Voting criteria

There are a number of different criteria which can be used for voting systems in an election, including the following

Condorcet criterion and similar criteria

Condorcet criterion

This section is an excerpt from Condorcet winner criterion.[ edit ]

A Condorcet (French: [kɔ̃dɔʁsɛ] , English: /kɒndɔːrˈseɪ/ ) winner is a candidate who would receive the support of more than half of the electorate 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 Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates. ^{[1] [2]}

The Condorcet winner is also called a majority winner, a majority-preferred candidate, ^{[3] [4] [5]} a beats-all winner, or tournament winner (by analogy with round-robin tournaments). A Condorcet winner may not necessarily always exist in a given electorate: it is possible to have a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox, ^[6] and is analogous to the counterintuitive intransitive dice phenomenon known in probability. However, the Smith set, a generalization of the Condorcet criteria that is the smallest set of candidates that are pairwise unbeaten by every candidate outside of it, will always exist.

If voters are arranged on a sole 1-dimensional axis, such as the left-right political spectrum for a common example, and always prefer candidates who are more similar to themselves, a majority-rule winner always exists and is the candidate whose ideology is most representative of the electorate, a result known as the median voter theorem. ^[7] However, in real-life political electorates are inherently multidimensional, and the use of a one- or even two-dimensional model of such electorates would be inaccurate. ^{[8] [9]} Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races. ^[10]

Systems that guarantee the election of a Condorcet winners (when one exists) include Ranked Pairs, Schulze's method, and the Tideman alternative method. Methods that do not guarantee that the Cordorcet winner will be elected, even when one does exist, include instant-runoff voting (often called ranked-choice in the United States), First-past-the-post voting, and the two-round system. Most rated systems, like score voting and highest median, fail the majority winner criterion.

Condorcet loser criterion

This section is an excerpt from Condorcet loser criterion.[ edit ]

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.

A voting system complying with the Condorcet loser criterion will never allow a Condorcet loser to win. A Condorcet loser is a candidate who can be defeated in a head-to-head competition against each other candidate. ^[11] (Not all elections will have a Condorcet loser since it is possible for three or more candidates to be mutually defeatable in different head-to-head competitions.)

Smith criterion

This section is an excerpt from Smith set.[ edit ]

The Smith set, ^{[note 1]} sometimes called the top-cycle, generalizes the idea of a Condorcet winner to cases where no such winner exists. It does so by allowing cycles of candidates to be treated jointly, as if they were a single Condorcet winner. ^[12] Voting systems that always elect a candidate from the Smith set pass the Smith criterion. The Smith set and Smith criterion are both named for mathematician John H Smith.

The Smith set provides one standard of optimal choice for an election outcome. An alternative, stricter criterion is given by the Landau set.

Consistency criterion

This section is an excerpt from Multiple districts paradox.[ edit ]

A voting system satisfies join-consistency (also called the reinforcement criterion) 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 (such as Instant-runoff voting or Condorcet methods) 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. ^[13]

There are three variants of join-consistency:

1. Winner-consistency: if two districts elect the same winner A, A also wins in the combined district.
2. Ranking-consistency: if two districts rank a set of candidates exactly the same way, then the combined district returns the same ranking of all candidates.
3. Grading-consistency: if two different districts assign a candidate the same overall grade to a candidate, the overall grade for the candidate must still be the same.

A voting system is winner-consistent if and only if it is a point-summing method; in other words, it must be a positional voting system or score voting (including approval voting). ^{[14] [15]}

As shown below under Kemeny-Young, whether a system passes reinforcement can depend on whether the election selects a single winner or a full ranking of the candidates (sometimes referred to as ranking consistency): in some methods, two electorates with the same winner but different rankings may, when added together, lead to a different winner. Kemeny-Young is the only ranking-consistent Condorcet method, and no Condorcet method can be winner-consistent. ^[15]

Homogeneity criterion

This section is an excerpt from Homogeneity criterion.[ edit ]

Homogeneity is a common property for voting systems. The property is satisfied if, in any election, the result depends only on the proportion of ballots of each possible type. That is, if every ballot is replicated the same number of times, then the result should not change. ^{[16] [17] [18]}

