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Spoiler effect

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For the same effect in sports, see Elimination from postseason contention § Spoiler effect.
"Independence of spoilers" redirects here. For the logical property in decision theory, see Independence of irrelevant alternatives.
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In social choice theory and politics, a spoiler is a losing candidate who affects the results of an election simply by participating,[1][2] a situation that is called a spoiler effect. If a major candidate is perceived to have lost an election because of a minor candidate, the minor candidate is called a spoiler candidate and the major candidate is said to have been spoiled. Often times the term spoiler will be applied to candidates or situations which do not meet the full definition, typically in real-world scenarios where the introduction of a new candidate can cause voters to change their opinions, either through their campaign or merely by existing. A voting system that is not affected by spoilers is called independent of irrelevant alternatives or spoilerproof.[3]

The frequency and severity of spoiler effects depends substantially on the voting method. Instant-runoff or ranked-choice voting (RCV), the two-round system (TRS), and especially first-past-the-post (FPP) without winnowing or primary elections[4] are highly sensitive to spoilers (though RCV and TRS less so in some circumstances), and all three rules are affected by center-squeeze and vote splitting.[5][6][7][8] Majority-rule (or Condorcet) methods are only rarely affected by spoilers, which are limited to rare[9][10] situations called cyclic ties.[11][12][13] Rated voting systems are not subject to Arrow's theorem. Whether such methods are spoilerproof depends on the nature of the rating scales the voters use to express their opinions.[14][15][16][17]

Spoiler effects can also occur in some methods of proportional representation, such as the single transferable vote (STV or RCV-PR) and the largest remainders method of party-list representation, where it is called a new party paradox. A new party entering an election causes some seats to shift from one unrelated party to another, even if the new party wins no seats.[18] This kind of spoiler effect is avoided by divisor methods and proportional approval.[18]: Thm.8.3 

Motivation

[edit]

In decision theory, independence of irrelevant alternatives is a fundamental principle of rational choice which says that a decision between two outcomes, A or B, should not depend on the quality of a third, unrelated outcome C. A famous joke by Sidney Morgenbesser illustrates this principle:[19]

A man is deciding whether to order apple, blueberry, or cherry pie before settling on apple. The waitress informs him that the cherry pie is very good and a favorite of most customers. The man replies "in that case, I'll have the blueberry."

Politicians and social choice theorists have long argued for the unfairness of spoiler effects. The mathematician and political economist Nicolas de Condorcet was the first to study the spoiler effect, in the 1780s.[20]

Manipulation by politicians

[edit]

Voting systems that violate independence of irrelevant alternatives are susceptible to being manipulated by strategic nomination. Such systems may produce an incentive to entry, increasing a candidate's chances of winning if similar candidates join the race, or an incentive to exit, reducing the candidate's chances of winning.

Some systems are particularly infamous for their ease of manipulation, such as the Borda count, which exhibits a particularly severe entry incentive, letting any party "clone their way to victory" by running a large number of candidates. This famously forced de Borda to concede that "my system is meant only for honest men,"[21][22] and eventually led to its abandonment by the French Academy of Sciences.[22]

Other systems exhibit an exit incentive. The vote splitting effect in plurality voting demonstrates this method's strong exit incentive: if multiple candidates with similar views run in an election, their supporters' votes will be diluted, which may cause a unified opposition candidate to win despite having less support. This effect encourages groups of similar candidates to form an organization to make sure they don't step on each other's toes.[23]

By electoral system

[edit]
Susceptibility to spoilers
Electoral system Spoiler effect
Plurality voting High
Runoffs or RCV Medium
Condorcet methods Low
Score or Medians Depends

Different electoral systems have different levels of vulnerability to spoilers. In general, spoilers are common with plurality voting, somewhat common in plurality-runoff methods, rare with majoritarian methods, and with a varying level of spoiler vulnerability with most rated voting methods.[note 1]

First-preference plurality

[edit]

In cases where there are many similar candidates, spoiler effects occur most often in first-preference plurality (FPP).[citation needed] For example, in the United States, vote splitting is common in primaries, where many similar candidates run against each other. The purpose of a primary election is to eliminate vote splitting among candidates from the same party in the general election by running only one candidate. In a two-party system, party primaries effectively turn FPP into a two-round system.[24][25][26]

