Two-round system

The two-round system (TRS or 2RS) is a voting method used to elect a single winner. In the United States, it is often called a jungle or nonpartisan primary. The system is also called runoff voting, though this term often means the closely-related exhaustive ballot and ranked-choice runoff systems (which tend to produce very similar results).

In a two-round system, both rounds are held using choose-one voting, where the voter marks their favorite candidate. The two candidates with the most votes in the first round proceed to a second round, where all other candidates are excluded. The two-round system is widely used in the election of legislative bodies and directly elected presidents. It is closely related to other members of the plurality-runoff family of methods (including plurality voting, ranked-choice voting, and plurality-with-primaries).

The two-round system first emerged in France, and has since become the most common single-winner electoral system worldwide.

In the United States, the system is used to elect most public officials in Mississippi, Georgia, Louisiana, and along the West Coast.



Origins
The French system of ballotage was first established as part of the reforms of the July Monarchy, with the term appearing in the Organic Decree of 2 February 1832 of the French government, and is defined as a second-round election "when none of the candidates obtains an absolute majority". The mechanism has since gained substantial popularity in South America, Eastern Europe, and Africa, where it is now the dominant system.

Some variants of the two-round system use slightly different rules for eliminating candidates in the first round, allowing more than two candidates to proceed to the second round in some cases. Under such systems, it is sufficient for a candidate to receive a plurality of votes (more votes than anyone else) to be elected in the second round.

2002 French presidential election
In the 2002 French presidential election, the two contenders described by the media as possible winners were Jacques Chirac and Lionel Jospin, who represented the largest two political parties in France at the time. However, 16 candidates were on the ballot, including Jean-Pierre Chevènement (5.33%) and Christiane Taubira (2.32%) from the Plural Left coalition of Jospin, who refused by excess of confidence to dissuade them.

With the left vote divided among a number of candidates, a third contender, Jean-Marie Le Pen, unexpectedly obtained slightly more than Jospin in the first round:


 * Jacques Chirac (Centre-right, Gaullist): 19.88%
 * Jean-Marie Le Pen (Far-right, National Front): 16.86%
 * Lionel Jospin (Centre-left, Socialist): 16.18%

The other candidates received smaller percentages on the first round.

Because no candidate had obtained an absolute majority of the votes in the first round, the top two candidates went into the second round. Most supporters of the parties which did not get through to the second round (and Chirac's supporters) voted for Chirac, who won with a very large majority:

Despite the, the final winner of the election, Jacques Chirac, was not a spoiler, and would have been elected under any other electoral system as well.
 * Jacques Chirac (Center-right, Gaullist): 82.21%
 * Jean-Marie Le Pen (Far-right, National Front): 17.79%

Nonpartisan blanket primary systems
Louisiana state elections since 1975 and federal elections since 1978 (with a brief return using a closed primary in 2010), is identical to a two-round system. There are no primary elections to choose each party's candidate, instead the general election allows voters to select any candidate, regardless of party affiliation. A candidate who receives 51% of the vote is elected, otherwise the top-two candidates are put in a later runoff election. This is sometimes called a "Louisiana primary" or a "jungle primary".

Washington state adopted a system similar to Louisiana's in 2008, which came into effect in 2010 after legal difficulties. California approved a similar system in 2010, coming into effect for the 36th congressional district election in February 2011. The system used in Washington and California is referred to as the nonpartisan blanket primary or top-two primary system. The first election (the primary) is held before the general election, with the top two candidates being entered in the general election. The general election is always held, even if a candidate gets 51%.

Court challenges have resulted in the rule that candidates are allowed to self-select any party label. This works around an interpretation of US constitutional law that would make the two-round systems illegal and is why they are called "nonpartisan".

Exhaustive ballot
The exhaustive ballot (EB) is similar to the two-round system, but involves more rounds of voting rather than just two. If no candidate receives an absolute majority in the first round, the candidate with the fewest votes is eliminated. This continues until one candidate has an absolute majority. Because voters may have to cast votes several times, EB is not used in large-scale public elections. Instead it is used in smaller contests such as the election of the presiding officer of an assembly; one long-standing example of its use is in the United Kingdom, where local associations (LCAs) of the Conservative Party use EB to elect their prospective parliamentary candidates (PPCs). Exhaustive ballot is also used by FIFA and the International Olympic Committee to select hosts.

