A little info on what Sony has said so far on Gaikai.
Gaikai streaming coming to PS4 and PS3 in 2014 | The Verge
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Firstly, the service won't be available immediately at launch, but rather sometime in 2014. When Sony does release it, first in the US, it's going to give PS4 and PS3 owners access to a limited library of PS3 games, before adding streaming functionality to the Vita sometime down the road. Also, the service is headed to the US first, before rolling out internationally, and Sony expects it to "evolve rapidly over the coming months," although what that means is anyone's guess.
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The Last of Us being a Sony owned IP seems like it would make it more likely to end up on Gaikai. How well it would play though depends a lot on how well your internet connection can handle it, if there's much input lag it could make playing the game very difficult.