8 CHAPTER 1 A Very Short Tour of Game Theory interactive situation. Solution concepts are proposed, and natural issues like exis- tence and uniqueness of the solution(s) are considered. For a given wireless scenario, it may be that no suitable solution concept is available in the current literature of game theory. This does not necessarily mean that game theory is not the right approach; it is more likely that the theory needs to be enriched. To this end, the authors would like to draw to the attention of the reader who is new to game theory, that it is not only about: 1. 2. 3. 4. 5. rational decision-makers; selfish decision-makers; non-cooperative games; Nash equilibrium as a solution concept; interactions due to the fact that the players' utility depends on the actions of the others. Some elements justifying these assertions are given throughout this chapter, espe- cially in Sec. 1.5. 1.3 REPRESENTATIONS AND CLASSIFICATION OF GAMES Now we have a rough idea of what game theory is about, the second natural question is: What is a game exactly? While everyone has an intuitive idea of what a game is, and knows many examples (e.g., board games, Olympic games, TV games, video games), game theory provides a mathematical definition of games. The definition depends on the representation considered. In this section, three 4 existing represen- tations of a game are described, and several classes of games are mentioned. The representations allow one to see how a game can be generically defined. In order to go further with the definitions (e.g., by defining the strategies precisely), the class of games under consideration has to be specified. 1.3.1 Representations of a Game There are three main types of representation of a game: · · · the normal or strategic form; the extensive form; the coalitional form. Coalitional games will not be addressed in this book. Indeed, unless explicitly stated otherwise, non-cooperative games will always be considered. This means that it will always be possible to identify the utility function of a particular player (or at least his preference order). 4 Here, only the three dominant representations are described. Other representations exist (e.g., the standard form which is used in the theory of equilibrium selection (Harsanyi and Selten, 2003)) but are used much more marginally.