Instant-runoff voting
Instant-runoff voting (IRV), like the exhaustive ballot, involves multiple reiterative counts in which the candidate with fewest votes is eliminated each time. Whilst the exhaustive ballot and the two-round system both involve voters casting a separate vote in each round, under instant-runoff, voters vote only once. This is possible because, rather than voting for only a single candidate, the voter ranks all of the candidates in order of preference. These preferences are then used to transfer the votes of those whose first preference has been eliminated during the course of the count. Because the two-round system and the exhaustive ballot involve separate rounds of voting, voters can use the results of one round to decide how they will vote in the next, whereas this is not possible under IRV. Because it is necessary to vote only once, IRV is used for elections in many places. For such as Australian general and state elections (called preferential voting). In the United States, it is known as ranked-choice voting and is used in a growing number of states and localities.

In Ireland it is known as single transferable vote (STV) and is used for presidential elections and parliamentary elections. However, STV as applied in multi-member districts is a proportional voting system, not a majoritarian one; and candidates need only achieve a quota (or the highest remaining fraction of a quota), to be elected. STV is used in Northern Ireland, Malta, the Australian senate and various other jurisdictions in Australia. It is often used for municipal elections in lieu of more party-based forms of proportional representation.

Contingent vote
The contingent or supplementary vote is a variant of instant-runoff voting that has been used in Queensland and was previously used in the United Kingdom to elect some mayors; it was ultimately abandoned as a result of its complex election administration. Under the contingent vote, voters rank the top-two candidates. However it involves only two rounds of counting and uses the same rule for eliminating candidates as the two-round system. After the first round all but the two candidates with most votes are eliminated. Therefore, one candidate always achieves an absolute majority in the second round. Because of these similarities, the contingent vote tends to elect the same winner as the two-round system and instant-runoff voting.

Two-party-preferred vote
In Australian politics, the two-party-preferred vote (TPP or 2PP) is the result of the final round of an election or opinion poll after preferences have been distributed to the highest two candidates, who in some cases can be independents. For the purposes of TPP, the Liberal/National Coalition is usually considered a single party, with Labor being the other major party. Typically the TPP is expressed as the percentages of votes attracted by each of the two major parties, e.g. "Coalition 45%, Labor 55%", where the values include both primary votes and preferences. The TPP is an indicator of how much swing has been attained/is required to change the result, taking into consideration later preferences.

A two-party vote is used for elections to the Bhutanese National Assembly, where the first round selects two parties that are allowed to compete in the second round. Then, a second round is held using single-member districts with first-past-the-post.

Compliance with voting method criteria
Most of the mathematical criteria by which voting methods are compared were formulated for voters with ordinal preferences. Some methods, like approval voting, request information than cannot be unambiguously inferred from a single set of ordinal preferences. The two-round system is such a method, because the voters are not forced to vote according to a single ordinal preference in both rounds.

If the voters determine their preferences before the election and always vote directly consistent to them, they will emulate the contingent vote and get the same results as if they were to use that method. Therefore, in that model of voting behavior, the two-round system passes all criteria that the contingent vote passes, and fails all criteria the contingent vote fails.

Since the voters in the two-round system do not have to choose their second round votes while voting in the first round, they are able to adjust their votes as players in a game. More complex models consider voter behavior when the voters reach a game-theoretical equilibrium from which they have no incentive, as defined by their internal preferences, to further change their behavior. However, because these equilibria are complex, only partial results are known. With respect to the voters' internal preferences, the two-round system passes the majority criterion in this model, as a majority can always coordinate to elect their preferred candidate. Also, in the case of three candidates or less and a robust political equilibrium, the two-round system will pick the Condorcet winner whenever there is one, which is not the case in the contingent vote model.

The equilibrium mentioned above is a perfect-information equilibrium and so only strictly holds in idealized conditions where every voter knows every other voter's preference. Thus it provides an upper bound on what can be achieved with rational (self-interested) coordination or knowledge of others' preferences. Since the voters almost surely will not have perfect information, it may not apply to real elections. In that matter, it is similar to the perfect competition model sometimes used in economics. To the extent that real elections approach this upper bound, large elections would do so less so than small ones, because it is less likely that a large electorate has information about all the other voters than that a small electorate has.

