Introduction
1Throughout the 2017 presidential campaign, commentators with knowledge of the workings of French politics discussed the chances that a victorious Emmanuel Macron would have of gaining a parliamentary majority in the legislative elections in June. The situation that had occurred in 1981 and 1988 following François Mitterrand’s dissolution of the National Assembly, and since 2002 thanks to the reduction of the presidential term in office from seven to five years and to the changes in the electoral calendar, seemed harder to envisage this time around. Emmanuel Macron’s task was more complicated than that of his predecessors for at least two reasons. First, it seemed unlikely that his party, La République en marche! (LREM), established only a year earlier, would be able to put forward a credible candidate in each of the country’s 577 constituencies. Second, the Parti socialiste (PS) and especially Les Républicains (LR) seemed able to limit Macron’s success due to the local presence of their candidates and, in particular, their outgoing deputies. The prospect of a hung parliament, or even a new coalition, seemed to be taking shape.
2The result of the legislative elections, held on June 11 and 18, 2017, therefore came as a surprise: Macron obtained a large parliamentary majority. This unexpected outcome was the result of the new president, like his predecessors, taking advantage of the associated effects of the electoral calendar and the majority vote. [1]
3The effects of proximity between elections, which are well known in the United States [2] under its party system, have now crossed the Atlantic. When they are held on the same day, as in the United States, the electoral options, the campaign, and the results of the legislative elections are thereby closely linked to the presidential election, via a process of “contamination” that the anglophone literature has named the “coattail effect.” [3] When legislative elections are held immediately after the presidential election, as has been the case in France since 2002, the ripple effect of the presidential election on the legislative elections may be even greater [4] and some have even deemed these election cycles to be “‘executive’ elections with four rounds” [5] or “fourround systems.” [6]
4Once again, and despite the uncertainties, the proximity effect played a key role and the majority vote had spectacular effects once more. Receiving 32.3% of the votes cast in the first round of the 2017 legislative elections, the LREMMoDem (La République en marche!Mouvement démocrate) coalition won 60% of the seats in the National Assembly (LREM 308 seats and MoDem 42 seats). When considered together, these two figures raise the question of how votes translate into seats and call for a review of the literature on the subject, especially since the mechanical effects are more complex in tworound elections than in oneround elections. Moreover, in 2017 this question can be seen in a new light, after Emmanuel Macron managed to successfully “demolish” the LeftRight divide. This situation leads us to reconsider how representation in the National Assembly is constructed, in other words, the analysis of the translation of votes into seats. This can no longer be carried out by looking at the results at a national level: instead, a study that looks at each constituency must be conducted and, more precisely, it must examine the configurations in the second round of voting in each constituency.
5In this light, it seems clear that the LREMMoDem coalition took advantage of the destructuring of the party system and the central position it occupied in the political arena in order to gain a large number of seats, despite being less successful than anticipated in the second round. The amplification effect of the singlemember district electoral system was fully instrumental in this, and the tworound system, as practiced in France, thus appeared to have a more brutal effect than ever before.
The voting system, the party system, and how representation in the National Assembly is constructed
6The question of the translation of votes into seats is now a wellestablished field of research that is part of the broader field of studies on the effects of voting systems, the theoretical framework of which was put forward by Maurice Duverger in 1951: the singlemember plurality system tends to produce a twoparty system (Duverger’s “law”), while proportional representation and the tworound system favor a multiparty system (Duverger’s “hypothesis”). [7]
7Duverger identifies two types of effect: mechanical effects and psychological effects. Mechanical effects arise after the election and relate to the way in which votes translate into seats. In the firstpastthepost election, the success of the majority party in terms of votes is thus considerably amplified in terms of seats. The leading party wins in a large number of constituencies, while the party that comes in third place is underrepresented. Psychological effects refer to the propensity of parties and voters to anticipate the mechanical effects of election rules. According to Duverger, these two effects must not be dissociated, since psychological effects are the product of mechanical effects: it is because they have little chance of winning the election locally that the thirdplace parties end up losing their voters. In the singlemember plurality system, mechanical and psychological effects are combined to favor or perpetuate a twoparty system.
8To understand the mechanical effects of a voting system in a given election, we can use the seatvote equation, working on the premise that the proportion of seats attributed to a given political party is a result of the proportion of votes that it has gained. This approach, initially used to assess the effects of firstpastthepost, was quickly extended to proportional representation. It is equally likely to be applied to tworound elections, especially when the party system is structured in two blocs, as was long the case in France, where the second round of the legislative elections in most constituencies consisted of a right/left runoff. [8]
9However, the calculation of the seatvote equation is based on the results—in votes and seats—collected by each party or bloc at national level and assumes that the range of electoral options is identical in all places. But this is not always the case. For example, in 1993 in Canada, the Bloc Québécois, which had only put forward candidates in Quebec, won 54 seats for only 13% of the votes cast nationally and emerged as the second largest political party in parliament, ahead of the Parti réformiste who, with more than 18% of the votes cast, had only won 52 seats. In this specific case, to understand the national construction of the assembly, it is necessary to “go down” to the provincial level and consider, province by province, how the vote to seat translation takes place depending on how the range of electoral options is configured.
10Mutatis mutandis, in France, the analysis of the translation of votes into seats is now inseparable from that of the range of electoral options in the second round. The “tripartition” of the French political landscape had already introduced a first disruption, which was particularly noticeable during the elections when, as in 1997, the Front national (FN) was able to hold its own in a large number of constituencies. The consequences of the “disruptive vote” of spring 2017 [9] are, for the issue at hand, of a completely different magnitude. The political explosion witnessed during the presidential election immediately led to a reconfiguration of the range of electoral options and a sudden change in the balance of power during the legislative elections. The parties or movements that had put forward the five main candidates in the presidential election together won 75.6 per cent of the votes cast in the first round of the legislative elections. In a multiparty system where no secondround coalition was established, the analysis of the second round is even more complicated and the logic of the vote to seat translation is inseparable from the study of secondround configurations. These configurations can take the form of twoway, threeway, or even four or fiveway contests, as long as the election rules allow more than two candidates to remain in the second round, as is the case in France where the qualification threshold is set at 12.5% of registered voters. However, even in the case of a twoway contest, the range of electoral options is likely to vary from one constituency to another. With at least five parties—or coalitions—likely to qualify, it can theoretically take ten different forms (Table 1), compared to only one when French political life was mainly structured into two blocs, the Left and the Right.
Secondround configurations: Five political coalitions/parties, ten theoretical types of twoway contest^{*}
Secondround configurations: Five political coalitions/parties, ten theoretical types of twoway contest^{*}
* previously called “Union pour un mouvement populaire,” (UMP)11The question of vote to seat translation can therefore no longer be considered globally, based on national data. Instead, we must examine the types of configuration actually encountered in the second round, which themselves result from the results of the first round.
12Because of the extremely low turnout in the first round (48.7%), only one constituency had a threeway contest in the second round. With the exception of the second round in the second constituency in the department of Aveyron, where only one candidate remained in the running after the withdrawal of the candidate who came second, there was a twoway contest in all the other constituencies that had not elected their candidate in the first round (Table 3).
13The second round was marked both by the virtual disappearance of the traditional left v. right contest (there were only 16 of these, compared to 443 in 2012), [10] and by the omnipresence of the LREMMoDem coalition. [11] Having won 32.3% of the votes cast in the first round, the LREMMoDem coalition was more than 13 points ahead of the LRUDI (Les Républicains Union des démocrates et indépendants) coalition, allowing it to qualify in nine out of ten constituencies, i.e., in 516 constituencies (Table 2). LREMMoDem was thus the central pivot of a second round whose three main aspects were the contests between LREMMoDem and the Right (273 constituencies), LREMMoDem and the Left (135 constituencies) and LREMMoDem and the FN (103 constituencies) (Table 3). In other words, although the FN had been disrupting the LeftRight divide for thirty years, Emmanuel Macron’s victory in the presidential election totally demolished it.
Firstround (R1) score and secondround (R2) presence of the five main political coalitions/parties
R1 score as a percentage of votes cast  R2: number of candidates who qualified  R2: Percentage of candidates who qualified  Percentage of constituencies where the candidate qualified  