Vote splitting is the most common cause of spoiler effects in FPP. In these systems, the presence of many ideologically-similar candidates causes their vote total to be split between them, placing these candidates at a disadvantage.[27][28] This is most visible in elections where a minor candidate draws votes away from a major candidate with similar politics, thereby causing a strong opponent of both to win.[27][29]

Runoff systems

[edit]

Plurality-runoff methods like the two-round system and RCV still experience vote-splitting in each round. This produces a kind of spoiler effect called a center squeeze. Compared to plurality without primaries, the elimination of weak candidates in earlier rounds reduces their effect on the final results; however, spoiled elections remain common compared to other systems.[28][30][31] As a result, instant-runoff voting still tends towards two-party rule through the process known as Duverger's law.[16][32] A notable example of this can be seen in Alaska's 2024 race, where party elites pressured candidate Nancy Dahlstrom into dropping out to avoid a repeat of the spoiled 2022 election.[33][34][35]

Tournament (Condorcet) voting

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Spoiler effects rarely occur when using tournament solutions, where candidates are compared in one-on-one matchups to determine relative preference. For each pair of candidates, there is a count for how many voters prefer the first candidate in the pair to the second candidate The resulting table of pairwise counts eliminates the step-by-step redistribution of votes, which is usually the cause for spoilers in other methods.[36] This pairwise comparison means that spoilers can only occur when there is a Condorcet cycle, where there is no single candidate preferred to all others.[36][37][38]

Theoretical models suggest that somewhere between 90% and 99% of real-world elections have a Condorcet winner,[37][38] and the first Condorcet cycle in a ranked American election was found in 2021.[39] Some systems like the Schulze method and ranked pairs have stronger spoiler resistance guarantees that limit which candidates can spoil an election without a Condorcet winner.[40]: 228–229 

Rated voting

[edit]

Rated voting methods ask voters to assign each candidate a score on a scale (e.g. rating them from 0 to 10), instead of listing them from first to last. Highest median and score (highest mean) voting are the two most prominent examples of rated voting rules. Whenever voters rate candidates independently, the rating given to one candidate does not affect the ratings given to the other candidates. Any new candidate cannot change the winner of the race without becoming the winner themselves, which would disqualify them from the definition of a spoiler. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.

The outcome of rated voting depends on the scale used by the voter or assumed by the mechanism.[41] If the voters use relative scales, i.e. scales that depend on what candidates are running, then the outcome can change if candidates who don't win drop out.[42] Empirical results from panel data suggest that judgments are at least in part relative.[43][44] Thus, rated methods, as used in practice, may exhibit a spoiler effect caused by the interaction between the voters and the system, even if the system itself passes IIA given an absolute scale.

Proportional representation

[edit]

Spoiler effects can also occur in some methods of proportional representation, such as the single transferable vote (STV or RCV-PR) and the largest remainders method of party-list representation, where it is called a new party paradox. A new party entering an election causes some seats to shift from one unrelated party to another, even if the new party wins no seats.[18] This kind of spoiler effect is avoided by divisor methods and proportional approval.[18]: Thm.8.3 

Spoiler campaign

[edit]
See also: List of elections involving vote splitting, Protest vote, and Single-issue politics

United States

[edit]
See also: Electoral fusion in the United States

A spoiler campaign in the United States is often one that cannot realistically win but can still determine the outcome by pulling support from a more competitive candidate.[45] The two major parties in the United States, the Republican Party and Democratic Party, have regularly won 98% of all state and federal seats.[46] The US presidential elections most consistently cited as having been spoiled by third-party candidates are 1844[47] and 2000.[48][49][50][47] The 2016 election is more disputed as to whether it contained spoiler candidates or not.[51][52][53] For the 2024 presidential election, Republican lawyers and operatives have fought to keep right-leaning third-parties like the Constitution Party off of swing state ballots[54] while working to get Cornel West on battleground ballots.[55] Democrats have helped some right-leaning third-parties gain ballot access while challenging ballot access of left-leaning third-parties like the Green Party.[56] According to the Associated Press, the GOP effort to prop up possible spoiler candidates in 2024 appears more far-reaching than the Democratic effort.[57] Barry Burden argues that they have almost no chance of winning the 2024 election but are often motivated by particular issues.[58]

Third party candidates are always controversial because almost anyone could play spoiler.[59][60] This is especially true in close elections where the chances of a spoiler effect increase.[61] Strategic voting, especially prevalent during high stakes elections with high political polarization, often leads to a third-party that underperforms its poll numbers with voters wanting to make sure their least favorite candidate is not in power.[46][62][63] Third-party campaigns are more likely to result in the candidate a third party voter least wants in the White House.[60] Third-party candidates prefer to focus on their platform than on their impact on the frontrunners.[60]

Notable unintentional spoilers

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An unintentional spoiler is one that has a realistic chance of winning but falls short and affects the outcome of the election.

