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In social choice theory and politics, a spoiler effect or spoiler paradox is a situation where a losing candidate's performance affects the results of an election. A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives and is sometimes called spoilerproof. Well-known examples include the 2022 Alaska special election (spoiled by Sarah Palin) and the 2000 presidential election (spoiled by Nader's overperformance). Arrow's theorem and the Condorcet paradox are well-known results showing all ranked voting systems are vulnerable to spoilers in some circumstances; however, the frequency and severity of spoilers depends substantially on the specific voting method.

The plurality-rule family of methods including both first-past-the-post (FPP) and ranked-choice voting (RCV) is highly sensitive to spoilers, and can manufacture spoiler effects even when doing so is not forced. The multi-round plurality rules used in practice in most U.S. states (as partisan two-round) and officially in Australia (as ranked-choice voting) produce generally similar results and create spoiler effects in a situation called a center squeeze, leading to similarly-erratic behavior and reducing political competition by causing potential challengers to drop out.

Condorcet (majority-rule) methods are typically not affected by spoilers, which are limited to rare situations called cyclic ties. Rated voting systems are not subject to Arrow's theorem, allowing many of them to be spoilerproof.

Spoiler effects can also occur in some methods of proportional representation, such as the single transferable vote (STV or RCV-PR) and the largest remainders method of party-list representation. Here, a new party entering an election can cause seats to shift from one unrelated party to another, even if the new party wins no seats; this is known as the new state (or party) paradox. This class of spoiler effect can be avoided by divisor methods and other rules based on vote reweighting, such as proportional approval voting.

Motivation

Social choice theorists have argued voting rules should be spoilerproof since the foundation of the field, with the Marquis de Condorcet being the first to study the effect in the 1780s.

Rational behavior

In decision theory, independence of irrelevant alternatives (IIA) is a fundamental principle of rationality, which says that which of two outcomes A or B is better, should not depend on how good another outcome (C) is. A famous joke by Sidney Morgenbesser illustrates this principle:

A man is deciding whether to order apple or blueberry pie before settling on apple. The waitress informs him that cherry pie is also an option, to which the man replies "in that case, I'll have the blueberry."

Social choice theorists argue it would be better to have a mechanism for making societal decisions that behaves rationally (or if this is not possible, one that is at least usually rational).

Manipulation

By controlling nominations

Voting systems that violate independence of irrelevant alternatives are susceptible to being manipulated by strategic control of the nomination process. Some systems are particularly infamous for their ease of manipulation, such as the Borda count, which lets any party "clone their way to victory" by running a large number of candidates. This famously forced de Borda to concede that "my system is meant only for honest men," and contributed to its abandonment by the French Academy of Sciences.

Vote-splitting systems like choose-one and ranked-choice voting have the opposite problem: because running many similar candidates at once makes it difficult for any of them to win the election, these systems tend to concentrate power in the hands of parties and political machines, which serve the role of clearing the field of potential spoilers. In the United States, this has lead to the adoption a de facto two-round system, where the top-two candidates are chosen by the major-party primaries before going on to receive the overwhelming majority of votes.

In some situations, a spoiler can extract concessions from other candidates by threatening to remain in the race unless bought off, typically with a promise of a high-ranking political position.

Fairness

Because a candidate's quality and popularity clearly do not depend on whether or not some other candidate runs for office, it seems intuitively unfair or undemocratic for a voting system to behave as if it does. A voting system that is objectively fair to candidates and their supporters should not behave like a lottery; it should select the highest-quality candidate regardless of factors outside of a candidate's control (like whether or not another politician decides to run).

Arrow's theorem

Arrow's impossibility theorem is a major result in social choice theory, which proves that every ranked-choice voting system is vulnerable to spoiler effects.

However, rated voting systems are not affected by Arrow's theorem. Approval voting, range voting, and median voting all satisfy the IIA criterion: if we disqualify or add candidates, without changing any of the existing ratings, the ratings of the winner remain unchanged, meaning the result is not affected.

