Can Le Pen win through differential turnout alone?

Probably not.

Last week Politico Europe and the FT ran stories about how Marine Le Pen could still win the French presidential election despite polls showing her substantially behind her most likely opponent in the second round.

Both stories cite Serge Galam, a sociophysicist at the CNRS. Galam's claims have received a lot of attention, because he claims that he predicted Trump's victory. (I say "claims", because the logic of Galam's model predicts a Trump popular vote win rather than an electoral college win).

Galam says something which is true, but tautologous. If people who say they intend to vote for Le Pen turn out at rates much higher than people who say they intend to vote for Macron, Le Pen can win even if more people say they intend to vote for Macron. In the extreme case, if 100% of Le Pen supporters vote, and no Macron supporters do, Le Pen wins.

Slightly more realistically: if 40% of people say they intend to vote for Le Pen, and 60% of people say they intend to vote for Macron, then Le Pen can win if her supporters turn out at 85% and Macron's supporters turn out at rates of 55%. That is, (0.4 * 0.85) / (0.4 * 0.85 + 0.6 * 0.55) > 0.5.

There are two problems with this analysis.

The first problem is that the analysis I've seen reported assumes that opinion polls report vote intention across the whole population, when in fact they reflect vote intention amongst those who say that they are certain to vote.

The second problem is that the size of the differential turnout required (85% minus 55% = 30% in the above example) is extremely large, being much larger than the differential turnout which polls already find, and much larger than most recent elections.

To demonstrate the first problem, it's enough to read the detailed notes which accompany most published opinion polls. Here, I've taken the three most recent polls listed on Wikipedia:

1. Odoxa (fieldwork: 29th to the 30th March; headline figures: Macron 59%, Le Pen 41%): "Les intentions de vote ont été établies sur la base des personnes sûres d’aller voter". 753 of 969 registered voters were certain to vote (77%).
2. Elabe (fieldwork: 28th to 29th March; headline figures: Macron 63%, Le Pen 37%): "Les rapports de forces électoraux présentés dans ce document sont calculés sur la base des personnes ayant exprimé une intention de vote et se disant certaines ou quasiment certaines d’aller voter, soit 600 à 700 personnes selon les hypothèses présentées dans ce document"
3. Ifop-Fiducial (fieldwork: 26th to 29th March; headline figures: Macron 60%, Le Pen 40%): Ifop do not state how they produce their headline figures, but they do report the proportion of voters in the first round who are certain to vote. 70% of those who most identified with En Marche! were certain to vote, compared to 74% of those who most identified with the Front National.

When Galam says that Le Pen could win thanks to differential turnout of the order of 20 to 30 percent, he either misunderstands vote intention polls, or means differential turnout in excess of that the polls already identify.

To demonstrate the second problem, we can either rely on indications of turnout given by the polling companies(!), or we can look across recent elections in other countries. If current polls are similar to the Ifop-Fiducial poll in indicating a differential turnout of around 4%, then asking for differential turnout of eight times that seems demanding.

If we want to look at recent elections in other countries, we need a cross-nationally comparable source of data. Here, I've looked at data from the fourth wave of the Comparative Study of Electoral Systems. For each country included in the analysis, I've grouped respondents by the party they most liked. (It's necessary to focus on the party which is most liked, rather than the party respondents intend to vote for, because CSES only asks about vote intention for respondents who intend to vote).

Within each country, and for each group of supporters, I have calculated the proportion of supporters who said that they intend to cast a vote (pre-election contact), or who say they have already cast a vote (post-election contact). I've then looked at the gap between the most popular party in the sample, and the second most popular party in the sample. I've exclude countries where voting is compulsory and enforced, but have included countries where voting is compulsory but where there are no enforced penalties for not voting.

I'll use as an example the 2014 Bulgarian elections, since this is the most extreme case in the data. I found that 445 out of Bulgarian 968 respondents most liked GERB, and 242 respondents most liked the Bulgarian Socialist Party. (These are all raw, or unweighted figures).

Of those who most liked GERB, 63% said that they either would cast a vote (in the cast of pre-election contact) or did cast a vote (in the case of post-election contact). Those who most liked the BSP were much more committed: 83% said that they either would or had already cast a vote. This makes for a difference of 83%–63%=20 percentage points.

By doing this for all elections covered in wave 4 of CSES, we can see that most differential turnout is small. The average differential turnout is 3% in favour of the second placed party. The average absolute value (4.3%) isn't very different.

It's important to note that this data is based on self-reported turnout intention. Self-reported turnout intention is almost always higher than actual turnout. This may reduce differences between groups, as high-reporting groups hit a ceiling of 100%.

These data do, however, indicate that a differential turnout between Le Pen and Macron supporters of between 25 and 44% is extremely unlikely, since differences of this magnitude haven't been seen in any of the elections included in wave 4 of the CSES, which spans a wide range of countries and types of election.

Even if we focus on a second round contest between Le Pen and Fillon (where the gap in vote intention is smaller), the rates of differential turnout required (high teens) are large.

I'm not sure why this story has been given as much attention, given that it is premised on something which, by itself, seems quite implausible. I suspect it's got traction because

• Galam is a physicist, and we expect physicists to understand numbers, and the range of plausible numbers
• Galam is not a modest man
• Publications are responding to Trump and Brexit by seriously investigating low-probability outcomes which are disruptive in the same way that Trump and Brexit have been disruptive.

Taking low probability outcomes seriously, however, doesn't mean indulging any possible route to a Le Pen victory. I think that a Le Pen victory is very unlikely (a one in ten or a one in twenty chance), but I think that the most likely route to a Le Pen victory doesn't involve polling failure or differential turnout, but rather things which happen over the course of the next thirty-five days which cause Le Pen to become more popular and Macron to become less popular. These kinds of events are harder to quantify, and even where we think we've recognised such an event, they often rebound in unexpected ways. Yet absent such an event, or some narrowing in the polls, it seems very difficult for Le Pen to win.

Notes: You can find my code at GitHub; data are from www.cses.org. Please do let me know if you find any errors in the analysis.