2009 Burlington mayoral election

[edit]

In Burlington, Vermont's second IRV election, spoiler Kurt Wright knocked out Democrat Andy Montroll in the second round, leading to the election of Bob Kiss, despite the election results showing most voters preferred Montroll to Kiss.[64] The results of every possible one-on-one election can be completed as follows:

Andy Montroll (D) 6262 (Montroll) –

591 (Simpson)

4570 (Montroll) –

2997 (Smith)

4597 (Montroll) –

3664 (Wright)

4064 (Montroll) –

3476 (Kiss)

4/4 Wins
Bob Kiss (P) 5514 (Kiss) –

844 (Simpson)

3944 (Kiss) –

3576 (Smith)

4313 (Kiss) –

4061 (Wright)

3/4 Wins RCV winner
Kurt Wright (R) 5270 (Wright) –

1310 (Simpson)

3971 (Wright) –

3793 (Smith)

2/4 Wins Spoiler for Montroll
Dan Smith (I) 5570 (Smith) –

721 (Simpson)

1/4 Wins
James Simpson (G) 0/4 Wins

This leads to an overall preference ranking of:

  1. Montroll – defeats all candidates below, including Kiss (4,064 to 3,476)
  2. Kiss – defeats all candidates below, including Wright (4,313 to 4,061)
  3. Wright – defeats all candidates below, including Smith (3,971 to 3,793)
  4. Smith – defeats Simpson (5,570 to 721) and the write-in candidates

Montroll was therefore preferred over Kiss by 54% of voters, over Wright by 56%, and over Smith by 60%. Had Wright not run, Montroll would have won instead of Kiss.[64][65]

Because all ballots were fully released, it is possible to reconstruct the winners under other voting methods. While Wright would have won under plurality, Kiss won under IRV, and would have won under a two-round vote or a traditional nonpartisan blanket primary. Montroll, being the majority-preferred candidate, would have won if the ballots were counted using ranked pairs (or any other Condorcet method).[66]

2022 Alaska's at-large congressional district special election

[edit]

In Alaska's first-ever IRV election, Nick Begich was eliminated in the first round to advance Mary Peltola and Sarah Palin. However, the pairwise comparison shows that Begich was the Condorcet winner while Palin was both the Condorcet loser and a spoiler:[67]

Pairwise comparison matrix by vote totals[68][67]
Begich Peltola Palin
Begich - 88,126 101,438
Peltola 79,486 - 91,375
Palin 63,666 86,197 -
Pairwise as a percentage
Winner Loser Winner Loser
Begich vs. Peltola 52.6% vs. 47.4%
Begich vs. Palin 61.4% vs. 38.6%
Peltola vs. Palin 51.5% vs. 48.5%

In the wake of the election, a poll found 54% of Alaskans, including a third of Peltola voters, supported a repeal of RCV.[69][70][71] Observers noted such pathologies would have occurred under Alaska's previous primary system as well, leading several to suggest Alaska adopt any one of several alternatives without this behavior.[72]

See also

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Notes

[edit]
  1. ^ Strategic voting can sometimes create additional spoiler-like behavior. However, this does not substantially affect the general order described here.

