The 81-year-old mogul has been unable to decide which child will succeed him and is setting the stage for an epic family battle after his death.

His daughter Elisabeth is a case in point. In a speech before the Edinburgh Television Festival, Murdoch's second daughter made the remarkable statement that "profit without purpose is a recipe for disaster." This is clearly a jab at her father and her younger brother James, the current favorite child who has been a focal point of the U.K. phone-hacking scandal. 

Elisabeth, who left the family business a decade ago, added that she has "absolutely no ambition" to be News Corp.'s next CEO. She has decided not to take a position on the board as the scandal around the Murdochs continues to intensify in the U.K.

Some analysts had speculated that Elisabeth would emerge as the new favorite after she sold her television production company Shine to News Corp for 415 million pounds  ($661 million) last year. She seemed to have the least baggage of any of the siblings. Her brother Lachlan was once the heir apparent, achieving the position of deputy chief operating officer at News Corp., but he resigned seven years ago, saying he "was fed up with his father's meddling and his patronising lieutenants," according to Australia's The Monthly. By all accounts, Lachlan seems happy to remain in Australia. 

James, of course, is damaged goods. The testimony he has given to Parliament has been contradicted by others. Elisabeth urged him to step away from the spotlight. Police in the U.K. are reportedly reviewing the cell records of a phone belonging to James that wasn't previously disclosed to them in early inquiries. 

The eldest daughter Prudence, who avoids the limelight, is a contender as well, since her husband Alasdair MacLeod is a top News Corp. executive.

For now,  News Corp.'s well-liked President Chase Carey reportedly enjoys the trust of the Murdoch family. One investor remarked that "in a perfect world" he would be the next CEO and Rupert would become chairman. Since that hasn't happened yet, however, it seems unlikely that it will while the elder Murdoch is alive.

Even if the family decides to back Carey after Murdoch's death, there is another huge problem waiting in the wings. Murdoch has two young daughters with his third wife Wendy Deng, Grace Helen (born 2001) and Chloe (born 2003). Once they come of age, who knows what demands they will make on the family business?

Jonathan Berr does not own shares of the listed companies. Follow him on Twitter@jdberr.