranked above B on that ballot. Also, if candidate A is ranked, and candidate B is not ranked at all on that ballot, then it counts as one vote for candidate A. The position of the other candidates is irrelevant to the pairwise contest between A and B.

If there is one candidate who wins all of their pairwise comparisons (which is more likely than it may sound), then he is a Condorcet winner, and he wins an election with any Condorcet method.

It is possible that there will be no Condorcet winner, for example if A wins her pairwise comparison against B, B wins his pairwise comparison against C, and C wins her pairwise comparison against A. This is called a “majority rule cycle.” There are several methods of breaking cycles, the most interesting of which are described in detail below.

But first some examples where a Condorcet winner does exist. Let’s take example 2 again, an example of a situation where IRV works well.

5%: Far Right > Right > Left > Far Left.

40%: Right > Far Right > Left > Far Left

36%: Left > Far Left > Right > Far Right

19%: Far Left > Left > Right > Far Right

The pairwise comparisons would look like this (I have put the pairwise victories here in bold, and left the defeats in plain text):

Far Right vs. Right = 5% vs. 95%

Far Right vs. Left = 45% vs. 55%

Far Right vs. Far Left = 45% vs. 55%

Right vs. Left = 45% vs. 55%

Right vs. Far Left = 45% vs. 55%

Left vs. Far Left = 81% vs. 19%

Left (who was also the IRV winner) has won all of his pairwise contests, and is therefore a Condorcet winner.

The same information can also be expressed as a matrix. The row marked “Far Right” represents Far Right’s score in her pairwise comparison with each other candidate. A candidate whose row consists only of victories, such as “Left” in this example, is a Condorcet winner.

Far Right Right Left Far Left

Far Right 5% 45% 45%

Right 95% 45% 45%

Left 55% 55% 81%

Far Left 55% 55% 19%

In this example, and surely many others, IRV and Condorcet’s method produce the same results. However, they do not always do so. Lets take example 3, An election between a “Right” candidate, a “Center” candidate, and a “Left” candidate.

33%: Left > Center > Right

16%: Center > Left > Right

16%: Center > Right > Left

35%: Right > Center > Left

Here is a diagram, which might make it easier to conceptualize:

Left Center Right

33% 32% 35%

<--------16% 16%---------->

33%----------> <-----------35%

The IRV tally would go like this:

Left Center Right

33% 32% 35%

round one: Center is eliminated, transferring 16% to each remaining candidate.

+16% ~~32%~~ +16%

49% 51%

Right wins the election using IRV. If, however, the losing candidate Left was deleted from the ballots, or withdrew from the race just before the election, Center would have beat Right with a crushing 65-35 majority. (Likewise, without Right in the race, Center would have soundly beat Left by 67-33.)

Indeed, those who voted Left > Center > Right will regret their votes given the above result, and wish that they had voted Center > Left > Right instead, which would have resulted in the election of their second choice (Center) rather than their last choice (Right).

If voters anticipate this sort of result before the election, then they have a strong incentive not to raise the position of a compromise candidate on the ballot, so that they can ensure that he or she is not eliminated early on. (This is an application of the compromising strategy.) While such strategy might produce an optimal result given good information prior to the election, the strategically altered vote no longer communicates the true preferences of the voters, and there is a danger that voters will make an unnecessary compromise which costs their sincere favorite the election.

Plus, if voters fail to anticipate such a result, then they will be left with a widely-regretted and unstable outcome like the one above.

If IRV voters are going to use the compromising strategy, it seems that it would be much better for them to raise the compromise candidate into an equal position with more preferred candidates, rather than a superior position, so that their preferences are less severely distorted. This is a good argument to allow equal ranking in IRV.

