Strategic or Expressive Voters? Evidence from a RDD in French Elections

Transcription

1 Strategic or Expressive Voters? Evidence from a RDD in French Elections Vincent Pons Clémence Tricaud February 2017 Abstract In French parliamentary and local elections, candidates ranked first and second in the first round automatically qualify for the second round, while a third qualifies only when selected by more than 12.5 percent of the registered citizens. Using a fuzzy regression discontinuity design around this threshold, we find that the third candidate attracts both loyal voters, who would have abstained or cast a blank or null vote if she were not present, and switchers, who would otherwise have voted for the top two candidates. Switchers vote for the third candidate even when she is very unlikely to win the election. This disproportionately harms the candidate ideologically closest to the third, and changes the winner in one fifth of the races, causing the loss of the Condorcet winner. These results suggest that a large fraction of voters value voting expressively over behaving strategically to ensure the victory of their second best. We rationalize our findings by a model in which voters trade off expressive and strategic motives. JEL Codes: D72, K16 Harvard Business School; BGIE group, Soldiers Field, Boston, MA 02163; vpons@hbs.edu; CREST, Ecole Polytechnique, Paris-Saclay University; Route de Saclay Palaiseau, France; clemence.tricaud@polytechnique.edu; +33 (0) We are grateful to Daron Acemoglu, Abhijit Banerjee, Pierre Boyer, Bruno Crépon, Torun Dewan, Rafael di Tella, Esther Duflo, Benjamin Olken, Julio Rotemberg, Kenneth Shepsle, Daniel Smith, James Snyder, and seminar participants at MIT, Harvard, the University of Chicago, Ecole Polytechnique, CREST for suggestions that have improved the paper. We thank Sebastian Calonico, Matias Cattaneo, Max Farrell and Rocio Titiunik for guiding us through the use of their RDD stata package ( and for sharing their upgrades regarding fuzzy design estimations, Nicolas Sauger for his help with the collection of the 1978 and 1981 French parliamentary election results, Abel François for sharing his data on 1993, 1997 and 2002 candidates campaign expenditures and Salomé Drouard and Eric Dubois for their excellent work of data entry and cleaning. 1

2 1 Introduction In indirect democracies, representatives vote and rule on behalf of the people. In theory, their representativeness comes from being elected by the people. In practice, it depends on the way in which voters translate their preferences into vote choices, and the way in which the voting rule maps vote choices into election outcomes. Under plurality rule, in elections with more than two candidates, supporters of lower-ranked candidates face a difficult trade-off: voting expressively for their favorite candidate, or voting strategically for a candidate with higher chances of winning. By voting expressively, voters reveal their true preference, but they may split their support over a large set of candidates and nominate less-preferred leaders. In his groundbreaking work on strategic behavior, Duverger (1954) posits that voters do not want to "waste their vote" and will thus exclusively vote for the two front-runners in general. Several game-theoric models emerged to formalize this intuition (e.g., Palfrey, 1989; Myerson and Weber, 1993; Cox, 1997). In these models, voters are instrumentally rational - they care only about who wins the election - and, as a result, a plurality election with multiple candidates boils down to a two-candidate race, where the candidate preferred by the majority wins the election. The division of the American political landscape between the Republicans and the Democrats is a famous illustration of Duverger s law. Yet, in other countries, more than two candidates frequently receive large fractions of the votes, contradicting the tendency toward two parties. In UK general elections, for instance, votes in many constituencies are split between the Labor, the Conservative, and the Liberal Democrat Parties, since the emergence of the latter party in the 1990 s. In addition, the presence of lower-ranked candidates can have a major impact on the outcome of the election even when they receive only a small fraction of the votes. In the 2000 U.S. Presidential election in Florida, the 3 percent of votes going to Ralph Nader were enough to sway the election in favor of George W. Bush. In this paper, we implement a Regression Discontinuity Design (RDD) in the context of French elections to compare electoral results when two versus three candidates are competing. In many races, we find that the third candidate steals votes away from the candidate who would have won a two way race, causing his defeat. Empirical studies usually measure the number of voters who vote strategically for a top two candidate instead of their favorite choice by comparing voters preferences and vote choices, relying on survey data (e.g., Blais et al., 2001; Blais, Young and Turcotte, 2005; Hillygus, 2007) or structural methods and natural experiments (e.g., Kawai and Watanabe, 2013; Spenkuch, 2015). But voters underlying preferences are fundamentally unobservable and the results of these studies hinge on the assumptions they make about them or on the reliability of survey responses. We use a different approach: instead of estimating preferences, we study strategic voting (or the lack thereof) by considering actual vote shares and election outcomes under two or three candidates. The number and types of competing candidates is not exogenous in general. In French local and parliamentary elections, however, the electoral rule creates exogeneity enabling us to identify the impact of electoral offer on voter behavior. Both elections are held under a two-round plurality voting rule. In most districts, no single candidate obtains the majority of votes in the first round, and a second round takes place 2

