Wikipedia

Center squeeze

Center squeezeis a kind ofindependence of irrelevant alternativesviolation seen in a number of election rules, such astwo-roundandinstant runoff,for example. In a center squeeze, theCondorcet winneris eliminated before they have the chance to face any of the other candidates in a one-on-one race (which by definition, they would win). The term can also refer[according to whom?]to tendency of such rules to encouragepolarizationamong elected officials.

Distribution of winners on a simulatedpolitical compass,showing how center-squeeze extends to more complex or multi-dimensional models. The number of winners is displayed as aheatmap.The bias ofFPTP,runoffsorprimaries,andRCV(center-left column) is clearly visible.

In a center squeeze, candidates focused on appealing to abaseofcore supporterscan squeeze the Condorcet winner out of the race by splitting thefirst-round votewith them, allowing a more-extreme alternative to win. This effect was first predicted bysocial choice theoristsin the 1940s and 50s, and has since been documented in various countries usingplurality-style electoral systems.

Famous examples of center squeezes include the2022 Alaska special election(whereNick Begich IIIwas eliminated in the first round bySarah Palin)[1]as well as the2007 French presidential election,where moderate liberalFrançois Bayrouwas eliminated byleft-wing populistcandidateSégolène Royal,allowingNicolas Sarkozyto win the second round.[2][3]

Overview

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Center squeezes are a kind ofindependence of irrelevant alternativesviolation in which the Condorcet winner is eliminated before the final round of an election.[4][5]Candidates focused on appealing to a smallbaseofcore supporterscan squeeze Condorcet winners out of the race, by splitting thefirst-round voteneeded to survive earlier rounds.[6][7]

The "center" in "center squeeze" refers to candidates who are close to the center of public opinion,and is not limited tocentristsalong a traditional, one-dimensionalpolitical spectrum.[8]A center squeeze can occur in any situation where voters prefer candidateswho hold views similar to their own.[9]

Center-squeeze has been observed inAustralia,[10][11]and variousUScities.[12]

Susceptibility by system

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Center squeeze is a major feature of two-party systems usingprimariesor other multiple-round systems.[6]In these methods, candidates must focus on appealing to theircore supportersto ensure they can make it past the first round, where only first-preferences count.[6][citation needed]

If voters assign scores to candidates based on ideological distance,score votingwill always select the candidate closest to somecentral tendencyof the voter distribution. As a result, while score voting does not always elect the candidate closest to the median voter, it often behaves like methods that do.[13][14]Under most models of strategic voting,spoilerproofcardinal methods tend to behave likeapproval votingand thus converge on the Condorcet winner.[15][16][17][18]

The opposite situation—a bias in favor of bland, inoffensive, or unknown candidates—is not common in any widely-used voting rules. However, it can occur for "negative" voting methods that choose candidates with the least opposition, likeanti-plurality,D21 – Janeček,orCoombs' method.[19]

Examples

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Start:Mid-Start:Mid-End:End:
33.4%17.2%13.4%35.9%

Alphabet explanation

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In Alphabet Land, voters are divided based on how names should be arranged on lists.thinks names should always be in alphabetical order;thinks they should be in reverse-alphabetical order; andthinks the order should be randomized. In this example, voters' happiness with the outcome fallslinearlywith the distance (number of letters) between the voter and the candidate.

Becauseis preferred to bothandin head-to-head match-ups,is themajority-preferred (Condorcet) winner.is thesocially-optimal winneras well.[14]Thus,is the "best" or "most popular" candidate under both common metrics of candidate quality insocial choice.[4]

Vote totals of an example first-past-the-post election, which is the same as the first round of an RCV election.

First-preference plurality (FPP)

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wins under a single-round of FPP, with 35.9% of voters choosing them as their favorite. However, over substantially more voters consideredto be their least favorite, with 63.1% of voters preferring.is elected, despite an overwhelming two-thirds majority preferring.

Ranked-choice runoff (Alternative, Two-round)

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Ranked-choice voting (RCV)tries to address vote-splitting in FPP by replacing it with a series of FPP elections, with the loser being eliminated in each round.[20][21]

The first round of the election is the same as the FPP election.has the leastfirst preferencesand is therefore eliminated. Their votes are reassigned toand,according to their ballot. In the second round, enough voters who preferredas their first choice tookas their second choice andwins the election. RCV thus fails to have a substantial moderating impact, instead causing only a swing from one extreme to the other.[22]

