Later, however, Frank starts withholding information from Zoe unless she has sex with him, which makes absolutely no sense. Frank is benefiting from the flow of information just as much as Zoe is -- jeopardizing all of that just to quench your thirsty boner is less the strategy of the most cold, cunning man in Washington, and more the desperate gambit of a man whose first priority is to have sex all the time.
In fact, the more you think about it, the more it looks like Frank is simply following the storied tradition of ruining his political career for the sake of booty. When he talks about his arrangement with Zoe with his shockingly understanding wife, he acts like sleeping with her is part of some Machiavellian scheme, but it absolutely isn't -- he's just having sex with a reporter who was low-level enough that trading skin-flapping nudity collisions with him in exchange for a steady source of career-making stories was a good deal for her. Ultimately, all Frank has done for himself is turn Zoe into an even bigger liability that he'll have to deal with later. He's putting his career in the hands of a kid working for what appears to be BuzzFeed's politics section, judging from her office.
At least they got to enjoy some fantastic paintings together.