LREMMoDem  32.3  516  45.0  90.0 
LRUDI  18.8  299  26.1  52.2 
FN  13.2  120  10.5  21.0 
FI  11,0  67  5.8  11.6 
PS  7.4  65  5.7  11.4 
Other political coalitions/parties*  17.3  79  6.9  
Total  100.0  1,146  100.0 
Firstround (R1) score and secondround (R2) presence of the five main political coalitions/parties
*DVD (Divers droite), DVG (Divers gauche), etc.The electoral configurations in the second round of the 2017 legislative elections (whole of France)^{(1),(2),(3)}
Types of electoral configurations*  

Seats elected in the first round  4  
Secondround contests  573  
Of which  1 LREMMoDem v. Right (1)  273 
2 LREMMoDem v. Left (2)  135  
3 LREMMoDem v. FN (3)  103  
4 Left v. Right  16  
5 Left v/ FN  6  
6 Right v. FN  11  
7 Other types of twoway contest  27  
8 Other cases (threeway contest, independent candidacy)  2  
Total number of constituencies  577 
The electoral configurations in the second round of the 2017 legislative elections (whole of France)^{(1),(2),(3)}
(1) Right: LR, UDI, DVD, DLF(2) Left: PS, PC, FI, Left Radicals, Ecologists, DVG
(3) Including Emmanuelle Ménard (4th constituency of Vaucluse)
The tworound system: Each round matters
14The very large majority obtained by the LREMMoDem coalition can be explained above all by its position in the political arena. As the “Condorcet winner,” it entered the second round in a favorable position, regardless of who its opponents were. However, as the name gives away, a tworound election also involves a second round, and this one had a dampening effect on Emmanuel Macron’s party’s announced victory.
Position matters: LREM, as close as possible to the median voter
15The LREMMoDem coalition garnered more than 50% of the votes cast, regardless of the secondround configuration (Tables 4a, 4b, and 4c), and took advantage of its central position to obtain an average of 52.7% of the votes cast when competing against the Right, 52.4% when competing against the Left, and 59.1% when competing against the Far Right. The results seen in each of these three configurations highlight the “mechanical effects” specific to the majority system(s): obtaining on average more than 50% of the votes cast meant that LREMMoDem was capable of winning in well over half of the constituencies.
LREMMoDem v. Right contests (273 constituencies)
LREMMoDem  Right  