Independence criteria

Independence of irrelevant alternatives

This section is an excerpt from Independence of irrelevant alternatives.[ edit ]

Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between ${\displaystyle A}$ and ${\displaystyle B}$ should not depend on the quality of a third, unrelated outcome ${\textstyle C}$. 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, ^[19] cognitive science, social choice, ^[19] fair division, rational choice, artificial intelligence, probability, ^[20] 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 behavioral economics, failures of IIA (caused by irrationality) are called menu effects or menu dependence. ^[21]

Independence of clones criterion

This section is an excerpt from Independence of clones criterion.[ edit ]

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. ^[22] It can be considered a weak form of the independence of irrelevant alternatives (IIA) criterion that nevertheless is failed by a number of voting rules. A method that passes the criterion is said to be clone independent. ^[23]

A group of candidates are called clones if they are always ranked together, placed side-by-side, by every voter; no voter ranks any of the non-clone candidates between or equal to the clones. In other words, the process of cloning a candidate involves taking an existing candidate C, then replacing them with several candidates C1, C2... who are slotted into the original ballots in the spot where C previously was, with the clones being arranged in any order. If a set of clones contains at least two candidates, the criterion requires that deleting one of the clones must not increase or decrease the winning chance of any candidate not in the set of clones.

Ranked pairs, the Schulze method, and systems that unconditionally satisfy independence of irrelevant alternatives are clone independent. Instant-runoff voting passes as long as tied ranks are disallowed. If they are allowed, its clone independence depends on specific details of how the criterion is defined and how tied ranks are handled. ^[24]

Rated methods like range voting or majority judgment that are spoilerproof under certain conditions are also clone independent under those conditions.

The Borda count, minimax, Kemeny–Young, Copeland's method, plurality, and the two-round system all fail the independence of clones criterion. Voting methods that limit the number of allowed ranks also fail the criterion, because the addition of clones can leave voters with insufficient space to express their preferences about other candidates. For similar reasons, ballot formats that impose such a limit may cause an otherwise clone-independent method to fail.

This criterion is very weak, as adding a substantially similar (but not quite identical) candidate to a race can still substantially affect the results and cause vote splitting. For example, the center squeeze pathology that affects instant-runoff voting means that several similar (but not identical) candidates competing in the same race will tend to hurt each other's chances of winning. ^[25]

Independence of Smith-dominated alternatives

This section is an excerpt from Independence of Smith-dominated alternatives.[ edit ]

Independence of Smith-dominated alternatives (ISDA, also known as Smith-IIA) is a voting system criterion which says that the winner of an election should not be affected by candidates who are not in the Smith set. ^[26]

Another way of defining ISDA is to say that adding a new candidate should not change the winner of an election, unless that new candidate would beats the original winner, either directly or indirectly (by beating a candidate who beats a candidate who... who beats the winner).^{[ citation needed ]}

Later-no criteria

Later-no-harm criterion

This section is an excerpt from Later-no-harm criterion.[ edit ]

Voting system

Name	Comply?
Plurality	Yes [note 2]
Two-round system	Yes
Partisan primary	Yes
Instant-runoff voting	Yes
Minimax Opposition	Yes
DSC	Yes
Anti-plurality	No[ citation needed ]
Approval	N/A
Borda	No
Dodgson	No
Copeland	No
Kemeny–Young	No
Ranked Pairs	No
Schulze	No
Score	No
Majority judgment	No

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. It is a common property in the plurality-rule family of voting systems.

For example, say a group of voters ranks Alice 2nd and Bob 6th, and Alice wins the election. In the next election, Bob focuses on expanding his appeal with this group of voters, but does not manage to defeat Alice—Bob's rating increases from 6th-place to 3rd. Later-no-harm says that this increased support from Alice's voters should not allow Bob to win. ^[27]

Later-no-harm may be confused as implying center squeeze, since later-no-harm is a defining characteristic of first-preference plurality (FPP) and instant-runoff voting (IRV), and descending solid coalitions (DSC), systems that have similar mechanics that are based on first preference counting. These systems pass later-no-harm compliance by making sure the results either do not depend on lower preferences at all (plurality) or only depend on them if all higher preferences have been eliminated (IRV and DSC), and thus exhibit a center squeeze effect. ^{[28] [29]} However, this does not mean that methods that pass later-no-harm must be vulnerable to center squeezes. The properties are distinct, as Minimax opposition also passes later-no-harm.

