Spoiler effect

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

Arrow's impossibility theorem is a well-known theorem showing that all rank-based voting systems are vulnerable to the spoiler effect. However, the frequency and severity of spoiler effects depends on the voting method.

Plurality and ranked choice (plurality-loser) are highly sensitive to spoilers, and can manufacture spoiler effects even when doing so is not forced. Majority-rule methods are usually not affected by spoilers, which are limited to rare situations called cyclic ties.

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

Motivation
Social choice theorists have long argued that voting methods should be spoiler-independent (at least so far as this is possible). The Marquis de Condorcet studied the same property going back to 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," and eventually led to its abandonment by the French Academy of Sciences.

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.

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.

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.

Plurality-runoff methods like the two-round system and instant-runoff voting 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, 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, somewhere between 90% and 99% of real-world elections have a Condorcet winner. 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.

Plurality voting
Vote splitting most easily occurs in plurality voting. In the United States vote splitting commonly occurs in primary elections. 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. 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.

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, though slightly less often than in plurality. 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). In Alaska's first-ever IRV election, Nick Begich was defeated in the first round by spoiler candidate Sarah Palin.

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
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 instant-runoff voting. Rated methods such as approval voting, range voting, and majority judgment also pass.

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.

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, 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.