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4 CHAPTER 1 A Very Short Tour of Game Theory birth of the theory of games. Since then, game theory has resounding success in many areas, such as biology, economics, and information sciences. It has been enriched by numerous brilliant game-theorists, such as Allais, Aumann, Debreu, Harsanyi, Maskin, Maynard-Smith, Myerson, Selten, to name just a few. The purpose of the theory of games is to model interactions between players, to define different types of possible outcome for the interactive situation under study (solution concepts), to pre- dict the solution(s) of a game under given information and behavior assumptions, to characterize all possible outcomes of the game, and to design good strategies to reach some of these outcomes. Before defining the main representations of a game (Sec. 1.3), which corresponds to defining a game mathematically, we now provide four examples, where the ideas of game theory can be used. 1.2 A BETTER UNDERSTANDING OF THE NEED FOR GAME THEORY FROM FOUR SIMPLE EXAMPLES Before defining the fundamental concepts presented in this chapter, it is useful to con- sider common situations where game theory is a natural paradigm. We have picked four examples. The first is not related to communication problems, but the other three correspond to models used in the area of wireless networks. All these situa- tions need a tool to predict or understand the outcome of the interactive situation under consideration. Example 1: A Bankruptcy Problem The problem is as follows. A man dies, leaving debts totaling more than the value of his estate. Assume that there are three creditors and the debts to them are 100, 200, and 300. How should the estate be divided among the creditors? About two thousand years ago, the Babylonian Talmud gave the answer (reported in the Table 1.1). While the recommendation in the case where the estate is 100 (the estate is divided equally) seems reasonable, those Table 1.1 A bankruptcy problem and the solution recommended by the Talmud: The estate is divided between three creditors, depending on how much the estate is worth and the claims of the creditors Claim 100 100 Estate 200 300 100 3 200 100 3 300 100 3 50 50 75 100 75 150