3 a week later where the candidate with the most votes wins the election. The two candidates who obtain the highest vote share in the first round automatically qualify for the second round. The other candidates qualify for the second round only if they receive a number of votes higher than 12.5 percent of all registered voters. Our identification strategy exploits the discontinuity generated by the qualification rule for the second round. We compare election results in the second round in districts where the third candidate obtains a first-round vote share just below the 12.5 percent threshold and is thus eliminated with districts where her vote share is just above and she is thus allowed to move on to the second round. For how unusual this setting may be, the second round races we consider are simply first-past-the-vote elections in which voters have access to a large amount of information. Indeed, first round results provide costless public information on the respective chances of the remaining competitors, which should facilitate coordination and strategic voting (Cox, 1997). This setting has several other advantages. First, the large number of districts at each election and the large number of elections we consider translate into a large number of districts and high statistical power. Second, the threshold is defined as a fraction of registered citizens rather than actual voters, making it particularly hard to manipulate. Third, as another consequence of this particular rule, the set of districts close to the threshold is very diverse. 1 This makes the external validity of our local average treatment effect estimates unusually high and it enables us to compare treatment effect size across a large variety of settings. Finally, the fact that the second round takes place only one week after the first round makes it highly unlikely that voters preferences change significantly between the two rounds, which enables voters to use first round results as signals about the distribution of preferences. It also gives little time for the two front-runners to adapt their campaign strategy to the qualification of the third candidate. 2 We first disentangle two types of voters who vote for the third candidate in the second round: loyal voters who would not have voted for any of the two other candidates if the third candidate was absent; and switchers, who would have and yet switch to the third when she is present. We find that the presence of the third candidate in the second round has a large and significant impact on participation. It increases voter turnout by 3.6 percentage points and reduces the share of blank and null votes by 3.6 percentage points, resulting in an overall increase of the share of candidate votes by 7.2 percentage points (13.0 percent). In addition, the presence of the third candidate decreases the vote share of the two front-runners (expressed as a fraction of registered citizens) by 6.0 percentage points (10.9 percent). Based on these results, we estimate that loyal voters account for 27.3 to 54.5 percent of the voters who vote for the third candidate and switchers for 45.5 to 72.7 percent. Our key results relate to the impact of switchers behaviors on the outcome of the elections. We find that switchers voting choice is often costly as it mainly impacts the vote share of the candidate among the top two that is closest to the third candidate on the left-right axis, and whom most switchers prefer to the other front-runner. The impact remains as strong and significant in elections where the third candidate is unlikely to become a front-runner. This has major consequences on the results: the presence of the third candidate in the second round changes the results in 19.2 percent of the elections close to the threshold, causing the 1 We define close to the threshold or close to the discontinuity as within 2 percentage points of the threshold. 2 The front-runners are defined as the two candidates in contention for victory in the second round and who ranked first and second in the first round. 3

4 victory of the candidate switchers dislike the most when their second best choice would have won otherwise. Moreover, in those elections, switchers second best is also the Condorcet winner - the candidate who would win in a two-candidate race against each of the other candidates. Hence, switchers voting choice is costly not only for them, as it leads to the loss of their second best choice, but also for the majority of the voters, as it leads to the loss of the Condorcet winner. These results are difficult to explain by instrumental voting models. Instead, they suggest that a large fraction of voters base their voting decisions on expressive motives rather than instrumental ones. To characterize the behavior of voters supporting the third candidate and explore the condition under which switchers behavior causes the loss of the Condorcet winner, we develop a model in which voters have to choose between three candidates. Supporters of the third candidate face the following tradeoff: voting for their second best increases the probability that she wins against the candidate they dislike the most, but induces an ideological cost. We assume that voters are group rule-utilitarian as defined in Feddersen and Sandroni (2006a) and Coate and Conlin (2004): they define a cutoff such that voters with ideological cost below the cutoff vote for their second best whereas voters with ideological cost above vote for their preferred candidate. We further model three different types of voters which differ by their motives and level of sophistication. Expressive (non-strategic) voters only value the expressive benefits of voting and thus always vote for their favorite candidate. Strategic-naive voters follow the rule but do not take into account the presence of expressive voters when designing it. Finally, strategic-sophisticated voters follow the rule and take into account the presence of expressive and strategic-naive voters when defining their cutoff. The model shows that the Condorcet winner loses when the fraction of sophisticated voters is too low to compensate for the behavior of expressive and naive voters. We show that our results are consistent with the existence of the three types of voters, and we provide an estimation of their relative fractions among voters who support the third candidate and vote whether she is present or not. 1.1 Contribution to the literature Social choice theory has shown that no electoral system or voting rule is uniformly best under all criteria (Arrow, 1951) 3, and that all voting systems are susceptible to strategic manipulation, meaning that voters always have incentives to misrepresent their true preferences in order to affect the outcome of an election (Gibbard, 1973; Satterthwaite, 1975). Following this result, a normative literature works on finding which rule best resists strategic manipulations and delivers outcomes that best represent populations preferences (e.g., Brams and Fishburn, 2002; Laslier, 2009; Balinski and Laraki, 2011; Posner and Weyl, 2015). Instead, we more directly contribute to the positive literature which studies how citizens behave under common voting rules and how their behavior impacts electoral outcomes. Although we would ideally like a system in which people have incentive to reveal their true preference (which has been proven impossible), under existing rules, electing leaders that correspond best to people preferences may require instead that a sufficiently large fraction of voters strategically misrepresent their preference. 4 Consider for instance an 3 Arrow s impossibility theorem says that, when there is more than two alternatives, there is no social welfare function that satisfies the Pareto property and the Independence of Irrelevant Alternatives and which is not a dictatorship. 4 For instance, Demeze, Moyouwou and Pongou (2016) find that in three-candidate elections, manipulation benefits from half to two-thirds of the population under simple plurality. 4