By electoral system

Different electoral systems have different levels of vulnerability to spoilers. As a rule of thumb, spoilers are very common with plurality voting, common in plurality-runoff methods, rare with paired counting (Condorcet), and impossible under most forms of rated voting.

Plurality-runoff methods like the two-round system, and ranked-choice voting still suffer from vote-splitting within rounds, and as a result, they do not eliminate the spoiler effect. The elimination of weak spoilers in earlier rounds reduces their effects on the results compared to single-round plurality voting, but spoiled elections remain common, moreso than in other systems.

Modern tournament voting eliminates vote splitting effects completely, because every one-on-one matchup is evaluated independently. If there is a Condorcet winner, Condorcet methods are completely invulnerable to spoilers. In practice, common estimates for the rate of Condorcet paradoxes in real elections tend to estimate rates of 1-10%. Many Condorcet rules (like ranked pairs) have other spoiler-resistance properties, reducing their severity even in cases where a Condorcet winner does not exist.

Cardinal voting methods can be immune to spoiler effects.

Pure FPP (e.g. United Kingdom)

Vote splitting most easily occurs in pure systems of first-past-the-post (FPP) voting that have many competing parties, as seen in the United Kingdom. In the United States, vote splitting most commonly occurs in primary elections.

Primary elections is to eliminate vote splitting among candidates in the same party before the general election. If primary elections or party nominations are not used to identify a single candidate from each party, the party that has more candidates is more likely to lose because of vote splitting among the candidates from the same party. In a two-party system, party primaries effectively turn plurality voting into a two-round system.

Multi-round systems (e.g. France, Australia, United States)

Spoilers also occur in the two-round system and ranked-choice voting at a substantially higher rate than for modern round-robin or rated voting methods, though less than in free-for-all style plurality votes. As a result, ranked-choice voting tends towards two-party rule.

In Burlington, Vermont's second IRV election, spoiler Kurt Wright knocked out Democrat Andy Montroll in the second round, leading to the election of Bob Kiss (despite the election results showing Montroll would have won a one-on-one election with Kiss). In Alaska's first-ever IRV election, Nick Begich was defeated as a result of vote-splitting in the first round by Sarah Palin. Since then, both major parties have aimed to minimize the number of candidates , with Democrats suing to keep Republicans pressuring candidate Nancy Dahlstrom into dropping out of the race.

Tournament (Condorcet) voting

Spoiler effects rarely occur when using tournament solutions, because each candidate's total in a paired comparison does not involve any other candidates. Instead, methods can separately compare every pair of candidates and check who would win in a one-on-one election. This pairwise comparison means that spoilers can only occur in the rare situation known as a Condorcet cycle.

For each pair of candidates, there is a count for how many voters prefer the first candidate (in the pair) to the second candidate, and how many voters have the opposite preference. The resulting table of pairwise counts eliminates the step-by-step redistribution of votes, which causes vote splitting in other methods.

Rated voting

Rated voting methods ask voters to assign each candidate a score on a scale (usually from 0 to 10), instead of listing them from first to last. The best-known of these methods is score voting, which elects the candidate with the highest total number of points. Because voters rate candidates independently, changing one candidate's score does not affect those of other candidates, which is what allows rated methods to evade Arrow's theorem.

While true spoilers are not possible under score voting, voters who behave strategically in response to candidates can create pseudo-spoiler effects (which can be distinguished from true spoilers in that they are caused by voter behavior, rather than the voting system itself).

Weaker forms

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Several weaker forms of independence of irrelevant alternatives (IIA) have been proposed as a way to compare ranked voting methods. Usually these procedures try to insulate the process from weak spoilers, ensuring that only a handful of candidates can change the outcome.

Local independence of irrelevant alternatives

Local independence from irrelevant alternatives (LIIA) is a weaker kind of independence that requires both of the following conditions:

1. If the option that finished in last place is deleted from all the votes, the winner should not change.
2. If the option that finished in first place is deleted from all the votes, the runner-up should win.

