# Game Theoretic Interaction and Decision: A Quantum Analysis

## TL;DR

This paper introduces a mathematical framework connecting interaction systems, cooperative game theory, and quantum systems, revealing their underlying similarities and providing tools like spectral analysis and Markov evolution.

## Contribution

It develops a unified mathematical approach to analyze interaction, cooperation, and quantum systems, including spectral representation and Markov dynamics, without requiring physics background.

## Key findings

- Spectral representation of interaction states.
- Natural emergence of cooperative game concepts.
- Framework encompasses quantum and classical Markov processes.

## Abstract

An interaction system has a finite set of agents that interact pairwise, depending on the current state of the system. Symmetric decomposition of the matrix of interaction coefficients yields the representation of states by self-adjoint matrices and hence a spectral representation. As a result, cooperation systems, decision systems and quantum systems all become visible as manifestations of special interaction systems. The treatment of the theory is purely mathematical and does not require any special knowledge of physics. It is shown how standard notions in cooperative game theory arise naturally in this context. In particular, Fourier transformation of cooperative games becomes meaningful. Moreover, quantum games fall into this framework. Finally, a theory of Markov evolution of interaction states is presented that generalizes classical homogeneous Markov chains to the present context.