A joint Politics and Economics series |
Social choice and electoral systems |
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Mathematicsportal |
In social choice, a no-show paradox is a pathology in some voting rules, where a candidate loses an election as a result of having too many supporters. [1] [2] More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. [3] Voting systems without the no-show paradox are said to satisfy the participation criterion. [4]
In systems that fail the participation criterion, a voter can end up effectively disenfranchised by the electoral system, because turning out to vote would make the result worse for them; such voters are sometimes referred to as having negative vote weights, particularly in the context of German constitutional law, where courts have ruled such a possibility violates the principle of one man, one vote. [5] [6] [7]
Positional methods and score voting satisfy the participation criterion. All deterministic voting rules that satisfy pairwise majority-rule [1] [8] can fail in situations involving four-way cyclic ties, though such scenarios are empirically rare, and the randomized Condorcet rule is not affected by the pathology. The majority judgment rule fails as well, but passes a weaker condition: giving a candidate the maximum (minimum) rating can never cause them to lose (win). [9] Ranked-choice voting (RCV) and the two-round system both fail the participation criterion with high frequency in competitive elections, typically as a result of a center squeeze. [2] [3] [10]
The no-show paradox is similar to, but not the same as, the perverse response paradox. Perverse response happens when an existing voter can make a candidate win by decreasing their rating of that candidate (or vice-versa). For example, under ranked-choice voting, moving a candidate from first-place to last-place on a ballot can cause them to win. [11]
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. [10] [3]
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. [12]
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 Center 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 |
Thus the increase in support for the Top party allows 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. [13]
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. [14] However, with more than 3 candidates, Hervé Moulin proved that every deterministic Condorcet method can sometimes fail participation. [14] [15] Similar incompatibilities have also been shown for set-valued voting rules. [15] [16] [17] The randomized Condorcet rule satisfies the criterion, but fails the closely-related monotonicity criterion in situations with Condorcet cycles. [18]
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. [19]
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. [20]
In fact, an even weaker property can be shown to be incompatible with the Condorcet criterion: it may be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly. [21]
Proportional representation systems using largest remainders for apportionment (such as STV or Hamilton's method) allow for no-show paradoxes. [5] [22]
In Germany, situations where a voter's ballot has the opposite of its intended effect (e.g. a vote for a party or candidate causes them to lose) are called Negatives Stimmgewicht (lit. 'negative voting weights'). An infamous example occurred in the 2005 German federal election, when an article in Der Spiegel laid out how CDU voters in Dresden's 2nd district would have to vote against their own party if they wished to avoid losing a seat in the Bundestag. [5] This led to a lawsuit by electoral reform organization Mehr Demokratie and Alliance 90/The Greens, joined by the neo-Nazi National Democratic Party, who argued the election law was undemocratic. [23]
The Federal Constitutional Court agreed with the plaintiffs, ruling that negative vote weights violate the German constitution's guarantee of equal and direct suffrage. The majority wrote that: [6] [24]
A seat allocation procedure that allows an increase in votes to lead to a loss of seats, or results in more seats being won if [proportionally] fewer votes are cast for it, contradicts the meaning and purpose of a democratic election[...]
Such nonsensical relationships between voting and electoral success not only impair the equality of the right to vote and the equal opportunities of the parties, but also the principle of a popular election, as it is no longer apparent to the voter how their vote results in the success or failure of a candidate.[...]
Negative vote weights cannot be accepted as constitutional on the premise that they cannot be predicted or planned, and thus can hardly be influenced by the individual voter. To what extent this is true can be set aside, as such arbitrary results make a mockery of the democratic competition for support from the electorate.
The ruling forced the Bundestag to abandon its old practice of ignoring overhang seats, and instead adopt a new system of compensation involving leveling seats. [23]
A common cause no-show paradoxes is the use of a quorum. For example, if a public referendum requires 50% turnout to be binding, additional "no" votes may push turnout above 50%, causing the measure to pass. A referendum that instead required a minimum number of yes votes (e.g. >25% of the population voting "yes") would pass the participation criterion. [25]
Many representative bodies have quorum requirements where the same dynamic can be at play. For example, the requirement for a two-thirds quorum in the Oregon Legislative Assembly effectively creates an unofficial two-thirds supermajority requirement for passing bills, and can result in a law passing if too many senators turn out to oppose it. [26] Deliberate ballot-spoiling strategies have been successful in ensuring referendums remain non-binding, as in the 2023 Polish referendum.
