Draft:Center squeeze

Center-squeezes are a class of elections where the majority-preferred and socially-best candidate tends to be eliminated by plurality rule and ranked-choice runoff (ranked-choice voting). In a center-squeeze, the candidates are arranged along an ideological spectrum. By the median voter theorem, the candidate who wins over the voter closest to the median will always be the majority-preferred candidate in this scenario.

However, in methods that strongly prioritize first preferences, majority-preferred candidates are often eliminated because they appeal to a broad coalition of voters, rather than to narrowly appealing to their party's base. Voting systems that suffer from the center-squeeze effect have a bias in favor of more extreme candidates, leading to unrepresentative winners and political polarization in the long run. Candidates in such systems are incentivized to avoid the political center.

Despite use of the term "center", the term does not refer to any particular political spectrum (such as the left-right spectrum). The effect is visible whenever voters prefer candidates who are similar to them along some trait (i.e. when they have single peaked preferences).

Election systems that exhibit center-squeeze candidates tend to elect leaders who are unrepresentative of the voting population. Candidates who draw support from across the voting population tend to perform worse in these elections, lacking enough staunch support to outcompete extreme candidates whose voters like fewer of the available options. Repeated poor results from consensus candidates discourages them from running in subsequent elections and voters abandon these candidates in order to form voting blocks with more extreme leaders.

Voting systems that have serious problems with center squeeze include first-preference plurality, two-round runoff, and ranked-choice runoff voting (RCV). 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, which shows that if candidates are arranged along a one-dimensional ideological spectrum, all Condorcet methods will select the candidate closest to the median voter. Rated voting systems like score or approval voting are also insulated from the pathology by closely-related results.

Examples
[[File:Center_Squeeze_Example_Normal_Distribution_With_Candidates.svg|left|thumb|446x446px|

]] Say there is a country using a three-letter alphabet. Voters are divided by alphabetical order of names. Candidate A thinks names should always be in alphabetical order; Candidate C believes they should be in reverse-alphabetical order; and Candidate B is in the middle, and thinks both sides should take turns alternating. Candidates and voters are therefore polarized along a political spectrum, as seen in the diagram left.

Voters with names near the beginning of the alphabet vote for A first, but are willing to accept B as a second choice. Similarly, voters who support C are willing to accept A as a second choice.

Because candidate B is preferred to both candidate A and candidate C in head-to-head matchups, candidate B is the majority-preferred (Condorcet) winner. If voters' utility falls linearly with respect to distance, the median minimizes the mean absolute error, making B the socially-optimal winner as well. Alternatively, this also holds true if the score for each candidate is a decreasing function of distance and the distribution of voters is roughly symmetric, in which case the median are equal. However, candidate B is experiencing a center-squeeze, losing first-preference votes to candidates A and C on either side.

First-past-the-post
Candidate C wins under a single-round of FPTP, with 1108 voters choosing them as their absolute preference. However, significantly more voters considered candidate C to be their least preferred candidate, with 1563 out of a total voters preferring either candidates A and B. With majority opposition, and a core group of supporters so far from the median voter, candidate C should be considered rather unrepresentative of the voting population.

Instant-runoff (Alternative vote, Ranked-choice voting)
Ranked-choice runoff tries to address the vote-splitting of first-past-the-post by replacing it with a series of first-past-the-post elections, where the loser is eliminated in each round. Voters submit ballots ranking their preferences, and each voter's highest-ranked candidate receives their support. The candidate in last place is eliminated and their votes redistributed according to each ballot's next preferred candidate. This repeats until all candidates except one have been eliminated.

The first round of the election proceeds exactly the same as the first-past-the-post election with candidate C having a slight lead. No candidate has a majority of the remaining votes, and so candidate B is eliminated in last place. Their votes are redistributed to both candidates A and C, according to their voter's ballot preferences. In the second round, enough voters who preferred candidate B as their first choice took candidate A as their second choice and candidate A wins the election. With near-majority opposition and a position far to the left of most voters, candidate A is also unrepresentative of the electorate. <!--

Borda
Center-squeeze is not inherent to every voting system. Borda count is one such method which does not suffer from this problem, and the same voting preferences used above can be easily interpreted with the Borda count method. Under this method each candidate is awarded one point for every candidate that they are ranked above on a voter's ballot. In this case first gets two points and second gets one. Third gets none as there are no more candidates to rank or leave off the ballot. Using the same preferences as described above the Borda count election results are seen below.

The first round of the election proceeds exactly the same as the first-past-the-post election with candidate C having a slight lead. No candidate has a majority of the remaining votes, and so candidate B is eliminated in last place. Their votes are redistributed to both candidates A and C, according to their voter's ballot preferences. In the second round, enough voters who preferred candidate B as their first choice took candidate A as their second choice and candidate A wins the election. With near-majority opposition and a position far to the left of most voters, candidate A is also unrepresentative of the electorate.

Systems that generally do well in center-squeeze elections include both Condorcet (majority) voting and rated voting. Methods based on plurality rule such as first-past-the-post, instant-runoff voting, and two-round runoff tend to do poorly.

Cardinal methods
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 exact way voters score candidates. different measures of central tendency minimize different distance metrics. This corresponds to the geometric median when each candidate's score falls off linearly with respect to ideological distance.

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.

Primary system
Center squeeze is also a feature of two-party systems which use a primary to select 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.

Cardinal methods
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 exact way voters score candidates. different measures of central tendency minimize different distance metrics.

Under 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. satisfy analogous results that allow them to prevent center-squeeze.

2022 Alaska Special Election
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, election scientists 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. 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. The final winner, Peltola, received no support from a majority of voters (having been ranked last on 52%).

Many social choice theorists criticized the ranked-choice runoff procedure for its pathological behavior. Along with being a center-squeeze, the election was also a negative voting weight event, where a voter's ballot has the opposite of its intended effect (i.e. a candidate being eliminated for having "too many votes"). In this race, Peltola won as a result of 5,200 ballots that ranked her last (after Palin then Begich), and would have lost if she had received more support from Palin voters.

2009 Burlington mayoral election
The 2009 Burlington mayoral election was held in March 2009 for the city of Burlington, Vermont. This was the second mayoral election since the city's 2005 change to ranked-choice runoff voting, after the 2006 mayoral election. In the 2009 election, incumbent Burlington mayor (Bob Kiss) won reelection as a member of the Vermont Progressive Party, 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). Several electoral reform advocates branded the election a failure after Kiss was elected despite 54% of voters voting for Montroll over Kiss, violating the principle of majority rule. 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.

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

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