2022 Alaska special election

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Main article:2022 Alaska's at-large congressional district special election
Center squeeze
Alaska's at-large congressional district
Turnout32.2%[23]
Candidate Mary Peltola Sarah Palin Nick Begich III
Party Democratic Republican Republican
First round 74,817
39.7% 		58,339
30.9% 		52,536
27.8%
Final round 91,266
51.5% 		86,026
48.5% 		Eliminated

The2022 Alaska special electionseat was an example of a center squeeze, whereNick Begich IIIwas eliminated in the first round byright-wingspoilerSarah Palin,[1][24]despite a majority of voters preferring Begich to either one of his opponents.[1][25]The ranked-choice runoff election involved oneDemocrat(Mary Peltola) and twoRepublicans(Sarah PalinandNick Begich III). Because the full ballot data for the race was released,social choice theoristswere able to confirm that Palin spoiled the race for Begich, with Peltola winning the race as a result of severalpathologicalbehaviors that tend to characterize center-squeeze elections.[25][1]

The election produced a winner opposed by a majority of voters, with a majority of voters ranking Begich above Peltola and Palin,[26][1][24]and more than half giving Peltola no support at all.[26][1]The election was also notable as ano-show paradox,where a candidate is eliminated as a result of votes cast insupportof their candidacy. In this case, ballots ranking Palin first and Begich second instead allowed Peltola to win.[1][27][24]

Manysocial choice theoristscriticized the ranked-choice runoff procedure for itspathological behavior.[28][1]Along with being a center squeeze, the election was anegative voting weight event,[27]where a voter's ballot has the opposite of its intended effect (e.g. where a candidate would needmorevotes to lose).[27][29]In this race, Peltola would have lost if she had received more support from Palin voters,[30][1][26]and won as a result of 5,200 ballots that ranked her last (after Palin then Begich).[27][1]However,social choice theoristswere careful to note the results likely would have been the same under Alaska's previous primary system as well. This led several to recommend replacing the system with any one of severalalternatives without these behaviors,such asSTAR,approval,orCondorcet voting.[25][26]

2009 Burlington mayoral election

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Main article:2009 Burlington mayoral election

The2009 Burlington mayoral electionwas held in March 2009 for the city ofBurlington, Vermont,and serves as an example of a four-candidate center squeeze. This was the second mayoral election since the city's 2005 change toranked-choice runoff voting,after the2006 mayoral election.[31]In the 2009 election, incumbentBurlington mayorBob Kisswon reelection as a member of theVermont Progressive Party,[32]defeating Kurt Wright in the final round with 48% of the vote.[33]

The election results were criticized bymathematiciansandvoting theoristsfor severalpathologiesassociated with RCV. These included ano-show paradox,where Kiss won only as a result of 750 votes ranking Kiss in last place.[34][35][36]Severalelectoral reformadvocates branded the election a failure after Kiss was elected despite 54% of voters voting for Montroll over Kiss,[37][38]violating themajority-rule principle.[39][40][41]Later analyses showed the race was spoiled, with Wright pulling moderate votes away from Montroll, who would have beat Kiss in a one-on-one race.[40][42]

The controversy culminated in a successful 2010initiativethat repealed RCV by a vote of 52% to 48%,[43][44]a 16-point shift from the 64% who had supported the 2005 ratification.[31]The results of every possible one-on-one election are as follows:

Party Candidate vs. Simpson vs. Smith vs. Wright vs. Kiss Win: Loss
Andy Montroll (D) 6262 (Montroll) –

591 (Simpson)

4570 (Montroll) –

2997 (Smith)

4597 (Montroll) –

3664 (Wright)

4064 (Montroll) –

3476 (Kiss)

4 W: 0 L
Bob Kiss (P) 5514 (Kiss) –

844 (Simpson)

3944 (Kiss) –

3576 (Smith)

4313 (Kiss) –

4061 (Wright)

3 W: 1 L
Kurt Wright (R) 5270 (Wright) –

1310 (Simpson)

3971 (Wright) –

3793 (Smith)

2 W: 2 L
Dan Smith (I) 5570 (Smith) –

721 (Simpson)

1 W: 3 L
James Simpson (G) 0 W: 4 L

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% of voters, over Smith by 60%, and over Simpson by 91% of voters.[45]

2016 United States presidential election

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Main article:2016 United States presidential election

Another possible example is the2016 United States presidential election,where polls found several alternatives includingBernie SandersandGary Johnsondefeating bothDonald TrumpandHillary Clintonunder amajority-orrated-voting rulesbut being squeezed out by both RCV and theprimary electionrules.[13][46]

2024 United States presidential election

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Main article:2024 United States presidential election

