Noncooperative Games
Nicolai N. Vorob’ev · Birkhäuser Basel eBooks · 1994
Definition. A noncooperative game is a triple 1.1 $$ \Gamma = \left\langle {I,\{ x_i \} _i \in _{I,} \{ H_i \} _i \in _I } \right\rangle , $$ where I is a finite set whose elements are called players (from now on, unless the contrary is stated, we take I = {1,2,... ,n}); x i (i ∈ I) are arbitrary pairwise disjoint sets, whose elements are called the strategies of the corresponding players i ∈ I. The ordered sets {x i }i∈I of strategies of the players, i.e., the elements of the Cartesian product 1.2 $$ x = \mathop \Pi \limits_{i \in I} x_i $$ are called situations (or, synonymously, n-tuples) in the game Γ; the bounded functions 1.3 $$ H_i :x \to {\text{R}} $$ corresponding to the players i ∈ I are called the payoff functions of the players, and their values on the situations are the payoffs in these situations.□