Condorcet produces a different result given this example. Here are the pairwise comparisons:

Left vs. Center = 33% vs. 67%

Left vs. Right = 49% vs. 51%

Center vs. Right = 65% vs. 35%

Or, as a matrix:

Left Center Right

Left 33% 49%

Center 67% 65%

Right 51% 35%

Center wins all of her pairwise comparisons (and quite easily, at that), and is therefore a Condorcet winner. IRV fails to elect Center because she is eliminated before any votes can be transferred to her, leaving a choice between the two wing candidates. Condorcet does not make the same mistake, since it doesn’t eliminate any candidates before it looks at the later preferences on the ballots.

In the examples used so far, a clear Condorcet winner has been present, and so there is no difference between results given by different Condorcet methods. Now let’s look at some of the different methods for choosing a winner when a cycle is present and no Condorcet winner exists.

I am presenting these methods as a sort of progression from the most simple Condorcet method, which is minimax, through Smith + minimax and Schwartz sequential dropping, to the more subtle Condorcet methods: beatpath and ranked pairs.

It is very rare that these more complex methods would produce a result different from minimax or Smith + minimax, and there are situations where a change from one of these to beatpath or ranked pairs would not be worth the added complexity. However, when a (single-winner) collective decision is very important and the resources to make the calculation are available, I would recommend the beatpath or ranked pairs methods.

I.A.3.a. minimax / successive reversal / Simpson[*]

Let’s take an example where no Condorcet winner exists, example 3.1. This is an imaginary election between Bush, Gore, and Nader, where Bush has lost a little bit of ground since 2000, and Nader has gained a lot of ground from Gore. This example involves truncated ballots, on which Bush voters refuse to rank either Gore or Nader, and some Gore voters rank neither Nader nor Bush.

Note that a ballot marked Nader > Gore is completely equivalent to a ballot marked Nader > Gore > Bush; the position of Bush in last place is implied, since he is the only other candidate. Hence such a ballot is not truncated in any meaningful way. A ballot which only indicates Bush as the first choice is equivalent to a ballot which indicates Bush > Gore = Nader.

45%: Bush

12%: Gore

14%: Gore > Nader

29%: Nader > Gore

Or, in diagram form:

Nader Gore Bush

29% 26% 45%

29%------------->

<-------------14%

The pairwise comparisons:

Nader vs. Gore = 29% vs. 26%

Nader vs. Bush = 43% vs. 45%

Gore vs. Bush = 55% vs. 45%

The same information expressed as a matrix:

Nader Gore Bush

Nader 29 43

Gore 26 55

Bush 45 45

In this example there is a cycle, which leaves no candidate unbeaten. The simplest way to resolve this cycle is to drop or disrecognize the weakest defeat, and to go on doing this until an unbeaten candidate emerges. This method is sometimes known as “successive reversal,” and because the resulting winner is the candidate whose worst loss is the least bad, “minimax.”

Actually, I can't recommend minimax for use in elections, because it fails the mutual majority criterion, the Condorcet loser criterion, and the Smith criterion. The Smith criterion corresponds to the Smith set, which is defined in the next section.

Criteria summary for minimax

Criteria passed: majority, Pareto, Condorcet, monotonicity

Criteria failed: mutual majority, Smith, participation, consistency, independence of clones, Condorcet loser, later preferences

Strategic vulnerability: Some compromising incentive and some burying vulnerability. The amount of each depends largely on defeat strength definition, but in general minimax should have worse burying problems than Smith-efficient methods with the same defeat strength definition, because of its mutual majority criterion failure.

I.A.3.aa. Winning votes versus margins

I wrote above that minimax drops the weakest defeat until an unbeaten candidate emerges. But how do we decide which defeat is the weakest? The two most common defeat strength definitions are margins and winning votes (WV).

The solution to example 3.1 above depends on which of these you choose. The margins of the three pairwise comparisons are 3% (29%-26%), 2% (45%-43%), and 10% (55%-45%). The smallest margin is 2%, which is the margin of Bush’s defeat of Nader. Using margin-based minimax, this defeat would be dropped, leaving Nader unbeaten and declaring him the winner.

