Vote splitting

Vote splitting is an electoral effect in which the distribution of votes among multiple similar candidates reduces the chance of winning for any of the similar candidates, and increases the chance of winning for a dissimilar candidate.
Vote splitting most easily occurs in plurality voting in which each voter indicates a single choice and the candidate with the most votes wins, even if the winner does not have majority support. For example, if candidate A1 receives 30% of the votes, similar candidate A2 receives another 30% of the votes, and dissimilar candidate B receives the remaining 40% of the votes, plurality voting declares candidate B as the winner, even though 60% of the voters prefer either candidate A1 or A2.
Under such systems vote pairing can mitigate the effect, but it requires two voters in different districts to agree, and identifying probabilities of candidates winning in those districts. A vote swap effectively preserves the total support for each party but moved it to where it is most effective. It is legal and practiced in US, Canadian & especially UK elections.
Cardinal voting methods are immune to vote splitting, since each candidate is rated independently of each other. Pairwise-counting Condorcet methods minimize vote splitting effects. Plurality-runoff voting methods still suffer from vote-splitting in each round, but can somewhat reduce its effects compared to single-round plurality voting.
A well-known effect of vote splitting is the spoiler effect, in which a popular candidate loses an election by a small margin because a less-popular similar candidate attracts votes away from the popular candidate, allowing a dissimilar candidate to win. As a result, the notion of vote splitting is controversial because it can discourage third party candidates.
Strategic nomination takes advantage of vote splitting to defeat a popular candidate by supporting another similar candidate.
Vote splitting is one possible cause for an electoral system failing the independence of clones or independence of irrelevant alternatives fairness criteria.

Vote splitting and electoral systems

Different electoral systems have different levels of vulnerability to vote splitting.

Plurality voting

Vote splitting most easily occurs in plurality voting because the ballots gather only the least bad preference of the voter. 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. Primary elections occur only within each party and so vote splitting can still occur between parties in the secondary election. In open primaries, vote splitting occurs between all candidates.
In addition to applying to single-winner voting systems, a split vote can occur in proportional representation methods that use election thresholds, such as in Germany, New Zealand and Turkey. In those cases, "fringe" parties that do not meet the threshold can take away votes from larger parties with similar ideologies.

Ordinal voting methods

When ranked ballots are used, voters can vote for a minor party candidate as their first choice and indicate their order of preference for the remaining candidates, without regard ro whether a candidate is in a major political party. For example, voters who support a very conservative candidate can select a somewhat-conservative candidate as their second choice, thus minimising the chance that their vote will result in the election of a liberal candidate.
Runoff voting is less vulnerable to vote splitting than plurality voting, but vote splitting can occur in any round of runoff voting.
Vote splitting rarely occurs when the chosen electoral system uses ranked ballots and a pairwise-counting method, such as a [Condorcet method. Pairwise counting methods do not involve distributing each voter's vote between the candidates. Instead, pairwise counting methods separately consider each possible pair of candidates, for all possible pairs. For each pair of candidates there is a count for how many voters prefer the first candidate to the second candidate and how many voters have the opposite preference. The resulting table of pairwise counts eliminates the step-by-step distribution of votes, which facilitates vote splitting in other voting methods.
Voting methods that are vulnerable to strategic nomination, especially methods that fail independence of clones, are vulnerable to vote splitting. Vote splitting also can occur in situations that do not involve strategic nomination, such as talent contests in which earlier rounds of voting determine the current contestants.

Cardinal voting methods

Cardinal voting methods require an independent score to be given to candidates, as opposed to a ranking. The three primary methods are approval voting, with a range between 0-1, score voting with an arbitrary range, and STAR voting.
All cardinal voting methods are immune to vote splitting, as each candidate is evaluated independently of each other candidate.

Historical examples

The spoiler effect is the effect of vote splitting between candidates or ballot questions who often have similar ideologies. One spoiler candidate presence in the election draws votes from a major candidate with similar politics, thereby causing a strong opponent of both or several to win. The minor candidate causing this effect is referred to as a spoiler. However, short of any electoral fraud, this presents no grounds for a legal challenge.
The spoiler effect is a problem in plurality voting systems because they enable a candidate to win with less than half of the vote.
The problem also exists in instant-runoff voting, also known as ranked-choice voting or alternative vote, though it is reduced, because voters are allowed to rank their candidate choices, with their vote transferring to their second choice if their first choice does not win, to their third choice if their second choice does not win, and so on. Other preferential voting or ranked ballot voting systems also suffer from variations of the spoiler effect, as they fail the independence of irrelevant alternatives criterion.
The problem does not exist in cardinal voting methods like approval voting, score voting, or majority judgment, since the rating of each candidate is independent of the ratings of other candidates.