Score in the second round (percentage of votes cast)  52.7  47.3 
Number of seats  164  109 
Proportion of seats  60.1  39.9 
Standard deviation  8.9  
N  3.8 
LREMMoDem v. Right contests (273 constituencies)
LREMMoDem v. Left contests (135 constituencies)
LREMMoDem  Left  

Score in the second round (percentage of votes cast)  52.4  47.6 
Number of seats  87  48 
Proportion of seats  64.4  35.6 
Standard deviation  8.0  
N  7.1 
LREMMoDem v. Left contests (135 constituencies)
LREMMoDem v. FN contests (103 constituencies)
LREMMoDem  FN  

Score in the second round (percentage of votes cast)  59.1  40.9 
Number of seats  94  9 
Proportion of seats  91.3  8.7 
Standard deviation  6.0  
N  6.4 
LREMMoDem v. FN contests (103 constituencies)
16However, in the firstpastthepost system, the translation of votes into seats is particularly sensitive to the geographical distribution of votes. The more nationalized the election is, in other words, the closer the standard deviation (calculated by constituency) of each party’s score is to 0, the more spectacular the mechanical effects of the tworound system. Thus, when two coalitions or parties clash, if the leading one obtains 51% of the votes with a standard deviation of 0 nationally, it will get 51% of the votes in each of the constituencies involved and will therefore obtain 100% of the seats. As shown in Tables 4a, 4b, and 4c, standard deviations are small, regardless of the type of contest involving the LREMMoDem coalition. The leverage effect is therefore very high. The LREMMoDem coalition thus gained 60.1% of the seats in the 273 constituencies where it was competing against the Right. Similarly, it won 64.4% of the seats in its contests with the Left, and 91.3% of the seats when it faced the Front national.
The second round matters, or the corrective ability of the second round
17Since 1958, there have been three types of scenario in second rounds of legislative elections. Most often, the second round is in line with the logic of the first round and confirms its verdict. But it sometimes amplifies the results of the first round (1993) [12] or, on the contrary, corrects them (1967, 2007). [13]
18On the eve of the first round, the survey institute IPSOS made a projection of the distribution of seats, based on estimates of the results of the first round, a delicate exercise that requires us to take the resulting projection with a pinch of salt. According to this projection, the LREMMoDem coalition could expect to obtain between 415 and 455 seats. [14] A few days later, another survey institute, OpinionWay, argued based on a survey conducted between the two rounds that the LREMMoDem coalition could obtain between 440 and 470 seats, [15] suggesting momentum between the two rounds (Table 5).
The second round: Comparison between the June 15 OpinionWay survey and the actual results on June 18
OpinionWay survey June 15, 2017  Secondround results, June 18, 2017  

