International Relations is currently under the culture of positivism, but the opposition to that culture remains significant. Post-positivists and anti-positivists criticizes the positivists belief that - we can study state behavior scientifically, quantitatively, and empirically. What I am not sure about is whether post-positivism and anti-positivism go as far as saying that game theory and formal models are not that important to understand the concepts of International Relations. If possible, please give the names of certain political scientists who purely adhere to qualitative methods.
To be honest, positivism and post-positivism have nothing to do with methods or methodology. They are philisophical-ontological worldviews about the nature of knowledge; they are claims about the possibility or desirability of "truth" and its pursuit. Methodology typically flows from the worldview one has, but this is not set in stone. If you see a construction problem from different angle, for instance, you are likely to use other tools in your toolbox because the old tools simply don't make a lot of sense given the way you see the problem.
Positivism begins with the notion that one has objective access to the external world. This is intuitive: you see or experience something, describe it, then manipulate or extrapolate from it (ie. game theory) to develop useful knowledge about the world. Positivism is an "epistemic" world view, whereby the mind and reality are insulated from one another, and science is practised to bridge the gap between these two realms.
Post-positivism problematizes the part pertaining to description. Much of this insight comes from the study of language, but the big point is that "there is no view from nowhere", meaning one is always trapped within a specific practical engagement with the world. There is no distinction between mind and reality; they are wrapped up in one another. This makes truth claims very problematic. In order to describe it, you do so relative to a scholarly practice. Robert Cox's famous line is theory is always for someone, for some purpose. This does not mean scholars have ulterior motives, but that it is literally impossible to describe something in isolation from some system of thinking. In this context a truth claim simply doesn't make sense. Some people interpret this to mean positivism is wrong because it can't get at the truth, while post-positivism can. Instead you should think of it in the sense that truth is an inappropriate ruler to measure statements about the world.
The former is far easier to understand than the latter. I've struggled with it for years.
You should read "The Conduct of Inquiry in International Relations" by Patrick Thaddeus Jackson. This goes over the world views issue (mind-world dualism versus mind-world monism). At least read the introduction, and then the corresponding chapter(s) that you become interested in after reading the intro. Constructivism is a good place to start if you want to explore non-quantitative study of IR. (Kratochwil, NOT Wendt... You'll lose your mind if you go for the weird positivist spin on constructivism)
On another note, I would say you could use methods typically associated with positivists as a post-positivist if you use Max Weber's notion of an ideal type. Here, descriptions of the world are always specified as fictions; they are not meant to be truthful, but useful. It recognizes the value-laden position of the researcher, and the power of a given ideal type is derived from a specification of its limitations. This way, your ideal-typical constructions can be judged on their pragmatic utility for dealing with a problem in the world, rather than whether or not they are "true" or "false".
Also, this book goes over a variety of methodologies: Porta and Keating - Approaches and Methodologies in the Social Sciences - A Pluralist Perspective
CBo's answer is very good, and should help you clear some doubts. If you are looking for an International Relations scholar who transcends the positivist-constructivist debate, I suggest you look at Robert Cox.
My area of research is Geopolitics and I face the same questions, in terms of Classical vs. Critical Geopolitics. We are both in allied disciplines. I also suggest the Critical Geopolitics Reader, ed. Gerard Toal, others. And, a research paper by Phil Kelly titled a Critique of Critical Geopolitics. You could also look at the textbook by Georg Sorensen and Robert Jackson, Introduction to IR. If you are interested, you can contact me and I can guide you towards non-Western IR, which is essential in the changing world.
And, yes, Game Theory is non-essential for IR.