Part A collects basic results that will be used in the book.

In view of the large variety of games that are introduced and studied, it is necessary to present a general setup that will cover all cases (in the normal and extensive forms).

Chapter I deals with normal form games.

The first three sections (I.1, I.2, I.3) offer a comprehensive treatment of the minmax theorem. We start with an analysis of the case of pure strategies, basically Sion's theorem (Theorem I.1.1 in this volume) and some variants. We further treat the case of mixed strategies (Proposition I.1.9). The basic tool is the separation theorem, which is briefly studied. Then we present extensions corresponding to topological regularization (continuity, compactness), measurability requirements leading to the general “mixed form” (Theorem I.2.4), and purification of mixed strategies (Proposition I.2.7). Next we study the case of ordered fields (Theorem I.3.6), and the elementary finite approach is presented in I.3 Ex.

The next section (I.4) is devoted to Nash equilibria (Theorem I.4.1), and several properties (manifold of equilibria, being semi-algebraic, fictitious play, etc.) are studied in I.4 Ex.

Chapter II defines extensive form games and treats successively the following topics:

Section II.1: The description of the extensive form, including the definition of pure, mixed, and behavioral strategies, linear games, and perfect recall (see also II.1 Ex.); Dalkey, Isbell, and Zermelo's theorems; and the measurable version of Kuhn's theorem (Theorem II.1.6).

Section II.2: The case of infinite games, first with perfect information, including Gale and Stewart's analysis and Martin's theorem (II.2.3) and then Blackwell's games (imperfect information) (Proposition II.2.8).

Section II.3: The notion of correlated equilibria, its properties (Aumann's theorem [Theorem II.3.2]), and several extensions: first, extensive form correlated equilibria, then communication equilibria (general formulation and properties; specific representation for finite games).

Section II.4: Games with vector payoffs and Blackwell's theorem (Theorem II.4.1).