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Draft:Center squeeze

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A center squeeze is a type of spoiler effect where a majority-preferred and socially-optimal candidate tends to be eliminated in favor of a more extreme candidate by plurality-runoff methods, like the two-round and ranked-choice runoff (RCV) rules.[1][2][3] In a center squeeze, the presence of several more-extreme candidates (with strong support from their base) squeezes any candidate trapped between them, starving them of the first-preference votes they need to survive in earlier rounds.[4][5]

The term does not refer to any particular political spectrum (such as the left-right spectrum).[6] Instead, a center squeeze can occur whenever candidates and voters are arranged in an ideological spectrum and prefer candidates similar to themselves (single-peaked preferences).[7] By the median voter theorem, the candidate who wins over the median voter will be the majority-preferred winner in this scenario, and as a result will be elected by any method compatible with majority-rule.[2][8]

However, in methods that strongly prioritize first preferences, majority-preferred candidates are often eliminated because they focus on building a broad appeal to many voters, rather than strong support in their party's base.[9][10][11] Voting systems that suffer from the center-squeeze effect have a bias in favor of more extreme candidates, incentivizing in such systems are incentivized to avoid the political center,[12][13] leading to unrepresentative winners and political polarization in the long run.[14][15] This effect, long predicted by social choice theorists, has been confirmed empirically in Australia,[16] California,[17][18] and Maine.[19]

Voting systems that have serious problems with center squeeze include first-preference plurality, plurality-with-primaries, two-round runoff,[2] and ranked-choice runoff voting (RCV).[2] By contrast, Condorcet and rated voting methods are not affected by such pathologies. Condorcet methods are insulated from center-squeezes by the median voter theorem, while rated voting systems like score or approval voting are protected by closely-related results.[20]

Susceptibility

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Primary elections

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Center squeeze is a major feature of two-party systems using primaries to elect candidates. In this case, the two parties tend to separate ideologically, and a "center" candidate, ideologically between the two, would find themselves unable to win a primary against another candidate closer to the centroid of the party. The center candidate would win in any one-on-one vote over the whole voting population, but will not win in the subset of the population represented by a party.[2] Surprisingly, this implies electoral reform at the primary level (in the absence of a multi-party system) will tend to have perverse effects and backfire, resulting in greater extremism, because candidates who are more representative of their political parties will tend to be more extreme compared to the population as a whole.

Cardinal and Condorcet methods

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If voters assign scores to candidates based on ideological distance, score voting will always select the candidate closest to some central tendency of the voter distribution. As a result, while score voting does not pass the median voter theorem per se, it tends to behave much like methods that do. The specific measure of central tendency minimized by the method depends on the precise way voters score candidates, with different measures of central tendency minimizing different distance metrics. Under the most common models of strategic voting, all spoilerproof cardinal methods will tend to behave like approval voting, and tend to converge on the Condorcet winner.[20]

Examples

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

Alphabet example

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In Alphabet Land, Voters are divided on how names should be arranged alphabetical order of names.   thinks names should always be in alphabetical order;   believes they should be in reverse-alphabetical order; and   thinks both sides should take turns alternating. Voters pick the candidate whose name is closest to theirs.

Because   is preferred to both   and   in head-to-head match-ups,   is the majority-preferred (Condorcet) winner. In this case, because voters' names are symmetrically-distributed,   is the socially-optimal winner as well.[21] Thus,   can be considered the "best" or "most popular" candidate by all commonly-used metrics in social choice, and as a result will be elected in the vast majority of electoral systems (including score voting, approval voting, and all Condorcet methods).

First-past-the-post

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

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

Ranked-choice runoff (Alternative, Two-round)

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Ranked-choice runoff voting (RCV) tries to address vote-splitting in first-past-the-post by replacing it with a series of first-past-the-post elections, where the loser is eliminated each round.[22][23] While this can prevent an unpopular minor candidate from spoiling a race, as in the 2000 Florida election, it is not able to prevent vote-splitting outside of two-party-dominant elections, as shown here.