Participation  46 %  42.6 %  
LREMMoDem  Others  LREMMoDem  Others  
LREMMoDem v. Left contests 135 constituencies  59 %  41 %  52.7 %  47.3 % 
LREMMoDem v. LRUDI contests 273 constituencies  58 %  42 %  52.4 %  47.6 % 
LREMMoDem v. FN contests 103 constituencies  60 %  40 %  59.1 %  40.9 % 
Total number of seats LREMMoDem  440 to 470  350 
The second round: Comparison between the June 15 OpinionWay survey and the actual results on June 18
19However, the second round of the 2017 legislative elections is clearly aligned with the notion of a “corrective” round. Its results are quite different from the seat projections made on the basis of the results of the first round and even more different from those made in surveys conducted between the two rounds. The turnout was even lower than in the first round (42.6%), and lower than predicted. The results of the LREM v. LR or LREM v. Left contests were much closer than expected and because of this the new presidential majority won significantly fewer seats than it had expected (Table 5).
20According to the abovementioned survey conducted by OpinionWay between the two rounds, 87% of French people believed that Emmanuel Macron would have an absolute majority at the end of the election (compared to 65% before the first round), but interest in the campaign declined further compared to the latest evaluations made by OpinionWay before the first round. The prospect of a major victory thus contributed to demobilizing those who had voted for LREM candidates, as well as slowing down the party’s electoral momentum by discouraging voters from other political parties to support LREM candidates in the second round.
21To understand the electoral shifts that occurred from one round to the next, it is possible, for each type of contest, to assume that the score of each of the two opposing coalitions/parties depends on their ability to keep their first round electorate mobilized and to attract voters who either abstained in the first round, cast a blank or null vote, or expressed their support for another candidate.
22For example, for the 273 LREMMoDem v. Right contest, the equation for the LREMMoDem secondround vote can be written as follows (calculation as a percentage of registered voters):
23LREMMoDem T2 = a + a1 ABS T1 + a2 BN T1 + a3 EXG T1 + a4 LEFT T1 + a5 REG+DIV T1 + a6 LREMMoDem T1+ a7 RIGHT T1 + a8 FN T1 + ∈
24A similar equation can, of course, be used for the rightwing vote in the second round. The same calculation can also be made for each type of contest.
25In operational terms, two types of method can be used to understand electoral movements from one round to the next. The first method consists of using individual data, from exit polls or postelection surveys, to estimate the mechanics behind the transfer of votes from the first to the second round. Many studies have thus enabled us to understand voting transfers from one round to another, or from one election to another. [16] The second method does not focus on individuals themselves but on aggregate data, i.e., the actual results of the two rounds of voting on a given scale. [17] In this case, however, we should avoid making ecological inferences. Strictly speaking, it is not possible to infer the behavior of voters (individuals) based on electoral results (aggregate data) even if, in the absence of data to the contrary, this suggests their likely behavior. [18] To prevent any confusion, in the following paragraphs we will use the terms “transfer of electorates” between the two rounds instead of “transfer of votes” between the two rounds.
26This second method is used here, particularly in the absence of robust individual data for each of the three types of secondround configurations studied here. Data from the two rounds of voting were collected at the level of each legislative constituency, allowing the construction of regression models on the transfer of electorates from the first to the second round. Several regression models were put into competition, each introducing (or not) a constraint on the estimators and/or the constant. [19] In the regression model used here, on the one hand, each coefficient is positive and ranges from 0 to 1; on the other hand, for the same type of contest, the sum of the coefficients assigned to the same firstround coalition or party is equal to 1.
27Thus, for the 273 LREMMoDem v. Right contests, the equation for the LREMMoDem secondround vote can be written as follows, not taking into account the scores of the political coalitions or parties that obtained a negligible score in the first round:
28LREMMoDem T2 = a + 0.01 ABS T1 + 0.44 BN T1 + 0.25 LEFT T1 + 0.74 LREMMoDem + 0.04 RIGHT T1 + 0.02 FN T1 + ∈ (Table 6a, column 4).
29This suggests, for example, that 25% of the left electorate in the first round voted for LREMMoDem in the second round in this type of contest.
30Five observations can be made from the three transfer matrices presented in Tables 6a, 6b and 6c. These make it possible to understand why the LREMMoDem coalition did not win the 450 seats that many promised it at the end of the first round.
31First, these matrices highlight a real demobilization in the second round of the LREMMoDem electorate in the first round. The observed transfer rate is approximately 75%, regardless of the configuration. This means that a quarter of the firstround LREMMoDem electorate abstained (or cast a null or blank vote) in the second round.
32Second, electorates for other political coalitions or parties who voted in the second round systematically mobilized better than the one who voted for the LREMMoDem coalition. 95% of the Far Right electorate in the first round voted for the FN in the second round. 96% of the Right electorate in the first round voted again for the Right in the second round. 80% of the Left electorate did the same in the second round, which deserves to be highlighted in view of the Left’s fragmentation (La France insoumise, Parti communiste, Parti socialiste, and Écologistes often each presenting a candidate in the first round) and the deep divisions it was beset with during the entire fiveyear term of François Hollande.
33Third, the LREMMoDem coalition derived only modest benefit from its central position in the political arena. In the LREMMoDem v. Right contests, the Leftwing electorate in the first round voted more often in the second round for LREMMoDem than they did for the Right, but only by a small amount: only a quarter of the firstround leftwing electorate voted for LREMMoDem in the second round, while more than half of it abstained or cast a null or blank vote (Table 6a). The case was exactly the same in the LREMMoDem v. Left contests, as the Rightwing electorate in the first round voted more often for LREMMoDem than for the Left, but in equally modest proportions: again, only a quarter of the firstround rightwing electorate voted for LREMMoDem in the second round, while more than half of it in this configuration abstained or cast a blank or null vote (Table 6b).
Matrix of transfer of votes from the first round to the second round. LREMMoDem v. Right contests (273 constituencies, in % of registered voters)
Abstention  Blank or null  LREMMoDem  Right  

Abstention R1  0.98  0.00  0.01  0.02 
Blank or null vote R1  0.00  0.44  0.44  0.13 
Left R1  0.33  0.31  0.25  0.11 
LREM + MDM R1  0.24  0.02  0.74  0.00 
Right R1  0.00  0.00  0.04  0.96 
Far Right R1  0.19  0.14  0.02  0.65 
Matrix of transfer of votes from the first round to the second round. LREMMoDem v. Right contests (273 constituencies, in % of registered voters)
Matrix of transfer of votes from the first round to the second round. LREMMoDem v. Left contests (135 constituencies)
Abstention  Blank or null  Left  LREMMoDem  

