Permissible extensions of classical to quantum games combining three strategies
Piotr Frąckiewicz, Marek Szopa
TL;DR
The paper addresses the problem that quantum extensions of classical games can depend on how the classical game is represented, leading to non-equivalent Nash equilibria. It uses the Eisert-Wilkens-Lewenstein framework to derive necessary invariance conditions on the added unitary strategy, proving the existence of three extension types and showing that only a subset preserves strong isomorphism with the input game (θ = $\frac{\pi}{2}$ and $\sin^2\alpha = \sin^2\beta$). Among these, Type I reproduces the original classical game, while Types II and III yield genuinely new quantum extensions that can alter equilibrium structure. As a practical demonstration, the Type II extension of Prisoner's Dilemma yields a unique mixed Nash equilibrium $\bigl( (\tfrac14,\tfrac14,\tfrac12), (\tfrac14,\tfrac14,\tfrac12) \bigr)$ with payoff $\tfrac{R+S+T+P}{4}$, illustrating how quantum extensions can move outcomes toward Pareto-optimality while remaining invariant to input form.
Abstract
We study the extension of classical games to the quantum domain, generated by the addition of one unitary strategy to two classical strategies of each player. The conditions that need to be met by unitary operations to ensure that the extended game is invariant with respect to the isomorphic transformations of the input game are determined. It has been shown that there are three types of these extensions, two of them are purely quantum. On the other hand, it has been demonstrated that the extensions of two versions of the same classical game by a unitary operator that does not meet these conditions may result in quantum games that are non-equivalent, e.g. having different Nash equilibria. We use the obtained results to extend the classical Prisoner's Dilemma game to a quantum game that has a unique Nash equilibrium closer to Pareto-optimal solutions than the original one.