The first round of the election is the same as the first-past-the-post election, with   having a slight lead.   has the least first preferences and is eliminated. Their votes are reassigned to   and  , according to their voter's ballot preferences. In the second round, enough voters who preferred   as their first choice took   as their second choice and   wins the election. RCV fails to have any moderating impact on the election, instead only causing a swing from one extreme to the other.

2022 Alaska Special Election

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Main article: 2022 Alaska's at-large congressional district special election
Alaska's at-large congressional district
Turnout32.2%[24]
       
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

The 2022 Alaska special election for the state's single House of Representatives seat was a prominent and conclusive example of a center squeeze. The ranked-choice runoff election involved one Democrat (Mary Peltola) and two Republicans (Sarah Palin and Nick Begich III). Because the full ballot data for the race was released, social choice theorists were able determine that Palin spoiled the race for Begich.

Begich was preferred to both Palin and Peltola in head-to-head matchups, but was eliminated in the first round after pulling slightly fewer first-preference votes than Peltola and Palin.[25][26] Of those who chose to support a second candidate, Begich's supporters were split roughly evenly between Palin and Peltola, whereas Palin's supporter's overwhelmingly preferred Begich to Peltola.[27] The final winner, Peltola, received no support from a majority of voters (having been ranked last on 52% of ballots).[25][28]

Many social choice theorists criticized the ranked-choice runoff procedure for its pathological behavior.[29][30] Along with being a center-squeeze, the election was a negative voting weight event,[28] where a voter's ballot has the opposite of its intended effect (i.e. a candidate winning as a result of "not having enough votes to lose").[28][31] In this race, Peltola would have lost if she had received more support from Palin voters,[32][33][34] and won as a result of 5,200 ballots that ranked her last (after Palin then Begich).[28][35]

2009 Burlington mayoral election

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Main article: 2009 Burlington mayoral election

The 2009 Burlington mayoral election was held in March 2009 for the city of Burlington, Vermont, and serves as an example of a four-candidate center squeeze. This was the second mayoral election since the city's 2005 change to ranked-choice runoff voting, after the 2006 mayoral election.[36] In the 2009 election, incumbent Burlington mayor (Bob Kiss) won reelection as a member of the Vermont Progressive Party,[37] defeating Kurt Wright in the final round with 48% of the vote (51.5% excluding exhausted ballots).

Some mathematicians and voting theorists criticized the election results as revealing several pathologies associated with ranked-choice runoff voting, including the monotonicity criterion, noting that Kiss was elected as a result of 750 votes cast against him (ranking Kiss in last place).[38][39][40] Several electoral reform advocates branded the election a failure after Kiss was elected despite 54% of voters voting for Montroll over Kiss,[41][42] violating the principle of majority rule.[43][44][45] Later analyses showed the race was spoiled, with Wright acting as a spoiler pulling moderate votes from Montroll, who would have beaten Kiss in a one-on-one race.[46][43]

The resulting controversy culminated in a successful 2010 initiative repealing RCV by a vote of 52% to 48%.[47][48]

Tournament matrix

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The results of every possible one-on-one election can be completed as follows:

Party Candidate vs. Simpson vs. Smith vs. Wright vs. Montroll 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, preferred over Wright by 56% of voters, over Smith by 60%, and over Simpson by 91% of voters.[49][50]

See Also

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References

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  This article incorporates text from this source, which is available under the CC BY-SA 4.0 license.

