Extension of 2 Person Zero Sum Game

Abstract

Let X and Y “be the pure strategy sets of 2 players Xavier and Yves. Then every payoff function g: X × Y → R defin es a 2 person zero sum game where Xavier maximize and Yves minimize. An extension of the games with pure strategy set X and Y is a “way of playing” the game, which associates to every payoff function g a “value”, in the duality interval

$$ \left[ {\begin{array}{*{20}{c}} {\sup \,\inf \,g\left( {x,y} \right)} \\ {x \in X\,y \in Y} \\ \end{array}, \,\begin{array}{*{20}{c}} {\inf \,\sup \,g\left( {x,y} \right)} \\ {y \in Y\,x \in X} \\ \end{array} } \right] $$

The classical example of extension is the mixed one. It will be shown it is the prototype of the “extensions with out exchange of information”. (part 2). We then study the general case (part 3) and typical example of Iterated games.