Spoiler effect

For the same effect in sports, see Elimination from postseason contention § Spoiler effect.

"Independence of spoilers" redirects here. For the logical property in decision theory, see Independence of irrelevant alternatives.

In social choice theory and politics, the spoiler effect or Arrow's paradox refers to a situation where a losing (that is, irrelevant) spoiler candidate affects the results of an election.^[1] A voting system that is not affected by spoilers satisfies independence of irrelevant alternatives or independence of spoilers.^[2]

Arrow's impossibility theorem posits that all rank-based voting systems^{[note 1]} are vulnerable to the spoiler effect. However, the frequency and severity of spoiler effects depends substantially on the voting method. Majority-rule methods are only rarely affected by spoilers, which are limited to rare^[3][4] situations called cyclic ties.^[5] Plurality is the most sensitive to spoilers, while ranked-choice voting (RCV-IRV) is still sensitive though less so in most scenarios.^[6][7][8]

Spoiler effects exist but may be less common in some methods of proportional representation, such as the single transferable vote (STV-PR 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 states paradox.

Motivation

Social choice theorists have long argued^[when?] that voting methods should be spoiler-independent (at least so far as this is possible).^{[citation needed]} The Marquis de Condorcet studied the same property going back to the 1780s.^[9]

Rational behavior

Main article: Independence of irrelevant alternatives

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 politicians

Voting systems that violate independence of irrelevant alternatives are susceptible to being manipulated by strategic nomination. 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,"^[10][11] and eventually led to its abandonment by the French Academy of Sciences.^[11]

Vote-splitting systems like choose-one and instant-runoff (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 and signalling a single candidate that voters should focus their support on; in many cases, this leads plurality voting systems to behave like a de facto two-round system, where the top-two candidates are nominated by party primaries.^{[citation needed]}

In some situations, a spoiler can extract concessions from other candidates by threatening to remain in the race unless they are bought off, typically with a promise of a high-ranking political position.^{[citation needed]}

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).^{[citation needed]}

Arrow's theorem

Main article: Arrow's impossibility 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 losing candidates, without changing ratings on votes, the score (and therefore winner) remains unchanged.^{[note 2]}

By electoral system

Different electoral systems have different levels of vulnerability to spoilers. As a rule of thumb, spoilers are extremely common with plurality voting, common in plurality-runoff methods, rare with paired counting (Condorcet), and impossible with rated voting.^{[note 3][citation needed]}

Spoiler effect by system

Electoral system	Spoiler effect
Plurality voting	High
Runoff/IRV/Ranked-choice	Medium
Proportional representation	Medium-low
Rated voting	Low
Deliberative demoracy	Low
Tournament voting	Low

Plurality-runoff methods like the two-round system^[12] and instant-runoff voting^[13] still suffer from vote-splitting in each round though they reduce the effect. 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.^{[citation needed]}

Modern tournament voting eliminates vote splitting effects completely, because every one-on-one matchup is evaluated independently.^[12] If there is a Condorcet winner, Condorcet methods are completely invulnerable to spoilers; in practice, somewhere between 90% and 99% of real-world elections have a Condorcet winner.^{[14][15][better source needed]} Some systems like ranked pairs have even stronger spoilerproofing guarantees that are applicable to most situations without a Condorcet winner.^{[citation needed]}

Plurality voting

Vote splitting most easily occurs in plurality voting.^{[citation needed]} In the United States vote splitting most commonly occurs in primary elections.^{[citation needed]} The purpose of 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.

Vote splitting is the most common cause of spoiler effects in the commonly-used plurality vote and two-round runoff systems. In these systems, the presence of many ideologically similar candidates causes their vote total to be split between them, placing these candidates at a disadvantage.^[16] This is most visible in elections where a minor candidate draws votes away from a major candidate with similar politics, thereby causing a strong opponent of both to win.^[16][17]

Runoff systems

Spoilers also occur in the two-round system and instant-runoff voting at a higher rate than for modern pairwise-counting or rated voting methods, though less often than in plurality.^[18][19][8] A 2022 paper suggests that Instant-runoff voting does increase support for third-party candidates.^[20][8]

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.^[21] This pairwise comparison means that spoilers can only occur in the rare situation^[14][15] known as a Condorcet cycle.^[21]

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.^{[citation needed]}

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).^{[citation needed]}

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

1. Heckelman, Jac C.; Miller, Nicholas R. (2015-12-18). Handbook of Social Choice and Voting. Edward Elgar Publishing. ISBN 9781783470730. A spoiler effect occurs when a single party or a candidate entering an election changes the outcome to favor a different candidate.
2. Miller, Nicholas R. (2019-04-01). "Reflections on Arrow's theorem and voting rules". Public Choice (journal). 179 (1): 113–124. doi:10.1007/s11127-018-0524-6. hdl:11603/20937. ISSN 1573-7101.
3. 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.
4. Van Deemen, Adrian (2014-03-01). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3): 311–330. doi:10.1007/s11127-013-0133-3. ISSN 1573-7101.
5. 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."
6. 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. In terms of the performance of the different election systems, we confirm the results of Merrill (1984, 1985, 1988) that in multicandidate elections run-off and sequential elimination systems perform far better than plurality elections, in that they are more likely to pick the Condorcet winner, and have a lower variance in their outcomes. This is true even if the distribution is skewed or bimodal. However, with skewed distributions, run-off and sequential elimination elections still have a bias away from the median in the direction of the mode, although this is typically smaller than that with plurality election.
7. 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
8. "Explainer: Instant runoff voting". MIT Election Lab. April 25, 2023. Retrieved 2024-08-29.
9. McLean, Iain (1995-10-01). "Independence of irrelevant alternatives before Arrow". Mathematical Social Sciences. 30 (2): 107–126. doi:10.1016/0165-4896(95)00784-J. ISSN 0165-4896.
10. Black, Duncan (1987) [1958]. The Theory of Committees and Elections. Springer Science & Business Media. ISBN 9780898381894.
11. McLean, Iain; Urken, Arnold B.; Hewitt, Fiona (1995). Classics of Social Choice. University of Michigan Press. ISBN 978-0472104505.
12. 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. ... [Condorcet] majority rule avoids such vote-splitting debacles because it allows voters to rank the candidates and candidates are compared pairwise
14. 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.
15. 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.
16. King, Bridgett A.; Hale, Kathleen (2016-07-11). Why Don't Americans Vote? Causes and Consequences: Causes and Consequences. ABC-CLIO. ISBN 9781440841163. Those votes that are cast for minor party candidates are perceived as taking away pivotal votes from major party candidates. ... This phenomenon is known as the 'spoiler effect'.
17. Buchler, Justin (2011-04-20). Hiring and Firing Public Officials: Rethinking the Purpose of Elections. Oxford University Press, USA. ISBN 9780199759965. a spoiler effect occurs when entry by a third-party candidate causes party A to defeat party B even though Party B would have won in a two-candidate race.
18. 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
19. 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
20. Simmons, Alan James; Gutierrez, Manuel; Transue, John E. (May 2022). "Ranked-Choice Voting and the Potential for Improved Electoral Performance of Third-Party Candidates in America". American Politics Research. 50 (3): 366–378. doi:10.1177/1532673X211072388. ISSN 1532-673X.
21. 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."