Abstention R1  0.96  0.00  0.03  0.01 
Blank or null vote R1  0.00  0.37  0.27  0.37 
Left R1  0.09  0.07  0.81  0.03 
LREM + MDM R1  0.21  0.02  0.00  0.77 
Right R1  0.43  0.14  0.18  0.25 
Far Right T1  0.12  0.25  0.59  0.05 
Matrix of transfer of votes from the first round to the second round. LREMMoDem v. Left contests (135 constituencies)
34Fourth, the border between LREMMoDem and the FN seems to have been impassable. The LREMMoDem coalition even appears to have been a real repulsion for FN voters. When a LREMMoDem candidate was competing against a rightwing candidate, 65% of the FN electorate voted for the rightwing candidate. When a LREMMoDem candidate was competing against a leftwing candidate, 59% of the FN electorate voted for the leftwing candidate.
35Finally, the logic of the republican front plays very imperfectly in the event of a LREMMoDem v. FN contest. Only half of the firstround rightwing electorate voted for LREMMoDem in the second round, while 30% voted for the FN and 20% abstained or cast a blank or null vote. Equally, less than half of the firstround leftwing electorate voted for LREMMoDem in the second round, while 15% voted for the FN. Above all, 40% abstained or cast a blank or null vote (Table 6c).
Matrix of transfer of votes from the first round to the second round. LREMMoDem/ FN contests (107 constituencies)
Matrix of transfer of votes from the first round to the second round. LREMMoDem/ FN contests (107 constituencies)
Seatvote equation(s): Measuring the mechanical effects of the tworound election from the perspective of the new configurations of the party system
36Several studies have aimed to model the mechanical effects of firstpastthepost to predict seat distribution, particularly in the United Kingdom, a country that has long been structured around two major parties.
37When two parties A and B—and only two parties—are in competition, the translation from votes to seats can be formulated as follows: S_{A}/S_{B} = (V_{A}/V_{B})^{n}. In this equation, S_{A} and S_{B} refer to the proportion of seats and V_{A} and V_{B} to the proportion of votes cast by A and B respectively. The exponent “n” measures the amplification effect specific to the voting system. [20]
38Applied to the very highly distorted British elections from 1950 to 1970, this equation showed that the amplification in terms of seats of the winning party took the form of a power exponent of 3, i.e., S(a)/S(b) = (Va/Vb)^{3}, subject to conditions, in particular a “neutral” division of the constituencies. [21] This result gave rise to the “cube law.” [22] In practice, when this exponent is equal to 3, this means that, when the distribution of the results constituency by constituency follows the normal law, the gain of one point in favor of one or the other of the two parties causes 3% of the seats to shift.
39The seatvote equation was again posited by Rein Taagepera in 1973 [23] and then extended to proportional representation. [24] But the logic of the “cube law” can also be applied to the tworound election, provided that the political landscape is structured around two blocs and that the second round pits one candidate from each of these blocs against the other in most constituencies, as was the case in France until the legislative elections in spring 2017. By this standard, the tworound system appeared to be the most brutal of all, with the seatvote equation then taking the form of the “law of four”: the ratio in seats between the two blocks—the Left and the Right—was thus equal to their ratio in votes measured in the second round, raised to the power of four. In concrete terms, the displacement of one point of the Right/Left balance of power measured in expressed votes was then likely to tip 4% of the seats, i.e., from 20 to 25 seats. [25]
40In 2017, the “n” amplification effect specific to the majority system was 3.8 for LREM v. Right contests (273 cases, Table 2). Indeed:
41S_{LREMMoDem}/S_{Right} = (V_{LREM}/V_{Right})^{n}
42Log [S_{LREMMoDem} /S_{Right}] = Log [V_{LREMMoDem} /V_{Right}] * n
43n= Log [S_{LREMMoDem} /S_{Right}]/Log [V_{LREMMoDem} /V_{Right}]
44n = Log [60.1 / 39.9] / Log[52.7/47.3]
45n = Log (1.50) / Log (1.11) = 0.41/0.11 = 3.8
46This means that, in practice, a shift of one point in the LREMMoDem v. Right power ratio (measured as a percentage of the votes cast) causes 3.8% of the seats to shift. Because the LREMMoDem v. Right configuration involves 273 constituencies, a shift of one point in the balance of power between the two parties is likely to cause 10 seats to shift (273* 0.038 = 10). The same calculation can be made for other types of contest. The “n” amplification effect is thus 7.1 for LREMMoDem v. Left contests and 6.4 for LREMMoDem v. Far Right contests (Tables 4a, 4b and 4c).
A more brutal voting system than ever before
47According to Taagepera, in a oneround election, “the largest party gets a bonus, the secondlargest breaks even, while the third party is heavily penalized.” [26]
48This “law” can undoubtedly be extended to the tworound election, as used in France, but its mechanical effects are even more brutal. In 2017, having won 32.3% of the votes cast in the first round of the 2017 legislative elections, the LREMMoDem coalition represented 45% of the candidates who ran in the second round and ended up winning seats in 60% of the constituencies. The LRUDI coalition won 19% of the votes cast in the first round, represented 26% of the candidates in the second round, and won 23% of the seats. The “second party” thus obtained a proportion of seats close to the proportion of votes it obtained in the first round. The “third” (the Front national), the “fourth” (La France insoumise) and the “fifth” (the Parti socialiste) were on the other hand heavily penalized, to use Rein Taagepera’s term. For example, the La France insoumise candidates obtained 11% of the votes cast in the first round, but they represented only 5.8% of the candidates in the second round and ended up winning seats in only 2.9% of the constituencies. While the leading party benefits significantly from the tworound election, parties beyond third place are negatively affected by attrition (Figure 1).
Score in the first round, proportion of candidates who qualified for the second round, and proportion of seats won by the five main parties (or electoral coalitions)
Score in the first round, proportion of candidates who qualified for the second round, and proportion of seats won by the five main parties (or electoral coalitions)
49These results once again raise the usual question about the distortions of representation induced by the voting systems and the tools to measure their extent. Arend Lijphart thus measured the gap between the proportion of seats and the proportion of votes obtained by the leading party. [27] More sophisticated indices have also been proposed to accurately measure the degree of disproportionality of a system, including the least squares index (LSq), also known as the Gallagher index (see Appendix 2).
50In 2017, the level of seatvote disproportionality is thus particularly high, especially compared to those of the French legislative elections held since 1958 (Table 7). It stands at 22.95, approaching the record set in the 1993 legislative elections, in which nearly 80% of the seats went to the Right (RPRUDI). Similarly, the effective number of parliamentary parties (ENPP), an index compiled by Laakso and Taagepera (see Appendix 2), was only 2.36—the lowest number since 1958, with the exception of 2002 (2.26), reflecting the imbalance in parliamentary representation (Table 7).
Election index: French legislative elections (1958–2017, whole of France)
The Gallagher index (LSq)  Effective number of electoral parties (ENEP)  Effective number of parliamentary parties (ENPP)  Total number of seats  