The participation criterion can also be justified as a weaker form of strategyproofness: while it is impossible for honesty to always be the best strategy (by Gibbard's theorem), the participation criterion guarantees honesty will always be an effective, rather than actively counterproductive, strategy (i.e. a voter can always safely cast a sincere vote). [1] This can be particularly effective for encouraging honest voting if voters exhibit loss aversion. Rules with no-show paradoxes do not always allow voters to cast a sincere vote; for example, a sincere Palin > Begich > Peltola voter in the 2022 Alaska special election would have been better off if they had not shown up at all, rather than casting an honest vote.
While no-show paradoxes can be deliberately exploited as a kind of strategic voting, systems that fail the participation criterion are typically considered to be undesirable because they expose the underlying system as logically incoherent or "spiteful" (actively seeking to violate the preferences of some voters). [27]
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 Y | A | B | C | D |
---|---|---|---|---|
A | 14 8 | 14 8 | 7 15 | |
B | 8 14 | 7 15 | 15 7 | |
C | 8 14 | 15 7 | 8 14 | |
D | 15 7 | 7 15 | 14 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 Y | A | B | C | D |
---|---|---|---|---|
A | 14 12 | 14 12 | 7 19 | |
B | 12 14 | 7 19 | 15 11 | |
C | 12 14 | 19 7 | 8 18 | |
D | 19 7 | 11 15 | 18 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.
Approval voting is a single-winner electoral system in which voters mark all the candidates they support, instead of just choosing one. The candidate with the highest approval rating is elected. Approval voting is currently in use for government elections in St. Louis, Missouri and Fargo, North Dakota.
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, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently self-contradictory. The result implies that it is logically impossible for any voting system to guarantee a winner will have support from a majority of voters: in some situations, a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties.
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 or Pairwise Majority Rule Winner (PMRW). The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.
Arrow's impossibility theorem is a key result in social choice, discovered by Kenneth Arrow, showing that no ranked voting rule can behave rationally. Specifically, any such rule violates independence of irrelevant alternatives (IIA), the idea that a choice between and should not depend on the quality of a third, unrelated option . The result is most often cited in election science and voting theory, where is called a spoiler candidate. In this context, Arrow's theorem can be restated as showing that no ranked voting rule can eliminate the spoiler effect.
The positive response, monotonicity, or nonperversitycriterion is a principle of social choice that says 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 said to show perverseresponse.
Independence of irrelevant alternatives (IIA) is a major axiom of decision theory which codifies the intuition that a choice between and should not depend on the quality of a third, unrelated outcome . There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski-Young theorem, and the money pump arguments.
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 median voter theorem in political science and social choice theory, developed by Duncan Black, states that if voters and candidates are distributed along a one-dimensional spectrum and voters have single-peaked preferences, any voting method that is compatible with majority-rule will elect the candidate preferred by the median voter. The median voter theorem thus shows that under a realistic model of voter behavior, Arrow's theorem, which essentially suggests that ranked-choice voting systems cannot eliminate the spoiler effect, does not apply, and therefore that rational social choice is in fact possible if the election system is using a Condorcet method.
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. Systems that fail the consistency criterion are susceptible to the multiple-district paradox, which allows for a particularly egregious kind of gerrymander: it is possible to draw boundaries in such a way that a candidate who wins the overall election fails to carry even a single electoral district.
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.
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 a higher-ranked candidate to lose.
In social choice theory, the independence of (irrelevant) clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a very weak form of the independence of irrelevant alternatives (IIA) criterion.
Instant-runoff voting (IRV), also known as ranked-choice voting (RCV), preferential voting (PV), or the alternative vote (AV), is a multi-round elimination method where the loser of each round is determined by the first-past-the-post method. In academic contexts, the term instant-runoff voting is generally preferred as it does not run the risk of conflating the method with methods of ranked voting in general.
Majority judgment (MJ) is a single-winner voting system proposed in 2010 by Michel Balinski and Rida Laraki. It is a kind of highest median rule, a cardinal voting system that elects the candidate with the highest median rating.
Ranked voting is any voting system that uses voters' orderings (rankings) of candidates to choose a single winner or multiple winners. More formally, a ranked rule is one that depends only on which of two candidates is preferred by a voter, and as such does not incorporate any information about intensity of preferences. Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties.
The later-no-help criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.
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.
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.
Of course, a method not satisfying participation will incentivize some strategic non-voting, as the voters in question will have an incentive not to vote (sincerely). But again, all voting methods incentivize strategic behavior[...] By contrast, we are troubled by failures of positive or negative involvement, as this shows that the method responds in the wrong way to unequivocal support for (resp. rejection of) a candidate.