In this book we have presented an extended version of game theory and its possible application to international relations. Game theory presents games as given; mechanism design theory presents the social planer as one who attempts to acquire complete information and then to decide in its light what game to play; by contradistinction, extended game theory rests on the supposition that a player makes the most important strategic decision when deciding what game to play. This includes, particularly, the decision what game not to play and, more particularly, what group of game to play (for example, a kind of trade), and with whom (for example, in what market). Obviously, it is much easier to choose what game not to play than what game to play. For, the readiness to play may depend on a few independent conditions, whereas for the readiness to avoid playing that game suffice it to cancel one of them.We recommend a methodology that renders game theory part-and-parcel of social science. Also, we distinguish pure game theory which is a branch of mathematics from applied game theory which is a part of the social sciences. Pure game theory is certain and does not refer to reality, whereas applied game theory refers to reality and is uncertain. Particularly uncertain is the answer the question what game is being played in a specific real situation. It is advisable to make it clear whether a given discussion belongs to pure or applied game theory. It is a big and harmful mistake to confuse them. The claim that a theory about society has the status of mathematics makes it closed for empirical testimonies, and then it is pseudo-scientific; it is particularly pernicious when used as a platform for recommendations that are against the public interest, including recommendations to make war.The interest of this book is in peace. We resist the aggressive recommendations that leading game theoreticians make, and even repeatedly. Some hawkish recommendation rest on arbitrary or distorted descriptions of real-life situations, and on the tacit assumption that a description is true and that the game in question is unchangeable. Our interest is to diffuse threats of war. To that end we recommend the following strategies. First, analyzing a real-life situation, we should ask, what game is being played. Being a question of fact, this question is not mathematical but social, and we should see to it that it be social-scientific, namely, testable: the answer to it should always be open for empirical examination. Game theory as social science, we claim, can help see how exactly a cooperative policy is much better in international relations, and why a policy is much better peaceful than aggressive, and how democracies are able to enjoy peace much more than dictatorships, and why freedom of information and transparency and public debates contribute to national security.