Mr. Trump says the 2016 national election is rigged – that the mainstream media won’t give him the time of day, that voting machines have been tampered with to favor Mrs. Clinton, and that Democrats are somehow registering millions of phantom voters, including undocumented immigrants and corpses. Notably, he has offered little evidence to support these allegations.

Unnoticed by most observers, and apparently even by Trump himself, is a new style of tampering available to high tech companies – Google and Facebook, in particular. By shaping the information we see on our computers and phones, they can shift large numbers of votes without anyone realizing what happened.

Compared to the election shenanigans of prior generations, new tech-enabled interventions are arguably more worrisome. For one, these tactics are not competitive; they require powerful allies which, at present, only the Democrats seem to have. Plus, they operate almost invisibly, evading current monitoring systems and leaving little trace for authorities to analyze. And some are so new that they are beyond the reach of existing laws and regulations.

The numbers are, as Mr. Trump would say, yuge. Suppose Facebook chooses to send day-of-election go-out-and-vote reminders only to supporters of one candidate. Extrapolating from a study published in 2012, we estimate that differential reminders would generate about 600,000 more votes for the favored candidate – half a percent of the likely national turnout of 130 million. If the company chooses to exercise this option on November 8th, it is unlikely that anyone outside Facebook would know.

If Google autocomplete suppressed negative search suggestions for one candidate for several months, we estimate that between 800,000 and 3.2 million undecided voters might shift toward that candidate. Manipulating autocomplete is particularly powerful because negative search suggestions draw more clicks than neutral or positive ones.

But Google can do more than adjust autocomplete. Search results can also favor one candidate over the other, boosting web pages that make one candidate look better. Based on published research, we estimate that search ranking manipulation could drive 2.6 to 10.4 million votes toward the favored candidate.

Other high-tech manipulations are also possible. In May, a whistleblower claimed that Facebook personnel were deliberately removing conservative posts from newsfeeds. Could biased newsfeeds tip the preferences of some undecided voters? Online services such as Tinder’s Swipe-the-Vote claim to be able to match people’s issue preferences with the right candidate. But who controls the matching algorithm? If the algorithm were biased toward one candidate, who would notice and when?

These new forms of influence are unprecedented in their effectiveness, scale, and invisibility. They work well mainly because people mistakenly believe that information they obtain from a computer is inherently impartial and objective. The power of these techniques also derives from the fact that behemoths like Google and Facebook have no competitors. If they chose to favor one candidate, it would be difficult if not impossible to counter the impact of their support. With no competitors to provide objective benchmarks, their dominance makes it difficult even to detect a manipulation.

These concerns are particularly urgent because tech titans strongly favor Democrats. Recent reports suggest that Silicon Valley donations favor Clinton over Trump by at least 25 to 1, and that doesn’t take into account a September gift to the Democrats by Dustin Moskovitz, Facebook’s co-founder, of $20 million.

Multiple investigative reports have documented Google’s strong support for both President Obama and Hillary Clinton. The President’s Chief Technology Officer, Megan Smith, is a former Google executive, as is Mrs. Clinton’s Chief Technology Officer, Stephanie Hannon. Since Mr. Obama became president, Google employees have visited the White House more than 420 times, and more than 250 people have swapped high positions between Google and the Obama administration.

In 2015, Eric Schmidt, Chairman of Alphabet (the holding company that owns Google), bankrolled a secretive company called The Groundwork that provides the Clinton campaign all manner of software engineering, including supporting both fundraising and get-out-the-vote efforts. If Schmidt had contributed through a PAC, he would have been prohibited from coordinating with Clinton’s campaign. By structuring his gift as a tech startup, he can make unlimited contributions as an “investment” and still coordinate with the campaign.

Tech titans have the power to influence elections, and apparently the interest. But new research goes further, establishing that they’re actually doing it. A 2015 study found that Google’s search rankings systematically favored Democrats, and a recent investigation has found that Google’s autocomplete was suppressing negative search suggestions related to Mrs. Clinton during summer 2016. A new WikiLeaks release reveals Schmidt’s close collaboration with Clinton’s campaign.

Perhaps we shouldn’t be surprised. These manipulations are legal, so companies with strong preferences have every reason to use them. But we can’t know the full extent of the problem until whistleblowers come forward, warrants are issued, or watchdog agencies establish comprehensive monitoring systems.

In the meantime, imagine if – and it is difficult to imagine otherwise – all these manipulations were being used by different high-tech companies to support the same presidential candidate. These tactics could shift more than 10 percent of the vote with no one the wiser.

For the record, we are both Clinton supporters, but we also believe in democracy, the integrity of which is now in jeopardy because of the power of Big Tech. Some Clintonites might welcome assistance from any source, worrying more about their candidate’s victory than the process that gets there, especially given the Clinton email scandal that never seems to go away. With the stakes so high, we understand this instinct. But what if, next time, Google and Facebook support a candidate who serves their needs more than the needs of society at large?

Worse still, if leader after leader ends up beholden to tech titans, how will government agencies be able to protect us from their misdeeds? In a 2012 antitrust investigation, FTC staff found clear evidence of bias in Google’s search rankings, recommending that the FTC pursue a formal complaint against Google. But as the agency’s commissioners were deciding how to proceed, Google’s chief lobbyist repeatedly visited the White House and discussed these very issues. Ultimately the commissioners overruled the staff recommendation and ended the investigation without a complaint or penalties. Would Google have gotten such special treatment if it were not so cozy with the Obama administration?

The upcoming election might be rigged, but not in the old-school ways Mr. Trump has suggested. The other elephant in the voting booth is not a party machine; it is an algorithm.

Robert Epstein (@DrREpstein) is senior research psychologist at the American Institute for Behavioral Research and Technology and the former editor-in-chief of Psychology Today. Benjamin Edelman (@BGEdelman) is associate professor of business administration at the Harvard Business School.