# Triangular election in France

In French politics, a triangular election (French: élection triangulaire) is a characteristic of the French electoral system, due to the two-round voting system for elections to the National Assembly. A triangular election occurs when three candidates from the first round have reached the retention threshold and do not withdraw in the second round. The winner of this election is determined by a relative majority. Depending on the elimination thresholds adopted for the first round, the second round may also give rise to a quadrangular election (4), or even a quinquangular (5) or sexangular election (6).
Triangular elections were particularly notable in the 2024 French legislative election between the Ensemble pour la République, New Popular Front, and the Union of the Far-Right.

## Origins

The first triangular elections appeared with the foundation of the two-round majority single-member constituency system. This electoral system was put in place during the legislative elections of 1852, then continued during the Second French Empire. The two-round system then continued under the French Third Republic, from the elections of 1876.
This voting method was originally considered favourable to the Legitimists, Orleanists, and Bonapartists, three political movements of the French right wing, who suffered from division. Indeed, in the case of an incomplete bipolarization, where a divided bloc withdraws in favour of the best of them during the second round, the triangular configuration is then more favourable to the political camp which has two qualified candidates.
Thus, this election characterizes the politics of France, as in order to be represented in the National Assembly without being able to win alone, you must have the support of other parties, in theory making the result more representative .

## In various elections

The presence of triangular races depends on the voting method of the elections, and more particularly on the qualification threshold for the second round:
* Triangular elections are observed less and less during legislative elections, which elects deputies to the National Assembly, where it is necessary to obtain at least 12.5% of registered voters to be able to remain (in the case where two other candidates have also crossed the 12.5% threshold). However as stated previously, a notable exception to this trend is the 2024 Legislative Election, where a record 306 triangulaires were observed in a tight race between three electoral coalitions.
* Triangular elections are impossible during a presidential election as the runoff only takes place between the top two candidates. However, the French Constitution allows withdrawals in favour of another candidate, which does not prevent strategies inspired by legislative elections.
* Triangular elections have become rarer in departmental elections [fr] because successive French governments have regularly increased the threshold for remaining in the second round, to their advantage, this being now placed at 12.5% of registered voters.
* In municipal elections, since 1983, half of the seats have been distributed proportionally, but there has been a threshold for access to the second round in cities with more than 1,000 inhabitants. This is set at 10% of the votes cast, thus allowing the phenomenon of triangular and quadrangular races in politically divided municipalities.
* In regional elections [fr], they previously had a single round, but the latest reform in 2003 modelled the municipal system, with a threshold for access to the second round of 10% of the votes cast, which gave rise to numerous three-way contests in 2004, 2010 and 2015.

## Number of triangular elections
### Legislative elections

Source:
* 1958: 235
* 1962: 129
* 1967: 65
* 1968: 49
* 1973: 96
* 1978: 1
* 1981: 1
* 1986: single-round proportional voting (no three-way race possible).
* 1988: 8
* 1993: 15
* 1997: 79 (105 before withdrawals)
* 2002: 10
* 2007: 1 (12 before withdrawals)
* 2012: 34 (46 before withdrawals)
* 2017:1
* 2022: 7 (8 before withdrawals)
* 2024: 89 (306 before withdrawals)

### Regional elections

In the 2021 French regional elections, there were several run-offs, with the second round preventing the National Rally from winning any regions.

## Influential factors

The number of three-way races in the various elections during the French Fifth Republic is thus relatively disparate; as it does not stabilize around any value. This is because many influencing factors, sometimes favouring the presence of three-way races, sometimes causing the number of second rounds with more than two candidates to fall, must be studied to better understand this evolution.

### The minimum threshold to be a candidate in the second round
#### Threshold

The higher the election threshold required to qualify for the second round, the fewer candidates there will be who will reach the second round and therefore the fewer three-way races there will be.