Tactical voting and strategic nomination
Runoff voting is intended to reduce the potential for eliminating "wasted" votes by tactical voting. Under the plurality voting system (also known as first past the post), voters are encouraged to vote tactically, by voting for only one of the two leading candidates, because a vote for any other candidate will not affect the result. Under runoff voting, this tactic, known as "compromising", it is sometimes unnecessary because even if a voter's favorite candidate is eliminated in the first round, they will still have an opportunity to influence the result of the election by voting for a more popular candidate in the second round. However the tactic of compromising can still be used in runoff voting—it is sometimes necessary to compromise as a way of influencing which two candidates will survive to the second round. In order to do this it is necessary to vote for one of the three leading candidates in the first round, just as in an election held under the plurality method it is necessary to vote for one of the two leading candidates.

Runoff voting is also vulnerable to another tactic called "push over". This is a tactic by which voters vote tactically for an unpopular "push over" candidate in the first round as a way of helping their true favorite candidate win in the second round. The purpose of voting for the push over, in theory, is to ensure that it is this weak candidate, rather than a stronger rival, who survives to challenge one's preferred candidate in the second round. But in practice, such a tactic may prove counter-productive. If so many voters give their first preferences to the "weak" candidate that it ends up winning the first round, it is highly likely they will gain enough campaign momentum to have a strong chance of winning the runoff, too, and with it, the election. At the very least, their opponent would have to start taking the so-called weak candidate seriously, particularly if the runoff follows quickly after the first round.

Runoff voting can be influenced by strategic nomination; this is where candidates and political factions influence the result of an election by either nominating extra candidates or withdrawing a candidate who would otherwise have stood. Runoff voting is vulnerable to strategic nomination for the same reasons that it is open to the voting tactic of compromising. This is because a candidate who knows they are unlikely to win can ensure that another candidate they support makes it to the second round by withdrawing from the race before the first round occurs, or by never choosing to stand in the first place. By withdrawing candidates a political faction can avoid the spoiler effect, whereby a candidate "splits the vote" of its supporters. A famous example of this spoiler effect occurred in the 2002 French presidential election, when so many left-wing candidates stood in the first round that all of them were eliminated and two right-wing candidates advanced to the second round. Conversely, an important faction may have an interest in helping fund the campaign of smaller factions with a very different political agenda, so that these smaller parties end up weakening their own agenda.

The intention of runoff voting is that the winning candidate will have the support of an absolute majority of voters. Under the first past the post method the candidate with most votes (a plurality) wins, even if they do not have an absolute majority (more than half) of votes. The two-round system tries to overcome this problem by permitting only two candidates in the second round, so that one must receive an absolute majority of votes.

Critics argue that the absolute majority obtained by the winner of runoff voting is an artificial one. Instant-runoff voting and the exhaustive ballot are two other voting methods that create an absolute majority for one candidate by eliminating weaker candidates over multiple rounds. However, in cases where there are three or more strong candidates, runoff voting will sometimes produce an absolute majority for a different winner than the candidate elected by the other two.

Advocates of Condorcet methods argue that a candidate can claim to have majority support only if they are the "Condorcet winner" – that is, the candidate who would beat every other candidate in a series of one-on-one elections. In runoff voting the winning candidate is only matched, one-on-one, with one of the other candidates. When a Condorcet winner exists, the candidate does not necessarily win a runoff election due to insufficient support in the first round.

Runoff advocates counter that voters' first preference is more important than lower preferences because that is where voters are putting the most effort of decision and that, unlike Condorcet methods, runoffs require a high showing among the full field of choices in addition to a strong showing in the final head-to-head competition. Condorcet methods can allow candidates to win who have minimal first-choice support and can win largely on the compromise appeal of being ranked second or third by more voters.

Impact on factions and candidates
Runoff voting encourages candidates to appeal to a broad cross-section of voters. This is because, in order to win an absolute majority in the second round, it is necessary for a candidate to win the support of voters whose favourite candidate has been eliminated. Under runoff voting, between rounds of voting, eliminated candidates, and the factions who previously supported them, often issue recommendations to their supporters as to whom to vote for in the second round of the contest. This means that eliminated candidates are still able to influence the result of the election. This influence leads to political bargaining between the two remaining candidates and the parties and candidates who have been eliminated, sometimes resulting in the two successful candidates making policy concessions to the less successful ones. Because it encourages conciliation and negotiation in these ways, runoff voting is advocated, in various forms, by some supporters of deliberative democracy.

