In French politics, a triangular election (French: '''élection triangulaire''') is an election characteristic of the French electoral system, due to the two-round voting system for elections to the National Assembly.[1] 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 French: [[Together (coalition)|Ensemble pour la République]], New Popular Front and the Union of the Far-Right.[2]
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.[3]
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.[4] 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 making the result more representative in theory.
The presence of triangular races depends on the voting method of the elections, and more particularly on the qualification threshold for the second round:
Source:[5]
In the 2021 French regional elections, there a number of run-offs, with the second round preventing the National Rally from winning any regions.[12]
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 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.
It must also be said, concerning the legislative elections, that this bar of one eighth of registered voters has not always been the same.[13]
The information below shows the different changes of this threshold:
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.[5]
Finally if the threshold for remaining in the second round is a percentage of votes cast, which is the case for 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.[16] [17]
Indeed, in the case of 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.[18] [19]
The voter turnout is a significant factor influencing the number of three-way races in 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 blankor 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 | |
However, from this graph we can see two things:
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.
Another factor influencing the number of triangular races is the presence of partisanship, which in France is known as 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:
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.
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 "".[20]
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.
To summarize the different factors impacting the number of triangular elections and their influence, here is a summary table:
| 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 |
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.
I'd like to point out that this article was largely made possible by the corresponding article in French.