References

[edit]
  1. ^ Heckelman, Jac C.; Miller, Nicholas R. (2015-12-18). Handbook of Social Choice and Voting. Edward Elgar Publishing. ISBN 9781783470730. A spoiler effect occurs when a single party or a candidate entering an election changes the outcome to favor a different candidate.
  2. ^ "The Spoiler Effect". The Center for Election Science. Retrieved 2024-03-03.
  3. ^ Miller, Nicholas R. (2019-04-01). "Reflections on Arrow's theorem and voting rules". Public Choice. 179 (1): 113–124. doi:10.1007/s11127-018-0524-6. hdl:11603/20937. ISSN 1573-7101.
  4. ^ Borodin, Allan; Lev, Omer; Shah, Nisarg; Strangway, Tyrone (2024-04-01). "Primarily about primaries". Artificial Intelligence. 329: 104095. doi:10.1016/j.artint.2024.104095. ISSN 0004-3702.
  5. ^ Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. ... Approval voting thus appears to solve the problem of vote splitting simply and elegantly. ... Range voting solves the problems of spoilers and vote splitting
  6. ^ Merrill, Samuel (1985). "A statistical model for Condorcet efficiency based on simulation under spatial model assumptions". Public Choice. 47 (2): 389–403. doi:10.1007/bf00127534. ISSN 0048-5829. the 'squeeze effect' that tends to reduce Condorcet efficiency if the relative dispersion (RD) of candidates is low. This effect is particularly strong for the plurality, runoff, and Hare systems, for which the garnering of first-place votes in a large field is essential to winning
  7. ^ McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002). "How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections". American Journal of Political Science. 46 (1): 134–147. doi:10.2307/3088418. ISSN 0092-5853. JSTOR 3088418. As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
  8. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it ... makes it less likely
  9. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
  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. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
  12. ^ 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.
  13. ^ Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election."
  14. ^ Morreau, Michael (2014-10-13). "Arrow's Theorem". Stanford Encyclopedia of Philosophy. Retrieved 2024-10-09. One important finding was that having cardinal utilities is not by itself enough to avoid an impossibility result. ... Intuitively speaking, to put information about preference strengths to good use it has to be possible to compare the strengths of different individuals' preferences.
  15. ^ Miller, Nicholas R. (2019-04-01). "Reflections on Arrow's theorem and voting rules". Public Choice. 179 (1): 113–124. doi:10.1007/s11127-018-0524-6. hdl:11603/20937. ISSN 1573-7101.
  16. ^ a b Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. ... Approval voting thus appears to solve the problem of vote splitting simply and elegantly. ... Range voting solves the problems of spoilers and vote splitting
  17. ^ "The Spoiler Effect". The Center for Election Science. 2015-05-20. Retrieved 2017-01-29.
  18. ^ a b c d Balinski, Michel L.; Young, H. Peyton (2001) [1982]. Fair Representation: Meeting the Ideal of One Man, One Vote. New Haven: Yale University Press. ISBN 0-300-02724-9.
  19. ^ Pearce, David. "Individual and social welfare: a Bayesian perspective" (PDF). Frisch Lecture Delivered to the World Congress of the Econometric Society.
  20. ^ McLean, Iain (1995-10-01). "Independence of irrelevant alternatives before Arrow". Mathematical Social Sciences. 30 (2): 107–126. doi:10.1016/0165-4896(95)00784-J. ISSN 0165-4896.
  21. ^ Black, Duncan (1987) [1958]. The Theory of Committees and Elections. Springer Science & Business Media. ISBN 9780898381894.
  22. ^ a b McLean, Iain; Urken, Arnold B.; Hewitt, Fiona (1995). Classics of Social Choice. University of Michigan Press. ISBN 978-0472104505.
  23. ^ Green-Armytage, James (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. ... if two or more candidates with similar views run in the same plurality election, then the voters who support those views will be divided among them, giving an advantage to other candidates with opposed views. Therefore, it is helpful for groups of fairly like-minded people to form some kind of association – that is, a political party – which fields only one candidate per election, and which provides some kind of process for deciding whom this one candidate should be – that is, a primary.
  24. ^ Santucci, Jack; Shugart, Matthew; Latner, Michael S. (2023-10-16). "Toward a Different Kind of Party Government". Protect Democracy. Archived from the original on 2024-07-16. Retrieved 2024-07-16. Finally, we should not discount the role of primaries. When we look at the range of countries with first-past-the-post (FPTP) elections (given no primaries), none with an assembly larger than Jamaica's (63) has a strict two-party system. These countries include the United Kingdom and Canada (where multiparty competition is in fact nationwide). Whether the U.S. should be called 'FPTP' itself is dubious, and not only because some states (e.g. Georgia) hold runoffs or use the alternative vote (e.g. Maine). Rather, the U.S. has an unusual two-round system in which the first round winnows the field. This usually is at the intraparty level, although sometimes it is without regard to party (e.g. in Alaska and California).
