This is an extract from the introductory chapter of A course in game theory by Martin J. Osborne and Ariel Rubinstein (MIT Press, 1994), © Copyright 1994 Massachusetts Institute of Technology.

Game Theory

Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue well-defined exogenous objectives (they are rational) and take into account their knowledge or expectations of other decision-makers' behavior (they reason strategically).

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.

Games and Solutions

A game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players' interests, but does not specify the actions that the players do take. A solution is a systematic description of the outcomes that may emerge in a family of games. Game theory suggests reasonable solutions for classes of games and examines their properties.

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.

Noncooperative and Cooperative Games

In all game theoretic models the basic entity is a player. A player may be interpreted as an individual or as a group of individuals making a decision. Once we define the set of players, we may distinguish between two types of models: those in which the sets of possible actions of individual players are primitives (Parts~ I, II, and III) and those in which the sets of possible joint actions of groups of players are primitives (Part IV). Sometimes models of the first type are referred to as "noncooperative", while those of the second type are referred to as "cooperative" (though these terms do not express well the differences between the models).

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.

Strategic Games and Extensive Games

In Part I we discuss the concept of a strategic game and in Parts II and III the concept of an extensive game. A strategic game is a model of a situation in which each player chooses his plan of action once and for all, and all players' decisions are made simultaneously (that is, when choosing a plan of action each player is not informed of the plan of action chosen by any other player). By contrast, the model of an extensive game specifies the possible orders of events; each player can consider his plan of action not only at the beginning of the game but also whenever he has to make a decision.

Games with Perfect and Imperfect Information

The third distinction that we make is between the models in Parts II and III. In the models in Part II the participants are fully informed about each others' moves, while in the models in Part III they may be imperfectly informed. The former models have firmer foundations. The latter were developed intensively only in the 1980s; we put less emphasis on them not because they are less realistic or important but because they are less mature.

Game Theory and the Theory of Competitive Equilibrium

To clarify further the nature of game theory, we now contrast it with the theory of competitive equilibrium that is used in economics. Game theoretic reasoning takes into account the attempts by each decision-maker to obtain, prior to making his decision, information about the other players' behavior, while competitive reasoning assumes that each agent is interested only in some environmental parameters (such as prices), even though these parameters are determined by the actions of all agents.

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.