Hi Shikhar,
Great that you developed an interest in Game Theory.
I think you should look into departments like Operations Research and Electrical Engineering. Computer Science is also an application and some departments might have professors working in the area. But I mostly know electrical engineering like MIT, Stanford, Berkeley. And then as I said OR is the main area where people would do such work.
Introductory game theory is usually taught in graduate level algorithm courses--stuff like zero-sum games, nash equilibrium and the kind. But more advanced stuff like stochastic games in more complex settings, multiplayer games and such are usually graduate level stuff in most places.
As for future prospects, it would have bright applications. So dont worry about that yet.
I am not sure about teachers at NSIT doing active work in the area. Ask around.
Best of luck