House of Cards has had a long running relationship with video games. Monument Valley, The Stanley Parable and Agar.io have featured in later series. However, his first appearance playing video games was with Killzone 3 in season one.
However, according to voice actor Troy Baker, that wasn't always the case. Speaking to me at the BAFTA Game Awards tonight said that his seminal title The Last of Us was meant to be that game. When talking about working with screen actors, he talked about his work with Kevin Spacey in Call of Duty: Advanced Warfare, before telling me:
He was meant to be playing The Last of Us in season one, when he was meant to be playing a game. [That game was Killzone 3]
This would make sense since Sony had a huge push during that season, with mention of Vita in the season too. For some reason that looked to have fallen through and instead the honor. Perhaps it was because the game wasn't released before the season.