Later-no-harm is also often confused with immunity to a kind of strategic voting called strategic truncation or bullet voting. ^[30] Satisfying later-no-harm does not provide immunity to such strategies. Systems like instant runoff that pass later-no-harm but fail monotonicity still incentivize truncation or bullet voting in some situations. ^{[31] [32] [33] : 401}

Later-no-help criterion

This section is an excerpt from Later-no-help criterion.[ edit ]

The later-no-help criterion (or LNHe, not to be confused with LNH) 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.^{[ citation needed ]}

Majority winner and loser

Majority winner criterion

This section is an excerpt from Majority winner criterion.[ edit ]

The majority criterion is a voting system criterion applicable to voting rules over ordinal preferences required that if only one candidate is ranked first by over 50% of voters, that candidate must win. ^[34]

Some methods that comply with this criterion include any Condorcet method, instant-runoff voting, Bucklin voting, plurality voting, and approval voting.

The mutual majority criterion is a generalized form of the criterion meant to account for when the majority prefers multiple candidates above all others; voting methods which pass majority but fail mutual majority can encourage all but one of the majority's preferred candidates to drop out in order to ensure one of the majority-preferred candidates wins, creating a spoiler effect. ^[35]

Majority loser criterion

This section is an excerpt from Majority loser criterion.[ edit ]

The majority loser criterion is a criterion to evaluate single-winner voting systems. ^{[36] [37] [38] [39]} The criterion states that if a majority of voters give a candidate no support, i.e. do not list that candidate on their ballot, that candidate must lose (unless no candidate is accepted by a majority of voters).

Either of the Condorcet loser criterion or the mutual majority criterion implies the majority loser criterion. However, the Condorcet criterion does not imply the majority loser criterion, since the minimax method satisfies the Condorcet but not the majority loser criterion. Also, the majority criterion is logically independent from the majority loser criterion, since the plurality rule satisfies the majority but not the majority loser criterion, and the anti-plurality rule satisfies the majority loser but not the majority criterion. There is no positional scoring rule which satisfies both the majority and the majority loser criterion, ^{[40] [41]} but several non-positional rules, including many Condorcet rules, do satisfy both. Some voting systems, like instant-runoff voting, fail the criterion if extended to handle incomplete ballots.

Monotonicity criterion

This section is an excerpt from Negative responsiveness.[ edit ]

A 4-candidate Yee diagram under IRV. The diagram shows who would win an IRV election if the electorate is centered at a particular point. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa. Black lines show the optimal solution (achieved by Condorcet or score voting).

In social choice, the negative responsiveness, ^{[42] [43]} perversity, ^[44] or additional support paradox ^[45] is a pathological behavior of some voting rules, where a candidate loses as a result of having "too much support" from some voters, or wins because they had "too much opposition". In other words, increasing (decreasing) a candidate's ranking or rating causes that candidate to lose (win). ^[45] Electoral systems that do not exhibit perversity are said to satisfy the positive response or monotonicity criterion. ^[46]

Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology. ^[47] Systems that allow for perverse response can create situations where a voter's ballot has a reversed effect on the election, thus treating the well-being of some voters as "less than worthless". ^[48] Similar arguments have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage. ^{[49] [50]} Negative response is often cited as an example of a perverse incentive, as voting rules with perverse response incentivize politicians to take unpopular or extreme positions in an attempt to shed excess votes.

