Nash equilibria in four-strategy quantum game extensions of the Prisoner's Dilemma

Abstract

This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.

Figures (2)

• Figure 1: The dependence of NE first player payoffs on the value of the parameter a (in the permissible range) for different strategy profiles of the exemplary PD (\ref{['pdgame']}) in the extension $A_1$ given by the matrix \ref{['A1_PD_example']}. Payoffs $\Delta^1_{22}= \Delta^1_{24}=\Delta^1_{42}=1$, which correspond to NE for $a=1$ are identical and depicted by a single dot.
• Figure 2: Dependence of the payoffs of the $A_1$ extension of the PD (\ref{['PDgame']}) on the payoffs $P$ and $R$ for $S=0$ and $T=5$ and the value of $a$ corresponding to the maximum and equal NE according to Table \ref{['A1_PD_NE_payoffs']}, where $P=1$ and $R=3$. For a better comparison, figures (d) and (e) show the relationships shown in (a), (b) and (c) from two different points of view. In all presented cases the payoffs $\Delta^{i}_{jk}$ are the same for both players $i\in\{1,2\}$

Theorems & Definitions (36)

• Definition 1
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• Definition 3
• Definition 4
• Theorem 1
• Example 1
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• Example 2