#### Development

It must also be said, concerning the legislative elections, that this bar of one eighth of registered voters has not always been the same.
The information below shows the different changes of this threshold:
* Following the ordinances of 1958 and 1959 which re-established among other things the ywo-round system, the minimum threshold of registered voters required to be in the second round was 5%
* After law number 66-1022 was passed on December 29, 1966, the minimum threshold of registered voters required to be in the ballot changed to 10 % ;
* After law number 76-665 was passed on July 19, 1976, the minimum threshold of registered voters required to be in the ballot increased to: 12.5%. This remains the case today.
The graph above is also very telling, as we see that the number of second rounds with more than two candidates collapses as the threshold is increased.
Furthermore, there were no more: sexangular; nor quinquangular after the threshold change of 1966, and the number of quadrangular elections also reduced drastically.

#### Depending on its nature (in % of registered or in % of expressed)

Finally if the threshold for remaining in the second round is a percentage of votes cast, which is the case for regional elections [fr] and municipal elections (for municipalities with more than 1,000 inhabitants) then it is then easier for a list or for a candidate to succeed in crossing this threshold, because too low a turnout would therefore have no influence on the election.
Indeed, in the case of departmental elections [fr] and legislative elections where a percentage threshold of registered voters is in force, candidates wishing to qualify for the second round are dependent on abstention.

### The participation rate

The voter turnout is a significant factor influencing the number of three-way races in departmental elections [fr] and legislative elections.
Indeed, within the framework of a threshold of maintenance set at (12.5%) of registered voters, for 3 candidates to qualify, the 3rd must therefore win at least one eighth of the votes of registered voters. But this then assumes that at least three eighths of registered voters (37.5%) go to the polls, because if the first two candidates receive, de facto, as many or more votes as the 3rd (25% of registered voters minimum between them).
Thus, an abstention rate exceeding 62.5% prohibits any possibility of a three-way race in the second round.
The calculation of the threshold for maintaining votes cast: divide one eighth by the participation rate.
From the participation rate, it is then possible to use the expression opposite to formulate the retention threshold as a percentage of votes cast..
In fact, we start from the retention threshold (12.5% or one eighth as presented in the formula) and divide it by the ratio of voters who expressed themselves to the number of registered voters.
The retention rates for certain scenarios are set out in the table below:
Calculation of the rate of maintenance of votes cast for 12.5% of registered voters :
| Registered voters who voted blank or abstained (%) | 0 | 5 | 10 | 15 | 20 | 25 | 30 | 33 | 35 | 40 | 45 | 50 | 62.5 |
| Registered voters who expressed their views (%) | 100 | 95 | 90 | 85 | 80 | 75 | 70 | 67 | 65 | 60 | 55 | 50 | 37.5 |
| Retention rate (% expressed) | 12.5 | 13.16 | 13.89 | 14.71 | 15.625 | 16.67 | 17.86 | 18.75 | 19.23 | 20.83 | 22.72 | 25 | 33.3 |
We then understand that the higher the participation, the more it favours the possibility of there being a triangular election because the maintenance rate, as a percentage of those expressed, is then lower.
However, when comparing the evolution of the participation rate and the number of second rounds with more than two candidates during the legislative elections under the Fifth Republic, the correlation is far from obvious, as shown in the graph opposite.
However, from this graph we can see two things:
* Turnout must be high enough to have many second rounds with more than two candidates (1958; 1962; 1973; 1997 and 2024)
* On the other hand, if participation is very high, this does not necessarily imply a large number of second rounds with more than two candidates (1978; 1981; 1993)
Thanks to these two curves, we can then better understand the role of participation in the frequency of appearance of triangular races: the higher it is, the more it favours the possibility of there being second rounds with more than two candidates (as was explained previously)
On the other hand, the fact that participation is very high does not systematically imply that the number of triangular races is skyrocketing, for the simple reason that other factors must be taken into account.