Most ranked methods (including Borda and all common round-robin rules) satisfy positive response, ^[46] as do all common rated voting methods (including approval, highest medians, and score). ^{[note 3]}

Perversity occurs in instant-runoff voting (IRV), ^[51] the single transferable vote, ^[44] and quota-based apportionment methods. ^[43] According to statistical culture models of elections, the paradox is especially common in RCV/IRV and the two-round system.^{[ citation needed ]} The randomized Condorcet method can violate monotonicity in the case of cyclic ties.

The participation criterion is a closely-related, but different, concept. While positive responsiveness deals with a voter changing their opinion (or vote), participation deals with situations where a voter choosing to cast a ballot has a reversed effect on the election. ^[52]

Proportionality for solid coalitions

This section is an excerpt from Proportionality for solid coalitions.[ edit ]

Proportionality for solid coalitions (PSC) is a criterion of proportionality for ranked voting systems. It is an adaptation of the quota rule to voting systems in which there are no official party lists, and voters can directly support candidates. The criterion was first proposed by the British philosopher and logician Michael Dummett. ^{[53] [54]}

PSC is a relatively minimal definition of proportionality. To be guaranteed representation, a coalition of voters must rank all candidates within the same party first before candidates of other parties. And PSC does not guarantee proportional representation if voters rank candidates of different parties together (as they will no longer form a solid coalition).

Participation criterion

This section is an excerpt from No-show paradox.[ edit ]

In social choice, a no-show paradox is a surprising behavior in some voting rules, where a candidate loses an election as a result of having too many supporters. ^{[55] [56]} More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. ^[57] Voting systems without the no-show paradox are said to satisfy the participation criterion. ^[58]

In systems that fail the participation criterion, a voter turning out to vote could 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. ^{[59] [60] [61]}

Positional methods and score voting satisfy the participation criterion. All deterministic voting rules that satisfy pairwise majority-rule ^{[55] [62]} 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. ^[63] 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. ^{[56] [57] [64]}

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 instant-runoff voting, moving a candidate from first-place to last-place on a ballot can cause them to win. ^[65]

Plurality criterion

Woodall 'splurality criterion is a voting criterion for ranked voting. It is stated as follows: ^{[66] [67]}

If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference [other than last], then A's probability of winning must be no less than B's.

Woodall has called the plurality criterion "a rather weak property that surely must hold in any real election" opining that "every reasonable electoral system seems to satisfy it."

Among Condorcet methods which permit truncation, whether the plurality criterion is satisfied depends often on the measure of defeat strength. When winning votes is used as the measure of defeat strength, plurality is satisfied. Plurality is failed when margins is used. Minimax using pairwise opposition also fails plurality.

When truncation is permitted under Borda count, the plurality criterion is satisfied when no points are scored to truncated candidates, and ranked candidates receive no fewer votes than if the truncated candidates had been ranked. If truncated candidates are instead scored the average number of points that would have been awarded to those candidates had they been strictly ranked, or if Nauru's modified Borda count is used, the plurality criterion is failed.

Resolvability criteria

This section is an excerpt from Resolvability criterion.[ edit ]

A voting system is called decisive, resolvable, or resolute if it ensures a low probability of tied elections. There are two different criterion that formalize this. ^[68]

• In Nicolaus Tideman's^{[ citation needed ]} version of the criterion, adding one extra vote (with no tied ranks) should make the winner unique.
• Douglas R. Woodall's^{[ citation needed ]} version requires that the probability of a tied vote under an impartial culture model gives a tie approaches zero as the number of voters increases toward infinity.

A non-resolvable social choice function is often only considered to be a partial electoral method, sometimes called a voting correspondence or set-valued voting rule. Such methods frequently require tiebreakers that can substantially affect the result. However, non-resolute methods can be used as a first stage to eliminate candidates before ties are broken with some other method. Methods that have been used this way include the Copeland set, the Smith set, and the Landau set.