5 election held under plurality rule where the support of the majority is divided between multiple candidates. In this case, if voters vote according to their true preference, they split their vote and may nominate a less preferred candidate. Hence, if voters do not coordinate, the plurality rule fails to satisfy a key measure of representativeness - the Condorcet criterion - which says that a voting system should always choose the Condorcet winner when one exists (Nurmi, 1983; Myerson and Weber, 1993). According to instrumental voting models (e.g., Palfrey, 1989; Myerson and Weber, 1993; Cox, 1997; Fey, 1997), voters care only about the winner of the election and thus all behave strategically. They choose whom to vote for after assessing the relative likelihood that each possible pair of candidates will be in contention for victory and, as a result, in most cases, two candidates receive all the votes in equilibrium and the one preferred by the majority of voters wins. 5 However, these models assume that voters have common knowledge of candidates platform and voters preferences, which is quite unrealistic in large elections. Departing from this assumption leads to quite different predictions. Myatt (2007) shows that when instrumentally rational voters only have private information about candidates support, more than two candidates may receive support in equilibrium. Bouton, Llorente-Saguer and Castanheira (2015) goes further and demonstrates that uncertainty on the level of candidates support can generate an equilibrium where all voters vote for their favorite candidate, which may lead to the victory of the Condorcet loser. Still, how voters behave, whether or not they vote strategically, and how often voting for a non top two candidate causes suboptimal outcomes is ultimately an empirical question. A large literature has found patterns consistent with strategic voting, starting with Cox (1997) s comprehensive study of strategic coordination across electoral systems. More recently, studies have compared elections results under different voting rules using RDD on population thresholds. In line with Duverger s prediction, they find that, in general, the top two candidates get more votes under simple plurality than under runoff or proportional elections (Fujiwara, 2011; Eggers, 2015 but Bordignon, Nannicini and Tabellini, 2016). However, the number and type of candidates may also differ under various electoral systems, raising doubt on whether these results can be uniquely attributed to changes in electoral rules. Evidence also suggests that voters use available information to coordinate: consistent with Myatt (2007) s predictions, Hall and Snyder (2015) find that higher levels of information in US primary elections decrease the number of votes and donations wasted on candidates unlikely to win. One of the important pieces of information used by voters to coordinate is candidates ranking in previous elections: using a RDD in first-past-the-post elections, Anagol and Fujiwara (2015) show that second-place candidates are substantially more likely than close third-place candidates to run in, and win, a subsequent election. Small-scale laboratory experiments also provide evidence of strategic behaviors (e.g., Forsythe et al., 1993; Rietz, 2008; Van der Straeten et al., 2010; Bouton, Llorente-Saguer and Castanheira, 2015). Although controlled environments are well suited to test how different electoral rules or pre-election signals affect coordination, it leaves open the question on whether strategic voting occurs in large electorate and whether voters act expressively when choosing among real candidates. 5 This result has been extended to a number of other settings. The model of Myerson and Weber (1993) applies to a wide range of single-winner electoral systems such as plurality rule, approval voting, and the Borda system. Cox (1994) extends the model to a multimember context. The case of dual ballot rule is studied in Bouton (2013) and Bouton and Gratton (2015). 5