### Bipolarization and tripolarization

Another factor influencing the number of triangular races is the presence of partisanship, which in France is known as bipolarisation or tripolarisation .
To better understand the subject, here are some historical explanations concerning the action of these phenomena on French parliamentary life between 1958 and 1997:
* Before the 1970s and 1980s, there were not, strictly speaking, two major political forces in opposition. Indeed, the SFIO (then the FGDS), the socialists, and the PCF the communists, did not stand down to help each other, although they were both left-wing. Similarly, the currents ranging from the radicals to the moderate right opposed the candidates of the Gaullist party. A form of multi-party democracy dominated at that time.
* The bipolarization of parliamentary life was, on the other hand, fully acquired between the elections of 1978 and 1986. Indeed, during this period, we find two large coalitions which alternately hold power: the parliamentary right (composed of the RPR and the UDF) and the parliamentary left (composed of the socialists and the communists)
* But from the 1986 elections, Jean-Marie Le Pen's National Front began to gain momentum during the legislative elections. More particularly in 1997, his party made a fairly significant breakthrough. Although the FN's score was respectable in the first round, it won very few seats in the National Assembly, in particular because of withdrawals in order to form
* "a Republican Front" against the extreme right. We cannot truly speak of tripolarization, because Jean-Marie Le Pen's party had too little presence in the chamber, but we can still point to the rise of a third camp in French political life.
The relationship between bipolarization, tripolarization and the number of second rounds is therefore as follows:
In the event of bipolarization, voters will massively shift to two political camps (in a more or less balanced way). In this situation, the other candidates will then receive very few votes: in general, they will not reach the minimum threshold of registered voters to qualify for the second round.
Bipolarization thus favours duels between two parties.
On the other hand, in the event of tripolarization, the votes of the French will be distributed in particular among 3 political groups (in a more or less balanced way). Thus, in this situation and with a fairly good turnout, the three candidates will be selected in the second round.
Tripolarization thus favours triangular elections.
By superimposing this influence factor on the curve of second rounds with more than two candidates during the legislative elections (1958 - 1997), the correlation is more accurate.
* During the period when multipartyism dominated, many second rounds took place with more than two candidates. Also, we note more precisely the presence of quadrangular, quinquangular and sexangular: this would not happen again during the century after the 1973 elections, the last marked by multipartyism
* During the period of bipolarization, the number of triangular races collapses, to only one occurrence for each election
* Finally, as the National Front began to gain momentum around 1986, the number of second rounds with more than two candidates rose again, reaching a fairly high figure in 1997. Indeed, during these elections, Jean-Marie Le Pen's party made a breakthrough: the number of three-way contests peaked at 79 (105 before the withdrawals)

### Withdrawals

Finally, a final crucial factor in understanding the number of triangular elections is the phenomenon of withdrawal from the second round.
Indeed, in the case of the 2024 French legislative election in which 89 triangular races were recorded. This number may be surprising because it represents only ten more than in the 1997 French legislative election, in a context where the National Front collected far fewer votes than the National Rally, and where the tripolarization of French political was similar to 2024.
To understand, you have to know that before the withdrawals, the number of three-way races was 306, a value in line with the context of tripolarization and high participation mentioned above. In the end, no less than two thirds of the three-way races were cancelled, because many candidates withdrew. The reason is as follows: while the National Rally came out on top in the first round in many constituencies, the presidential majority and the new popular front decided to withdraw their candidates who came in third place, in order to "block the extreme right" and form a "Republican Front".
This process of withdrawing the least well-placed candidates to prevent the first from winning the election had already happened before, but in a much less widespread way.
The lowest placed candidates having withdrawn, this explains why the number of real three-way races is not as high as one might think.

### Synthesis

To summarize the different factors impacting the number of triangular elections and their influence, here is a summary table:
| Influencing factor | Nature of influence |
| Minimum threshold required to qualify for the 2nd round in % of votes cast | increase |
| Minimum threshold required to qualify for the 2nd round in % of registrants | decrease |
| Lowering of the minimum threshold required to qualify for the 2nd round | increase |
| Increase in the minimum threshold required to qualify for the 2nd round | decrease |
| High participation | increase* |
| High abstention | decrease* |
| Bipolarization | decrease |
| Tripolarization or multipartyism | increase |
| Disclaimers | decrease |
*Note that in the case of regional and municipal elections (of more than 1000 inhabitants) where the retention threshold is calculated from the number of votes cast, the participation rate has no influence on the number of three-way races.

## More than three candidates

Beyond the triangular election scenario, quadrangular, quinquangular, sexangular, heptangular or even octangular elections may also occur, during which four, five, six, seven or eight candidates respectively remain in the second round. These situations, more common during regional and municipal elections, remain extremely rare during legislative elections.