Election law scholarNed Foleycriticized thetwo-round systemvariant used in the United States, which has been described as a first round ofprimariesbefore ade-factorunoff,[47][48]for creating a center squeeze in the 2024 presidential election and thus contributing topolitical polarization.Foley argued that both the primary system and a hypotheticalinstant-runoff votingsystem led to the election ofDonald Trumpby eliminatingNikki Haley,the majority-preferred (or Condorcet) candidate according to polling.[49][50]

See also

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References

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  1. ^abcdefghijGraham-Squire, Adam; McCune, David (2022-09-11). "A Mathematical Analysis of the 2022 Alaska Special Election for US House". p. 2.arXiv:2209.04764v3[econ.GN].Since Begich wins both... he is the Condorcet winner of the election... AK election also contains a Condorcet loser: Sarah Palin.... she is also a spoiler candidate
  2. ^Balinski, Michel; Laraki, Rida (2011), Dolez, Bernard; Grofman, Bernard; Laurent, Annie (eds.),"Election by Majority Judgment: Experimental Evidence",In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections,New York, NY: Springer, pp. 13–54,doi:10.1007/978-1-4419-7539-3_2,ISBN978-1-4419-7539-3,retrieved2024-10-31
  3. ^Balinski, Michel; Laraki, Rida (2020-03-01)."Majority judgment vs. majority rule".Social Choice and Welfare.54(2): 429–461.doi:10.1007/s00355-019-01200-x.ISSN1432-217X.
  4. ^abMerrill, Samuel (1984)."A Comparison of Efficiency of Multicandidate Electoral Systems".American Journal of Political Science.28(1): 23–48.doi:10.2307/2110786.ISSN0092-5853.JSTOR2110786.However, squeezed by surrounding opponents, a centrist candidate may receive few first-place votes and be eliminated under Hare.
  5. ^Atkinson, Nathan; Ganz, Scott C. (2022-10-30)."The flaw in ranked-choice voting: rewarding extremists".The Hill.Retrieved2023-05-14.However, ranked-choice voting makes it more difficult to elect moderate candidates when the electorate is polarized. For example, in a three-person race, the moderate candidate may be preferred by a majority of voters to each of the more extreme candidates. However, voters with far-left and far-right views will rank the candidate in second place rather than in first place. Since ranked-choice voting counts only the number of first-choice votes (among the remaining candidates), the moderate candidate would be eliminated in the first round, leaving one of the extreme candidates to be declared the winner.
  6. ^abcMerrill, 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.ISSN0048-5829.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.ISSN0092-5853.JSTOR3088418.As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
  8. ^Davis, Otto A.; Hinich, Melvin J.; Ordeshook, Peter C. (1970-01-01). "An Expository Development of a Mathematical Model of the Electoral Process".The American Political Science Review.64(2): 426–448.doi:10.2307/1953842.JSTOR1953842.S2CID1161006.Since our model is multi-dimensional, we can incorporate all criteria which we normally associate with a citizen's voting decision process — issues, style, partisan identification, and the like.
  9. ^Lewyn 2012:"third place Candidate C is a centrist who is 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"
  10. ^Mussel, Johanan D.; Schlechta, Henry (2023-07-21)."Australia: No party convergence where we would most expect it".Party Politics.doi:10.1177/13540688231189363.ISSN1354-0688.
  11. ^Lewyn, Michael (2012)."Two Cheers for Instant Runoff Voting".6 Phoenix L. Rev.117.Rochester, NY.SSRN2276015.
  12. ^Vishwanath, Arjun (2021)."Electoral Institutions and Substantive Representation in Local Politics: The Effects of Ranked Choice Voting".SSRN Electronic Journal.doi:10.2139/ssrn.3802566.ISSN1556-5068.
  13. ^abPotthoff, Richard F.; Munger, Michael C. (November 2021)."Condorcet Loser in 2016: Apparently Trump; Condorcet Winner: Not Clinton?".American Politics Research.49(6): 618–636.doi:10.1177/1532673X211009499.ISSN1532-673X.
  14. ^abDeGroot, Morris H. (1970).Optimal Statistical Decisions.McGraw-Hill Book Co., New York-London-Sydney. p. 232.ISBN9780471680291.MR0356303.
  15. ^Laslier, Jean-François (January 2009)."The Leader Rule: A Model of Strategic Approval Voting in a Large Electorate".Journal of Theoretical Politics.21(1): 113–136.doi:10.1177/0951629808097286.ISSN0951-6298.
