Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme

Abstract

Classical game theory is an important field with a long tradition of useful results. Recently, the quantum versions of classical games, such as the prisoner’s dilemma (PD), have attracted a lot of attention. This game variant can be considered as a specific type of game where the player’s actions and strategies are formed using notions from quantum computation. Similarly, state machines, and specifically finite automata, have also been under constant and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the quantum finite automata, have been studied extensively. In this work, we examine well-known conditional strategies that have been studied within the framework of the classical repeated PD game. Then, we try to associate these strategies to proper quantum finite automata that receive them as inputs and recognize them with a probability of 1, achieving some interesting results. We also study the quantum version of PD under the Eisert–Wilkens–Lewenstein scheme, proposing a novel conditional strategy for the repeated version of this game.

QKD Based on Symmetric Entangled Bernstein-Vazirani

“…As for the Eisert-Wilkens-Lewenstein scheme, it proved fruitful in providing many interesting results. For example, it led to quantum adaptations of the famous prisoners’ dilemma in which the quantum strategies are better than any classical strategy [ 22 ], as well as extensions of the classical repeated prisoners’ dilemma conditional strategies to a quantum setting [ 25 ].…”

DKPRG or How to Succeed in the Kolkata Paise Restaurant Game via TSP

“…It is quite straightforward to convince ourselves that playing a 3-stop game is better than playing a a 2-stop game. A precise quantitative analysis of the resulting advantages can be performed by considering the exact formulas ( 19) - (26). Nonetheless, we believe it is expedient to showcase the difference with the following example.…”

“…Since the pioneering works of Meyer [22] and Eisert et al [23], quantum versions for a plethora of well-known classical games have been studied in the literature. Staring from the most famous of all games, the Prisoners' Dilemma [23], [24], [25], [26], many researchers have sought to achieve better solutions by employing quantumness (see the recent [27] and [26] and references therein), or other tools, such as automata ( [28]). It not surprising that unconventional approaches to classical games are undertaken because they promise clear advantages over the classical ones.…”

“…This last example can be considered as a compelling argument that advocates the importance of topological analysis for the network of restaurants. The above examples were studied using the exact formulas ( 19) - (26). Tables 1 -5 reflect the daily evolution of the above five instances of the m-DKPRG according to rigorous mathematical description provided by formulas ( 19) - (26).…”

The Connection between the PQ Penny Flip Game and the Dihedral Groups

“…Several quantum adaptations of the famous prisoners' dilemma have been defined and studied, giving quantum strategies that are better than any classical strategy ( [5]). Some recent results were presented in [12], where the correspondence of typical conditional strategies, used in the classical repeated prisoners' dilemma game, to languages accepted by quantum automata was established, and in [13], where the Eisert-Wilkens-Lewenstein scheme was extended. Quantum games, especially coin tossing, have also been fruitfully utilized in many quantum cryptographic protocols.…”