6 Beyond the identification of patterns consistent with strategic voting, another strand of the empirical literature seeks to determine the actual proportion of strategic voters in the electorate as a whole, or in specific subgroups. These studies typically compare people s preferences and voting choice to measure the fraction of voters who strategically vote for a candidate other than their preferred one. They use different methodologies to address the fact that voters true preferences are unobserved. A large body of empirical studies (e.g., Blais et al., 2001; Blais, Young and Turcotte, 2005; Hillygus, 2007; Eggers and Vivyan, 2016) 6 rely on surveys and compare people s self-reported preferences to their voting choice. These studies typically obtain low estimates of strategic behavior ranging from 3 to 17 percent. 7 A potential concern is that respondents may misreport their true preference and voting behaviors. They often overreport voting for the winner (Wright, 1990, 1992; Atkeson, 1999; Campbell, 2010), which would lead to overestimating strategic behavior. Alternatively, to avoid cognitive dissonance (Festinger, 1962), they may adjust their stated preference to their voting choice, which would lead to underestimating it. Finally, the shortage of survey data at the district level makes it difficult to measure the impact of strategic voting on the election results (Herrmann, Munzert and Selb, 2016). Cognizant of surveys limitations, other studies rely on structural models and calibrate them using elections data at the district level. They use this method to estimate the number of voters who did not vote for their preferred candidate (Kawai and Watanabe, 2013) or the impact of strategic voting on the number of seats won by a party (Myatt and Fisher, 2002), but it requires imposing strong assumptions on voter preferences. Another branch of the literature analyses voting behaviors in split-ticket voting systems and compare votes cast for party lists under a proportional system (for which voters have an incentive to follow their true preference) with votes for individual candidates under plurality rule (for which they have an incentive to be tactical). Assuming that preferences for parties and candidates are perfectly correlated, Spenkuch (2014) infers that strategic considerations matter for only one third of the voters. Closer to our empirical strategy, Spenkuch (2015) exploits a natural experiment in Germany 2005 elections and compares electoral outcomes in two different situations: in one district where a party may gain seats by receiving fewer votes versus all the other districts where voters do not face such reverse incentive. Under the common trend assumption, he concludes that at least 8 percent of voters did not vote sincerely in this particular district. Finally, a last branch of empirical studies measures strategic voting by comparing electoral outcomes in the first and in the second round of runoff elections, where more than two candidates can run in the second round (Blais, Dolez and Laurent, 2017 using French municipal elections; Kiss, 2015 using Hungarian elections). They identify strategic voters as the ones who vote for a non top two candidate in the first round but who vote for a front-runner in the second round, relying on the debatable assumption that voters reveal their true preference in the first round of a runoff election (see Piketty, 2000; Martinelli, 2002; Dolez and Laurent, 2010 but Bouton and Gratton, 2015). In this paper, instead of comparing preferences with voting choices, we compare electoral outcomes when two versus three candidates are competing. We find that a large fraction of voters are ready to vote 6 See Alvarez and Nagler (2000) for a review of past studies. 7 One exception is Kiewiet (2013). Using data from surveys as well as aggregate results, he estimates that, on average, one third of the supporters of nonviable candidates voted strategically. 6

7 for the third candidate even when she has a low foreseeable chance of becoming a front-runner, and at the cost of causing the victory of the candidate they dislike the most over the Condorcet winner. Our results are unlikely to be driven by voters uncertainty. Instead, they suggest that a large fraction of voters value voting expressively for their favorite candidate over contributing to the victory of their second best choice. To characterize the behavior of voters supporting the third candidate, we develop a model built on group rule-utilitarian models of turnout, which assume that voters base their voting decisions on a group rule rather than on pivot probabilities. In Feddersen and Sandroni (2006a), ethical voters supporting each candidate define a cutoff such that those facing a lower voting cost turn out whereas the others abstain. We extend this framework to the vote choice of supporters of the third candidate, who have to decide between voting for her or for their second-best choice. Methodologically, our paper draws on previous empirical studies that exploit vote shares thresholds to estimate causal effects of interest, such as the incumbency effect (Lee, 2008), the effect of having a governor of the same party in office on a presidential candidate s chance of victory in a given U.S. state (Erikson, Folke and Snyder, 2015), or the impact of electing a woman on women s future participation in politics (Broockman, 2013). 8 We adopt a non-parametric approach following Calonico, Cattaneo and Tiriunik (2014) and use the optimal bandwidth defined by Calonico et al. (2016). We also show the robustness of our results to Imbens and Kalyanaraman (2012) s optimal bandwidths. The remainder of the paper is organized as follows. We describe the data we use in Section 2 and our empirical framework in Section 3. Sections 4 and 5 present our main empirical results. Section 6 develops a model to rationalize our empirical findings. Section 7 concludes. 2 Research setting 2.1 French parliamentary and local elections Our sample includes two types of elections, those for the national legislature ( parliamentary elections) and those for local councils ( local elections). Parliamentary elections elect the representatives of the French National Assembly, the lower house of French Parliament. France is divided into 577 constituencies, each of which elects a Member of Parliament every five years. Local elections determine the members of the departmental councils. France is divided into 101 departments, which have authority over education, social assistance, transportation, housing, culture, local development, and tourism. 9 Each department is further divided into small constituencies, the cantons, which elect members of the departmental councils for a length of six years. In elections held before 2013, each canton elected one departmental council member. 10 From the 2013 reform, each canton elects a ticket composed of 8 See de la Cuesta and Imai (2016) for a more comprehensive list of recent studies exploiting vote shares thresholds. 9 In France, local government expenditures accounted for 20.6 percent of general government expenditures in 2011 (OECD, 2013). 10 All French territories participate in local elections, except for Paris and Lyon (where the departmental council is elected during municipal elections) and some French territories overseas. 7