  1. ^ Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786. However, squeezed by surrounding opponents, a centrist candidate may receive few first-place votes and be eliminated under Hare.
  2. ^ a b c d e 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
  3. ^ Atkinson, Nathan; Ganz, Scott C. (2022-10-30). "The flaw in ranked-choice voting: rewarding extremists". The Hill. Retrieved 2023-05-14. 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.
  4. ^ 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
  5. ^ 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.
  6. ^ 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. JSTOR 1953842. S2CID 1161006. 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.
  7. ^ Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". 6 Phoenix L. Rev. 117. Rochester, NY. SSRN 2276015. third place 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, C would prevail over both Candidates A ... and B ... in a one-on-one runoff election. Yet, C would not prevail under IRV because he or she finished third and thus would be the first candidate eliminated.
  8. ^ Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23–48. doi:10.2307/2110786. ISSN 0092-5853. JSTOR 2110786. However, squeezed by surrounding opponents, a centrist candidate may receive few first-place votes and be eliminated under Hare.
  9. ^ Ogren, Marcus (2024-08-01). "Candidate incentive distributions: How voting methods shape electoral incentives". Electoral Studies. 90: 102799. doi:10.1016/j.electstud.2024.102799. ISSN 0261-3794.
  10. ^ 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
  11. ^ Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". 6 Phoenix L. Rev. 117. Rochester, NY. 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
  12. ^ Robinette, Robbie (2023-09-01). "Implications of strategic position choices by candidates". Constitutional Political Economy. 34 (3): 445–457. doi:10.1007/s10602-022-09378-6. ISSN 1572-9966.
  13. ^ 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.
  14. ^ Ogren, Marcus (2024-08-01). "Candidate incentive distributions: How voting methods shape electoral incentives". Electoral Studies. 90: 102799. doi:10.1016/j.electstud.2024.102799. ISSN 0261-3794.
  15. ^ Merrill, Samuel (1984). "A Comparison of Efficiency of Multicandidate Electoral Systems". American Journal of Political Science. 28 (1): 23. doi:10.2307/2110786. ISSN 0092-5853. However, squeezed by surrounding opponents, a centrist candidate may receive few first-place votes and be eliminated under Hare.
  16. ^ Mussel, Johanan D.; Schlechta, Henry (2023-07-21). "Australia: No party convergence where we would most expect it". Party Politics. doi:10.1177/13540688231189363. ISSN 1354-0688.
  17. ^ Kousser, Thad; Phillips, Justin; Shor, Boris (2016). "Reform and Representation: A New Method Applied to Recent Electoral Changes*". Political Science Research and Methods. 6 (4): 809–827. doi:10.1017/psrm.2016.43. ISSN 2049-8470. SSRN 2260083. neither the Citizens Redistricting Commission nor the top-two primary immediately halted the continuing partisan polarization of California's elected lawmakers or their drift away from the average voter
  18. ^ Kousser, Thad. "California's jungle primary sets up polarized governor's race for November". The Conversation. Retrieved 2018-06-23. The idea was that by opening up primaries to all voters, regardless of party, a flood of new centrist voters would arrive. That would give moderate candidates a route to victory .. Candidates did not represent voters any better after the reforms, taking positions just as polarized as they did before the top two. We detected no shift toward the ideological middle.
  19. ^ "The Effect of Ranked-Choice Voting in Maine | MIT Election Lab". electionlab.mit.edu. Retrieved 2024-07-25. One of the main claims made by reformers about RCV is that it will make campaigns more civil, as campaigns will have an incentive to seek the second-place vote of supporters of different candidates. To study this claim, I first conducted a difference-in-differences analysis on independent expenditures for and against candidates. In this analysis, I found that negative spending increased significantly in Maine following the implementation of ranked-choice voting, casting doubt on the claim that RCV makes campaigns more civil. To provide more evidence, I also created a dataset of all Facebook advertisements that mentioned any congressional candidates for 2018, the first year that RCV was used in Maine. I then conducted a sentiment analysis to find each advertisement's sentiment (whether it was negative or not). I then used genetic matching to approximate an experiment to find the impact of RCV on civility. In doing so, I found that the 2018 campaign was even more negative than in paired districts around the country.
