James Green-Armytage
Voting methods: definitions and criteria
Contents:
PRELIMINARY DEFINITIONS
Pairwise comparison, pairwise defeat,
pairwise tie
DEFINITIONS AND CRITERIA RELATING TO MAJORITY RULE
Majority criterion
Mutual majority criterion
Condorcet winner,
Condorcet criterion
Condorcet loser, Condorcet loser criterion
Majority rule cycle
Condorcet
completion method
Minimal dominant set and Smith criterion
Summary of majority rule criteria
DEFINITIONS
RELATING TO DEFEAT STRENGTH
Margins
Winning votes
Cardinal pairwise
Immune set
DEFINITIONS
RELATING TO STRATEGY
Compromising strategy
Burying strategy
Paradoxical strategy
Compression and reversal
Counterstrategy
CONTINUITY CRITERIA
Monotonicity criterion
Participation criterion
Consistency
Continuity
criteria summary
MISCELLANEOUS CRITERIA
Pareto
Later
preferences criterion
Independence of clones
Miscellaneous criteria summary
Criteria definitions for non-ranked methods
See all summary tables together
Preliminary note on > and = symbols: I use these in two slightly different ways. For example, “A>B” can mean that an individual voter or a specific set of voters ranks A above B, and it can also mean that A has a pairwise victory over B. “A=B” can signify an equal ranking of A and B, or a pairwise tie between A and B. The meaning will be made clear by the context.
Pairwise comparison, pairwise defeat, pairwise tie: A pairwise comparison uses ranked ballots to simulate head-to-head contests between different candidates.
Given two candidates A and B, there is a pairwise defeat of B by A if and only if A is ranked above B on more ballots than B is ranked above A.
If the number of A>B ballots is equal to the number of B>A ballots, then there is a pairwise tie between A and B. For the purpose of a pairwise tally, all candidates who are not ranked on a given ballot are considered to be tied for last place on that ballot.
Example 1, preferences:
49 voters: A>B>C
24 voters: B>A>C
24 voters: B>C>A
3 voters: C>B>A
Imagine that 100 votes were cast as above, between candidates A, B, and C. We want to do pairwise comparisons between all candidates, so that means comparing A with B, A with C, and B with C. The table below illustrates how these pairwise comparisons are performed.
|
|
A>B |
B>A |
A>C |
C>A |
B>C |
C>B |
|
49: A>B>C |
49 |
|
49 |
|
49 |
|
|
24: B>A>C |
|
24 |
24 |
|
24 |
|
|
24: B>C>A |
|
24 |
|
24 |
24 |
|
|
3: C>B>A |
|
3 |
|
3 |
|
3 |
|
total |
49 |
51 |
73 |
27 |
97 |
3 |
You can see that 49 voters prefer A to B, whereas 51 voters prefer B to A. Hence, B wins its pairwise comparison with A, 52-48. Likewise, A beats C (73-27), and B beats C (97-3). I often express this information as follows:
Example 1, pairwise comparisons:
B>A : 51-49
A>C : 73-27
B>C : 97-3
The same data can also be expressed as a matrix:
| A | B | C | |
| A | 49 | 73 | |
| B |
51 |
97 | |
| C |
27 |
3 |
|
DEFINITIONS AND CRITERIA RELATED TO
MAJORITY RULE
Majority criterion (MC): If more than half of the voters rank candidate X over every other candidate, then the winner should be candidate X.
Some methods that pass MC: Smith/minimax, sequential dropping, ranked pairs, beatpath, river, cardinal pairwise, minimax, plurality [*], IRV, two round runoff
Some methods that fail MC: approval [*], ratings summation, Borda
Mutual majority criterion (MMC): If there is a single majority of the voters who rank every candidate in a set over every candidate outside that set, then the winner should always be a member of the set.
Example 2, preferences:
45: A>B>C
30: B>C>A
25: C>B>A
The smallest mutual majority set is {B, C}, because the 30 B>C>A voters and the 25 C>B>A voters rank the candidates in that set above candidate A. Methods that pass the mutual majority criterion will not elect candidate A in this example.
