2 - Games and Coalitions

Abstract

The majority of investigations on coalitions deal with the mathematical tool of game theory. This chapter presents the basic results of this theory, focusing on the major differences between the two-person games and the n-person games for which n ≥ 3. It is the games of chance that prompted the first meditations upon the theory and calculus of probabilities. Blaise Pascal set about his computations of the probabilities of events starting from the letter of a dicer. The theory of two-person competitive games can be treated satisfactorily from a mathematical point of view. In a two-person game, two strategies are in equilibrium if neither player is to gain by a unilateral change of strategy. The outcome corresponding to this pair is termed the equilibrium point. There can be several equilibrium points, but if there are, they will have the same outcome. This is true for the zero-sum games. One can also speak of equilibrium strategies or equilibrium points in the case of n-person games or nonzero-sum games, but they will not always lead to the same outcome, the choosing of the most favorable situation being difficult.