The winning vote totals for the defeats are 29%, 45%, and 55%. The weakest defeat in a WV method is Nader’s defeat of Gore, with a magnitude of 29%. So, using WV-based minimax, this defeat is disrecognized, and Gore is declared the winner.

I prefer Condorcet methods that are based on winning votes rather than margins, because I believe that margins methods do not allow for stable counterstrategies, potentially causing very serious strategic turmoil. (Also, it seems odd to me that Nader should win in the example above, since he never achieves more than 43% of the vote in any of his comparisons, whereas Bush has 45% of the vote in all of his.) I write more about the strategic vulnerability of margins here.

If all of the voters rank all of the candidates, then margin-based results will be identical to magnitude-based results, because a defeat that has a greater magnitude will also have a correspondingly greater margin. There will be no difference between magnitude and margin results in any of the other examples below.

(Notice that IRV gives the victory in this case to Bush, an outcome that seems unfair, and problematic in terms of third party participation. (That is, Nader’s presence in the race once again has a sort of “spoiler effect,” in that Gore would have won the election instead of Bush if Nader had been deleted from the ballots.))

I.A.3.b. Minimal dominant set (Smith set) // minimax

A fairly simple variation on the basic minimax method of breaking cycles is to first exclude candidates who are not a member of the top cycle in the first place. One way of doing this is to only include members of the minimal dominant set, also known as the Smith[*] set, or the GeTChA set (which stands for Generalized Top Choice Axiom)[*].

The minimal dominant set is the smallest possible set of candidates such that every candidate inside the set beats every candidate outside of the set.

Here is an example where eliminating non-members of the Smith set will make a difference, example 4. The preference rankings and the resulting pairwise comparison matrix:

6 voters: A>B>C>D

6 voters: D>C>A>B

6 voters: B>C>A>D

5 voters: D>A>B>C

4 voters: C>A>B>D

4 voters: D>B>C>A

2 voters: B>C>D>A

2 voters: A>C>B>D

1 voter: A>C>D>B

A B C D

A 24 14 19

B 12 23 20

C 22 13 21

D 17 16 15

When conceptualizing Condorcet cycles, I often find it helpful to draw diagrams like the one below. If an arrow is drawn from A-->B, it means that A beats B in pairwise comparison. The number assigned to the arrow is the magnitude of the defeat. When possible, I put the numbers on the outer edge of the line to avoid crowding. Otherwise, I try to put them close to the point of the arrow, such as the B-->C defeat with 23 magnitude below. Later on, a double-sided arrow will symbolize a pairwise tie.

Using these sorts of diagrams, it is more readily apparent that A B and C all beat D, and that they form a cycle with each other.

In this case, the Smith set consists of A, B, and C, but not D, because A, B, and C each beat D in pairwise comparisons.

Plain minimax will in fact choose D as the winner, like so: first his defeat by A is dropped, that being the weakest defeat. There is still no unbeaten candidate, so his defeat by B is dropped, and then finally his defeat by C, leaving D unbeaten and therefore the winner of the election.

D is what is known as a Condorcet loser, that is a candidate who loses all of their pairwise contests. It seems undesirable for a Condorcet loser to win an election, and excluding non-members of the Smith set prevents this.

With a Smith set + minimax combination, D is eliminated first in this example. There is still no unbeaten candidate, so A’s defeat by C, which is the weakest, is dropped, leaving A as the winner.

Criteria summary for Smith//minimax

Criteria passed: Smith, majority, mutual majority, Pareto, Condorcet, Condorcet loser, monotonicity

Criteria failed: participation, consistency, independence of clones, later preferences

Strategic vulnerability: Some compromising incentive and some burying vulnerability. The amount of each depends largely on defeat strength definition, but in general Smith-efficient methods should have better strategic resistance than their Smith-failing counterparts.

I.A.3.bb. Sequential dropping

There is a subtle but important difference between the sequential dropping rule and the minimax rule. The minimax rule is to drop the weakest defeat until there is an unbeaten candidate.