8 a man and a woman to ensure gender equality in departmental councils. To leave the total number of council members unchanged, the reform further reduced the number of cantons from 4035 to Any French citizen over eighteen years old, registered on the voter rolls and not sentenced to ineligibility, may be a candidate for a given legislative election. Candidates for departmental councils must also live in the department they mean to represent. Both elections are held under a two-round plurality voting rule. In order to win directly at the first round, a candidate needs to obtain a number of votes larger than 50 percent of the candidate votes and 25 percent of the registered citizens. In the vast majority of districts, no candidate wins at the first round and a second round takes place one week later. In the second round, the election is decided by simple plurality: the candidate who receives the largest vote share in the second round wins the election while the other candidates are left with nothing. The two candidates who obtain the highest vote share in the first round automatically qualify for the second round. Other candidates qualify only if they obtain a first round vote share higher than 12.5 percent of the registered citizens. Importantly, this threshold only determines which candidates are eligible for the second round. Our sample includes all parliamentary and local elections using this qualification threshold: the eight parliamentary elections which took place since 1978 as well as the 2011 and 2015 local elections. 12 As a result of this high threshold, at most three candidates qualify for the second round, in general, in all but a handful of districts, which is ideal for our study design. In the elections we consider, the third candidate received more than 12.5 percent of the votes in 1,215 districts (16.7 percent of our sample). 13 All candidates qualified for the second round can decide to drop out off the race between rounds. Dropouts result from strategic considerations and local and national alliances. For instance, left-wing parties commonly ask their candidates to drop out off the race if they ranked lower than another left candidate. In our sample, when the third candidate qualifies for the second, she decides to dropout in around 50 percent of the cases. 2.2 Data In French local and parliamentary elections, voters have to insert a pre-printed ballot with the name of the candidate they want to vote for in an envelope and put the envelope in the ballot box. A vote is valid when the voter inserts a unique ballot in the envelope or when she lets the envelop empty. Candidate votes is the 11 This new rule applied to the 2015 local election which is included in our sample. As the two candidates organize a common electoral campaign, run in the election under the same label, and get (or not) elected together, we consider a ticket as a single candidate in the analysis. 12 Each of the 10 elections we consider took place at a different date. Moreover, the local and parliamentary elections we study were never held at the same date as other types of elections such as presidential, mayoral or regional ones. 13 The threshold required to qualify for the second round was lower in previous elections, resulting in a large number of constituenceis in which more than three candidates qualified. In local elections, the required vote share was 10 percent of the registered citizens until 2010, when the threshold was increased to 12.5 percent. One exception was made in the 2011 local elections, where the threshold remained at 10 percent in the 9 cantons belonging to Mayottte department (0.6 percent of 2011 observations). The threshold in parliamentary elections was first increased from 5 percent to 10 percent in 1966 and from 10 percent to 12.5 percent in The official goal of these reforms was to reduce the number of candidates competing in the second round, so that the winning candidate would obtain a larger vote share, increasing her legitimacy ( 8

9 total number of voters who insert one ballot in the envelope (i.e., who vote for one of the candidate running). A vote is null when the voter writes something on the ballot or inserts multiple ballots in the envelope. A vote is blank when the voter puts an empty envelope in the ballot box. Official results of local and parliamentary elections were obtained from the website of the French Interior Ministry ( for most elections, and digitalized from printed booklets for the 1978 and 1981 parliamentary elections. For each election, round, and district we observe the number of registered voters, the number of voters that turned out, the number of candidate votes, the number of votes received by each candidate, and their political label. 14 After excluding districts where only one round took place or with less than three candidates in the first round, our sample includes a total of 7,257 observations: 3,458 (47.7 percent) from local elections and 3,799 (52.3 percent) from parliamentary elections. 15 Table 1 gives the breakdown of the sample data by election type and year. Table 1: Elections used in the sample Year Nb of observations Parliamentary elections Total 3,799 Local Elections , ,897 Total 3,458 Total 7,257 Table 2 presents some descriptive statistics on our sample. In the average district, 7.78 candidates compete in the first round and turnout is 58.2 percent. On average 56.2 percent of the registered citizens express a valid vote for one of the candidates. Turnout in the second round is slightly higher than in the first round (58.8 percent on average) but the fraction of candidate votes over registered voters slightly lower (55.4 percent on average). The average number of candidates in the second round is 2.04 and there are three candidates in the second round in 453 districts (6.2 percent of the sample). We further allocate each candidate to one of six political families: far-right, right, center, left, far-left, and independent. In the elections we consider, the candidate who ranked third in the first round was on the 14 We observe the number of null and blank votes separately for the 2015 local elections only, and, for all the other elections, we measure the total number of null and blank votes together as the number of citizens who turned out minus the number of candidate votes. 15 We also excluded 3 elections where the second and third candidates in the first round obtained exactly the same vote share. In this case, they are both allowed to move on to the second round, regardless of the number of votes they obtained in the first round. Hence, in these elections, the 12.5 qualification rule for the third candidate does not apply. 9