1958  21.22  6.09  3.45  465 
1962  14.99  4.93  3.43  465 
1967  10.03  4.56  3.76  470 
1968  19.21  4.31  2.49  470 
1973  11.01  5.68  4.52  473 
1978  6.57  5.08  4.20  474 
1981  16.04  4.13  2.68  474 
1986  7.23  4.65  3.90  556 
1988  11.84  4.40  3.07  555 
1993  25.25  6.89  2.86  577 
1997  17.69  6.56  3.54  577 
2002  21.95  5.22  2.26  576 
2007  13.58  4.32  2.49  577 
2012  17.66  5.27  2.83  577 
2017  22.95  5.57  2.36  577 
Election index: French legislative elections (1958–2017, whole of France)
51The literature has shown that seatvote distortions are determined by the electoral formula: they are more important in (one or two round) majority elections than proportional elections, the latter tending to penalize small parties less. [29] But the high indicators observed under the Fifth Republic tend to confirm that the tworound election “produces the most disproportionate results in Western democracies.” [30]
52* * *
53By the end of the 2017 electoral cycle, France had gone from having a twobloc system to having a true multiparty system. The analysis of the translation of votes into seats during the second round of the 2017 legislative elections shows that, in this context, the tworound election has the same mechanical effects as the oneround election: The coalition that won the first round (LREMMoDem) was overrepresented in seats after the second round; the coalition that came second (LRUDI) obtained a proportion of seats equivalent to the proportion of votes it received in the first round; and the other coalitions or parties are now underrepresented in the National Assembly. This result can be explained by several factors, some of which are specific to the voting system. For example, coalitions or parties in third place (or lower) rarely qualify for the second round and are even less often elected. This means that they are subject to twostage attrition. Other factors are linked to the central position occupied by the LREMMoDem coalition in the political arena (Condorcet winner), which allowed it to win a large number of seats, even if it was less successful than anticipated in the second round.
54The translation of votes into seats is therefore not the only result of the electoral system, even if, as Duverger very early on pointed out, each one has its own effects. Other parameters, such as the position of the parties in the political arena and the current context, also contribute together to explaining how the translation from votes into seats plays out in a tworound election.
55The question of the interactions between election rules and their environment is at the heart of the concept of “embedded institutions,” coined by Shaun Bowler and Bernard Grofman [31] who favor a global approach, based in particular on the articulation between the voting system, the political system, and the party system. More specifically, the effects of several factors are also worth mentioning: the size of the Assembly, which has an effect on the party system, [32] as well as the number of decisive elections and the nature of the regime (presidential and semipresidential versus parliamentary.) [33]
56To return to the French situation, all these elements invite us to question both the very principle of the presidential election and the electoral calendar, that is, the temporal arrangement between the presidential election and the legislative elections, which, as indicated above, results in the presidential election having significant ripple effects on the legislative elections.
57Recognizing that the voting system is “embedded” makes it easier to understand why the tworound majority system has systematically produced a large parliamentary majority since 2002, despite fragmentation or, in 2017, the fragmentation of the party landscape. It also enables us to understand that the introduction of a dose of proportional representation in the next legislative elections will be offset by the reduction in the size of the Assembly.
58Above all, recognizing this embeddedness enables us to understand that, other than revisiting the very principle of the presidential election, “true” electoral reform would involve at least modifying an electoral calendar that is today, more than the voting system itself, the main matrix at the source of all the “unlocatable” majorities.
Appendix 1: Models
59Several regression models were put into competition. The basic model placed no constraints on the estimators and the constant. Subsequently, constraints were gradually introduced into the model.
60For the coefficients, a positivity constraint was first added (coefficients > 0), then the coefficients were normalized (> 0 and < 1). Finally, their inline sum was reduced to 1 and similar operations were performed for the constant. Three types of models were tested: the first where the constant was equal to 0; the second with a free constant ( ∞ + ∞) and the last where the constancy was between 1 and + 1.
61For reasons related both to the quality of the correlation coefficients between observed and estimated results (see below) and to the “goodness of fit,” for all the regressions published in this article we have chosen to use the model where: 1/ each coefficient is positive and ranges between 0 and 1; 2/ for the same type of twoway contest, the sum of the coefficients assigned to the same first round coalition or party must be equal to 1; 3/ the constant varies between  1 and + 1.
62Correlation coefficients between the estimated value and the observed value, by district:
In LREMMoDem v. Right contests
Coefficients  Abstention  Blank or null  LREMMoDem  Right 