Runoff voting is designed for single-seat constituencies. Therefore, like other single-seat methods, if used to elect a council or legislature it will not produce proportional representation (PR). This means that it is likely to lead to the representation of a small number of larger parties in an assembly, rather than a proliferation of small parties. In practice, runoff voting produces results very similar to those produced by the plurality method, and encourages a two-party system similar to those found in many countries that use plurality. Under a parliamentary system, it is more likely to produce single-party governments than are PR methods, which tend to produce coalition governments. While runoff voting is designed to ensure that each individual candidate elected is supported by a majority of those in their constituency, if used to elect an assembly it does not ensure this result on a national level. As in other non-PR methods, the party or coalition which wins a majority of seats will often not have the support of an absolute majority of voters across the nation.

Practical implications
In smaller elections, such as those in assemblies or private organizations, it is sometimes possible to conduct both rounds in quick succession. More commonly, however, large-scale public elections the two rounds of runoff voting are held on separate days, and so involve voters going to the polls twice and governments conducting two elections.

Costs
One of the strongest criticisms against the two-round voting system is that the cost to the taxpayer required to conduct two ballots can be nearly twice that of holding a single ballot. The reason the cost is probably not quite double is that an instant-runoff ballot method, for example, may have more-expensive voting machines (at least until more places adopt this method) and the longer vote-counting time could pose additional costs to election administrators.

Political instability
The two-round voting system also has the potential to cause political instability between the two rounds of voting, adding further to economic impact.

Simple to tally
In runoff voting, the counting of votes in each round is simple and occurs in the same way as under the plurality methods. Ranked-choice runoff voting in particular involves a much longer and more complicated count, as it is not possible to tally results locally.

Usage
The two-round system is the most common way used to elect heads of state (presidents) of countries worldwide, a total of 84 countries elect their heads of state directly with a two-round system as opposed to only 21 countries that used single-round plurality (first-past-the-post).

Heads of state elected by TRS in direct popular elections

 * 🇩🇿 Algeria
 * 🇦🇷 Argentina
 * 🇦🇹 Austria
 * 🇦🇿 Azerbaijan
 * 🇧🇾 Belarus
 * 🇧🇯 Benin
 * 🇧🇴 Bolivia
 * 🇧🇷 Brazil
 * 🇧🇬 Bulgaria
 * 🇧🇫 Burkina Faso
 * 🇧🇮 Burundi
 * 🇨🇻 Cape Verde
 * 🇨🇫 Central African Republic
 * 🇹🇩 Chad
 * 🇨🇱 Chile
 * 🇨🇴 Colombia
 * Republic of the Congo
 * 🇰🇲 Comoros
 * 🇨🇷 Costa Rica
 * 🇭🇷 Croatia
 * 🇨🇾 Cyprus
 * 🇨🇿 Czech Republic
 * 🇩🇯 Djibouti
 * 🇩🇴 Dominican Republic
 * East Timor
 * 🇪🇨 Ecuador
 * 🇪🇬 Egypt
 * 🇸🇻 El Salvador
 * 🇫🇮 Finland
 * 🇫🇷 France
 * 🇬🇭 Ghana
 * 🇬🇹 Guatemala
 * 🇬🇳 Guinea
 * 🇬🇼 Guinea-Bissau
 * 🇭🇹 Haiti
 * 🇮🇩 Indonesia
 * 🇮🇷 Iran
 * Ivory Coast
 * 🇰🇿 Kazakhstan
 * 🇰🇪 Kenya
 * 🇰🇬 Kyrgyzstan
 * 🇱🇷 Liberia
 * 🇱🇾 Libya
 * 🇱🇹 Lithuania
 * 🇲🇬 Madagascar
 * 🇲🇼 Malawi
 * 🇲🇻 Maldives
 * 🇲🇱 Mali
 * 🇲🇷 Mauritania
 * 🇲🇩 Moldova
 * 🇲🇳 Mongolia
 * 🇲🇪 Montenegro
 * 🇲🇿 Mozambique
 * 🇳🇦 Namibia
 * 🇳🇪 Niger
 * 🇳🇬 Nigeria
 * North Macedonia
 * Northern Cyprus
 * 🇵🇼 Palau
 * 🇵🇪 Peru
 * 🇵🇱 Poland
 * 🇵🇹 Portugal
 * 🇷🇴 Romania
 * 🇷🇺 Russia
 * São Tomé and Príncipe
 * 🇸🇳 Senegal
 * 🇷🇸 Serbia
 * 🇸🇨 Seychelles
 * 🇸🇱 Sierra Leone
 * 🇸🇰 Slovakia
 * 🇸🇮 Slovenia
 * 🇸🇩 Sudan
 * Syria
 * 🇹🇯 Tajikistan
 * 🇹🇬 Togo
 * 🇹🇳 Tunisia
 * 🇹🇷 Turkey
 * 🇹🇲 Turkmenistan
 * 🇺🇬 Uganda
 * 🇺🇦 Ukraine
 * 🇺🇾 Uruguay
 * 🇺🇿 Uzbekistan
 * 🇾🇪 Yemen
 * 🇿🇲 Zambia
 * 🇿🇼 Zimbabwe

