On Quaternions, Octonions, and the Quantization of Games

The study of quantum games is fairly new, arising from a seminal paper of David A. Meyer. In the following decade, quantum game theory has gained a considerable interest. This book presents a mathematical formalism for quantum games generalizing the classical notion of mixing strategies. It begins with an exposition of the fundamentals of quantum information theory and proceeds to introduce games played under “quantum mediated communication”, that is, the “quantum” internet. The particular focus is on two and three player games where each player has precisely two pure strategic choices. The main result of this work is the co-ordinatization of the “quantum” payoff functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then exploited to analyze and potentially classify the Nash equilibria in the quantized games. This book is intended for students and scholars who are interested in quantum information and game theory. Although the book is sufficiently self-contained, it is assumed that the reader has already had some exposure to linear algebra and quantum information theory.