10 right in 19.6 percent of the districts, on the left in 37.1 percent, on the far-right in 36.7 percent, and from another political family in the remaining 6.6 percent of the districts. 16 Table 2: Summary statistics Mean Sd Min Max Obs. Panel A. 1st round Registered voters 45,964 30, ,384 7,257 Turnout ,257 Candidate votes ,257 Number of candidates ,257 Panel B. 2nd round Turnout ,257 Candidate votes ,257 Number of candidates , Vote share of the third candidate Candidates who came in third place in the first round can be expected to have lower chances of winning the second round or finishing second than the candidates who ranked first and second in the first round. In most cases, voters casting a ballot for the third candidate should expect their vote to be wasted. Strikingly, however, third candidates who qualify and compete in the second round gather more votes than in the first round on average and get a remarkably high vote share: 29.0 percent of the candidate votes, on average. This result is not driven by any particular configuration: the vote share obtained by the third candidate in the second round is large when she is on the left (32.8 percent), the right (34.2 percent), and the far-right (21.7 percent). Voters who choose the third candidate instead of voting for the two front-runners may belong to two different types: loyal voters who do not vote for any candidate when the third candidate is absent (they simply do not vote or vote blank or null) and who vote for the third candidate when she is present; and switchers who vote for one of the top two candidates when the third candidate is absent but switch to voting for the third candidate when she is present. To disentangle these two types, we use a Regression Discontinuity Design framework which is described in the next section. 16 Before the election takes place, the French Interior Ministry attributes a political label to each candidate, which is based on several indicators: candidates self-reported political affiliation, party endorsement, past candidacies, public declarations, local press, etc. Candidates can ask to revise their political label before the definitive list of first round candidates has been released. We allocated candidates political labels to 6 political families, mainly based on the allocation chosen by political analysts such as Laurent de Boissieu in his blog France Politique : We also refer to public declarations made by the candidates. Appendix F gives the allocation of the different labels into the six political families for each election. For instance, for the 2015 departmental election: Far-left candidates were running under the label Extrême Gauche. -wing labels includes Parti Socialiste, Radicaux de Gauche, Front de Gauche, Parti de Gauche, Parti Communiste, Europe Ecologie Les Verts, Union de Gauche, left but unaffiliated with any specific party Divers Gauche. Center labels include Modem and Union du Centre. Right-wing labels include Union pour la Majorité Présidentielle, Union des Démocrates Indépendants, Debout la France, and Union Démocrate, on the right but unaffiliated with any specific party Divers Droite. All far right candidates were from Front National. Independent were running under the label Divers. 10

11 3 Empirical strategy 3.1 Identification We exploit the 12.5 percent vote share threshold, which determines whether the third-highest-ranking candidate qualifies for the second round, to estimate the impact of her presence on voters behavior. Following Rubins s potential-outcome framework for causal inference (Rubin, 1974), we define Y i (0) (respectively Y i (1)) as the potential electoral outcome in district i if the third-highest-ranking candidate in the first round is absent (resp. is present) in the second round. Next, we define the rating variable X i as the qualifying margin of this candidate in the first round: X i is the difference between her vote share, expressed as a fraction of the number of registered citizens, and the 12.5 percent threshold. The third candidate is allowed to compete in the second round if and only if X i is positive. The assignment variable D i thus takes two values: D i = 0 if X i < 0 and D i = 1 if X i 0. Finally, let T i indicates the treatment status of district i: T i = 1 if the third candidate is present in the second round, and T i = 0 if she is absent. The identification assumption is that the distribution of potential confounders changes continuously around the 12.5 percent vote share threshold, so that the only discrete change occurring at this threshold is the shift in treatment status. The growing use of RDD exploiting vote share thresholds spawned a methodological debate on the validity of this source of identification (Caughey and Sekhon, 2011; Hainmueller, Hall and Snyder, 2015; Eggers et al., 2015). As emphasized by de la Cuesta and Imai (2016), sorting of candidates across the threshold only threatens the validity of the RD design if it occurs at the cutoff, with potential losers pushed just above the threshold or potential barely winners pushed just below. This is unlikely in general, as it requires the ability to predict election outcomes and deploy resources with extreme accuracy. Even when candidates accurately predict the results of the first round and exert extra effort to fall above the threshold (or prevent others from doing so) when they expect to be near it, weather conditions on Election Day and other unpredictable events will still make the outcome of the election uncertain. In our setting, manipulation of the threshold is perhaps even more unlikely than in other RDDs around vote share thresholds. First, information available to candidates about voters intentions in French parliamentary and local races is very limited. District-level polls are very rare during parliamentary elections, and inexistent during local elections, due to small district size and limited campaign funding. In addition, the threshold is defined as a share of registered voters. Manipulating it would thus require to accurately predict and manipulate both the fraction of registered voters turning out and the share of candidate votes going to the third candidate, making it unusually difficult. To bring empirical support for the identification assumption, we check if there is a jump in the density of the rating variable at the threshold (McCrary, 2008). As Figure 1 shows, we do not observe any. In Section 3.3 we run placebo tests on baseline variables and provide additional evidence supporting the identification assumption. 11

