How to start Game theory?
"A course in game theory" by Martin J. Osborne and Ariel Rubinstein is probably the standard more mathematical starting point. A more concise, more modern, and slightly CS-leaning text is "Essentials of Game Theory -- A Concise, Multidisciplinary Introduction" by Kevin Leyton-Brown and Yoav Shoham.
I asked this same question about a year ago, so I'm very slightly ahead of you. Here's what I know:
As you probably know, there are two major branches for game theory. There's (for lack of a better term) "economics" game theory dealing with real world situations, economics, politics and the like. I know next to nothing about that. However, I do know a decent amount about combinatorial game theory, which is a little bit more ground in mathematics, and deals with two player games such as Go, Chess, Nim, or Tic-Tac-Toe.
The best introductory text is going to be Conway's Winning Ways, any of the volumes 1-4. These are the books to read to get into any other subset of combinatorial games, in my opinion.
My personal specialization thus far is generalizations of Tic-Tac-Toe called achievement games, which you can read about (along with much more) in Tic-Tac-Toe Theory.
However, if you want to go even further in these studies, you are a little bit out of luck. What's very exciting to me about combinatorial game theory is that it's pretty much a brand new field of mathematics, and right now the best techniques we have to study it are educated guessing/brute-forcing and a little bit of discrete mathematics. Although it's disenchanting sometimes, this also means that there is potentially a world of possible links and connections to other branches of math that we don't know about, and is just out there waiting to be discovered.
There is a new (well, the English translation is) book that treats both noncooperative and cooperative (but not combinatorial) game theory on a high level, is extremely well written, mathematically rigorous and fairly comprehensive: Game Theory by Michael Maschler, Eilon Solan, and Shmuel Zamir. For someone who knows some undergraduate real analysis and linear algebra, the book should be self contained (with a few exceptions, where reference literature is recommended in the book). The book doesn't contain everything (there is very little on refinements), but it contains enough to get one near the frontier of research fast.