Relationship with other effects

The spoiler candidate takes votes away from a more viable candidate or candidates, a common effect called vote splitting. If one opposing candidate is ideologically or politically similar and therefore receives far fewer votes than other opposing candidates to the spoiler candidate, vote splitting has a spoiler effect.
In some cases, even though spoiler candidates cannot win themselves, their influence upon the voters may enable the candidate to determine deliberately the more viable candidate who wins the election, a situation known as a kingmaker scenario. With a first-past-the-post voting system, that is particularly feasible when spoiler candidates recommend tactical voting or run on a false manifesto to bolster the prospects for their secretly-preferred candidate to win.
In a preferential voting system, voters can feel more inclined to vote for a minor party or independent as their first choice and can record a preference between the remaining candidates, whether they are in a major or established party or not. For example, voters for a minor left-wing candidate might select a major left-wing candidate as their second choice, thus minimizing the probability that their vote will result in the election of a right-wing candidate, or voters for an independent candidate perceived as libertarian, or simply as the voter prefers that ideology might select a particular libertarian candidate as their second choice, thus minimising the probability of an authoritarian candidate being elected. Approval voting and proportional representation systems can also reduce the spoiler effect.
One of the main functions of political parties is to mitigate the effect of spoiler-prone voting methods by winnowing on a local level the contenders before the election. Each party nominates at most one candidate per office since each party expects to lose if they nominate more than one. In some cases, a party can expect to "lose" by "suffering a rival elected opponent" if they nominate more than zero, where two opponents exist and one is considered a candidate they can "work with" — a party may prefer the candidate who would win if the party nominates zero.
Thus, empirical observations of the frequency of spoiled elections do not provide a good measure of how prone to spoiling a particular voting method is, since the observations omit the relevant information about potential candidates who did not run because of not wanting to spoil the election.

Mathematical definitions

Possible mathematical definitions for the spoiler effect include failure of the independence of irrelevant alternatives axiom, and vote splitting.
Arrow's impossibility theorem states that rank-voting systems are unable to satisfy the independence of irrelevant alternatives criterion without exhibiting other undesirable properties as a consequence. However, different voting systems are affected to a greater or lesser extent by IIA failure. For example, instant runoff voting is considered to have less frequent IIA failure than First Past the Post. The independence of Smith-dominated alternatives criterion is much weaker than IIA; unlike IIA, some ranked-ballot voting methods can pass ISDA.
A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates. If X would have won had S not been one of the nominees, and if the voters who prefer S over W also prefer X over W, then S is a spoiler. Here is an example to illustrate: Suppose the voters' orders of preference are as follows:
The voters who prefer S over W also prefer X over W. W is the winner under Plurality Rule, Top Two Runoff, and Instant Runoff. If S is deleted from the votes then X would be the winner. Thus S is a spoiler with these three voting methods.

Spoiler effect in American elections

A 2014 analysis by Philip Bump for the Washington Post found that 1.5% of general election races in the U.S. from 2006 to 2012 were spoiled by third-party candidates.
Historically, the Democratic and Republican parties have benefited from the alleged spoiler effect created by the existing U.S. plurality voting system. This benefit is based in the theory that not voting for other parties and for independents and that third parties and independent candidates themselves declining to run, means to avoid "wasting votes" or splitting the vote causing an election result not wanted.
There are third-party candidates who have been accused of denying victory to a major nominee in U.S. Presidential Elections; a notable case among these is the 1912 election, where Progressive Party candidate Theodore Roosevelt won almost 700,000 votes more than did the Republican incumbent, William Howard Taft, and thus it could be said that Taft was the spoiler for Roosevelt in that election. This argument worried Republicans, who, after Woodrow Wilson won the 1912 election, became concerned that Roosevelt might return to split the Republican vote again.
In 1968, George Wallace ran for president as the American Independent Party's nominee. He received numerous votes from Southern demographics that typically voted for Democratic candidates, thereby undercutting the candidacy of that election's Democratic nominee, Hubert Humphrey.

Bush, Gore, and Nader (2000 U.S. presidential election)

The 2000 U.S. Presidential election is often cited as an example of the spoiler effect. In that election, Al Gore, the Democratic candidate, received more popular votes than George W. Bush, the Republican candidate, but lost in the electoral college. In the state of Florida, the final certified vote count showed Bush with just 537 more votes than Gore. Because Bush defeated Gore in Florida, he won the state, received more votes in the electoral college, and became president of the United States.
Gore supporters argued that had candidate Ralph Nader, a progressive, not run in the election, the majority of the 97,421 votes he received in Florida would have been cast for Gore. Thus, they contend that Nader's candidacy spoiled the election for Gore by taking away enough votes from Gore in Florida to swing the election to Bush. Their argument is bolstered by a poll of Nader voters, asking them for whom they would have voted had Nader not run, which said 45 percent of Nader voters would have voted for Gore, 27 percent would have voted for Bush, and the rest would not have voted.
Nader himself and many of his supporters argued that most Nader voters would either have chosen another minor party candidate or abstained from voting, had Nader not been on the ballot. All other third party candidates on the ballot in Florida received more than the 537 vote difference between Bush and Gore. Still, some observers began to refer to the spoiler effect as the "Nader effect" after the 2000 election. A 2006 study found that at least 40% of Nader voters in Florida would have voted for Bush if Nader had not run, while the other 60% would have voted for Gore. The study concluded that this 60% "did indeed spoil the 2000 presidential election for Gore but only because of highly idiosyncratic circumstances, namely, Florida’s extreme closeness."

