An Introduction to Complex Game Theory

TL;DR

This paper extends two-player zero-sum game theory into complex space, providing a complete framework, new proofs, and methods for calculating equilibria using complex strategies and linear systems.

Contribution

It introduces a comprehensive complex space theory for zero-sum games, including new proofs and solution methods for equilibria calculation.

Findings

A complete compact theory for complex zero-sum games.

A new constructive proof of the Minimax Theorem in complex space.

A simplified solution method using complex linear systems.

Abstract

The known results regarding two-player zero-sum games are naturally generalized in complex space and are presented through a complete compact theory. The payoff function is defined by the real part of the payoff function in the real case, and pure complex strategies are defined by the extreme points of the convex polytope $S_\alpha^m:=\{z\in\mathbb{C}^m:$ $|argz|\leqq\alpha,\;\sum_{i=1}^{m}z_i=1\}$ for "strategy argument" $\alpha$ in $(0,\frac{\pi}{2})e$. These strategies allow definitions and results regarding Nash equilibria, security levels of players and their relations to be extended in $\mathbb{C}^{m}$. A new constructive proof of the Minimax Theorem in complex space is given, which indicates a method for precisely calculating the equilibria of two-player zero-sum complex games. A simpler solution method of such games, based on the solutions of complex linear systems of the form $Bz=b$, is also obtained.