Legislative chambers exclusively elected by TRS in single-member districts

 * 🇧🇭 Bahrain (lower house only)
 * 🇰🇲 Comoros (unicameral)
 * Republic of the Congo (lower house only)
 * 🇨🇿 Czech Republic (upper house only)
 * 🇫🇷 France (lower house only)
 * 🇬🇦 Gabon (both houses; lower house elected directly but upper house elected indirectly)
 * 🇭🇹 Haiti (both houses)
 * 🇲🇱 Mali (unicameral)
 * 🇺🇿 Uzbekistan (lower house only)

Sub-national legislatures:

 * Georgia (United States) (both houses)
 * Mississippi (United States) (both houses)
 * 🇲🇸 Montserrat (United Kingdom) (unicameral)
 * 🇳🇨 New Caledonia (France) (unicameral)
 * Texas (United States) (both houses)
 * Louisiana (United States) (both houses)

Legislatures elected by TRS in multi-member districts (block voting)

 * 🇮🇷 Iran – modified; 25% required to win in first round (unicameral)
 * 🇰🇮 Kiribati (unicameral)
 * 🇲🇳 Mongolia – modified; 28% required to win in first round (unicameral)
 * Vietnam (unicameral)

Sub-national legislatures:

 * 🇵🇫 French Polynesia (France) – two-round majority bonus system (unicameral)
 * 🇮🇹 Italy – City councils (Majority bonus system)
 * 🇮🇹 Italy – City councils (Majority bonus system)

Legislatures partially elected by TRS (mixed systems)

 * 🇪🇬 Egypt (both houses; single-member and small multi-member districts, alongside party-list proportional representation in large multi-member districts)
 * 🇫🇷 France (upper house only; indirect elections in single-member and small multi-member districts, alongside party-list proportional representation in large multi-member districts)
 * 🇱🇹 Lithuania (unicameral; single-member districts alongside party-list proportional representation nationwide)
 * 🇹🇯 Tajikistan (unicameral; single-member districts alongside party-list proportional representation nationwide)
 * 🇭🇺 Hungary (until 2010) (unicameral; single-member districts alongside party-list proportional representation in multi-member districts and compensatory seats nationwide)
 * 🇺🇸United States – Used in Louisiana and Georgia. Variant used in California and Washington.

Other examples of use
Two-round voting is also used in French departmental elections. In Italy, it is used to elect mayors, but also to decide which party or coalition receives a majority bonus in city councils.

Most U.S. States officially use the first-past-the-post system to elect candidates in the general election. However, the use of primary elections to select the two major-party nominees, followed by a general election where these two nominees are the only viable candidates, can be considered "an unusual two-round system", though it appears to cause greater partisanship than a single-round system.

Historically it was used to elect the Reichstag in the German Empire between 1871 and 1918 and the Storting of Norway from 1905 to 1919, in New Zealand in the 1908 and 1911 elections, and in Israel to elect the Prime Minister in the 1996, 1999 and 2001 elections.