The sequential dropping rule is to drop the weakest defeat that's in a cycle until there is an unbeaten candidate.

Sequential dropping naturally passes the Smith criterion without having to add a special provision as in Smith/minimax. Sequential dropping may be the reasonably good base method that is easiest to define and explain.

Criteria summary for sequential dropping

Criteria passed: Smith, majority, mutual majority, Pareto, Condorcet, Condorcet loser

Criteria failed: participation, consistency, independence of clones, later preferences, monotonicity

I.A.3.c. Union of minimal undominated sets (Schwartz set)

The union of minimal undominated sets is the same as the minimal dominant set, as long as there are no pairwise ties (the odds of which should be statistically negligible in a public election, but may come in to play when a smaller group is voting). The union of minimal undominated sets is also known as the Schwartz set[*], or the GOCHA set (Generalized Optimal CHoice Axiom)[*].

An undominated set is a set of candidates not beaten by any candidates outside the set. A minimal undominated set does not contain other undominated sets. It is possible for more than one minimal undominated set to exist at once, so the complete Schwartz set is the union of all of them.

The Schwartz sets is always a subset of the Smith set, that is it may be the entire Smith set, or only one or a few members of the Smith set, but it will not include candidates outside the Smith set. Hence if any of the two sets is smaller, it will be the Schwartz set.

Here is an example where they are different, example 5. I will omit the preference rankings this time. The double-sided arrow in the diagram indicates a tie (the magnitude of which is not important).

A B C D

A 54 56 50

B 46 58 52

C 44 42 60

D 50 48 40

Here, the minimal dominant set is all the candidates, because there is no smaller set of candidates who beats all of the other candidates.

There is however a single minimal undominated set, which consists only of A. Hence A is the only member of the Schwartz set.

Like the Smith set, the Schwartz set usually contains more than one candidate if no Condorcet winner exists. So, it is not a satisfactory method in itself for finding a single winner, but it is useful as a tool and a criterion for other methods.

I.A.3.d. Schwartz sequential dropping

If a Condorcet winner does not exist, Schwartz sequential dropping first excludes non-members of the Schwartz set. Next it drops the weakest defeat, that is, it replaces the weakest defeat with a pairwise tie. If there is still no unbeaten candidate, it recalculates the Schwartz set and excludes non-members, and then drops the weakest remaining defeat within the set. This process continues until there is an unbeaten candidate, who is then declared the winner.

Here is an example where this is different from Smith + minimax, example 6:

A B C D E

A 108 106 102 90

B 92 88 120 114

C 94 112 84 118

D 98 80 116 104

E 110 86 82 96

In this case the Smith set is all five candidates. minimax will then go on dropping the weakest defeats until an unbeaten candidate emerges. In this case, the minimax winner is A, whose worst loss is least bad. (A’s worst loss is 110-90, while the other candidates’ worst losses are 112-88, 116-84, 120-80, and 118-82.)

The Schwartz set is also all five candidates, and Schwartz sequential dropping also begins the same way, by dropping the weakest defeats one by one. However, look what happens when we get to this point:

A B C D E

A -- -- -- 90

B -- 88 120 114

C -- 112 84 118

D -- 80 116 --

E 110 86 82 --

The Smith set would still be all five candidates, because there is no smaller set that beats all candidates outside that set. However, the Schwartz set at this point is reduced to only B, C, and D. B, C, and D constitute a minimal undominated set in that none of them are beaten by A or E, and there is no smaller set of undominated candidates within B, C, and D. Both A and E have at least one defeat by the B C D set, so they cannot qualify as an undominated set. The whole set of candidates A, B, C, D, and E does not qualify as a minimal undominated set because it contains the smaller undominated set B, C, and D.

So, A and E are eliminated at this point because they are no longer part of the Schwartz set. The matrix of the remaining candidates would look like this:

B C D

B 88 120

C 112 84