Some methods that pass MMC: Smith/minimax, sequential dropping, ranked pairs, beatpath, river, cardinal pairwise, IRV, Raynaud
Some methods that fail MMC: approval [*], ratings summation, Borda, minimax, plurality [*], two round runoff
Condorcet winner (CW), Condorcet criterion: A Condorcet winner, also called a ‘dominant candidate,’ is a candidate that wins all of its pairwise comparisons.
If a voting method always elects a Condorcet winner when one exists, the method is Condorcet-efficient, and passes the Condorcet criterion.
Repeating example 1, pairwise comparisons:
B>A : 51-49
A>C : 73-27
B>C : 97-3
Candidate B is the Condorcet winner, because it wins all of its pairwise comparisons.
Some methods that pass the Condorcet criterion: Smith/minimax, sequential dropping, ranked pairs, beatpath, river, cardinal pairwise, minimax
Some methods that fail the Condorcet criterion: approval [*], ratings summation, plurality [*], IRV, Borda, two round runoff
Condorcet loser, Condorcet loser criterion: A candidate that loses all of its pairwise comparisons is known as a Condorcet loser.
A method that never selects a Condorcet loser passes the Condorcet loser criterion.
Repeating example 1, pairwise comparisons:
B>A : 51-49
A>C : 73-27
B>C : 97-3
Candidate C is the Condorcet loser, because it loses all of its pairwise comparisons.
Some methods that pass the Condorcet loser criterion: Smith/minimax, sequential dropping, ranked pairs, beatpath, river, cardinal pairwise, IRV, Borda, two round runoff
Some methods that fail the Condorcet loser criterion: minimax, approval [*], ratings summation, plurality [*]
Majority rule cycle: A circular series of pairwise defeats (e.g. A beats B, B beats C, C beats A) that leaves no single candidate unbeaten.
Example 2, preferences:
35: A>B>C
33: B>C>A
32: C>A>B
Example 2, pairwise comparisons:
A>B : 67-33
B>C : 68-32
C>A : 65-35
You can see that each candidate is defeated in one pairwise comparison. A is defeated by C, C is defeated by B, and B is defeated by A. There is no Condorcet winner (or Condorcet loser) in this example.
Condorcet completion method: A method that chooses the Condorcet winner when one exists, and that also includes a rule for choosing a winner when no Condorcet winner exists. Dozens of Condorcet completion methods have been proposed, including minimax, Smith/minimax, sequential dropping, Schwartz sequential dropping, beatpath, ranked pairs, river, cardinal pairwise, approval-weighted pairwise, Black, Nanson, Copeland, and so on.
Minimal dominant set: Also known as the Smith set and the GeTChA set. The smallest set of candidates such that every candidate within the set has a pairwise victory over every candidate outside the set.
Smith criterion: A method that always chooses from the minimal dominant set is Smith-efficient, and passes the Smith criterion.[*]
Some methods that pass the Smith criterion: Smith/minimax, sequential dropping, ranked pairs, beatpath, river, cardinal pairwise
Some methods that fail the Smith criterion: IRV, minimax, approval [*], ratings summation, Borda, plurality [*], two round runoff
Summary
of majority rule criteria:
The table below summarizes the compliance/noncompliance of 6 methods with the 5 majority rule criteria (the majority criterion, mutual majority criterion, Condorcet criterion, Condorcet loser criterion, and Smith criterion).
|
|
plurality |
approval |
Borda |
IRV |
minimax |
ranked pairs |
|
MC |
pass |
fail |
fail |
pass |
pass |
pass |
|
MMC |
fail |
fail |
fail |
pass |
fail |
pass |
|
Condorcet |
fail |
fail |
fail |
fail |
pass |
pass |
|
Condorcet loser |
fail |
fail |
pass |
pass |
fail |
pass |
|
Smith |
fail |
fail |
fail |
fail |
fail |
pass |
The non-compliances of approval voting as shown above (and elsewhere on the page) are shared by the ratings summation method. The compliances of ranked pairs as shown above are shared by several Condorcet completion methods, including beatpath, river, cardinal pairwise, sequential dropping, and so on.
