Game algebra: algebraic system of strategic-form games

Abstract

In game theory, a 'game' is a system in which two or more decision makers interact with each other. This article proposes and investigates the properties of a new approach, which we call 'game algebra', which represents the strategic form (or normal form) of a game as an algebraic entity. In game theory, the representation of games as mathematical models allows them to be treated as algebraic entities in a way similar to the algebra of relational database theory. In game algebra, we regard the coupling of two games in strategic form as an algebraic operation. We prove that these operations preserve various game solution properties, such as strategic dominance and Nash equilibria. We show that a class of n-person games in strategic form are a commutative (abelian) group and a commutative monoid.