Reversal symmetry

This section is an excerpt from Best-is-worst paradox.[ edit ]

In social choice theory, the best-is-worst paradox occurs when a voting rule declares the same candidate to be both the best and worst possible winner. The worst candidate can be identified by reversing each voter's ballot (to rank candidates from worst-to-best), then applying the voting rule to the reversed ballots find a new "anti-winner". ^{[69] [70]}

Rules that never exhibit a best-is-worst paradox are said to satisfy the reversal criterion, which states that if every voter's opinions on each candidate are perfectly reversed (i.e. they rank candidates from worst to best), the outcome of the election should be reversed as well, meaning the first- and last- place finishers should switch places. ^[70] In other words, the results of the election should not depend arbitrarily on whether voters rank candidates from best to worst (and then select the best candidate), or whether we ask them to rank candidates from worst to best (and then select the least-bad candidate).

Methods that satisfy reversal symmetry include the Borda count, ranked pairs, Kemeny–Young, and Schulze. Most rated voting systems, including approval and score voting, satisfy the criterion as well. Best-is-worst paradoxes can occur in ranked-choice runoff voting (RCV) and minimax. A well-known example is the 2022 Alaska special election, where candidate Mary Peltola was both the winner and anti-winner.

Unrestricted domain

This section is an excerpt from Unrestricted domain.[ edit ]

In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem.

With unrestricted domain, the social welfare function accounts for all preferences among all voters to yield a unique and complete ranking of societal choices. Thus, the voting mechanism must account for all individual preferences, it must do so in a manner that results in a complete ranking of preferences for society, and it must deterministically provide the same ranking each time voters' preferences are presented the same way.

See also

• Voting system
• Experimental metrics
• Comparison of electoral systems
• Arrow's Impossibility theorem

Notes

1. Many authors reserve the term "Schwartz set" for the strict Smith set described below.
2. Plurality voting can be thought of as a ranked voting system that disregards preferences after the first; because all preferences other than the first are unimportant, plurality passes later-no-harm as traditionally defined.
3. Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.

