# Center squeeze

Center squeeze is a kind of independence of irrelevant alternatives violation in which the Condorcet winner is eliminated before the final round of an election.
In a center squeeze, the majority-preferred Condorcet winner candidate is eliminated or otherwise rendered irrelevant before they have the chance to face any of the other candidates in a one-on-one race, even though (by definition) the voters would have preferred the Condorcet winner to the actual winner.
This situation may occur if candidates that strongly appeal to a base of core supporters can squeeze Condorcet winner candidates out of the race, by winning higher first-preference votes than them. This effect was first predicted by social choice theorists in the 1940s and 50s, and has since been identified to be a fin a number of election rules, and documented among countries using plurality-style electoral systems.
Real-life examples of center squeezes include the 2022 Alaska special election (where Nick Begich III was eliminated in the first round by Sarah Palin) as well as the 2007 French presidential election, where moderate liberal François Bayrou was eliminated by left-wing candidate Ségolène Royal, allowing Nicolas Sarkozy to win the second round.

## Overview

The "center" in "center squeeze" refers to candidates who are close to the center of the electorate, and is not necessarily limited to centrists along a traditional, one-dimensional political spectrum. A center squeeze can occur in any situation where voters prefer candidates who hold views similar to their own.
Center-squeeze has been observed in Australia, and various US cities.

### Susceptibility by system

Center squeeze is a major feature of two-party systems using primaries or other multiple-round systems. In these methods, candidates must focus on appealing to their core supporters to ensure they can make it past the first round, where only first-preferences count.
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 always elect the candidate closest to the median voter, it often behaves like methods that do. Under most models of strategic voting, spoilerproof cardinal methods tend to behave like approval voting and thus converge on the Condorcet winner.
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, like anti-plurality, D21 – Janeček, or Coombs' method.

## Example

In Alphabet Land, voters are divided based on how names should be arranged on lists. $A$ thinks names should always be in alphabetical order; $Z$ thinks they should be in reverse-alphabetical order; and $M$ thinks the order should be randomized. In this example, voters' happiness with the outcome falls linearly with the distance (number of letters) between the voter and the candidate.
Because $M$ is preferred to both $A$ and $Z$ in head-to-head match-ups, $M$ is the majority-preferred (Condorcet) winner. $M$ is the socially-optimal winner as well. Thus, $M$ is the "best" or "most popular" candidate under both common metrics of candidate quality in social choice.

### First-preference plurality (FPP)

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

### Ranked-choice runoff (Alternative, Two-round)

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.
The first round of the election is the same as the FPP election. $M$ has the least first preferences and is therefore eliminated. Their votes are reassigned to $A$ and $Z$, according to their ballot. In the second round, enough voters who preferred $M$ as their first choice took $A$ as their second choice and $A$ wins the election. RCV thus fails to have a substantial moderating impact, instead causing only a swing from one extreme to the other.

## Real-world occurrences
### 2022 Alaska special election

The 2022 Alaska special election seat was an example of a center squeeze, where Nick Begich III was eliminated in the first round by right-wing spoiler Sarah Palin, despite a majority of voters preferring Begich to either one of his opponents. 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 to confirm that Palin spoiled the race for Begich, with Peltola winning the race as a result of several counter-intuitive behaviors that tend to characterize center-squeeze elections.
The election produced a winner opposed by a majority of voters, with a majority of voters ranking Begich above Peltola and Palin, and more than half giving Peltola no support at all. The election was also notable as a no-show paradox, where a candidate is eliminated as a result of votes cast in support of their candidacy. In this case, ballots ranking Palin first and Begich second instead allowed Peltola to win.
Many social choice theorists criticized the ranked-choice runoff procedure for its counter-intuitive behavior. Along with being a center squeeze, the election was a negative voting weight event, where a voter's ballot has the opposite of its intended effect (e.g. where a candidate would need more votes to lose). In this race, Peltola would have lost if she had received more support from Palin voters, and won as a result of 5,200 ballots that ranked her last (after Palin then Begich). However, social choice theorists were 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 several alternatives without these behaviors, such as STAR, approval, or Condorcet voting.

### 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. 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.
The election results were criticized by mathematicians and voting theorists for several paradoxes associated with RCV. These included a no-show paradox, where Kiss won only as a result of 750 votes 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 majority-rule principle. 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.
The controversy culminated in a successful 2010 initiative that repealed RCV by a vote of 52% to 48%, a 16-point shift from the 64% who had supported the 2005 ratification. The results of every possible one-on-one election are as follows:
| 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:
* Montroll – defeats all candidates below, including Kiss (4,064 to 3,476)
* Kiss – defeats all candidates below, including Wright (4,313 to 4,061)
* Wright – defeats all candidates below, including Smith (3,971 to 3,793)
* 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.

## Possible real-world occurrences revealed by polling
### 2016 United States presidential election

Another possible example is the 2016 United States presidential election, where polls found several alternatives including Bernie Sanders and Gary Johnson defeating both Donald Trump and Hillary Clinton under a majority- or rated-voting rules but being squeezed out by both RCV and the primary election rules.

### 2024 United States presidential election

Election law scholar Ned Foley criticized the two-round system variant used in the United States, which has been described as a first round of primaries before a de-facto runoff, for creating a center squeeze in the 2024 presidential election and thus contributing to political polarization. Foley argued that both the primary system and a hypothetical instant-runoff voting system led to the election of Donald Trump by eliminating Nikki Haley, the majority-preferred (or Condorcet) candidate according to polling.