12 Figure 1: McCrary test of the density of the rating variable Notes: This Figure tests for a jump in the density of the rating variable (the qualifying margin of the third-highestranking candidate in the first round) at the threshold. The solid line represents the density of the rating variable. Thin lines represent the confidence intervals. 3.2 Evaluation framework As discussed in Section 2, candidates are allowed to drop out off the race between the two rounds of the election and, as shown in Figure 2, the third candidate decided to drop out in around 40 percent of the elections where she qualified, close to the discontinuity (more information on dropouts in Appendix B). Hence, the treatment status is not a deterministic function of the rating variable, making our Regression Discontinuity Design fuzzy. Following Imbens and Lemieux (2008) and Calonico, Cattaneo and Tiriunik (2014), our main specification uses a non-parametric approach, which amounts to fitting two linear regressions on districts respectively close to the left, and close to the right of the threshold. To deal with imperfect compliance and avoid selection bias, we instrument the treatment status T with the assignment variable D. We estimate the two following equations: 1st stage: T i = α 0 + γd i + δ 1 X i + δ 2 X i D i + ε i (1) 2nd stage: Y i = α 1 + τt i + β 1 X i + β 2 X i T i + µ i (2) Where T i in equation [2] is replaced by the predicted value obtained from equation [1]. The coefficients of interest are ˆγ and ˆτ. ˆγ gives the probability, at the threshold, that a district receives the treatment (the third candidate is present in the second round) given that it has been assigned to it (the third candidate qualified for the second round). ˆτ estimates the treatment impact on outcome Y at the threshold in the complying districts (in districts where the third candidate qualified and is present in the second round). We test the 12

13 robustness of our results to a quadratic specification, including X 2 i and its interaction with D i in equation [1] and X 2 i and its interaction with T i in equation [2]. Figure 2: Imperfect Compliance Treatment status Rating variable Notes: Dots represent the local averages of the treatment status (y-axis). Averages are calculated within quantilespaced bins of the rating variable (x-axis). The rating variable (the qualifying margin of the third-highest-ranking candidate in the first round) is measured as percentage points. Continuous lines are a linear fit. Our estimation procedure follows Calonico, Cattaneo and Tiriunik (2014), which provides robust confidence interval estimators. Our preferred specification uses the MSERD bandwidths developed by Calonico et al. (2016), which reduce potential bias the most. We also test the robustness of the main results to using the optimal bandwidths computed according to Imbens and Kalyanaraman (2012). The bandwidths used for the estimations are data driven and therefore vary depending on the outcomes we consider. Thanks to the size of our sample (7,257 districts), the number of districts where the vote share of the third candidate in the first round is close to the threshold is itself very large in general, generating high statistical power. For instance, the vote share of the third candidate is less than 2 percentages points above or below the threshold in more than 1,800 districts. Using the specifications we just described, Table 3 provides the formal estimates for the first stage. Columns (1) and (2) show the results obtained under the MSRED and IK optimal bandwidths, using a local linear regression. Columns (3) and (4) present the results using a quadratic specification. All four estimates are significant at the 1 percent level. In our preferred specification (column 1), we find that the probability to receive the treatment jumps from 0 to approximately 55.2 percent at the threshold. 13

