How can we apply the game theory to understand the dynamics of International Politics?
Are there any crucial decisions taken at the international level that were the results of game theory's application to a significant extent?
Expanding on what @Tom said:
Besides of zero-sum games and win-win situations that may lead to some Pareto equilibrium, Nash equilibrium or other types of equilibrium. We can also find lose-lose situations, and the nature of relations may vary depending on the scope as in tragedy of the commons, by introducing in the system some externalities, which may correspond to some winning or losing that was not accounted for.
Back to the question, the application of games theory and decision theory to politics is a fundamental approach (often found combined with operations research) that consists on modeling the relations and decisions made in politics by assigning a value to the expected utility of some outcome or action. These values are assigned as a quantification of how desirable the outcome or the actions are. Then the mathematical apparatus is used to reach some conclusion about the decisions that have to be made by an agent (decision theory) or a set/system of agents (games theory) to maximize the expected utility, which means obtaining the most desirable outcome for the agent or agents.
The specific way in which relations and decisions are modeled depends on the specific political problem that is faced. This can be considered analogous (to some extent) to modeling moving objects, their masses, etc. in physics and then applying maths. In this case we model agents, their preferences, their possibilities, etc. and then apply maths, like the maths referred in the wikipedia links I've included in this answer.
WRT to the application in the real world in a explicit way. Unfortunately this is not common, voters often lack the education to understand this kind of matters and it is much easier for politics to simply ignore these aspects and focus on what is more profitable for them, in terms of votes (reputation) or money.
This is applied in an implicit way, though, in every decision. Politics ignore games theory because they find a greater expected utility in the decisions that do not need to be explained in terms of games theory. Voters, on the other hand, don't usually appreciate any utility in this kind of explanations, they require some time investment to be understood, and that time has a greater utility for many people than this understanding and validation of political decisions.
There are only a few aspects in which the application of games theory is somewhat direct, as when refusing to negotiate with terrorists. In this case the outcome would have a lower utility when negotiating than when not negotiating in the long term. Not to extend this answer more, I'll leave to an external source how to model terrorism from a game theoretical perspective. I haven't read it by myself, but Google ranks it high, so I guess it will state right things in clear ways, feel free to comment on that PDF if there is something confusing in it.
Game theory relates to the study of decision making processes. http://en.wikipedia.org/wiki/Game_theory
Specifically, it relates to the mathematical modelling of decision "branches" as if they were a "game," or possibly a computer program. This is used in the study of economics and political science.
There are two basic types of "games" representing two types of political situations. The first is called a zero sum game, in which my gain is your loss, and vice versa. These are the hardest political situations to navigate because someone has to lose.
The second type of situation is called a positive sum game. This is often referred to as a "win-win" situation. That is, if I do something for you, and you reciprocate, we can both come out ahead. These are relatively easy political situations to manage, unless one of the parties feels that the relationship is too one-sided, and refuses to "play" because he is only a little better off, and the other party is much better off.
So game theory models whether two or more political agents will compete or co-operate in a given situation, and if so how. One important political application of game theory is voting systems. Another is bargaining postures. A third is what motivates parties to form alliances, e.g. why was there a "Triple Alliance" vs. a Triple Entente before World War I? Or how and why did India informally "align" with the Soviet Union against Pakistan, China, and the U.S. https://history.stackexchange.com/questions/8736/how-did-india-join-the-soviet-bloc-during-the-cold-war/8779#8779