Other US elections

In the German presidential election of 1925, Communist Ernst Thälmann refused to withdraw his candidacy although it was extremely unlikely that he would have won although the leadership of the Communist International urged him not to run. In the second round of balloting, Thälmann won 1,931,151 votes, or 6.4 percent. Centrist candidate Wilhelm Marx won 13,751,605, or 45.3 percent. The independent candidate Paul von Hindenburg, endorsed by the German National People's Party and German People's Party shared 14,655,641 votes, or 48.3 percent. If most of Thälmann's supporters had voted for Marx, instead of Hindenburg, Marx would have won the election. The election of 1925 had great significance because in 1933 Hindenburg appointed Adolf Hitler as chancellor and then died the following year, giving Hitler unchecked control of the German government. By contrast, Marx, who was dedicated to republican democracy, lived until 1946.
In New Zealand, there have been two notable cases of the spoiler effect. In the 1984 general election, the free-market New Zealand Party deliberately ran for office to weaken support former Prime Minister Robert Muldoon, the incumbent. Later, the 1993 general election saw the New Zealand Labour Party's vote split by The Alliance, which has been attributed to the vagaries of the plurality vote. In response to these problems, New Zealand has since adopted mixed-member proportional representation.
Likewise, in France, the 2002 presidential elections have been cited as a case of the spoiler effect: the numerous left-wing candidates, such as Christiane Taubira and Jean-Pierre Chevènement, both from political parties allied to the French Socialist Party, or the three candidates from Trotskyist parties, which altogether totalled around 20%, have been charged with making Lionel Jospin, the Socialist Party candidate, lose the two-round election in the first round to the benefit of Jean-Marie Le Pen, who was separated from Jospin by only 0.68%. Some also cite the case of some districts in which the moderate right and the far right had more than half of the votes together, but the left still won the election; they accuse the left of profiting from the split.
In Hong Kong,vote-splitting is very common for the pro-democracy camp, which caused it to suffer greatly in many elections, including the 2016 Hong Kong legislative election and the 2015 Hong Kong local elections. Pro-democracy supporters typically have different ideologies and suffer from factional disputes that are exacerbated after the advent of localist groups. However, many have wider aggregate support fewer seats are earned than the pro-Beijing camp, an example being in Kowloon East in which pro-democracy parties got over 55% of cast ballots but won only 2 seats out of 5.
In Canada, in the 2004 federal election, the right-wing Conservative Party won power with 29.63% of the popular vote. Its two major left-wing competitors, the Liberal Party and the New Democratic Party, won respectively 36.73% and 15.68% of the vote. Similar vote splits between the two major left-wing parties continued to assist the Conservative Party in maintaining power in the 2006, 2008, and 2011 federal elections, despite most of the popular vote going to left-wing parties in each race.
Also in Canada, the 2015 provincial election in Alberta saw the left-wing New Democratic Party win 62% of the seats with 40.6% of the province's popular vote after a division within the right-wing Progressive Conservative Party, which left it with only 27.8% of the vote, and its breakaway movement, the Wildrose Party, with 24.2% of the vote. In 2008, the last election in which the Progressive Conservative Party had been unified, it won 52.72% of the popular vote.
In Greece, Antonis Samaras was the Minister for Foreign Affairs for the liberal conservative government of New Democracy under Prime Minister Konstantinos Mitsotakis but ended up leaving and founding the national conservative Political Spring in response to the Macedonia naming dispute, resulting in the 1993 Greek legislative election where PASOK won with its leader Andreas Papandreou making a successful political comeback, which was considered to be responsible for the Greek government debt crisis.
In the run up to 2019 UK General Election, the Brexit Party, led by former UKIP leader Nigel Farage, initially put up candidates in 600 seats after a strong showing for the newly-formed party in the 2019 European Elections, but days later, he reversed his position after Conservative British Prime Minister Boris Johnson stated that he would not consider an electoral pact with the Brexit Party. That was seen as benefiting the Conservative Party and disadvantaging the Labour Party. Farage later encouraged voters not to vote for the Labour Party in areas that traditionally favoured it but voted to leave in the 2016 EU Membership Referendum but instead to vote tactically. After the Conservatives' decisive victory, it was suggested by some media outlets and political analysts that Farage had acted as "kingmaker" and stalking horse and effectively won the election for the Tories, as Farage's decision avoided splitting the vote.