Spoiler effect

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

Contents

By Arrow's theorem, all ranked-choice voting systems [note 1] are vulnerable to spoiler effects. [3] However, the susceptibility of different ranked systems varies greatly. Plurality, two-round, and instant-runoff systems suffer frequent spoiler effects that can substantially affect the outcome of the race. [2] [4] Majority-rule systems are usually not affected by spoilers, which are limited to rare [5] [6] situations known as cyclic ties. [7]

Rated voting systems are not subject to Arrow's theorem; as a result, many satisfy independence of irrelevant alternatives (sometimes called spoilerproofness). [3] [8]

Motivation

Social choice theorists have long argued that voting methods should be spoiler-independent, at least as far as this is possible, since at least the 1950s (with work by economists and mathematicians such as Kenneth Arrow and John von Neumann). The Marquis de Condorcet studied similar properties at least as far back as 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 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," [9] [10] leading to its abandonment by the French Academy of Sciences. [10]

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 runoff system, where the top-two candidates are nominated by party primaries.

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.

Fairness

Because a candidate's quality and popularity clearly do not depend on whether an unpopular 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 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]

Plurality-runoff methods like the two-round system [11] and instant-runoff voting [8] still suffer from vote-splitting in each round. As a result, they do not eliminate the spoiler effect. The elimination of weak spoilers in earlier rounds somewhat reduces their effects on the results compared to single-round plurality voting, [12] but spoiled elections remain common, moreso than in other systems. [3]

Modern tournament voting eliminates vote splitting effects completely, because every one-on-one matchup is evaluated independently. [11] [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. [5] [6] Some systems like ranked pairs have even stronger spoilerproofing guarantees that are applicable to most situations without a Condorcet winner.

Cardinal voting methods can be fully immune to spoiler effects. [3] [8]

Plurality voting

Vote splitting most easily occurs in plurality voting. [13] In the United States vote splitting commonly occurs in primary elections. [12] 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. [14] 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. [14] [15]

Runoff systems

Spoilers also occur in the two-round system and instant-runoff voting at a substantially higher rate than for modern pairwise-counting or rated voting methods, [3] though slightly less often than in plurality. [4] [16] As a result, instant-runoff voting still 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). [17] In Alaska's first-ever IRV election, Nick Begich was defeated in the first round by spoiler candidate Sarah Palin. [18]

Tournament (Condorcet) voting

Spoiler effects rarely occur when using tournament methods, 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. [7] This pairwise comparison means that spoilers can only occur in the extremely rare situation [5] [6] of a Condorcet cycle. [7]

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

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

A weaker criterion proposed by H. Peyton Young and A. Levenglick is called local independence from irrelevant alternatives (LIIA). [19] LIIA requires that both of the following conditions always hold:

  1. If the option that finished in last place is deleted from all the votes, then the order of finish of the remaining options must not change. (The winner must not change.)
  2. If the winning option is deleted from all the votes, the order of finish of the remaining options must not change. (The option that finished in second place must become the winner.)

In other words, for any group of candidates who are listed consecutively in the finish order, eliminating every candidate who is not part of that group shouldn't change how the method orders the members of that group.

LIIA is satisfied by only a few voting methods. These include Kemeny-Young and ranked pairs, but not Schulze or instant-runoff voting. Rated methods such as approval voting, range voting, and majority judgment also pass.

Independence of worst candidates

One may also consider a weakening of LIIA where only the first point is required to hold - that eliminating the n worst candidates does not alter the order of the remaining candidates.

This criterion is passed by methods that eliminate losers one at a time, because eliminating the worst candidates is part of its natural procedure.[ citation needed ]

If a method passes this criterion and reversal symmetry, it also passes LIIA. As a consequence, common elimination methods like instant-runoff voting almost invariably fail reversal symmetry.

Some social choice authors distinguish between methods that select a single winner and methods that determine a rank of finish. They then construct the latter from the former by repeatedly eliminating winners and using the elimination order as order of finish. If a method constructed in this manner passes independence of worst candidates, it also passes LIIA.

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 (by beating some candidate A who defeats B).

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 equally on every ballot. The criterion is satisfied by instant-runoff voting, all systems that satisfy independence of irrelevant alternatives (including cardinal systems), and most tournament solutions.

However, it is worth noting this criterion is extremely fragile, as even a single voter expressing a preference for one candidate over the other (or placing another candidate between them) can nullify a system's protection.

Examples by system

Borda count

In a Borda count, 5 voters rank 5 alternatives [A, B, C, D, E].

3 voters rank [A>B>C>D>E]. 1 voter ranks [C>D>E>B>A]. 1 voter ranks [E>C>D>B>A].

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

Now, the voter who ranks [C>D>E>B>A] instead ranks [C>B>E>D>A]; and the voter who ranks [E>C>D>B>A] instead ranks [E>C>B>D>A]. They change their preferences only over the pairs [B, D], [B, E] and [D, E].

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

The social choice has changed the ranking of [B, A] and [B, C]. 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 [B, C].

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, [5] [6] 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).

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

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.

Related Research Articles

<span class="mw-page-title-main">Condorcet method</span> Pairwise-comparison electoral system

A Condorcet method is an election method that elects the candidate who wins a majority of the vote in every head-to-head election against each of the other candidates, whenever there is such a candidate. A candidate with this property, the pairwise champion or beats-all winner, is formally called the Condorcet winner. The head-to-head elections need not be done separately; a voter's choice within any given pair can be determined from the ranking.

Arrow's impossibility theorem is a key impossibility theorem in social choice theory, showing that no ranked voting rule can produce a logically coherent ranking of more than two candidates. Specifically, no such rule can satisfy a key criterion of rational choice called independence of irrelevant alternatives: that a choice between and should not depend on the quality of a third, unrelated outcome .

