Hofer led the first round with 35.1 percent versus 21.3 percent for Van Der Bellen. In that first round, four other candidates divided the rest of the votes: 18.9 percent for Irmgard Griss (Independent); 11.3 percent for Rudolf Hundstorfer (Social Democratic Party); 11.1 percent for Andreas Khol (Austrian People’s Party); and 2.3 percent for Richard Lugner (Independent). The Social Democratic Party, on the center-left, and the Austrian People’s Party, on the center-right, had been the dominant parties.

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The Freedom Party lodged a legal challenge to the final election results, alleging “failures and irregularities” and claiming a “systemic failure” in the counting of mail-in ballots especially. The Constitutional Court of Austria apparently agreed, and ordered today a rerun of the election.

We use these methods to check on election fraud

Our methods are available for others to use at our Election Forensics Toolkit. Election forensics uses statistical tests on reported election data. Reported vote totals and turnout counts for polling stations, constituencies or districts should exhibit particular characteristics where irregularities or frauds are present compared to locales without such irregularities.

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Our Toolkit applies three kinds of statistical methods using computational techniques specially designed by Walter Mebane and Kirill Kalinin, two members of our team. No one statistic is definitive in demonstrating fraud. However, when several statistics differ significantly from what we would expect to see in a normal election process, fraudulent behavior is likely the cause.

Here’s the background. The Green and Freedom parties have been historically minor parties. But 265,221 more votes were cast in the election’s second round than in the first round. In other words, most voters in the second round typically vote for one of the major parties, not the Freedom Party or the Greens – and so cast their ballots for parties to which they do not have long-term allegiances.

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What probably happened is this. Many citizens, on both left and right, were worried about the fact that one of two minor parties on opposing ideological edges was going to win the presidency. And so more people voted than might have if the contenders were the usual centrist candidates.

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Further, many people were probably motivated to vote because they expected that many others would vote, and most voters were voting for a party that wasn’t their most preferred party. Such strategic behavior — behavior in which reasonable expectations about what others will do affect voters’ actions — can produce turnout numbers and results that resemble the patterns produced by such fraudulent acts as ballot-box stuffing, voter intimidation and vote-buying.

In Austria, various levels of governmental bodies report presidential election data. Throughout the country, there are community (Gemeinde) level report results. In Vienna, where the “community” level is very large, there’s a level below that one — the election district level — that also reports results. Finally, mail-in ballots are reported at a level above the Gemeinde, at a larger district level that’s akin to counties in the United States; this level typically includes several communities.

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Our election forensics found no fraud in the Austrian election

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We found no signs of anomalies, when checking most of the statistical indicators using Gemeinde (and, where appropriate, district) counts of eligible voters and cast votes, in either the first or second round of the election.

There was one exception: One technique detected the presence of a small anomaly in round two. But the patterns that model detects are also consistent with strategic behavior, as Mebane shows in his research.

Even if we assume that these anomalies do indicate fraud, our best estimate is that only 3,870 potentially fraudulent votes occurred, or .087 percent of the vote total. That’s not enough to change the outcome. But let us emphasize that these anomalies are consistent with what happens during periods of intense mobilization: supporters rallying in favor of Van der Bellen. Further, only Vienna showed that small number of anomalous turnout results. It stands to reason that Vienna, a heavily anti-Hofer city with many immigrants and a diverse population, saw intensive and unusual turnout activity.

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The other statistical tests show no evidence of unusual patterns in the election data.

We find no forensic evidence to support the Freedom Party’s claims of fraud.

What we found when we analyzed Turkey’s 2015 elections

In a previous article for the Monkey Cage, several of us summarized analyses on Turkey’s 2015 parliamentary elections. We found evidence consistent with election fraud in parts of Turkey. That article also described the fraud-detection methods available as part of the Election Forensics Toolkit. Readers can refer to that article for more details on our methods.

Walter R. Mebane, Jr. is a professor of political science, professor of statistics and research associate at the Center for Political Studies at the University of Michigan.

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Allen Hicken is an associate professor of political science and a research associate professor at the Center for Political Studies at the University of Michigan.

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Kirill Kalinin is a doctoral candidate in political science at the University of Michigan.

Ken Kollman is director of the Center for Political Studies and professor of political science at the University of Michigan.

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