References

1. Lepelley, Dominique; Merlin, Vincent (1998). "Choix social positionnel et principe majoritaire". Annales d'Économie et de Statistique (51): 29–48. doi:10.2307/20076136. ISSN   0769-489X. JSTOR   20076136.
2. Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN   0036-1399. JSTOR   2100704.
3. Brandl, Florian; Brandt, Felix; Seedig, Hans Georg (2016). "Consistent Probabilistic Social Choice". Econometrica. 84 (5): 1839–1880. arXiv: 1503.00694 . doi:10.3982/ECTA13337. ISSN   0012-9682.
4. Sen, Amartya (2020). "Majority decision and Condorcet winners". Social Choice and Welfare. 54 (2/3): 211–217. doi:10.1007/s00355-020-01244-4. ISSN   0176-1714. JSTOR   45286016.
5. Lewyn, Michael (2012), Two Cheers for Instant Runoff Voting (SSRN Scholarly Paper), Rochester, NY, SSRN   2276015 , retrieved 2024-04-21
6. Fishburn, Peter C. (1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030. ISSN   0036-1399.
7. Black, Duncan (1948). "On the Rationale of Group Decision-making". The Journal of Political Economy. 56 (1): 23–34. doi:10.1086/256633. JSTOR   1825026. S2CID   153953456.
8. Alós-Ferrer, Carlos; Granić, Đura-Georg (2015-09-01). "Political space representations with approval data". Electoral Studies. 39: 56–71. doi:10.1016/j.electstud.2015.04.003. hdl: 1765/111247 . The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space.
9. Black, Duncan; Newing, R.A. (2013-03-09). McLean, Iain S. [in Welsh]; McMillan, Alistair; Monroe, Burt L. (eds.). The Theory of Committees and Elections by Duncan Black and Committee Decisions with Complementary Valuation by Duncan Black and R.A. Newing. Springer Science & Business Media. ISBN   9789401148603. For instance, if preferences are distributed spatially, there need only be two or more dimensions to the alternative space for cyclic preferences to be almost inevitable
10. Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN   1573-7101.
11. https://arxiv.org/pdf/1801.05911 "We say that an alternative is a Condorcet loser if it would be defeated by every other alternative in a kind of one-on-one contest that takes place in a sequential pairwise voting with a fixed agenda4.– Condorcet loser criterion (CLC), [...] we say that a social choice procedure satisfies the Condorcet loser criterion (CLC) provided that a Condorcet loser is never among the social choices."
12. Soh, Leen-Kiat (2017-10-04). "Voting: Preference Aggregating & Social Choice [CSCE475/875 class handout]" (PDF).
13. Franceschini, Fiorenzo; Maisano, Domenico A. (2022-06-01). "Analysing paradoxes in design decisions: the case of "multiple-district" paradox". International Journal on Interactive Design and Manufacturing (IJIDeM). 16 (2): 677–689. doi: 10.1007/s12008-022-00860-x . ISSN   1955-2505.
14. Balinski, Michel; Laraki, Rida (2011-01-28). Majority Judgment. The MIT Press. doi:10.7551/mitpress/9780262015134.001.0001. ISBN   978-0-262-01513-4.
15. Young, H. P.; Levenglick, A. (1978). "A Consistent Extension of Condorcet's Election Principle" (PDF). SIAM Journal on Applied Mathematics. 35 (2): 285–300. doi:10.1137/0135023. ISSN   0036-1399. JSTOR   2100667.
16. Smith, John H. (November 1973). "Aggregation of Preferences with Variable Electorate". Econometrica. 41 (6): 1027–1041. doi:10.2307/1914033. JSTOR   1914033.
17. Woodall, Douglas R. (1996). "Monotonicity and single-seat election rules". Voting matters. 6: 9–14.
18. Homogeneity and monotonicity of distance-rationalizable voting rules. 2 May 2011. pp. 821–828. ISBN   978-0-9826571-6-4.
19. Peters, Hans; Wakker, Peter (1991). "Independence of Irrelevant Alternatives and Revealed Group Preferences". Econometrica. 59 (6): 1787–1801. doi:10.2307/2938291. hdl: 1765/23217 . ISSN   0012-9682. JSTOR   2938291.
20. Cerreia-Vioglio, Simone; Lindberg, Per Olov; Maccheroni, Fabio; Marinacci, Massimo; Rustichini, Aldo (2021-09-01). "A canon of probabilistic rationality". Journal of Economic Theory. 196: 105289. arXiv: 2007.11386 . doi:10.1016/j.jet.2021.105289. ISSN   0022-0531.
21. Sniderman, Paul M.; Bullock, John (2018). "A Consistency Theory of Public Opinion and Political Choice: The Hypothesis of Menu Dependence". In Saris, Willem E.; Sniderman, Paul M. (eds.). Studies in Public Opinion: Attitudes, Nonattitudes, Measurement Error, and Change. Princeton University Press. pp. 337–358. doi:10.2307/j.ctv346px8.16. ISBN   978-0-691-18838-6. JSTOR   j.ctv346px8.16.