For every electoral method, it is possible to construct an order-of-finish that ranks candidates in terms of strength. This can be done by first finding the winner, then repeatedly deleting them and finding a new winner. This process is repeated to find which candidates rank 3rd, 4th, etc. As a result, LIIA can also be thought of as indicating independence from the weakest alternative, i.e. the alternative who would not win unless every other candidate dropped out.

Despite being a very weak form of spoiler-resistance (requiring that only the last-place finisher is unable to affect the outcome), LIIA is satisfied by only a few voting methods. These include Kemeny-Young and ranked pairs, but not Schulze or ranked-choice voting. Rated methods such as approval voting, range voting, and majority judgment also pass, by passing the stronger IIA criterion.

Condorcet independence criteria

Besides its interpretation in terms of majoritarianism, the Condorcet criterion can be interpreted as a kind of spoiler-resistance. In general, Condorcet methods are highly resistant to spoiler effects. Intuitively, this is because the only way to dislodge a beats-all champion is by beating them, so spoilers can only exist when there is no beats-all champion (which is rare). This property, of stability for Condorcet winners, is a major advantage of Condorcet methods.

Smith-independence is another kind of spoiler-resistance for Condorcet methods. This criterion says that a candidate should not affect the results of an election, unless they have a "reasonable claim" to the title of Condorcet winner (fall in the Smith set). Smith candidates are ones who can defeat every other candidate either directly or indirectly (e.g. if A can defeat B, who in turn defeats C).

Independence of clones

Independence of clones is the most commonly-fulfilled spoiler-resistance criterion, and says that "cloning" a candidate—adding a new candidate identical to an existing one—should not affect the results. Two candidates are considered identical if they are ranked side-by-side on every ballot; in other words, if there is no other candidate ranked in between them. The criterion is satisfied by ranked-choice runoff voting, all systems that satisfy independence of irrelevant alternatives (including cardinal systems), and most tournament solutions.

This criterion is very weak, as adding a substantially similar (but not quite identical) candidate to a race can still substantially affect the results, causing vote splitting. For example, the center squeeze pathology that affects RCV means that several similar (but not identical) candidates competing in the same race will tend to hurt each others' chances of winning.

Examples by rule

Borda count

In a Borda count, 5 voters rank 5 alternatives .

3 voters rank . 1 voter ranks . 1 voter ranks .

Borda count (a=0, b=1): C=13, A=12, B=11, D=8, E=6. C wins.

Now, the voter who ranks instead ranks ; and the voter who ranks instead ranks . They change their preferences only over the pairs , and .

The new Borda count: B=14, C=13, A=12, E=6, D=5. B wins.

The social choice has changed the ranking of and . The changes in the social choice ranking are dependent on irrelevant changes in the preference profile. In particular, B now wins instead of C, even though no voter changed their preference over .

Condorcet methods

A single example is enough to show that every Condorcet method must fail independence of irrelevant alternatives. Say that 3 candidates are in a Condorcet cycle. Label them Rock, Paper, and Scissors. In a one-on-one race, Rock loses to Paper, Paper to Scissors, etc. Without loss of generality, say that Rock wins the election with a certain method. Then, Scissors is a spoiler candidate for Paper: if Scissors were to drop out, Paper would win the only one-on-one race (Paper defeats Rock). The same reasoning applies regardless of the winner.

This example also shows why Condorcet elections are rarely (if ever) spoiled: spoilers can only happen if there is no Condorcet winner. Condorcet cycles are rare in large elections, and the median voter theorem shows cycles are impossible whenever candidates are arrayed on a left-right spectrum.

Plurality

Plurality voting is a degenerate form of ranked-choice voting, where the top-rated candidate receives a single point while all others receive none. The following example shows a plurality voting system with 7 voters ranking 3 alternatives (A, B, C).

