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Although Le Pen lost the election, she got a higher proportion in the second round than in the first round: 33% versus 21%. Given that all the other parties were either neutral or supported Macron, and she was branded an enemy of the republic, how did she gain an addition 12% or so?

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    There were 5 candidates with a vote share above 5% in the first and 2 in the second. – Bradley Wilson May 8 '17 at 11:34
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    @BradleyWilson 11 candidates in the first round. In fact, one of them who was 6th (Nicolas dupont aignan) with just under 5% did rally le pen. – user5751924 May 8 '17 at 11:36
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    Imagine a world in which some proportion of people prefer McDonalds to Wendys, some prefer Wendys to Burger King, and some prefer Burger King to McDonalds. Play around with different percentages, and then simulate multi-round runoff elections of groups of people trying to decide where to go for lunch. You will quickly find that these sorts of intransitivities make for occasionally counterintuitive outcomes. – Eric Lippert May 8 '17 at 16:43
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    I'm not understanding the mystery here. There were more other options for people to chose from the first round, and only two in the second. You'd expect both candidates to pick up some votes from the candidates who were no longer on the ballot. If one of them had lost support, that would be unusual. – T.E.D. May 8 '17 at 18:36
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    The parties may have been neutral or supported Macron, but who cares about them! It's the people who voted! – curiousdannii May 9 '17 at 4:02
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Given that all the other parties were either neutral or supported Macron, and was branded an enemy of the republic, how did she gain an addition 12% or so?

Voters aren't beholden to parties.

It's quite possible that 12% of the voters preferred François Fillon, Jean-Luc Mélenchon, etc. in the first round but preferred Marine Le Pen to Emmanuel Macron in the second round. That's not so much.

It would also be possible that some people voted in the first round but not in the second. That of course is the problem with being neutral. It's like splitting your vote in two, half for Macron and half for Le Pen.

Looking at the actual numbers, it seems more the first than the second. Overall about five million fewer people voted in the second round than the first. But Le Pen still received about two and a half million more votes in the second round than the first. So roughly 8% of first round voters switched to Le Pen. The number went up more due to the smaller base.

This suggests to me that they should have used ranked voting (IRV, Condorcet, even Range or Approval) in the first round, as many voters don't seem engaged with their actual choices.

  • Even with ranked voting, you might just have got 5 million exhausted ballots. Perhaps from people who found it inconveivable that neither of the two main parties would make it to the final round, and therefore wouldn't have bothered ranking beyond one of those. Obviously that's kind of negligent, but no more so than not voting at all in the second round, which is what those 5 million people actually did! – Steve Jessop May 8 '17 at 16:12
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Those percentages are the percentages in expressed votes. They do not take into account the blank or null votes (11.5 percent of the votes casted, 4.07 million votes, all-time highest score), as well as the participation (down to 74.6%, lowest in quite a long time). Those are not really relevant to study "additional votes", even though it is the criteria which determines the president.

You can however look at the number of votes Marine le Pen got at each round. First round: 7.7 million; Second Round : 10.6 million

Nicolas Dupont Aignan, the only candidate from the first round who rallied her, had 1.7 million votes in the first round. Of course, not all of his voters followed his lead. According to a polling institute, Marine le Pen got:

  • 30 % of Nicolas dupont aignan voters

  • 20 % of françois fillon voters

  • 7 % of Jean Luc Mélenchon voters

  • 2 % of Benoit Hamon voters

To these, you can add almost all of her original voters.

Sources: IPSOS, Le Monde

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Even without looking at the actual numbers in this election, the confusion seems to come, as others have indicated, form you looking at percentages as actual votes.

Let's say there are 5 candidates and a hundred voters, and 3 candidates get 19 votes, one gets 21 and one gets 22 votes. Now the absolute numbers are equal to the percentages, so the candidates with 21% and 22% go through to round two.

Let's say that in round two, everybody that voted for the other three candidates decide to stay home. The rest votes the same way they did earlier, so now we have again 21 versus 22 votes, but this time it is roughly 49% versus 51%.

Both candidates more than doubled their percentage in the second round, without a single voter changing their vote!

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