if big media is liberal, why are they all run by conservatives?

From the LA Weekly's Column: Deadline Hollywood" by Nikki Finke

If big media look like they’re propping up W’s presidency, they are. Because doing so is good for corporate coffers — in the form of government contracts, billion-dollar tax breaks, regulatory relaxations and security favors. At least that wily old codger Sumner Redstone, head of Viacom, parent company of CBS, has admitted what everyone already knows is true: that, while he personally may be a Democrat, “It happens that I vote for Viacom. Viacom is my life, and I do believe that a Republican administration is better for media companies than a Democratic one.”

When it comes to NBC’s parent company, GE’s No. 1 and No. 2, Jeffrey Immelt and Bob Wright, are avowed Republicans, as are Time Warner’s Dick Parsons (CNN) and News Corp.’s Rupert Murdoch (Fox News Channel).
...
As for Immelt, he publicly wishes his MSNBC could be a clone of FNC. Not surprising, since he let his network and cable news cheerlead the run-up to the Iraqi war without ever bothering to tell viewers GE had billions in contracts pending. More than half of Iraq’s power grid is GE technology. It was also under Immelt that GE installed a former adviser to W and Condi...as NBC Universal’s executive vice president of communications.

...in October 2004, the Republican-controlled House and Senate and White House okayed a $137 billion corporate-tax bill — dubbed “No Lobbyist Left Behind” — that gave a huge $8 billion tax break to GE, which had bankrolled a record $17 million lobbying effort for it. (Meanwhile, in that same bill, House Republicans at the last minute stripped the movie studios of about $1 billion worth of tax credits because of Hollywood’s near-constant support of the Democratic Party and its candidates.)

Disney, parent company of ABC, has turned most of its extensive radio network and owned-and-operated stations into a 24/7 orgy of right-wing talk... And let’s not forget that Michael Eisner pulled the distribution plug on Fahrenheit 9/11.