• 3 voters rank (A>B>C)
• 2 voters rank (B>A>C)
• 2 voters rank (C>B>A)

In an election, initially only A and B run: B wins with 4 votes to A's 3, but the entry of C into the race makes A the new winner.

The relative positions of A and B are reversed by the introduction of C, an "irrelevant" alternative.

See also

• Electoral fusion
• Electoral threshold
• List of elections involving vote splitting
• Protest vote
• Single-issue politics

Notes

1. In election science, ranked voting systems include plurality rule, which is equivalent to ranking all candidates and selecting the one with the most first-place votes.
2. Results can still be irrational if voters fail independence of irrelevant alternatives, i.e. if they change their ballots in response to another candidate joining or dropping out. However, in this situation, it is the voters, not the voting rule, that generates the incoherence; the system still passes IIA.
3. Strategic voting can sometimes create the appearance of a spoiler for any method (including rated methods). However, this does not greatly affect the general ordering described here, except by making cardinal and Condorcet methods closer to even.

References

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6. 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
7. ^ Sen, Amartya; Maskin, Eric (2017-06-08). "A Better Way to Choose Presidents" (PDF). New York Review of Books. ISSN 0028-7504. Retrieved 2019-07-20. plurality-rule voting is seriously vulnerable to vote-splitting ... runoff voting ... as French history shows, it too is highly subject to vote-splitting. ... majority rule avoids such vote-splitting debacles because it allows voters to rank the candidates and candidates are compared pairwise
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9. ^ Poundstone, William. (2013). Gaming the vote : why elections aren't fair (and what we can do about it). Farrar, Straus and Giroux. pp. 168, 197, 234. ISBN 9781429957649. OCLC 872601019. IRV is subject to something called the "center squeeze." A popular moderate can receive relatively few first-place votes through no fault of her own but because of vote splitting from candidates to the right and left. ... Approval voting thus appears to solve the problem of vote splitting simply and elegantly. ... Range voting solves the problems of spoilers and vote splitting
10. 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. As with simple plurality elections, it is apparent the outcome will be highly sensitive to the distribution of candidates.
11. 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
12. Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it ... makes it less likely
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18. Holliday, Wesley H.; Pacuit, Eric (2023-02-11), Stable Voting, arXiv:2108.00542, retrieved 2024-03-11. "This is a kind of stability property of Condorcet winners: you cannot dislodge a Condorcet winner A by adding a new candidate B to the election if A beats B in a head-to-head majority vote. For example, although the 2000 U.S. Presidential Election in Florida did not use ranked ballots, it is plausible (see Magee 2003) that Al Gore (A) would have won without Ralph Nader (B) in the election, and Gore would have beaten Nader head-to-head. Thus, Gore should still have won with Nader included in the election."
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33. ^ "The Spoiler Effect". The Center for Election Science. 2015-05-20. Retrieved 2017-01-29.
34. ^ Gehrlein, William V. (2002-03-01). "Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*". Theory and Decision. 52 (2): 171–199. doi:10.1023/A:1015551010381. ISSN 1573-7187.
35. ^ Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice (journal). 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
36. T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.
37. "Top 5 Ways Plurality Voting Fails". The Center for Election Science. 2015-03-30. Retrieved 2017-10-07. You likely have opinions about all those candidates. And yet, you only get a say about one.
38. Borgers, Christoph (2010-01-01). Mathematics of Social Choice: Voting, Compensation, and Division. SIAM. ISBN 9780898716955. Candidates C and D spoiled the election for B ... With them in the running, A won, whereas without them in the running, B would have won. ... Instant runoff voting ... does not do away with the spoiler problem entirely, although it ... makes it less likely
39. Poundstone, William (2009-02-17). Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It). Farrar, Straus and Giroux. ISBN 9781429957649. IRV is excellent for preventing classic spoilers-minor candidates who tip the election from one major candidate to another. It is not so good when the 'spoiler' has a real chance of winning
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• Voting theory