Part of the Politics and Economics series |
Electoral systems |
---|
![]() |
![]() ![]() |
The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election as a result of receiving too many votes in support. [1] [2] More formally, it says that adding more voters who prefer Alice to Bob should not cause Alice to lose the election to Bob. [3]
Voting systems that fail the participation criterion exhibit the no-show paradox, [4] where a voter is effectively disenfranchised by the electoral system because turning out to vote would make the outcome worse. In such a scenario, these voters' ballots are treated as "less than worthless", actively harming their own interests by reversing an otherwise-favorable outcome. [5]
Positional methods and score voting satisfy the participation criterion. All methods satisfying paired majority-rule [4] [6] can fail in situations involving four-way cyclic ties. Most notably, instant-runoff voting and the two-round system often fail the participation criterion in competitive elections. [1] [7] [2]
The most common cause of no-show paradoxes is the use of instant-runoff (often called ranked-choice voting in the United States). In instant-runoff voting, a no-show paradox can occur even in elections with only three candidates, and occur in 50%-60% of all 3-candidate elections where the results of IRV disagree with those of plurality. [7] [2]
A notable example is given in the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place. [8]
An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, say that a group of pro-Top voters joins the election, making the electorate more supportive of the Top party, and more strongly opposed to the Bottom party. This increase in the number of voters who rank Bottom last causes the moderate candidate to lose to the Bottom party:
More-popular Bottom | Less-popular Bottom | |||||
---|---|---|---|---|---|---|
Round 1 | Round 2 | Round 1 | Round 2 | |||
Top | ![]() | +6 | Top | 31 | 46 | |
Center | 30 | 55![]() | Center | ![]() | ||
Bottom | 39 | 39 | Bottom | 39 | 54 ![]() |
Here, the increase in support for the Top party allowed it to defeat the Center party in the first round. This makes the election an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulty electing the majority-preferred candidate because of vote-splitting in the first round. [9]
When there are at most 3 major candidates, Minimax Condorcet and its variants (such as Ranked Pairs and Schulze's method) satisfy the participation criterion. [4] However, with more than 3 candidates, every resolute and deterministic Condorcet method can sometimes fail participation. [4] [6] Similar incompatibilities have also been proven for set-valued voting rules. [6] [10] [11]
Studies suggest such failures may be empirically rare, however. One study surveying 306 publicly-available election datasets found no participation failures for methods in the ranked pairs-minimax family. [12]
Certain conditions weaker than the participation criterion are also incompatible with the Condorcet criterion. For example, weak positive involvement requires that adding a ballot in which candidate A is one of the voter's most-preferred candidates does not change the winner away from A. Similarly, weak negative involvement requires that adding a ballot in which A is one of the voter's least-preferred does not make A the winner if it was not the winner before. Both conditions are incompatible with the Condorcet criterion. [13]
In fact, an even weaker property can be shown to be incompatible with the Condorcet criterion: it can be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly. [14]
Proportional representation systems using largest remainders for apportionment (such as STV or Hamilton's method) do not pass the participation criterion. This happened in the 2005 German federal election, when CDU voters in Dresden were instructed to vote for the FDP, a strategy that allowed the party an additional seat. [15] As a result, the Federal Constitutional Court ruled that negative voting weights violate the German constitution's guarantee of the one man, one vote principle. [16]
One common failure of the participation criterion in elections is not in the use of particular voting systems to elect candidates to office, but in simple yes or no measures that place quorum requirements. A public referendum, for example, if it required majority approval and a certain number of voters to participate in order to pass, would fail the participation criterion, as a minority of voters preferring the "no" option could cause the measure to fail by simply not voting rather than voting no. In other words, the addition of a "no" vote may make the measure more likely to pass. A referendum that required a minimum number of yes votes (not counting no votes), by contrast, would pass the participation criterion. [17] Many representative bodies have quorum requirements where the same dynamic can be at play.