  16. ^Laslier, Jean-François; Sanver, M. Remzi, eds. (2010).Handbook on Approval Voting.Studies in Choice and Welfare. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 2.doi:10.1007/978-3-642-02839-7.ISBN978-3-642-02838-0.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.
  17. ^Cox, Gary W. (1985)."Electoral Equilibrium under Approval Voting".American Journal of Political Science.29(1): 112–118.doi:10.2307/2111214.ISSN0092-5853.JSTOR2111214.
  18. ^Myerson, Roger B.; Weber, Robert J. (1993)."A Theory of Voting Equilibria".The American Political Science Review.87(1): 102–114.doi:10.2307/2938959.ISSN0003-0554.JSTOR2938959.
  19. ^Shankar, Karthik H. (2022-12-01)."Normed Negative Voting to Depolarize Politics".Group Decision and Negotiation.31(6): 1097–1120.doi:10.1007/s10726-022-09799-6.ISSN1572-9907.
  20. ^"Ranked Choice Voting".FairVote.Retrieved2024-07-22.
  21. ^"Avoid Vote-Splitting and Weak Plurality Results".RCV Resources.Ranked Choice Voting Resource Center. 2020-11-09.Retrieved2024-07-22.
  22. ^Atkinson, Nathan; Foley, Edward B.; Ganz, Scott (2023)."Beyond the Spoiler Effect: Can Ranked Choice Voting Solve the Problem of Political Polarization?".SSRN Electronic Journal.doi:10.2139/ssrn.4411173.ISSN1556-5068.
  23. ^"State of Alaska | 2022 Special General Election | Election Summary Report | August 16, 2022 | Official Results"(PDF).Alaska Division of Elections.September 2, 2022.Archived(PDF)from the original on August 17, 2022.RetrievedSeptember 2,2022.
  24. ^abcAtkinson, Nathan; Ganz, Scott C. (2022-10-30)."The flaw in ranked-choice voting: rewarding extremists".The Hill.Retrieved2023-05-14.However, ranked-choice voting makes it more difficult to elect moderate candidates when the electorate is polarized. For example, in a three-person race, the moderate candidate may be preferred to each of the more extreme candidates by a majority of voters. However, voters with far-left and far-right views will rank the candidate in second place rather than in first place. Since ranked-choice voting counts only the number of first-choice votes (among the remaining candidates), the moderate candidate would be eliminated in the first round, leaving one of the extreme candidates to be declared the winner.
  25. ^abcMaskin, Eric;Foley, Edward B. (2022-11-01)."Alaska's ranked-choice voting is flawed. But there's an easy fix".Washington Post.ISSN0190-8286.Retrieved2024-02-09.
  26. ^abcdClelland, Jeanne N. (2023-02-28). "Ranked Choice Voting And the Center Squeeze in the Alaska 2022 Special Election: How Might Other Voting Methods Compare?". p. 6.arXiv:2303.00108v1[cs.CY].
  27. ^abcdGraham-Squire, Adam; McCune, David (2024-01-02)."Ranked Choice Wackiness in Alaska".Math Horizons.31(1): 24–27.doi:10.1080/10724117.2023.2224675.ISSN1072-4117.
  28. ^Maskin, Eric;Foley, Edward B. (2022-11-01)."Opinion: Alaska's ranked-choice voting is flawed. But there's an easy fix".Washington Post.ISSN0190-8286.Retrieved2024-02-09.
  29. ^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.ISSN0092-5853.JSTOR2110496.
  30. ^Hamlin, Aaron (2022-09-16)."RCV Fools Palin Voters into Electing a Progressive Democrat".The Center for Election Science.Retrieved2024-07-11.It's a good thing for Peltola that she didn't attract more Palin voters—she'd have lost [...] The strangeness continues. Peltola could have actually gotten more 1st choice votes in this election and caused herself to lose. How's that? Let's look. [...] Imagine if Peltola reached across the aisle and spoke directly to Palin voters. Imagine that she empathized with their position and identified issues they cared about that Palin and even Begich ignored. And let's say that as a consequence, Peltola got the first-choice votes of between 5,200 and 8,500 voters who would have otherwise ranked only Palin. What happens as a result? Palin would have gotten eliminated in the first round and Peltola would still not be able to beat Begich.
  31. ^ab"Frequently Asked Questions about Burlington's use of Instant Runoff Voting".Archived fromthe originalon January 30, 2008.RetrievedJuly 21,2024.4. How did this change to IRV come about?
    Over 64% of Burlington voters voted in favor of the IRV Charter amendment in March 2005, and it went into effect on May 12, 2005, when the governor signed the ratification bill, H.505, which had been passed by both the House and Senate.