  20. ^ a b Laslier, Jean-François; Sanver, Remzi, eds. (2010). Handbook on Approval Voting. Studies in Choice and Welfare. Heidelberg: Springer Berlin. doi:10.1007/978-3-642-02839-7. ISBN 978-3-642-02839-7.
  21. ^ DeGroot, Morris H. (1970). Optimal Statistical Decisions. McGraw-Hill Book Co., New York-London-Sydney. p. 232. ISBN 9780471680291. MR 0356303.
  22. ^ "Ranked Choice Voting". FairVote. FairVote. Retrieved 2024-07-22.
  23. ^ "Avoid Vote-Splitting and Weak Plurality Results". RCV Resources. Ranked Choice Voting Resource Center. 2020-11-09. Retrieved 2024-07-22.
  24. ^ "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. Retrieved September 2, 2022.
  25. ^ a b Graham-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
  26. ^ Atkinson, Nathan; Ganz, Scott C. (2022-10-30). "The flaw in ranked-choice voting: rewarding extremists". The Hill. Retrieved 2023-05-14. 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.
  27. ^ Clelland, 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].
  28. ^ a b c d Graham-Squire, Adam; McCune, David (2024-01-02). "Ranked Choice Wackiness in Alaska". Math Horizons. 31 (1): 24–27. doi:10.1080/10724117.2023.2224675. ISSN 1072-4117.
  29. ^ Maskin, Eric; Foley, Edward B. (2022-11-01). "Opinion: Alaska's ranked-choice voting is flawed. But there's an easy fix". Washington Post. ISSN 0190-8286. Retrieved 2024-02-09.
  30. ^ Graham-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
  31. ^ 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.
  32. ^ Hamlin, Aaron (2022-09-16). "RCV Fools Palin Voters into Electing a Progressive Democrat". The Center for Election Science. Retrieved 2024-07-11. 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.
  33. ^ Graham-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
  34. ^ Clelland, 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].
  35. ^ Graham-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
  36. ^ 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.
  37. ^ "Mayor Bob Kiss". City of Burlington. Archived from the original on November 29, 2007. Retrieved November 16, 2007.
  38. ^ 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. ISSN 0165-4896. A display of non-monotonicity under the Alternative Vote method was reported recently, for the March 2009 mayoral election in Burlington, Vermont.
  39. ^ 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. ISSN 0048-5829. S2CID 30833409. 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
  40. ^ 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 from the original on July 26, 2011.
  41. ^ Gierzynski, Anthony; Hamilton, Wes; Smith, Warren D. (March 2009). "Burlington Vermont 2009 IRV mayoral election". RangeVoting.org. Retrieved October 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
  42. ^ 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. doi:10.1007/s10602-023-09393-1.
  43. ^ a b Lewyn, Michael (2012). "Two Cheers for Instant Runoff Voting". Phoenix L. Rev. 6: 117. SSRN 2276015. election where Democratic candidate for mayor was Condorcet winner but finished third behind Republican and 'Progressive'
  44. ^ Ellenberg, Jordan (May 29, 2014). How Not to Be Wrong: The Power of Mathematical Thinking. Penguin. p. 385. ISBN 9780698163843. 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.
  45. ^ 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. SSRN 2670462. K was elected even though M was a clear Condorcet winner and W was a clear Plurality winner.
  46. ^ Laatu, Juho; Smith, Warren D. (March 2009). "THE RANK-ORDER VOTES IN THE 2009 BURLINGTON MAYORAL ELECTION".
  47. ^ "Instant run-off voting experiment ends in Burlington : Rutland Herald Online". Rutlandherald.com. April 27, 2010. Archived from the original on March 4, 2016. Retrieved April 1, 2016.
  48. ^ "Official Results Of 2010 Annual City Election" (PDF). City of Burlington. March 2, 2010.
  49. ^ Stensholt, Eivind (2015). "What Happened in Burlington?". SSRN Electronic Journal. Elsevier BV: 10–12. doi:10.2139/ssrn.2670462. hdl:11250/2356264. ISSN 1556-5068.
  50. ^ "IRV and Core Support". The Center for Election Science. Retrieved December 4, 2019.
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