0.96  0.92  0.91  0.93 
In LREMMoDem v. Right contests
In LREMMoDem v. Left contests
Coefficients  Abstention  Blank or null  LREMMoDem  Right 

0.97  0.87  0.88  0.95 
In LREMMoDem v. Left contests
Appendix 2: Indices used
63The Gallagher index, sometimes called the least squares index (LSq) or least squares method, measures the degree of disproportionality of an electoral system in a given election. It is based on the difference between the percentages in terms of votes received and seats won by a given party after an election. Its formula is the following E where V_{i} is the percentage of votes received from a party i and S_{i} is the percentage of seats won by that same party. [34]
64The “effective number of parties” (ENP) established by Markku Laakso and Rein Taagepera, is the most commonly used index to measure the dispersion of votes. It enables the number of competitors to be counted, as well as taking into account, by weighting, their relative strength. It can be calculated in votes, effective number of electoral parties: ENEP = 1/Σvi^{2}), or in seats (effective number of parliamentary parties: ENPP = 1/Σsi^{2}), where v_{i} is the percentage of votes cast (or s_{i} is the percentage of seats) collected by list i. The higher the number, the greater the political pluralism. [35]
Notes

[1]
Pierre Martin, “Les élections législatives des 11 et 18 juin 2017,” Commentaire 159 (2017): 525–534,; Bernard Dolez and Annie Laurent, “La logique implacable des élections séquentielles,” Revue politique et parlementaire 1083–1084 (2017): 127–142.

[2]
Matt Golder, “Presidential Coattails and Legislative Fragmentation,” American Journal of Political Science 50, no. 1 (2006): 34–48.

[3]
Golder, “Presidential Coattails,” 34–48.

[4]
Annie Laurent, “Des effets de l’inversion du calendrier électoral sur la fragmentation du système partisan français (1967–2012),” in Institutions, élections, opinion. Mélanges en l’honneur de JeanLuc Parodi, eds. Yves Déloye, Alexandre Dézé, and Sophie Maurer (Paris: Presses de Sciences Po, 2014), 119–138. Translator’s note: Our translation here. Unless otherwise stated, all translations of cited foreign language material in this article are our own.

[5]
JeanLuc Parodi, “L’ancrage d’une curiosité française: l’élection ‘exécutive’ à quatre tours,” Revue française de science politique 57, nos. 3–4 (2007): 285–291.

[6]
Élisabeth Dupoirier and Nicolas Sauger, “Four Rounds in a Row: The Impact of Presidential Election Outcomes on Legislative Elections in France,” French Politics 8, no. 1 (2010): 21–41.

[7]
Maurice Duverger, Les partis politiques (Paris: Armand Colin, 1951).

[8]
Bernard Dolez, “Les mystères de la chambre bleue: Des voix aux sièges lors des élections législatives de juin 2002,” Revue française de science politique 52, no. 5 (2002): 577–591; Bernard Dolez and Annie Laurent, “The Seat–Vote Equation in French Legislative Elections (1978–2002),” French Politics 3, no. 2 (2005): 124–141.

[9]
Pascal Perrineau, ed., Le vote disruptif. Les élections présidentielle et législatives de 2017 (Paris: Presses de Sciences Po, 2017).

[10]
Pierre Martin, “Les élections législatives des 10 et 17 juin 2012,” Commentaire 139, (2012): 853–864.

[11]
The information used in this article has come from the Ministry of the Interior (http://www.data.gouv.fr/) and from Jean Chiche (Cevipof), to whom we are grateful.

