Algebras of Interaction and Cooperation

Abstract

Systems of cooperation and interaction are usually studied in the context of real or complex vector spaces. Additional insight, however, is gained when such systems are represented in vector spaces with multiplicative structures, i.e., in algebras. Algebras, on the other hand, are conveniently viewed as polynomial algebras. In particular, basic interpretations of natural numbers yield natural polynomial algebras and offer a new unifying view on cooperation and interaction. For example, the concept of Galois transforms and zero-dividends of cooperative games is introduced as a nonlinear analogue of the classical Harsanyi dividends. Moreover, the polynomial model unifies various versions of Fourier transforms. Tensor products of polynomial spaces establish a unifying model with quantum theory and allow to study classical cooperative games as interaction activities in a quantum-theoretic context.