Following Bush’s reelection in 2004, Robert F. Kennedy Jr. wrote a 2006 article in Rolling Stone claiming that Bush’s campaign had stolen the election. The culprit for many Democrats was electronic voting machines that could be hacked to flip Democratic votes to the Republican column.

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Trailing in the polls to Barack Obama during the 2008 campaign, John McCain warned in a debate that the advocacy organization ACORN was engaging in voter registration fraud and was “on the verge of maybe perpetrating one of the greatest frauds in voter history in this country, maybe destroying the fabric of democracy.” When Obama defeated McCain and Mitt Romney in 2012, Republicans became more likely to express doubts about the legitimacy of those elections.

To some extent, these patterns are predictable. People whose preferred candidate loses are less likely to believe an election was fair than those who supported the winner, something we call “loser’s regret” or a “hurrah effect.” In an article published by two of us (Stewart and Sances), we showed how important the “hurrah effect” really is.

The figure below shows the trend in voter confidence from the end of 2000 to the end of 2012, using every public opinion poll we could find. The top line in each chart shows how confident respondents said they were in their own vote. The lower line shows how confident people were in the national vote. We present the results for all voters, as well as for Republicans and Democrats separately.

First the good news: Despite a small dip in confidence in 2006, the percentage of respondents expressing confidence that their own vote (the solid circles) was counted as cast hovered consistently around 75 percent for over a decade.

The bad news, however, is that the percentage of respondents who were very confident that the national vote (the hollow circles) was counted as cast dropped by 30 percentage points, from around 50 percent in 2000 to around 20 percent in 2012.

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Partisan divisions grew during the decade, as they have in so many other areas of American political life.

Between 2000 and 2006, during the Bush presidency, Republicans were very confident in the nationwide vote count. After the 2008 election, GOP confidence declined steadily. Democrats, meanwhile, moved in the opposite direction: lower confidence (compared to Republicans) from 2000 to 2006, then an increase during the Obama years.

This seesaw pattern isn’t necessarily concerning, as long as partisans give at least grudging support to the proposition that the outcome was fairly decided.

But in research we conducted in 2012, we discovered that a substantial number of Republicans were reluctant to even acknowledge that Obama had won the election legitimately.

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As part of the postelection wave of the 2012 Cooperative Congressional Election Study, two of us (Gronke and Stewart) asked respondents: “Taking everything into account concerning the 2012 presidential election, indicate which statement most closely describes how you believe the outcome was decided?” They were asked to choose among the following:

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Votes were counted accurately nationwide. The man who actually received the most votes was elected president in a fair election.

There was a lot of fraud in counting the votes after the election. Nonetheless, the man who actually received the most votes nationwide was elected president.

There was a lot of fraud in counting the votes after the election. Because of that, the man who actually received the most votes nationwide was denied the presidency.

Among those expressing an opinion, 31 percent of Romney voters chose the last response, endorsing the idea that Obama was not the legitimate winner. Among those with “very positive” views of the tea party, 51 percent held that perspective. But 22 percent of Romney voters with a less positive view of the tea party also questioned the legitimacy of the election, suggesting that this belief was not restricted to a fringe of the party.

Dark views about the fairness of the electoral process have persisted. In May, the AP-NORC Center for Public Affairs Research found that only 36 percent of respondents said that they had a great deal of confidence that their vote would be counted accurately. While we don’t know the partisan breakdown, past patterns suggest that confidence among Republicans is lower, which would be less than we have seen in more than a decade’s worth of survey data.

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These data raise the concern that Trump’s supporters may be especially receptive to allegations of fraud. Republican voters, especially those most inclined to be anti-establishment, may be primed to believe that the American political and economic system is rigged against them.

And Trump’s rhetoric has also been unusually stark, comparing U.S. elections to those in Russia and Zimbabwe. Candidates and parties in the past decade have been content to warn their supporters to be vigilant against possible shady practices, but in a system that is otherwise fair. The GOP nominee’s allegations have been described as “unprecedented.”

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Can such unfounded charges be effectively challenged? One might hope that critical scrutiny by the media of Trump’s claims will tamp down anxiety among the public. But voter fraud appears to be clickbait for the media, and in any case academic research suggests that efforts to debunk rumors can sometimes worsen their effects.

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The best hope is for a combination of voices, both partisan and nonpartisan, to remind Americans of the mechanisms in place to ensure that votes are counted fairly. This, along with redoubled efforts by state and local election officials to make the coming election as transparent as possible, is probably the best means to contain the damage that has been done.

Paul Gronke is professor political science at Reed College and director of the Early Voting Information Center.

Michael W. Sances is assistant professor of political science at the University of Memphis.