  32. ^"Mayor Bob Kiss".City of Burlington.Archived fromthe originalon November 29, 2007.RetrievedNovember 16,2007.
  33. ^"ChoicePlus Pro 2009 Burlington Mayor Round Detail Report".July 25, 2011. Archived fromthe originalon July 25, 2011.RetrievedJanuary 3,2018.
  34. ^Felsenthal, Dan S.; Tideman, Nicolaus (2014). "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.ISSN0165-4896.A display of non-monotonicity under the Alternative Vote method was reported recently, for the March 2009 mayoral election in Burlington, Vermont.
  35. ^Ornstein, Joseph T.; Norman, Robert Z. (October 1, 2014). "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.ISSN0048-5829.S2CID30833409.Although the Democrat was the Condorcet winner (a majority of voters preferred him in all two way contests), he received the fewest first-place votes and so was eliminated... 2009 mayoral election in Burlington, VT, which illustrates the key features of an upward monotonicity failure
  36. ^Baruth, Philip (March 12, 2009)."Voting Paradoxes and Perverse Outcomes: Political Scientist Tony Gierzynski Lays Out A Case Against Instant Runoff Voting".Vermont Daily Briefing. Archived fromthe originalon July 26, 2011.
  37. ^Gierzynski, Anthony; Hamilton, Wes; Smith, Warren D. (March 2009)."Burlington Vermont 2009 IRV mayoral election".RangeVoting.org.RetrievedOctober 1,2017.Montroll was favored over Republican Kurt Wright 56% to 44%... and over Progressive Bob Kiss 54% to 46%... In other words, in voting terminology, Montroll was a 'beats-all winner,' also called a 'Condorcet winner'... However, in the IRV election, Montroll came in third!... voters preferred Montroll over every other candidate... Montroll is the most-approved
  38. ^Bristow-Johnson, Robert (2023). "The failure of Instant Runoff to accomplish the purpose for which it was adopted: a case study from Burlington Vermont".Constitutional Political Economy.34(3): 378–389.doi:10.1007/s10602-023-09393-1.
  39. ^Ellenberg, Jordan (May 29, 2014).How Not to Be Wrong: The Power of Mathematical Thinking.Penguin. p.385.ISBN9780698163843.a majority of voters liked the centrist candidate Montroll better than Kiss, and a majority of voters liked Montroll better than Wright... yet Montroll was tossed in the first round.
  40. ^abLewyn 2012,p. 122, note 19: "election where Democratic candidate for mayor was Condorcet winner but finished third behind Republican and 'Progressive,'"
  41. ^Stensholt, Eivind (October 7, 2015). "What Happened in Burlington?".NHH Dept. Of Business and Management Science.Discussion Paper No. 2015/26.doi:10.2139/ssrn.2670462.hdl:11250/2356264.SSRN2670462.K was elected even though M was a clear Condorcet winner and W was a clear Plurality winner.
  42. ^Laatu, Juho; Smith, Warren D. (March 2009)."The Rank-Order Votes in the 2009 Burlington Mayoral Election".
  43. ^"Instant run-off voting experiment ends in Burlington: Rutland Herald Online".Rutlandherald.com.April 27, 2010. Archived fromthe originalon March 4, 2016.RetrievedApril 1,2016.
  44. ^"Official Results Of 2010 Annual City Election"(PDF).City of Burlington.March 2, 2010.
  45. ^Stensholt, Eivind (2015). "What Happened in Burlington?".SSRN Electronic Journal.Elsevier BV: 10–12.doi:10.2139/ssrn.2670462.hdl:11250/2356264.ISSN1556-5068.
  46. ^Igersheim, Herrade; Durand, François; Hamlin, Aaron; Laslier, Jean-François (2022)."Comparing Voting Methods: 2016 US Presidential Election".European Journal of Political Economy.71.doi:10.1016/j.ejpoleco.2021.102057.
  47. ^Santucci, Jack; Shugart, Matthew; Latner, Michael S. (October 16, 2023)."Toward a Different Kind of Party Government".Protect Democracy.Archivedfrom the original on July 16, 2024.RetrievedNovember 15,2024.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).
  48. ^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, New York: Springer. pp. 135–146.doi:10.1007/978-0-387-09720-6_9.ISBN978-0-387-09720-6.RetrievedNovember 15,2024.In effect, the primary system means that the USA has a two-round runoff system of elections.
  49. ^Foley, Edward B. (January 20, 2024)."Haley's Campaign Demonstrates the Need for Electoral Reform".Common Ground Democracy.RetrievedNovember 11,2024.
  50. ^Foley, Edward B. (November 6, 2024)."The Case for Condorcet Voting Is Clear".Common Ground Democracy.RetrievedNovember 15,2024.
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