Quantum Nash Equilibria and The Re-emergence of Quaternions
Computer Futures, Inc.
Southern New Hampshire University
Last modified: October 23, 2007
In a 2006 paper we reviewed the emerging discipline of quantum game theory, beginning with David Meyer's and Eisert, Wilkins and Lewenstein’s ( Phys. Rev. Lett. 83, 1999), original formulations of the quantum prisoner’s dilemma, and continuing through Landsburg’s generalized form of quantum games. After reviewing various refinements of the theory introduced by Flitney et al, we then examined Huberman and Hogg’s realizable quantum Nash equilibrium schema. Finally, we reviewed elements of Cheon and Tsutsui’s mapping of quantum Nash equilibria in an extra-dimensional projective plane (“Classical and Quantum Contents of Solvable Game Theory on Hilbert Space”, Physics Letters, A346).
The classical Nash equilibrium has already been shown to display a great deal of hitherto unexpected complex, emergent structure. At ICCS 6, in 2006 we were fortunate enough to find out in conversation with John Nash how he became interested in Polytope decision structures, the foundation of the Nash equilibrium, some thirty years after polytopes had become an area of little interested to most mathematicians. In a similar fashion quantum Nash equilibria have revived interest in quaternions a virtually all quantum Nash equilibrium solutions have a quaternionic structure. In this paper we explore the role of quaternions in quantum Nash equilibrium as well as selected other areas of physics involving higher order terms normally considered to be outside the domain of classical physics.