Negative vote weight refers to an effect that occurs in certain elections where votes can have the opposite effect of what the voter intended. A vote for a party might result in the loss of seats in parliament, or the party might gain extra seats by not receiving votes. This runs counter to the intuition that an individual voter voting for an option in a democratic election should only increase the chances of that option winning the election overall, compared to not voting (a no-show pathology) or voting against it (a monotonicity or negative response pathology).[ citation needed ]
This example shows that majority judgment violates the participation criterion. Assume two candidates A and B with 5 potential voters and the following ratings:
Candidates | # of voters | |
---|---|---|
A | B | |
Excellent | Good | 2 |
Fair | Poor | 2 |
Poor | Good | 1 |
The two voters rating A "Excellent" are unsure whether to participate in the election.
Assume the 2 voters would not show up at the polling place.
The ratings of the remaining 3 voters would be:
Candidates | # of voters | |
---|---|---|
A | B | |
Fair | Poor | 2 |
Poor | Good | 1 |
The sorted ratings would be as follows:
Candidate |
| |||||||||
A | ||||||||||
B | ||||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Poor". Thus, A is elected majority judgment winner.
Now, consider the 2 voters decide to participate:
Candidates | # of voters | |
---|---|---|
A | B | |
Excellent | Good | 2 |
Fair | Poor | 2 |
Poor | Good | 1 |
The sorted ratings would be as follows:
Candidate |
| |||||||||
A | ||||||||||
B | ||||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is the majority judgment winner.
This example shows how Condorcet methods can violate the participation criterion when there is a preference paradox. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:
Preferences | # of voters |
---|---|
A > D > B > C | 8 |
B > C > A > D | 7 |
C > D > B > A | 7 |
This gives the pairwise counting method:
X | |||||
---|---|---|---|---|---|
A | B | C | D | ||
Y | A | [X] 14 [Y] 8 | [X] 14 [Y] 8 | [X] 7 [Y] 15 | |
B | [X] 8 [Y] 14 | [X] 7 [Y] 15 | [X] 15 [Y] 7 | ||
C | [X] 8 [Y] 14 | [X] 15 [Y] 7 | [X] 8 [Y] 14 | ||
D | [X] 15 [Y] 7 | [X] 7 [Y] 15 | [X] 14 [Y] 8 | ||
Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
Pair | Winner |
---|---|
A (15) vs. D (7) | A 15 |
B (15) vs. C (7) | B 15 |
B (7) vs. D (15) | D 15 |
A (8) vs. B (14) | B 14 |
A (8) vs. C (14) | C 14 |
C (14) vs. D (8) | C 14 |
Result: A > D, B > C and D > B are locked in (and the other three can't be locked in after that), so the full ranking is A > D > B > C. Thus, A is elected ranked pairs winner.
Now, assume an extra 4 voters, in the top row, decide to participate:
Preferences | # of voters |
---|---|
A > B > C > D | 4 |
A > D > B > C | 8 |
B > C > A > D | 7 |
C > D > B > A | 7 |
The results would be tabulated as follows:
X | |||||
---|---|---|---|---|---|
A | B | C | D | ||
Y | A | [X] 14 [Y] 12 | [X] 14 [Y] 12 | [X] 7 [Y] 19 | |
B | [X] 12 [Y] 14 | [X] 7 [Y] 19 | [X] 15 [Y] 11 | ||
C | [X] 12 [Y] 14 | [X] 19 [Y] 7 | [X] 8 [Y] 18 | ||
D | [X] 19 [Y] 7 | [X] 11 [Y] 15 | [X] 18 [Y] 8 | ||
Pairwise results for X, won-tied-lost | 1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
Pair | Winner |
---|---|
A (19) vs. D (7) | A 19 |
B (19) vs. C (7) | B 19 |
C (18) vs. D (8) | C 18 |
B (11) vs. D (15) | D 15 |
A (12) vs. B (14) | B 14 |
A (12) vs. C (14) | C 14 |
Result: A > D, B > C and C > D are locked in first. Now, D > B can't be locked in since it would create a cycle B > C > D > B. Finally, B > A and C > A are locked in. Hence, the full ranking is B > C > A > D. Thus, B is elected ranked pairs winner by adding a set of voters who prefer A to B.
Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approval voting, but also lets voters give partial (in-between) approval ratings to candidates.
In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing candidate affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.
A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
The Copeland or Llull method is a ranked-choice voting system based on counting each candidate's pairwise wins and losses.
The positive response, monotonicity, or nonperversitycriterion is a principle of social choice theory that says that increasing a candidate's ranking or rating should not cause them to lose. Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters ; rules that violate positive response are called perverse.
Ranked Pairs (RP) is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.
In an election, a candidate is called a majority winner or majority-preferred candidate if more than half of all voters would support them in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the majority-rule principle, because they extend the principle of majority rule to elections with multiple candidates.
The majority favorite criterion is a voting system criterion that says that, if a candidate would win more than half the vote in a first-preference plurality election, that candidate should win. Equivalently, if only one candidate is ranked first by a over 50% of voters, that candidate must win. It is occasionally referred to simply as the "majority criterion", but this term is more often used to refer to Condorcet's majority-rule principle.
A voting system satisfies join-consistency if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion, which only requires join-consistency when one of the sets of votes unanimously prefers A over B.
The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.
In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.
Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.
Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause this higher-ranked candidate to lose.
The Borda count is a family of positional voting rules which gives each candidate, for each ballot, a number of points corresponding to the number of candidates ranked lower. In the original variant, the lowest-ranked candidate gets 0 points, the next-lowest gets 1 point, etc., and the highest-ranked candidate gets n − 1 points, where n is the number of candidates. Once all votes have been counted, the option or candidate with the most points is the winner. The Borda count is intended to elect broadly acceptable options or candidates, rather than those preferred by a majority, and so is often described as a consensus-based voting system rather than a majoritarian one.
Schulze STV is a draft single transferable vote (STV) ranked voting system designed to achieve proportional representation. It was invented by Markus Schulze, who developed the Schulze method for resolving ties using a Condorcet method. Schulze STV is similar to CPO-STV in that it compares possible winning candidate pairs and selects the Condorcet winner. It is not used in parliamentary elections.
Instant-runoff voting (IRV), also known as ranked-choice voting or the alternative vote (AV), combines ranked voting together with a system for choosing winners from these rankings by repeatedly eliminating the candidate with the fewest first-place votes and reassigning their votes until only one candidate is left. It can be seen as a modified form of a runoff election or exhaustive ballot in which, after eliminating some candidates, the choice among the rest is made from already-given voter rankings rather than from a separate election. Many sources conflate this system of choosing winners with ranked-choice voting more generally, for which several other systems of choosing winners have also been used.
The 2009 Burlington mayoral election was held in March 2009 for the city of Burlington, Vermont. This was the second mayoral election since the city's 2005 change to instant-runoff voting (IRV), after the 2006 mayoral election. In the 2009 election, incumbent Burlington mayor won reelection as a member of the Vermont Progressive Party, defeating Kurt Wright in the final round with 48% of the vote.
A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.
STAR voting is an electoral system for single-seat elections. The name stands for "Score then Automatic Runoff", referring to the fact that this system is a combination of score voting, to pick two finalists with the highest total scores, followed by an "automatic runoff" in which the finalist who is preferred on more ballots wins. It is a type of cardinal voting electoral system.
The sincere favorite or no favorite-betrayal criterion is a property of some voting systems, that says voters should have no incentive to vote for someone else over their favorite. It protects voters from having to engage in a kind of strategy called lesser evil voting or decapitation.
By eliminating the squeezing effect, Approval Voting would encourage the election of consensual candidates. The squeezing effect is typically observed in multiparty elections with a runoff. The runoff tends to prevent extremist candidates from winning, but a centrist candidate who would win any pairwise runoff (the "Condorcet winner") is also often "squeezed" between the left-wing and the right-wing candidates and so eliminated in the first round.