Mathematical Game Theory: A New Approach

TL;DR

This paper introduces a unified mathematical framework for game theory inspired by physics, linking classical, combinatorial, and quantum models to analyze various strategic interactions.

Contribution

It presents a novel approach that models game states as system sequences and connects game theory with quantum mechanics through Hilbert space representations.

Findings

Unified framework for classical, combinatorial, and quantum game models

Application of Hilbert space to represent game systems

Insights into equilibria and cooperative games

Abstract

These lecture notes attempt a mathematical treatment of game theory akin to mathematical physics. A game instance is defined as a sequence of states of an underlying system. This viewpoint unifies classical mathematical models for 2-person and, in particular, combinatorial and zero-sum games as well as models for investing and betting. n-person games are studied with emphasis on notions of utilities, potentials and equilibria, which allows to subsume cooperative games as special cases. The represenation of a game theoretic system in a Hilbert space furthermore establishes a link to the mathematical model of quantum mechancis and general interaction systems. The notes sketch an outline of the theory. Details are available as a textbook elsewhere.