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The company at the heart of the Facebook scandal boasted of having “vast amounts of data” that could sway the 2016 referendum on membership of the European Union.

A 10-page document written by Cambridge Analytica, headed “Big Data Solutions for the EU Referendum”, claimed it could single out Brexiteers among voters, donors, politicians and even journalists.

MPs today called for a wider investigation into the firm, which has been accused of obtaining 50 million people’s private details harvested from Facebook, amid questions over the role it may have played in the referendum.

The vote was won by Leave by a four-point margin. Cambridge Analytica has denied wrongdoing.

The document was made for Leave.EU, one of the two main Brexit campaign groups, in a pitch for business. It bragged: “We use vast amounts of data, including consumer histories, lifestyle information... and “state-of-the-art psychological analysis.”

Cambridge Analytica said it could pick out likely Leave supporters by identifying their “top-line issues” and “voters’ personality traits” using its own data. It said Leave.EU would have to pass over its own Facebook data to get the full benefit.

Explaining the power of its analysis, Cambridge Analytica (CA) said: “Voters and businesses alike see the coming referendum as an opportunity to voice their concerns over issues caused by Britain’s membership of the EU.

"Whether it is regulation, border controls or Britain’s international profile, British people have real worries.”

Dominic Grieve, the Tory ex-attorney general, said there was a case for a wide-ranging investigation into CA’s activities. “The more one hears about this case, the more public disquiet there must be about these allegations,” he told the Standard. “They need to be seriously investigated.”

Three official investigations involving Cambridge Analytica (CA) are under way. The Information Commissioner, Elizabeth Denham, is seeking a warrant to see if its computers contain data that was allegedly obtained without proper consent.

This follows the 2014 Facebook incident, when people were paid to take a personality test which unlocked their own details and those of their friends.

In addition, the Electoral Commission is probing whether Leave.EU correctly declared that it received no services, paid or unpaid, from CA.

Meanwhile, the House of Commons culture committee, chaired by Damian Collins, has recalled CA’s suspended chief executive, Alexander Nix, for an inquiry into fake news. CA’s 2015 document, written a year after the Facebook data harvesting, was placed on the committee’s page of the Parliament website as part of its evidence-gathering.

Contradictory statements appear to have been made by CA and senior figures in Leave.EU about whether or not the pitch was accepted. Mr Nix was quoted in 2016 in Campaign magazine saying: “We have already helped supercharge Leave.EU’s social media campaign.”

However, in evidence to the select committee, he assured Mr Collins: “We did not work with them.” Mr Nix, an Old Etonian, was suspended by CA on Wednesday after he was secretly recorded by Channel 4 allegedly discussing potential bribery and entrapment. He denies any wrongdoing.

The founder of Leave.EU, Arron Banks, referred in his book The Bad Boys Of Brexit to CA being “hired” in October 2015. But he told the committee this simply referred to an early meeting and an intention to work together if Leave.EU won lead status, entitling it to spend up to £7 million, get a free mailshot, TV broadcasts and £600,000 public funds, in the referendum campaign. He insisted the group “devised and implemented its own social media strategy ... without any input from Cambridge Analytica”.

Confusingly, a CA staffer, Brittany Kaiser, appeared on the platform of a Leave.EU press conference in November 2015 alongside Mr Banks, seemingly to present their campaign plans. She has since left the data company.

Whatever the role played in the referendum, some MPs believe the introduction of micro-targeting, in which millions of voters may be sent individual private messages, raises serious questions for a healthy democracy.

For example, some believe that voters identified as being worried about “borders” were sent the notorious false claims that Turkey would join the EU in 2020, flooding the UK with millions of migrants.

The Tory former business minister Anna Soubry said micro-targeting meant voters could be bombarded with claims that sidestepped normal scrutiny, challenges and fact-checking by the Press. “This really is dabbling in the dark arts,” she said. “If this takes place ‘under the radar’ then there is no way for people to complain or question what they are seeing in the privacy of reading their phone in their living room.”

One Conservative believes a form of micro-targeting triggered death threats sent to her and fellow MPs. Vicky Ford told the Commons in a debate about CA this week: “I have been the victim of false news stories being micro-targeted at Facebook accounts in my constituency to deliberately undermine me and cause hate.”

The Chelmsford MP was one of 15 Conservatives branded “mutineers” by sections of the media last year for considering rebelling against Theresa May’s attempt to fix a Brexit date in law. Afterwards, a Facebook advert was sent to Conservatives in her constituency accusing her of “trying to sabotage Brexit” and, above an unflattering photograph, urged the recipients: “We need you to hold this woman to account.” Soon after this Ms Ford says she received an anonymous message warning: “We think you should all be hanged.”

The adverts bore the name Voter Consultancy Ltd, a firm run by Thomas Borwick, the son of the former Tory MP for Kensington, Victoria Borwick. Mr Borwick was chief technology officer at the Vote Leave campaign group in the referendum, responsible for planning a central database system for 43 million voters.

He told the Standard that CA had used a doorstep canvassing app that he developed. He confirmed that at Vote Leave he “worked with” Aggregate IQ, a Canadian data firm that has been linked in the media with CA and which earned £3.5 million working on the referendum.

Asked about the ads, Mr Borwick defended them as “entirely factual, merely highlighting the way that Vicky Ford voted on a particular issue”. He said: “This advertisement was sent to Facebook users who self-declared as Conservatives in her constituency.

“Voter Consultancy spent £0.67p and the advertisement was viewed by 87 people. It is noted that Vicky and the Conservative party spent about £2,000 on advertising in the constituency and perhaps much more in the most recent elections.”

He scorned the idea that the adverts triggered death threats to Ms Ford and others, including Ms Soubry and the former education secretary Nicky Morgan. Mr Borwick said: “I condemn abhorrent threats of this nature against MPs and any insinuation that this advertisement was responsible for these threats is farcical.”