22. Tideman, T. N. (1987). "Independence of clones as a criterion for voting rules" (PDF). Social Choice and Welfare. 4 (3). Springer Science and Business Media LLC: 185–206. doi:10.1007/bf00433944. ISSN   0176-1714.
23. Schulze, Markus (2010-07-11). "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method" (PDF). Social Choice and Welfare. 36 (2). Springer Science and Business Media LLC: 267–303. doi:10.1007/s00355-010-0475-4. ISSN   0176-1714.
24. Delemazure, Théo; Peters, Dominik (2024-04-17). "Generalizing Instant Runoff Voting to Allow Indifferences". arXiv: 2404.11407 [cs.GT].
25. J. Green-Armytage (2014). "Strategic voting and nomination". Social Choice and Welfare. 42 (1). Springer: 111–138. doi:10.1007/s00355-013-0725-3. ISSN   0176-1714. JSTOR   43663746. S2CID   253847024 . Retrieved 2024-02-23. Figure 4 on page 137 shows instant-runoff voting having exit incentive despite being clone independent.
26. Green-Armytage, J. (2011). "Four Condorcet-Hare hybrid methods for single-winner elections" (PDF). Voting Matters (29): 1–14. S2CID   15220771. Smith-IIA [ISDA] Definition: Removing a candidate from the ballot who is not a member of the Smith set will not change the result of the election. ('IIA' here stands for 'independence of irrelevant alternatives'.)
28. Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". 6 Phoenix L. Rev. 117. Rochester, NY. SSRN   2276015. third place Candidate C is a centrist who is in fact the second choice of Candidate A's left-wing supporters and Candidate B's right-wing supporters. ... In such a situation, Candidate C would prevail over both Candidates A ... and B ... in a one-on-one runoff election. Yet, Candidate C would not prevail under IRV because he or she finished third and thus would be the first candidate eliminated
29. Stensholt, Eivind (2015-10-07). "What Happened in Burlington?". Discussion Papers: 13. There is a Condorcet ranking according to distance from the center, but Condorcet winner M, the most central candidate, was squeezed between the two others, got the smallest primary support, and was eliminated.
30. The Non-majority Rule Desk (July 29, 2011). "Why Approval Voting is Unworkable in Contested Elections - FairVote". FairVote Blog. Retrieved 11 October 2016.
31. Graham-Squire, Adam; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv: 2301.12075 . doi:10.1080/00344893.2023.2221689. ISSN   0034-4893.
32. Brams, Steven (1982). "The AMS nomination procedure is vulnerable to 'truncation of preferences'". Notices of the American Mathematical Society. 29. American Mathematical Society: 136–138. ISSN   0002-9920.
33. Fishburn, Peter C.; Brams, Steven J. (1984-01-01). "Manipulability of voting by sincere truncation of preferences". Public Choice. 44 (3): 397–410. doi:10.1007/BF00119689. ISSN   1573-7101.
34. Pennock, Ronald; Chapman, John W. (1977). Due Process: Nomos XVIII. NYU Press. p. 266. ISBN   9780814765692. if there is some single alternative which is ranked first by a majority of voters, we shall say there exists a majority will in favor of that alternative, according to the absolute majority (AM) criterion.
35. Kondratev, Aleksei Y.; Nesterov, Alexander S. (2020). "Measuring Majority Power and Veto Power of Voting Rules". Public Choice. 183 (1–2): 187–210. arXiv: 1811.06739 . doi:10.1007/s11127-019-00697-1. S2CID   53670198.
36. Lepelley, Dominique; Merlin, Vincent (1998). "Choix social positionnel et principe majoritaire". Annales d'Économie et de Statistique (51): 29–48. doi:10.2307/20076136. JSTOR   20076136.
37. Sertel, Murat R.; Yılmaz, Bilge (1999-09-01). "The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable". Social Choice and Welfare. 16 (4): 615–627. CiteSeerX   10.1.1.597.1421 . doi:10.1007/s003550050164. ISSN   0176-1714. S2CID   128357237.
38. Felsenthal, Dan S; Nurmi, Hannu (2018). Voting procedures for electing a single candidate : proving their (in)vulnerability to various voting paradoxes. Cham, Switzerland: Springer. ISBN   978-3-319-74033-1.
39. Kondratev, Aleksei Y.; Nesterov, Alexander S. (2020). "Measuring Majority Power and Veto Power of Voting Rules". Public Choice. 183 (1–2): 187–210. arXiv: 1811.06739 . doi:10.1007/s11127-019-00697-1. S2CID   53670198.
40. Sanver, M. Remzi (2002-03-01). "Scoring rules cannot respect majority in choice and elimination simultaneously". Mathematical Social Sciences. 43 (2): 151–155. doi:10.1016/S0165-4896(01)00087-7.
41. Woeginger, Gerhard J. (December 2003). "A note on scoring rules that respect majority in choice and elimination" (PDF). Mathematical Social Sciences. 46 (3): 347–354. doi:10.1016/S0165-4896(03)00050-7.
