EMMANUEL MACRON once longed for a more “Jupiterian” presidency. This April he may realise his dream by securing a rare second term in the Elysée. At the launch of The Economist’s statistical model for the French election, Mr Macron’s lead in the polls suggests that he has a 79% chance to come away victorious. In political terms, there is still an aeon until April 10th, when the first round among a dozen candidates takes place, let alone the two-candidate run-off to be held on April 24th. Updated daily, our model will keep track of his and his rivals’ fortunes until the votes are cast. If Mr Macron succeeds, he will be the first president to win re-election since Jacques Chirac achieved the feat two decades ago.

Although there are many rich traditions in France, sophisticated modelling of elections, common in America and Britain, is not yet one of them. That is not because it is impossible. The essential input—reputable polls—have existed there for decades. Since 1965, when our historical data on pre-election polls begins, polls taken in the final week of the presidential campaign for either round have deviated from the actual vote shares achieved by candidates by an average of just 2.5 percentage points. Our efforts rest on this rich record of polling, both past and present.

As several recent elections have shown, polls cannot eliminate uncertainty. Neither can our model. We try to take the fullest account possible of the error in French election polling, but our model offers no guarantees. It uses statistics to put candidates’ positions in the polls today in the context of the evolution of past campaigns and the errors of past polling—no more and no less. Our assessment of Mr Macron, for example, does not assure him victory. It simply observes that, four times out of five, presidential candidates with a lead of this magnitude ten weeks from election day have gone on to win. That is roughly the chance that a professional footballer successfully scores a penalty kick.

For many people, election modelling is a black box, and its statistical methods a dark art. Ours is not, in a literal sense, since we are releasing the underlying code powering our predictions. The thinking behind our methods is readily intelligible, too. The underlying approach can be summarised simply: we combine the current polls with historical error to create probabilities. The rest of this walk-through fills in the details.

We start by aggregating all current polls for the presidential campaign. We then construct a polling average for every candidate, which is more accurate than any individual poll. We use a modestly fancy statistical tool to do this using a curve of best fit called a “spline”, which resembles the linear regressions familiar to statistics students. So that we do not put too much weight on the past and build overconfident models, we use a machine-learning technique called cross-validation.

The next step is to calculate the historical error of the polls. We use a “timeline” method popularised by the political scientists Will Jennings and Christopher Wlezien, whose remarkable polling database we also rely on. As one gets nearer and nearer to an election day, the difference between the polling estimate and the final result narrows. Six months before the election date, for example, French presidential polls deviate an average of five percentage points from the final result. We can compute this average “absolute error” for every point in the campaign: three months from the election, say, up to the day before.

This is a different sort of measure from the “margin of error” that is sometimes reported by pollsters, which reflects the chance that the people surveyed are unrepresentative of the French electorate. The margin of error also does not account for equally worrisome potential pitfalls, such as the chance that certain kinds of voters are systematically shying away from pollsters, that questions are poorly worded or that pollsters’ methods for correcting for known biases are faulty. The historically rooted “absolute error” method therefore gives a better sense of overall accuracy.

Like manufacturing paint or whisky, the art of blending is the all-important final step for election models. With a starting point (the candidate’s position in the polls on a given day) and a guide (the amount of uncertainty that history implies) we are able to make conjectures about the destination. Our preferred method when the maths get complicated, as it does here, is to employ simulation and let a computer answer our questions by brute force. Where the simulations ought to start is provided by today’s polls. How widely spread apart they ought to be is provided by the historical error.

There is one final quirk—the large number of candidates—to straighten out. When elections only feature two candidates, statisticians can conveniently simulate error by taking votes from one candidate and transferring them directly to the other (such is the nature of a zero-sum system). But in a race with many candidates, this process is not so simple. One candidate’s lost supporters are not equally distributed among the other options. For example, Mr Macron’s standing in the polls has tended to fall as support for Valérie Pécresse, the centre-right Republican candidate, has risen. We solve this by observing the trade-offs in polling averages over the campaign and estimating a covariance matrix—a table that stipulates how each candidate’s votes move in relation to everyone else’s.

At last, we are able to quantify these relationships between all pairs of candidates and simulate 10,000 potential first-round outcomes based on these critical factors: candidates’ current estimated standing, the observed fluctuations between pairs of candidates and the historical uncertainty in the polls. With these simulations in hand for the first round, we repeat the process for the (much simpler) second round. For every single first-round simulation, we generate 1,000 second-round simulations using the patterns in second-round polling for the top two vote-getters. Together, we get 10m total simulations for the French election—refreshed each day.

Bonne chance

As we launched our model, Emmanuel Macron won in about 7.9m of those simulated elections—giving him the commanding (but by no means certain) probability of being re-elected of close to 79%. Mrs Pécresse is the second likeliest to win, albeit with a relatively slender chance of 11%. Marine Le Pen, a far-right candidate, trails in third with a chance closer to 8%. According to the polls today, the other candidates are extreme longshots, splitting the remaining 2% among themselves.

George Box, an eminent statistician, once quipped that “All models are wrong but some are useful”. For an exercise as complicated as an election, a model like ours must necessarily be simplified. But it will prove useful if it correctly measures uncertainty. Mr Macron may like his starting chances, but he should not take false confidence. Who hasn’t seen a professional footballer miss a penalty kick? ■

Sources: “Election polling errors across time and space”, by Will Jennings and Christopher Wlezien, Nature and Human Behaviour, 2018; Alexandre Léchenet, NSPPolls; The Economist