Algebras of Interaction and Cooperation

TL;DR

This paper explores the representation of systems of cooperation and interaction within algebraic structures, offering new insights through polynomial algebras, Galois transforms, and connections to quantum theory.

Contribution

It introduces a novel algebraic framework for cooperation and interaction, unifying polynomial models, transforms, and quantum perspectives in the study of cooperative systems.

Findings

Polynomial algebras provide a unifying view of cooperation.

Galois transforms generalize classical dividends in cooperative games.

Tensor products link classical cooperation with quantum theory.

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.