DEFEAT STRENGTH
When there is a majority rule cycle, many pairwise methods choose the winner by assigning a strength value to the different pairwise defeats, in order to decide which defeat(s) to overturn.
Here are a few definitions of defeat strength. Note that margins and winning votes use only ordinal (ranking) information, but cardinal pairwise uses cardinal (rating) information as well.
Margins: Defines the strength of an A>B pairwise defeat as the number of A>B voters minus the number of B>A voters.
Winning votes (WV): Defines the strength of an A>B pairwise defeat as the number of A>B voters.
Margins-based and winning votes-based methods produce the same result unless some voters give equal rankings to some candidates (or truncate their ballots).
Example 2.1, preferences:
45: A>B=C
12: B>A=C
14: B>C>A
29: C>B>A
Pairwise comparisons:
A>C : 45-43 (45 winning votes, with a margin of 2 votes)
C>B : 29-26 (29 winning votes, with a margin of 3 votes)
B>A : 55-45 (55 winning votes, with a margin of 10 votes)
In margins-based methods, the weakest defeat is A>C, with a margin of 2 votes. In winning votes-based methods, the weakest defeat is C>B, with 29 winning votes. Thus, C wins most margins-based methods, whereas B wins most winning votes-based methods.
Cardinal pairwise: Defines the strength of an A>B pairwise defeat as follows: For each A>B voter, and only for A>B voters, subtract B’s rating from A’s rating, to get the rating differential. Sum the A>B rating differentials to get the A>B defeat strength.[*]
Immune set: The set of candidates such that any defeat against a candidate within the set is countered by a string of stronger (or equally strong) defeats leading back to the defeating candidate. There is an immune set definition for each possible defeat strength definition, e.g. the WV-defined immune set, the margins-defined immune set, the CWP-defined immune set.[*]
Repeating example 2, pairwise comparisons:
A>B : 67-33
B>C : 68-32
C>A : 65-35
{A} is both the WV-defined immune set and the margins-defined immune set for this example. It’s true that C beats A, but A beats B, B beats C, and the A>B and B>C defeats are both stronger than the C>A defeat.
The cardinal pairwise immune set cannot be defined for this example because no rating information is available.
STRATEGY
Compromising strategy: Insincerely ranking (or rating) an option higher in order to decrease the probability that a less-preferred option will win. For example, if my sincere preferences are R>S>T, a compromising strategy would be to vote S>R>T or R=S>T, raising S’s ranking in order to decrease T’s chances of winning. (The drawback is that this often decreases R’s chances of winning as well.)[*]
Condorcet-efficient methods minimize the incentive for the compromising strategy. All resolvable voting methods that satisfy the mutual majority criterion have a compromising incentive when there is a majority rule cycle. But voters in Condorcet-efficient methods never have an incentive to use the compromising strategy when there is a Condorcet winner. This is an important property because, in the absence of a majority rule cycle, it allows me to vote my R>S preference without worrying that it will undermine my S>T preference. This is a more complete way of curtailing the “lesser of two evils” problem, that is, decreasing the extent to which voters have to worry about earlier choices drawing support away from later choices.
Example 3, sincere preferences:
36: Right>Center>Left
15: Center>Right>Left
15: Center>Left>Right
34: Left>Center>Right
In IRV, Center is eliminated first, and Right is the winner. However, if at least 5 of the Left voters vote Left=Center>Right or Center>Left>Right, then Left will be eliminated first, and their second choice, Center, will win instead of their third choice, Right. This would constitute a successful compromising strategy.
Burying strategy: Insincerely ranking (or rating) an option lower in order to increase the probability that a more-preferred option will win. For example, if my sincere preferences are R>S>T, a burying strategy would be to vote R>T>S or R>S=T, lowering S’s ranking in order to increase R’s chances of winning. (The drawback is that this often increases T’s chances of winning as well.)[*]
Burying strategies can only be successful in methods that fail the later preferences criterion (defined below).
Example 4a, sincere preferences:
46: A>B>C
44: B>A>C
5: C>A>B
5: C>B>A
Example 4b, pairwise comparisons:
A>B 51-49
A>C 90-10
B>C 90-10
Candidate A is the sincere Condorcet winner, and thus wins with any Condorcet completion method if votes are sincere. However, in many Condorcet completion methods, it is possible for the B supporters to change the result to their advantage.