Strategic nomination refers to the entry of a candidate into an election with the intention of changing the ranking of other candidates. The name is an echo of ‘tactical voting’ and is intended to imply that it is the candidates rather than the voters who are seeking to manipulate the result in a manner unfaithful to voters’ true preferences.

Ranked pairs, sometimes called the Tideman method, is a tournament-style system of ranked-choice voting first proposed by Nicolaus Tideman in 1987.

In an election, a candidate is called a Condorcet, beats-all, or majority-rule winner if more than half of voters support them in any one-on-one matchup with another candidate. Such a candidate is also called an undefeated, or tournament champion, by analogy with round-robin tournaments. Voting systems where a majority-rule winner will always win the election are said to satisfy the majority-rule principle, also known as the Condorcet criterion. Condorcet voting methods extend majority rule to elections with more than one candidate.

The majority criterion is a voting system criterion. The criterion states that "if only one candidate is ranked first by a majority of voters, then that candidate must win."

The mutual majority criterion, also known as majority for solid coalitions or the generalized majority criterion, is a voting system criterion that says that if a majority of voters ranks a certain group of candidates at the top of their ballot, then one of these candidates should win the election.

The Borda count electoral system can be combined with an instant-runoff procedure to create hybrid election methods that are called Nanson method and Baldwin method. Both methods are designed to satisfy the Condorcet criterion, and allow for incomplete ballots and equal rankings.

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.

Reversal symmetry is a voting system criterion which requires that if candidate A is the unique winner, and each voter's individual preferences are inverted, then A must not be elected. Methods that satisfy reversal symmetry include Borda count, ranked pairs, Kemeny–Young method, and Schulze method. Methods that fail include Bucklin voting, instant-runoff voting and Condorcet methods that fail the Condorcet loser criterion such as Minimax.

The later-no-harm criterion is a voting system criterion first formulated by Douglas Woodall. Woodall defined the criterion by saying that "[a]dding a later preference to a ballot should not harm any candidate already listed." For example, a ranked voting method in which a voter adding a 3rd preference could reduce the likelihood of their 1st preference being selected, fails later-no-harm.

In voting systems theory, the independence of clones criterion measures an election method's robustness to strategic nomination. Nicolaus Tideman was the first to formulate this criterion, which states that the winner must not change due to the addition of a non-winning candidate who is similar to a candidate already present. It is a relative criterion: it states how changing an election should or shouldn't affect the outcome.

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Instant-runoff voting (IRV), also known as plurality with elimination or plurality loser, is a ranked-choice voting system that modifies plurality by repeatedly eliminating the last-place winner until only one candidate is left. In the United Kingdom, it is generally called the alternative vote (AV). In the United States, IRV is often referred to as ranked-choice voting (RCV), by way of conflation with ranked voting systems in general.

<span class="mw-page-title-main">Ranked voting</span> Family of electoral systems

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The later-no-help criterion is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.


A major branch of social choice theory is devoted to the comparison of electoral systems, otherwise known as social choice functions. Viewed from the perspective of political science, electoral systems are rules for conducting elections and determining winners from the ballots cast. From the perspective of economics, mathematics, and philosophy, a social choice function is a mathematical function that determines how a society should make choices, given a collection of individual preferences.

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A top-four primary or top-four ranked-choice voting is an election method using a nonpartisan blanket primary where up to four candidates, those with the most votes, advance from a first round of FPTP voting, regardless of the political party. The round two (general) election, held some weeks later, uses instant-runoff voting to confirm a winner among the top set of candidates.

Descending Solid Coalitions (DSC) is a ranked-choice voting system. It is designed to preserve the advantages of instant-runoff voting, while satisfying monotonicity. It was developed by voting theorist Douglas Woodall as an improvement on (and replacement for) the use of the alternative vote.

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. 1 2 "The Spoiler Effect". The Center for Election Science. Retrieved 2024-03-03.
  3. 1 2 3 4 5 "The Spoiler Effect". The Center for Election Science. 2015-05-20. Retrieved 2017-01-29.
  4. 1 2 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
  5. 1 2 3 4 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.
  6. 1 2 3 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.
  7. 1 2 3 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."
  8. 1 2 3 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
  9. ↑ Black, Duncan (1987) [1958]. The Theory of Committees and Elections. Springer Science & Business Media. ISBN   9780898381894.
  10. 1 2 McLean, Iain; Urken, Arnold B.; Hewitt, Fiona (1995). Classics of Social Choice. University of Michigan Press. ISBN   978-0472104505.
  11. 1 2 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
  12. 1 2 3 Ending The Hidden Unfairness In U.S. Elections explains why plurality and runoff voting methods are vulnerable to vote splitting.
  13. ↑ "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.
  14. 1 2 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'.
  15. ↑ 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.
  16. ↑ 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
  17. ↑ Stensholt, Eivind (2015-10-07). "What Happened in Burlington?". Discussion Papers: 13. There is a Condorcet ranking according to distance from the center, but Condorcet winner M, the most central candidate, was squeezed between the two others, got the smallest primary support, and was eliminated.
  18. ↑ Clelland, Jeanne N. (2023-02-28), Ranked Choice Voting And the Center Squeeze in the Alaska 2022 Special Election: How Might Other Voting Methods Compare?, arXiv: 2303.00108
  19. ↑ Young, Peyton (1995-02-01). "Optimal Voting Rules". Journal of Economic Perspectives. 9 (1): 51–64. doi:10.1257/jep.9.1.51. ISSN   0895-3309.
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