14 Table 3: First-stage (1) (2) (3) (4) Outcome Treatment status Assignment status 0.552*** 0.611*** 0.509*** 0.566*** (0.042) (0.030) (0.051) (0.043) Observations 1,541 3,579 2,141 3,579 Polyn. order Bandwidth Band. method MSERD IK MSERD IK Outcome mean Notes: Standard errors are in parentheses. ***, **, * indicate significance at 1, 5 and 10%. Each column reports the results from a separate local polynomial regression. The outcome is a dummy equal to 1 if the third candidate is present in the second round. The dependent variable is a dummy equal to 1 if the third candidate gathered more than 12.5 percent of the registered votes in the first round. The polynomial order is 1 in columns 1 and 2 and 2 in columns 3 and 4. The bandwidths are derived under the MSERD (columns 1 and 3) and IK (columns 2 and 4) procedures. We look at the impact of the presence of the third candidate on electoral outcomes to compare voters behaviors in elections with two versus three candidates. Indeed, in general, when the third candidate is present, voters have to choose between three candidates whereas when she is absent, they have to choose between only two candidates. But there are two exceptions to this rule: elections where the second candidate drops out off the race and elections where the fourth candidate qualifies and runs in the second round. In the first case, the third candidate is present but only two candidates are running; in the second case, the third candidate always drops out meaning that she is absent and yet three candidates are running. As shown in Appendix E, it only concerns a very small fraction of the observations and we provide evidence that our results are not driven by these particular cases. 3.3 Placebo tests We perform a series of placebo tests which examine whether there is a discontinuity in any of the following first-round variables at the cutoff: voter turnout, number of registered voters, number of candidates, and closeness (defined as the difference between the vote shares obtained by the two top candidates). As shown in Figure 3, there is no significant jump at the cutoff for any of these variables. The formal estimation confirms the absence of treatment effect. Columns 1 through 4 of Table A1 in Appendix present the results obtained for these four outcomes under our preferred specification. None of the estimates is statistically significant at the standard levels. Hence, we cannot reject the null hypothesis that the treatment has no effect on these baseline variables. 14

15 Figure 3: Placebo tests on baseline variables Turnout 1st round Rating variable Number of registered citizens Rating variable Number of candidates 1st round Distance 1st-2nd 1st round Rating variable Rating variable Notes: Dots represent the local averages of the baseline variable (y-axis). Averages are calculated within 0.2 percentage point wide bins of the rating variable (x-axis). The rating variable (qualifying margin of the third-highest-ranking candidate in the first round) is measured as percentage points. Continuous lines are a quadratic fit. In addition, we conduct the general following test for imbalance. We regress the assignment variable D on a larger set of first-round variables, including the four aforementioned variables as well as share of candidate votes, vote share of each of the top three candidates, political label of the three candidates, political family of the three candidates, number of candidates from the left, right, far-right, far-left and center. We then use the coefficients from this regression to predict assignment status, and test whether it jumps at the threshold. As shown in Figure A1, the assignment status predicted by baseline variables increases continuously as a function of the rating variable and does not show any discontinuity at the threshold. This suggests that there is no systematic discontinuity in the preexisting observable districts characteristics at the threshold. The formal estimate in Column 5 of Table A1 confirms this result: the coefficient is small (1.5 percentage points) and non significant. If not otherwise specified, in the rest of the analysis, all outcomes, including vote shares, use the number of registered voters as the denominator instead of the number of citizens who vote, as the latter may be affected by the treatment while the former remains unchanged between the two rounds. In addition, our 15

16 outcomes are defined as a simple difference between the second and first rounds. Using simple differences helps account for a large share of the latent variance of our outcomes. 4 Loyal voters and switchers To the extent that the third candidate attracts new voters, we should see more citizens participating when she in the race; and to the extent that she attracts voters away from the two other candidates, we should see less votes going to the two front-runners when she is present. To test these predictions, we estimate the impact of the presence of the third candidate on participation and on the votes going to the top two candidates. We then use these results to estimate bounds on the fractions of loyal voters and switchers among voters who vote for the third candidate. 4.1 Impact on participation and candidate votes We consider three outcomes related to participation: turnout, the share of null and blank votes, and the share of candidate votes. We begin with a graphical analysis before presenting formal estimates of treatment effects. The graphs in Figure 4 depict the impact of the presence of the third candidate on our three outcomes. Each dot represents the average value of the outcome within a given bin of the rating variable. To facilitate visualization, a quadratic polynomial is fitted on each side of the 12.5 percent threshold. The graphical evidence shows a clear discontinuity at the cutoff for each outcome: the presence of the third candidate has a large and positive impact on the share of registered citizens who vote and on the share of citizens who vote for one of the competing candidates rather than casting a blank or a null vote. Table 4 provides the formal estimates of the impacts using our preferred specification. On average, the presence of the third candidate at the second round increases turnout by 3.6 percentage points (6.0 percent), reduces the share of null and blank votes by 3.6 percentage points (75.0 percent), 17 and increases the share of candidate votes by 7.2 percentage points (13.0 percent). All effects are significant at the 1 percent level. As should be expected, the impact on the share of candidate votes corresponds to the sum of the absolute value of the impacts on the turnout rate and the share of blank and null votes. To probe the robustness of the results to specification and bandwidth choices, Table 5 estimates the treatment effect on the share of candidate votes using four different specifications. Columns (1) and (2) show the results obtained under the MSERD and IK optimal bandwidths, using a local linear regression. Instead, columns (3) and (4) use a quadratic specification. The estimates obtained using these different specifications are all significant at the 1 percent level and very close in magnitude. 17 Blank and null votes were counted together until the 2015 local elections. In these elections, the last in our sample, the impact on both the share of null votes and the share of blank votes is negative (Figure A2 and Table A2 in Appendix). 16