Example 4b, includes some insincere preferences:
46: A>B>C
44: B>C>A (sincerely B>A>C)
5: C>A>B
5: C>B>A
Example 4b, pairwise comparisons:
A>B 51-49
C>A 54-46
B>C 90-10
Now there is a majority rule cycle. Margins and WV define the A>B defeat to be the weakest in the cycle, and hence many margins and WV methods elect B as a result of the altered preferences. This constitutes a successful burying strategy.
Paradoxical strategy: Insincerely ranking an option lower in order to increase the probability that the same option will win, or insincerely ranking an option higher in order to decrease the probability that the same option will win.
Paradoxical strategies can only be successful in methods that fail the monotonicity criterion.
Example 5a, sincere preferences:
38: A>B>C
10: A>C>B
9: B>A>C
18: B>C>A
25: C>B>A
Given sincere votes, B is the winner in IRV. However, supporters of candidate A can use a paradoxical strategy to their advantage.
Example 5b, some insincere preferences:
38: A>B>C
5: A>C>B
5: C>A>B (sincerely A>C>B)
9: B>A>C
18: B>C>A
25: C>B>A
Now, B is eliminated first instead of C, and the 9 B>A>C votes transfer to A, allowing it to defeat B 52-48 in the runoff.
Note that paradoxical strategies in IRV tend to be more difficult to use than burying strategies in most Condorcet completion methods, in that coordination and information usually need to be extremely precise in the former, and the risk/reward ratios tend to be higher.
Compression strategy: Any strategy that involves insincerely ranking two candidates equally (includes strategic truncation).
Reversal strategy: Any strategy that involves insincerely reversing the relative order in which two candidates are ranked.
Compromising-compression: A compromising strategy that involves insincerely giving two candidates an equal ranking.
Compromising-reversal: A compromising strategy that involves insincerely reversing the order of two candidates on the ballot.
Burying-compression: A burying strategy that involves insincerely giving two candidates an equal ranking.
Burying-reversal: A burying strategy that involves insincerely reversing the order of two candidates on the ballot.
For example, if my sincere preferences are R>S>T...
1. If I vote R=S>T in order to decrease T’s chances of winning, this is a compromising-compression strategy.
2. If I vote S>R>T in order to decrease T's chances of winning, this is a compromising-reversal strategy.
3. If I vote R>S=T in order to increase R's chances of winning, this is a burying-compression strategy.
4. If I vote R>T>S in order to increase R's chances of winning, this is a burying-reversal strategy.
Counterstrategy: A strategy by one group of voters that is intended to render ineffective a strategy by another group of voters.
Compromising counterstrategy: Insincerely ranking (or rating) an option higher in order to protect it from a burying strategy.
Burying/deterrent counterstrategy: Insincerely ranking (or rating) an option lower in order to make that option less likely to win as a result of a burying strategy.
For an example of these two counterstrategies, see part 7.d. (stable counterstrategies) of my cardinal pairwise paper.
The table below is an attempt to visually summarize the vulnerability of different methods to different types of strategic manipulation.
I have given each method a score from 0 to 6 for each type of strategy, with 6 being the most vulnerable, and 0 being invulnerable. I didn't use a particular mathematical formula in deciding which scores to enter in which cells, so the entries are largely subjective. I welcome debate on any and all entries in the table. I tried to take into account both the frequency of incentives and severity of difficulties caused by the strategy if attempted.
|
|
plurality |
approval |
Borda |
nER-IRV |
ER-IRV |
margins |
WV | cardinal pairwise |
|
compromising-c |
disallowed | c forced | 6 | disallowed | 4 | 2 | 2 | 1 |
|
compromising-r |
6 | 0 | 6 | 5 | 3 | 2 | 1 | 1 |
|
burying-c |
disallowed | c forced | 6 | disallowed | 0 | 5 | 1 | 1 |
|
burying-r |
0 | 0 | 6 | 0 | 0 | 6 | 4 | 1 |
| paradoxical | 0 | 0 |