The models of game theory are highly abstract representations of classes of real-life situations. Their abstractness allows them to be used to study a wide range of phenomena. For example, the theory of Nash equilibrium (Chapter 2) has been used to study oligopolistic and political competition. The theory of mixed strategy equilibrium (Chapter 3) has been used to explain the distributions of tongue length in bees and tube length in flowers. The theory of repeated games (Chapter 8) has been used to illuminate social phenomena like threats and promises. The theory of the core (Chapter 13) reveals a sense in which the outcome of trading under a price system is stable in an economy that contains many agents.
The boundary between pure and applied game theory is vague; some developments in the pure theory were motivated by issues that arose in applications. Nevertheless we believe that such a line can be drawn. Though we hope that this book appeals to those who are interested in applications, we stay almost entirely in the territory of "pure" theory. The art of applying an abstract model to a real-life situation should be the subject of another tome.
Game theory uses mathematics to express its ideas formally. However, the game theoretical ideas that we discuss are not inherently mathematical; in principle a book could be written that had essentially the same content as this one and was devoid of mathematics. A mathematical formulation makes it easy to define concepts precisely, to verify the consistency of ideas, and to explore the implications of assumptions. Consequently our style is formal: we state definitions and results precisely, interspersing them with motivations and interpretations of the concepts.
The use of mathematical models creates independent mathematical interest. In this book, however, we treat game theory not as a branch of mathematics but as a social science whose aim is to understand the behavior of interacting decision-makers; we do not elaborate on points of mathematical interest. From our point of view the mathematical results are interesting only if they are confirmed by intuition.
We study four groups of game theoretic models, indicated by the titles of the four parts of the book: strategic games (Part I), extensive games with and without perfect information (Parts II and III), and coalitional games (Part IV). We now explain some of the dimensions on which this division is based.
The numbers of pages that we devote to each of these branches of the theory reflect the fact that in recent years most research has been devoted to noncooperative games; it does not express our evaluation of the relative importance of the two branches. In particular, we do not share the view of some authors that noncooperative models are more "basic" than cooperative ones; in our opinion, neither group of models is more "basic" than the other.
To illustrate the difference between the theories, consider an environment in which the level of some activity (like fishing) of each agent depends on the level of pollution, which in turn depends on the levels of the agents' activities. In a competitive analysis of this situation we look for a level of pollution consistent with the actions that the agents take when each of them regards this level as given. By contrast, in a game theoretic analysis of the situation we require that each agent's action be optimal given the agent's expectation of the pollution created by the combination of his action and all the other agents' actions.