# Spoiler Resistance Analysis

The "spoiler effect" occurs when the addition of a weaker candidate in the race draws votes at the expense of a strong contender, causing the stronger candidate to lose. When this happens, the election is said to experience "vote-splitting," and the weaker of the two candidates is often labelled the "spoiler."

A voting method that is vulnerable to the spoiler effect, like plurality voting, restricts the range of voices and choices in the voting booth, because

- voters are reluctant to vote for minor candidates for fear of electing the front-runner they most dislike;
- minor candidates are reluctant to run for fear of being shamed as spoilers; and
- major candidates will exclude minor candidates from live debates for risk of being spoiled.

The end result is fewer options on the ballot, a more limited range of debate, and fewer reasons for voters to turn out on election day.

## Formal Definitions of the Spoiler Effect

There are formal criteria that attempt to capture the "spoiler" phenomenon in mathematical terms.

### Independence of Irrelevant Alternatives

[explanation as to why IIA is an inadequate formalization of the "spoiler effect" in progress ...]

### Independence of Clones

In 1987, Professor Nicolaus Tideman formulated a property known as *Independence of Clones*, a criterion that nicely captures the phenomenon of vote-splitting between very similar candidates. As defined by Tideman, set of "clones" is a set of at least two candidates such that no ballot ranks any candidate outside the set as either (a) tied with any candidate in the set or (b) between any two candidates inside the set. In other words, all the clones are either tied with or ranked immediately above or below each other on the ballot.

A voting method is said to be "independent of clones" if adding or removing clones from the election does not change the outcome for candidates outside the set. If a candidate outside the set of clones wins an election, then adding or removing clones should not cause that candidate to lose; and if a clone wins the election, then a clone should win under any election when clones are added or removed, although a different clone may win in this case.

### Independence of Spoilers

While Independence of Clones captures the idea of vote-splitting between very similar candidates — candidates that are more or less interchangeable — it does not capture the more general idea of the "spoiler effect," when one candidate may be distinctly "weaker" than another. The following property more fully captures the notion of the the "spoiler effect."

A candidate *S* is a spoiler of a candidate *C* if and only if the following two properties hold:

- S draws its votes at the expense of C. Specifically, on every ballot, either S is tied or ranked immediately above C, or C is ranked higher than S. This allows S and C to be clones, but it also allows candidates to be ranked between C and S, so long as C is higher.
- S is a weaker contender than C. Specifically, S finishes after C (i.e. S is lower in the social ordering than C) under the voting method.

A voting method is "independent of spoilers" when adding or removing spoilers does not change the winner. A voting method that is independent of spoilers is necessarily independent of clones; thus, a method that is not independent of clones is not independent of spoilers.

## Evaluation of Spoiler Resistance

### Ranked Choice Voting: High Resistance

Ranked Choice Voting satisfies both the *Independence of Clones* and the stronger *Independence of Spoilers* criteria, so it is free of the spoiler effect.

### Condorcet methods: High Resistance

Condorcet methods vary in their compliance with *Independence of Clones*, but the Schulze method (probably the most popular Condorcet method in use today) does satisfy it, as does the Ranked Pairs method. We are not aware of any Condorcet methods that satisfy the stronger *Independence of Spoilers* property, but Condorcet methods will only be vulnerable to spoilers when there exists a majority rule cycle, which is rare.

### Two-round runoff: Medium Resistance

Two-round Runoff is not independent of clones. Consider an election between 3 candidates — A, B, and C — where 35% of voters prefer A > C, 35% prefer C > A, and 30% prefer B > C. Under Two-Round Runoff, A and C advance to the runoff and C wins. However, imagine instead that candidate D, a clone of candidate C, enters the race, splitting the 35% that preferred C > A into 25% that prefer C > D > A and 10% that prefer D > C > A. Now A and B enter the top-two runoff, changing the outcome to A. Two-Round Runoff will nevertheless eliminate most spoilers in practice, a reality which we can infer from the fact that the winner of Ranked Choice Voting elections is very often one of the top-two finishers in first choice preferences.

### Approval Voting and Range Voting: Medium Resistance

Neither approval voting nor range voting is independent of clones. Consider an election between two candidates, A and B, where out of 100 voters, 54 prefer A and 46 prefer B. In an Approval election, we can reasonably expect that A voters will vote for only A, and B voters for only B, leading to the election of A. However, now candidate C, a clone of candidate A, enters the race and splits the 54% majority into 30% that prefer A > C and 24% that prefer C > A. In addition, candidate C differentiates themselves from both A and B with respect to a particular set of issues that many A voters care deeply about. While 1/2 of C>A voters vote for both C and A, the other half vote, unwilling to express equal favorability for A on the ballot, vote for only C. Now B wins with 46 votes to A's 42 votes.

Approval and Range are more resistant to spoilers than plurality, because voters can in theory give their favorite front-runner the top score to prevent that candidate from being "spoiled." However, the expectation that voters will behave in this fashion depends on three assumptions, which are often true but also often not. First, voters need to *know* who the front-runners are, so they require access to accurate polling data in advance. Second, there must be no more than 2 clear front-runners, otherwise the question of how best to vote to avoid spoilers is further complicated. Third, voters must be comfortable insincerely giving a front-runner the same score as their true favorite. Whenever any of these 3 assumptions are not true, the "spoiler effect" remains.

### Plurality Voting: Low Resistance

Plurality voting is not independent of clones, and unlike approval or range, it offers no way to show support for a spoiler without losing the ability to have a say between the front-runners. Consider an election between two candidates, A and B, where 55% prefer A and 45% prefer B, so A should therefore wins. However, now candidate C, a clone of candidate A, enters the race and splits the 55% majority into 40% that prefer A > C and 15% that prefer C > A. Now B wins with 45% of the vote.