Standing before political and business leaders in New York last fall, Alexander Nix promised a revolution. Many companies compete in the market for political microtargeting, using huge data sets and sophisticated software to identify and persuade voters. But Mr. Nix’s little-known firm, Cambridge Analytica, claimed to have developed something unique: “psychographic” profiles that could predict the personality and hidden political leanings of every American adult. “Of the two candidates left in the election, one of them is using these technologies,” Mr. Nix said, referring to Donald J. Trump. Capitalizing on its work for the man who is now president, Cambridge has pitched potential clients in the United States ranging from MasterCard and the New York Yankees to the Joint Chiefs of Staff. Ahead of this year’s elections in Europe, Mr. Nix is promoting the four-year-old United States-based company abroad, too. Cambridge Analytica’s rise has rattled some of President Trump’s critics and privacy advocates, who warn of a blizzard of high-tech, Facebook-optimized propaganda aimed at the American public, controlled by the people behind the alt-right hub Breitbart News. Cambridge is principally owned by the billionaire Robert Mercer, a Trump backer and investor in Breitbart. Stephen K. Bannon, the former Breitbart chairman who is Mr. Trump’s senior White House counselor, served until last summer as vice president of Cambridge’s board.