42. May, Kenneth O. (1952). "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision". Econometrica. 20 (4): 680–684. doi:10.2307/1907651. ISSN   0012-9682. JSTOR   1907651.
43. Pukelsheim, Friedrich (2014). Proportional representation: apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
44. Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN   0092-5853. JSTOR   2110496.
46. D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters , Issue 6, 1996
47. Felsenthal, Dan S.; Tideman, Nicolaus (2014-01-01). "Interacting double monotonicity failure with direction of impact under five voting methods". Mathematical Social Sciences. 67: 57–66. doi:10.1016/j.mathsocsci.2013.08.001. ISSN   0165-4896. it is generally agreed among social choice theorists that a voting method that is susceptible to any type of monotonicity failure suffers from a particularly serious defect.
48. Arrow, Kenneth J. (2017-12-13). Social Choice and Individual Values. p. 25. doi:10.12987/9780300186987. ISBN   978-0-300-18698-7. Since we are trying to describe social welfare and not some sort of illfare, we must assume that the social welfare function is such that the social ordering responds positively to alterations in individual values, or at least not negatively. Hence, if one alternative social state rises or remains still in the ordering of every individual without any other change in those orderings, we expect that it rises, or at least does not fall, in the social ordering.
49. Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
50. dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
51. Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under Instant Runoff Voting: estimates based on a spatial model of elections". Public Choice. 161 (1–2): 1–9. doi:10.1007/s11127-013-0118-2. ISSN   0048-5829. S2CID   30833409.
52. Dančišin, Vladimír (2017-01-01). "No-show paradox in Slovak party-list proportional system". Human Affairs. 27 (1): 15–21. doi:10.1515/humaff-2017-0002. ISSN   1337-401X.
53. Dummett, M.: Voting procedures. Oxford Clarendon Press (1984).
54. D. R. Woodall: Monotonicity of single-seat preferential election rules. Discrete Applied Mathematics 77 (1997), p. 83–84.
55. Moulin, Hervé (1988-06-01). "Condorcet's principle implies the no show paradox". Journal of Economic Theory. 45 (1): 53–64. doi:10.1016/0022-0531(88)90253-0.
56. Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN   0092-5853. JSTOR   2110496.
57. Ray, Depankar (1986-04-01). "On the practical possibility of a 'no show paradox' under the single transferable vote". Mathematical Social Sciences. 11 (2): 183–189. doi:10.1016/0165-4896(86)90024-7. ISSN   0165-4896.
58. Woodall, Douglas (December 1994). "Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994".
59. Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN   978-3-319-03855-1.
60. dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN   0044-2070 . Retrieved 2024-05-02.
61. BVerfG, Urteil des Zweiten Senats vom 03. Juli 2008 - 2 BvC 1/07, 2 BvC 7/07 -, Rn. 1-145. 121BVerfGE- 2 BvC 1/07, 2 BvC 7/07 - (BVerfG 03 July 2008). ECLI:DE:BVerfG:2008:cs20080703.2bvc000107
62. Brandt, Felix; Geist, Christian; Peters, Dominik (2016-01-01). "Optimal Bounds for the No-Show Paradox via SAT Solving". Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems. AAMAS '16. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 314–322. arXiv: 1602.08063 . ISBN   9781450342391.
63. Markus Schulze (1998-06-12). "Regretted Turnout. Insincere = ranking" . Retrieved 2011-05-14.
64. McCune, David; Wilson, Jennifer (2024-04-07). "The Negative Participation Paradox in Three-Candidate Instant Runoff Elections". arXiv: 2403.18857 [physics.soc-ph].
65. Fishburn, Peter C.; Brams, Steven J. (1983-01-01). "Paradoxes of Preferential Voting". Mathematics Magazine. 56 (4): 207–214. doi:10.2307/2689808. JSTOR   2689808.
66. D R Woodall, "Properties of Preferential Election Rules", Voting matters , Issue 3 (1994), pp. 8–15.
67. D R Woodall, "Monotonicity and Single-Seat Election Rules", Voting matters, Issue 6 (1996), pp. 9–12.
68. Schulze, Markus (2011). "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method". Social Choice and Welfare. 36 (2): 267–303. doi:10.1007/s00355-010-0475-4. ISSN   0176-1714. JSTOR   41108129.
69. Schulze, Markus (2024-03-03), The Schulze Method of Voting, arXiv: 1804.02973
70. Saari, Donald G. (2012-12-06). Geometry of Voting. Springer Science & Business Media. ISBN   978-3-642-48644-9.