[12]
Colette Ysmal, “Les logiques d’un choix sous contraintes: Le second tour,” in Le vote sanction. Les élections législatives des 21 et 28 mars 1993 eds. Philippe Habert, Pascal Perrineau, and Colette Ysmal (Paris: Presses de Sciences Po, 1993), 229–250.

[13]
François Goguel “Les élections législatives des 5 et 12 mars 1967” Revue française de science politique 17, no. 3 (1967): 429–467; Élisabeth Dupoirier, “Le parti socialiste et la gauche: l’implacable spirale de l’échec,” in Le vote de rupture. Les élections présidentielle et législatives d’avriljuin 2007, ed. Pascal Perrineau (Paris: Presses de Science Po, 2008), 145–174.

[14]
Ipsos/Sopra Steria survey for France Télévisions and Radio France.

[15]
OpinionWay, Légitrack, cinquième vague (fifth wave), June 15, 2017, http://opinionlab.opinionway.com/dokumenty/LegiTrackOpinionWayOrpiCinquiemevague15juin2017.pdf, accessed December 15, 2018.

[16]
See Jacques Capdevielle, Élisabeth Dupoirier, and Colette Ysmal, “Tableau des électorats en mars 1978,” in France de gauche, vote à droite, eds. Jacques Capdevielle, Élisabeth Dupoirier, Gérard Grunberg, Etienne Schweisguth, and Colette Ysmal (Paris: Presses de Sciences Po, 1981), in particular 86–89.

[17]
Alain Lancelot and Pierre Weil, “Les transferts de voix du premier au second tour des élections de mars 1967 : une analyse de régression,” in Les élections législatives de mars 1967 (Paris: Armand Colin, 1971), 373–388 ; Colette Ysmal, “Le second tour: Le prix de l’isolement de la droite modérée,” in Le vote surprise. Les élections législatives des 25 mai et 1er juin 1997, eds. Pascal Perrineau and Colette Ysmal (Paris: Presses de Sciences Po, 1998), 289, n. 1.

[18]
Luana Russo, “Estimating Floating Voters: A Comparison between the Ecological Inference and the Survey Methods,” Quality & Quantity 48, no 8 (2014), 1667–1683.

[19]
For more information on the competing models, see Appendix 1.

[20]
This equation, based on the calculation of probabilities, was empirically established at the very beginning of the twentieth century by the economist and statistician Francis Ysidro Edgeworth, based on the results of the 1909 British elections.

[21]
Maurice Georges Kendall and Alan Stuart, “The Law of the Cubic Proportion on Election Results,” British Journal of Sociology 1, no. 3 (1950): 183–197.

[22]
Kendall and Stuart, “The Law of the Cubic Proportion,” 183–197.

[23]
Rein Taagepera, “Seats and Votes: A Generalization of the Cube Law of Elections,” Social Science Research 2, no. 3 (1973): 257–275.

[24]
Rein Taagepera, “Reformulating the Cube Law for Proportional Representation Elections,” The American Political Science Review 80, no. 2 (1986): 489–504.

[25]
Dolez, “Les mystères de la chambre bleue”; Dolez and Laurent, “The Seat–Vote Equation in French Legislative Elections.”

[26]
Rein Taagepera, Predicting Party Sizes. The Logic of Simple Electoral Systems, (Oxford: Oxford University Press, 2007), 205.

[27]
Arend Lijphart, Electoral Systems and Party Systems. A Study of TwentySeven Democracies, 1945–1990, (Oxford: Oxford University Press, 1994).

[28]
Michael Gallagher, <https://www.tcd.ie/Political_Science/people/michael_gallagher/ElSystems/>, 2017.

[29]
Duverger, Les partis politiques.

[30]
See the ACE project: http://aceproject.org/acefr/topics/es/esd/esd01/esd01e/esd01e.

[31]
Shaun Bowler and Bernard Grofman, eds. Elections in Australia, Ireland, and Malta under the Single Transferable Vote. Reflections on an Embedded Institution (Ann Arbor: The University of Michigan Press, 2000).

[32]
Rein Taagepera has thus shown that in the voting systems he describes as “simple” (i.e. when the allocation of all seats is made on the basis of votes collected in constituencies of similar size, whether it is a proportional list vote or a majority vote), the party system is based on what it calls the “seat factor” (MS), which is the product of magnitude (M)—i.e., the number of seats to be filled per constituency—and the total number of seats to be filled (S), i.e. the size of the Assembly. See Rein Taagepera, “Le macroagenda duvergérien, à demiachevé,” Revue internationale de politique comparée 17, no. 1 (2010): 93–109, here 94.

[33]
Allen Hicken and Heather Stoll, “Are All Presidents Created Equal? Presidential Powers and the Shadow of Presidential Elections,” Comparative Political Studies 46, no. 3 (2013): 291–319.

[34]
Michael Gallagher, “Proportionality, Disproportionality and Electoral Systems,” Electoral Studies 10 (1) (1991): 33–51.

[35]
See Markku Laakso and Rein Taagepera, “‘Effective’ Number of Parties: A Measure with Application to West Europe,” Comparative Political Studies 12, no. 1 (1979): 3–27.