Where do you get the idea that Antiva doesn't have the Game? The fact that it's home to the Crows tells a different story. The Crows are basically the same as Bards.
I get the feeling that the "game" played in Antiva isn't the same as the Grand Game of Orlais itself.
The Crows are career assassins. Everyone is pretty blunt on that point in Antiva, considering they're a political force in their own right. The Crows are paid, the target is killed, the assassin cleans the blood off his boots, and we all move on with our day. Killing people on contract is what the Crows DO.
In Orlais, bards are hired as well, but they're not killers per say. They are sent to spy, to influence, to entertain. Even if every noble in Orlais is aware of what bards do, how dangerous they can be, they're working under the assumption that more is going on than an attempt on their life. The fun in the Game is uncertainty; having to keep your wits about you, read every subtle cue, figure out whether the bard means to to kill you in your sleep or just glean information. You have to outwit the person trained to outwit YOU. Also, it's rarely a bard's main objective to kill a specific target, their jobs are a little more subterfuge-oriented.
Anyway, my point is that while the two read similarly, The Grand Game is the far more complicated, developed, and intricate of the two(and SILLY. Oh so silly.). It has rules, and etiquette, and is played like a game. Antivan Crows, on the other hand, are not toys of the nobility in the way that bards are.
So our darling Josephine may have an understanding of the Game from observation and experience, but if she ever did play a game in Antiva, it was on a much lower level than anything the Grand Game can dish out.
Political maneuvering and intrigue is hardly exclusive to Orlais. In Origins, Arle Howe does plenty of that. Antiva has its own system too, no doubt. The difference is that Orlais openly admits it.
It's true that political machinations of this sort happen everywhere in Thedas, from Orzammar to Tevinter, and in that regard Orlais is hardly unique. The Grand Game, however, is a system of political intrigue that is unique to Orlais, developed over ages, openly practiced, with an unspoken set of rules and etiquette to be followed. Other countries may play at politics if they wish, but Orlais has made a bona fide GAME of it. Nowhere else is their particular brand of insanity practiced.
Think "The Court of Louis XIV" on steroids. Er, with maybe a tad more murder involved. And less of that whole "Lady-In-Waiting-Assigned-To-Remove-The-Queen's-Undergarments" thing.