  25. ^ Gallagher, Michael; Mitchell, Paul (2005-09-15). "The American Electoral System". The Politics of Electoral Systems. OUP Oxford. p. 192. ISBN 978-0-19-153151-4. American elections become a two-round run-off system with a delay of several months between the rounds.
  26. ^ Bowler, Shaun; Grofman, Bernard; Blais, André (2009), "The United States: A Case of Duvergerian Equilibrium", Duverger's Law of Plurality Voting: The Logic of Party Competition in Canada, India, the United Kingdom and the United States, New York, NY: Springer, pp. 135–146, doi:10.1007/978-0-387-09720-6_9, ISBN 978-0-387-09720-6, retrieved 2024-08-31, In effect, the primary system means that the USA has a two-round runoff system of elections.
  27. ^ a b King, Bridgett A.; Hale, Kathleen (2016-07-11). Why Don't Americans Vote? Causes and Consequences: Causes and Consequences. ABC-CLIO. ISBN 9781440841163. Those votes that are cast for minor party candidates are perceived as taking away pivotal votes from major party candidates. ... This phenomenon is known as the 'spoiler effect'.
  28. ^ a b Sen, Amartya; Maskin, Eric (2017-06-08). "A Better Way to Choose Presidents" (PDF). New York Review of Books. ISSN 0028-7504. Retrieved 2019-07-20. plurality-rule voting is seriously vulnerable to vote-splitting ... runoff voting ... as French history shows, it too is highly subject to vote-splitting. ... [Condorcet] majority rule avoids such vote-splitting debacles because it allows voters to rank the candidates and candidates are compared pairwise
  29. ^ Buchler, Justin (2011-04-20). Hiring and Firing Public Officials: Rethinking the Purpose of Elections. Oxford University Press, USA. ISBN 9780199759965. a spoiler effect occurs when entry by a third-party candidate causes party A to defeat party B even though Party B would have won in a two-candidate race.
  30. ^ Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it ... makes it less likely
  31. ^ Poundstone, William (2009-02-17). Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It). Farrar, Straus and Giroux. ISBN 9781429957649. IRV is excellent for preventing classic spoilers-minor candidates who tip the election from one major candidate to another. It is not so good when the 'spoiler' has a real chance of winning
  32. ^ Drutman, Lee (2024-09-12). "We need more (and better) parties". Undercurrent Events. Retrieved 2024-09-19.
  33. ^ Volić, Ismar (2024-04-02). "Duverger's law". Making Democracy Count. Princeton University Press. Ch. 2. doi:10.2307/jj.7492228. ISBN 978-0-691-24882-0.
  34. ^ Strassel, Kimberly A. (2024-08-27). "Ranked Choice May Die in Alaska". The Wall Street Journal.
  35. ^ Early, Wesley (2024-09-05). "Why candidates are withdrawing from Alaska's general election". Alaska Public Media. Anchorage, Alaska: NPR. Retrieved 2024-09-20.
  36. ^ a b Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election."
  37. ^ a b Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
  38. ^ a b Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice (journal). 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
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  40. ^ Schulze, Markus (2018-03-15). "The Schulze Method of Voting". arXiv:1804.02973 [cs.GT]. The Smith criterion and Smith-IIA (where IIA means "independence of irrelevant alternatives") say that weak alternatives should have no impact on the result of the elections ... the Schulze method, as defined in section 2.2, satisfies Smith-IIA.
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  46. ^ a b Masket, Seth (Fall 2023). "Giving Minor Parties a Chance". Democracy. 70.
  47. ^ a b Green, Donald J. (2010). Third-party matters: politics, presidents, and third parties in American history. Santa Barbara, Calif: Praeger. pp. 153–154. ISBN 978-0-313-36591-1.
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  49. ^ Herron, Michael C.; Lewis, Jeffrey B. (April 24, 2006). "Did Ralph Nader spoil Al Gore's Presidential bid? A ballot-level study of Green and Reform Party voters in the 2000 Presidential election". Quarterly Journal of Political Science. 2 (3). Now Publishing Inc.: 205–226. doi:10.1561/100.00005039. Pdf.
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  51. ^ Devine, Christopher J.; Kopko, Kyle C. (2021-09-01). "Did Gary Johnson and Jill Stein Cost Hillary Clinton the Presidency? A Counterfactual Analysis of Minor Party Voting in the 2016 US Presidential Election". The Forum. 19 (2): 173–201. doi:10.1515/for-2021-0011. ISSN 1540-8884. S2CID 237457376. The perception that Johnson and Stein 'stole' the 2016 presidential election from Clinton is widespread...Our analysis indicates that Johnson and Stein did not deprive Clinton of an Electoral College majority, nor Trump the legitimacy of winning the national popular vote.
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